Extracted Text
design_axial_flow_co2_laser.pdf
DESIGN AND CONSTRUCTION OF AXIAL SLOW FLOW CONTINUOUS WAVE
FOLDED CARBON DIOXIDE LASER
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
THE MIDDLE EAST TECHNICAL UNIVERSITY
BY
NECMETTİN KENAR
IN PARTIAL FULFILLEMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
THE DEPARTMENT OF PHYSICS
AUGUST 2003
ii
Approval of the Graduate School of Natural and Applied Sciences
______________________________
Prof. Dr. Canan ÖZGEN
Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of
Master of Science.
_____________________________
Prof. Dr. Sinan BİLİKMEN
Head of Department
This is to certify that we have read this thesis and that in our opinion it is fully
adequate, in scope and quality, as a thesis for the degree of Master of Science
___________________________
Assoc. Prof. Dr. Gülay ÖKE
Superviser
Examining Committee Members
Prof. Dr. Sinan BİLİKMEN ___________________________
Assoc. Prof. Dr. Serhat ÇAKIR _________________________
Assoc. Prof. Dr. Akif ESENDEMİR ___________________________
Dr. Ali ALAÇAKIR ____________________________
Assoc. Prof. Dr. Gülay ÖKE ____________________________
iii
ABSTRACT
DESIGN AND CONSTRUCTION OF AXIAL SLOW FLOW CONTINUOUS
WAVE FOLDED CARBON DIOXIDE LASER
KENAR, Necmettin
M.S., Department of Physics
Supervisor: Assoc. Prof. Dr. Gülay ÖKE
August 2003, 106 pages
Design and realization of a conventional carbon dioxide laser was performed.
Gas composition and gas pressure effects on laser output power were studied. Effects
of input electrical power and current on laser power were also investigated. Beam
profiling of the laser beam was performed by pinhole method. Laser beam
parameters like beam divergence, beam propagation factor were measured. These
properties were extracted from focusing a laser beam in near field and performing a
number of cuts across the beam cross-section and measuring the beam diameter at
these points. Diameter measurements were obtained by knife edge method. Laser
beam parameters were obtained for three different power laser beams in two axes
across the beam. Found parameters were compared with regard to beam power and
beam cross-section axis. Also possibility of using the obtained laser beam in material
processing was investigated.
Keywords: Carbon dioxide laser, beam profiling, laser beam propagation
factor.
iv
ÖZ
YAVAŞ EKSENEL AKIŞLI SUREKLİ DALGA BÜKÜK KARBON D İOKSİT
LASERİ TASARIM VE YAPIMI
KENAR, Necmettin
Yüksek Lisans, Fizik Bölümü
Tez Yöneticisi: Doç. Dr. Gülay ÖKE
Ağustos 2003, 106 sayfa
Geleneksel karbon dioksit laseri tasarımı ve gerçekleştirilmesi çalışması
yapılmıştır. Gaz karışım oranlarının ve gaz basıncının laserin çıkış gücü üzerine
etkileri çalışılmıştır. Elektrik giriş gücü ve akımının laserin gücü üzerine etkileride
ayrıca incelenmiştir. Laser demeti kesitinde üç boyutlu güç şiddeti profili iğne deliği
yöntemiyle çıkartılmıştır. Laser demeti parametreleri , demet genişleme ve demet
yayılma katsayısı ölçülmüştür. Bu özellikler yakın mesafede laser demeti
odaklanarak, demetten çok sayıda kesit alınarak ölçülen demet çaplarından
çıkartılmıştır. Çap ölçümleri bıçak kenarı yöntemi ile yapılmıştır. Laser demet
parametreleri üç ayrı güçte laser demetinin her iki kesit ekseni için elde edilmiştir.
Bulunan parametreler demet gücü ve eksen açısından değerlendirilmiştir. Ayrıca elde
edilen laser demetinin malzeme işlemede kullanılma imkanları araştırılmıştır.
Anahtar Kelimeler: Karbon dioksit laseri, laser demeti güç şiddeti profili,
laser demeti yayılma katsayısı.
v
In Memory of Dr. Iulian GUTU
vi
ACKNOWLEDGEMENTS
I would like to express my sincere thanks to my advisor Assoc. Prof. Dr. Gülay
ÖKE for her support and careful reading of manuscript.
I would like to express my sincere thanks to Prof. Dr. Sinan BİLİKMEN for
the opportunity to study on CO
2 lasers as a member of Laser Laboratory.
I am very grateful to late Dr. Iulian GUTU for the discussions on CO
2 lasers
and laser material processing. He was the person who established my basics on the
subject.
I am grateful to Dr. Ali Alacakır for tolerating my faults and his help during
experiments.
I am also grateful to Mr. Oğuz PERVAN for his encouragement and help
during experiments.
I thank to Dr. Hilal GÖKTAŞ for allowing me to use his instruments.
Special thanks go to İlker YILDIZ and Ertan ERYILMAZ for their friendship
and help during my work.
I express my thanks to our technicians İsmail DOĞRU, Muharrem KUZU and
Murat AYDIN for their help.
I would like to thank to Mr. Semih ÖZEL from Genel Makina Tasarım Ltd. Şti.
for manufacturing of beam delivery unit and 3-axis stage, without his kind help this
work could not be completed.
I would like to thank to Mr. Hikmet ÖZGÜR from Makina Dizayn Ltd. Şti. for
spending his valuable time in supplying material that I could not find,
Also I thank to Mr. İbrahim DİNÇER from Dinçer Medikal Ltd. Şti. for his
valuable discussions and encouragement.
At last but not least I would like to express my sincere thanks to my family for
their patient and support during my work.
vii
TABLE OF CONTENTS
ABSTRACT……………………… ………………………………………………iii
ÖZ…………………………………………… ……… ……………….……………iv
ACKNOWLEDGMENT …………………………… ………………...…...…….vi
TABLE OF CONTENTS……… ………… ……………………………...……vii
LIST OF TABLES……………………………………………………………………………. .x
LIST OF FIGURES…………………………………… … … ………………….… …….….xi
CHAPTER
1.
INTRODUCTION………………………………………….…….………1
2.
FUNDAMENTALS OF CO 2 LASER……………….………….….....3
2.1-
History……………………………………………………….…...……3
2.2-
Slow Axial Flow CO2 Laser……………………….………..4
2.2.1- General Design………………………………..……….…………5
2.2.2- Gas Composition and Gas Flow……………………..……...……6
2.2.3- Tube Diameter…………………………………………...……….8
2.2.4- Pressure and Current…………………………...……….…..….…9
2.2.5- Discharge Energy and Gas Temperature……………….……….10
2.2.6- Resonator Configuration and Output Coupling………….…...…14
3.
THEORY OF CO2 LASER . . . . . . . . . . . . . . . . . . . . . . . . . . ……. . . . 18
3.1
- Carbon Dioxide Molecule . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 18
viii
3.2-
Excitation of Carbon Dioxide . . . . . . . . . . . . . . . .. . . . . ………. . . . . . . 20
3.3-
Relaxation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4
- General Energy Transfer Processes in CO2 Laser . . . . . . . . . 21
4.
LASER BEAMS AND THEIR PROPERTIES . . . . . . ……….. . . 25
4.1-
Transverse Modes of Laser Resonators . . . . . . . . . . . . . . .. . . . 25
4.1.1- Intensity Distribution of Transverse Modes . . . . . . . . . . . ... . . . 25
4.1.2- Characteristics of Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.3- Laser Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.4- Stability of a Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
4.2
- Beam Propagation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3-
Focusing of a Laser Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1- Diffraction Limited Spot Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2- Effects of M
2
on Focusing. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .36
4.3.4- Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.5- Depth of Focus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 39
5.
DESIGN OF SLOW AXIAL FLOW CO 2 LASER. ……….... . . . .40
5.1-
Laser Applications and Specification of Laser Output Beam
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2-
Determination of Laser Resonator Parameters ……………….41
5.2.1- Resonator Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
5.2.2- Discharge and Resonator Lengths . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.3- Determination of End Mirror Curvature and Stability Check . . 42
5.2.4- Determination of Laser Tube Bore Diameter. . . . . . . . . . . . . . . .43
5.2.5–Determination of Output Coupler Transmittance . . . . . . . .. . . . . 43
5.2.6–Selection of Mirror Material. . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
5.2.6.1- Material Selection for Output Coupler . ........................ 46
5.2.6.2- Material Selection for Full Reflector Mirrors. . . . . . . . 48
6
. EXPERIMENTS CARRIED OUT ON THE DESIGNED
LASER
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1-
Experiments on Laser Gas and Electrical Feeding. . . . . . . . . 50
ix
6.1.1- Laser System Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1.2- Current - Partial Pressure Characteristics of Laser Discharge .. 52
6.1.3- Effects Partial Gas Pressures on Output Laser Power. . . . . . .. . .53
6.1.4- Effects of Gas Composition on Output Power. . . . . . . . . . . . . . .54
6.1.5-Laser Discharge with Constant Gas Composition . . . . . . . .. . . . . 56
6.1.6- Maximum Power Obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
6.2-
Experiments on Laser beam Properties . . . . . . . . . . . . . . . . . . 61
6.2.1- Laser Beam Profiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
6.2.2- Beam Propagation Parameter and Divergence . . . . .. . . . . . . . . 66
6.3-
Experiments on Material Processing . . . . . . . . . . . . . . . . . . . . 71
7.
DISCUSSION and CONCLUSION ……………….….……………...74
REFERENCES…………………………………………………….……….…….77
APPENDICES
I. LASER DESIGN CALCULATIONS . . . . . . . . . . . . . . . . ... . . . . . . ……..79
II.
REALIZED DESIGN AND TECHNICAL DRAWINGS . ... …………88
III.
BEAM PROFILING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . ................93
IV.
KNIFE EDGE METHOD FOR LASER BEAM DIAMETER
MEASUREMENTS
. ……………….... . . . . . . . . . . . . . .. . . . . . . . . . . ……96
V. DETERMINATION OF BEAM PROPERTIES . . . . . . . . . . . …..……..99
VI
. BEAM DELIVERY UNIT AND NOZLE DRAWING . . . …………105
x
LIST OF TABLES
TABLE
6.1- Results on Laser Beam Properties
………………………………………...… 70
A1.1- Material Properties of possible Output Coupler Materials………...…85
A1.2- Material Properties of Possible Materials for Total Reflector
Mirrors………………………………………………………...………87
xi
LIST OF FIGURES
FIGURES
2.1- Schematic diagram of a conventional CO
2 Laser………………………………..5
2.2- Gain versus total mixture pressure……………………………………………....6
2.3- Excitation Mechanism of CO
2 Laser………………………………………….....7
2.4- Gain versus CO
2 flow rate for CO2:N2: He mixture at near optimum pressure
and mixture ratio………………………………………………………………..8
2.5- Gain versus tube diameter for optimum current density………………………...9
2.6- Gain versus discharge current for various flowing gas media at optimum
mixture ratios and a constant flow rate in 12 and 37 mm bore amplifier tubes...10
2.7- Optimum current and total pressures for maximum cw oscillator output power
as a function of tube diameter…………………………………………………11
2.8- Spatial variation of gain for a 3% CO
2: 10%N2: 87% He mixture at 100 torr.
Two gain distributions are shown; the one on the left corresponds to energy
input of 120 J/l atm. The other gain distribution corresponds to an input
energy of 350 J/l atm. Representative gas temperatures are also shown……...12
2.9- Spatial variation of gain for a 10%CO
2: 90% He mixture at 150 Torr…...…...12
2.10- Carbon-monoxide production in a 10%CO
2: 90% He mixture at 150 Torr.…13
2.11- Small-signal gain and gas temperature as a function of energy……………...13
2.12- Resonator Configurations…………………………………………………….15
2.13- Graph of output power versus window reflectivity…………………………..16
2.14- Optimum reflectivity as a function of the ratio of laser length to diameter….16
3.1- The fundamental modes of vibration of the CO
2 molecule…………………....19
3.2- Schematic diagram of the CO
2 laser mechanism showing relaxation times for
typical gas mixture (1:1:8 of CO
2:N2:He) at a total pressure of 15 torr and a
xii
gas-kinetic temperature of 420
o
K. Level populations shown are equilibrium
vibrational populations in the absence of laser action……………………..…..22
3.3- Diagram of some of the CO
2 and N2 vibrational levels involved in the
analysis…………………………………………………………………………24
4.1- Intensity profile of higher transversal modes of stable laser resonators with
circular symmetry…………………………………………………………….26
4.2- Contour of a Gaussian beam……………………………………………….…...28
4.3- Mode parameters of interest for a resonator with mirrors of unequal
curvature………………………………………………………………………..29
4.4- Stability diagram. Unstable resonator systems lie in the shaded regions……....32
4.5- Divergence determined by measuring the waist size behind a focusing lens…33
4.6- Diagram illustrating the diffraction limited spot size……………………….….35
4.7- Spherical aberration of a single lens focusing a parallel beam…………………37
4.8- A graph of spherical aberration for lenses of different shape but the same
focal length………………………………………...…………………………...38
5.1- Output Coupling Efficiency versus transmittance plot………………………...44
5.2- Optical distortion figures of merit compare substrate materials for CO
2 laser
output coupler…………........…………………………………………………47
5.3- Optical distortion figures of merit compare substrate materials for CO
2 laser
full reflector mirrors…………………………………………………….…….48
6.1- Experimental setup…………………………………………………………….51
6.2- Current -partial pressure behavior of laser gas constituents…………….……..52
6.3- Effect of partial pressures on output power……………………………………54
6.4- Gas composition effects on output power……………………………….…….55
6.5- Effect of current on laser power ……………………………………………….56
6.6- Efficiency versus current graph ……………………………………………….57
6.7- Efficiency versus laser power graph……………………………………..…….58
6.8- Effects of input power on output laser power……………………………...…..58
6.9- Effects of pressure on output laser power…………………………………..…59
6.10- Contour plot of combined effects of pressure and input power on output
laser power…………………………………………………………………...59
6.11- Maximum power obtained for single tube operation…………………………60
xiii
6.12- Beam brofiling method of laser beam…………………………..……………62
6.13- Beam intensity profiles obtained from different laser powers, a) 10 W, b) 20
W, and c) 30 W……………………………………………………………….64
6.14- Burn patterns of laser beam on wood for 10W, 20 W and 30 W…………….65
6.15-Radial intensity distribution for TEM
00, TEM01 and sum TEM00+TEM01…....66
6.16- Experimental setup for knife edge method………………………….………..67
6.17- Curve fitted to data of 10 W laser beam……………………………….……..69
6.18- Processed materials; a) Teflon processing, b) Teflon processed, c) Paper
processed, d) Cut performed in plexiglas, e) Stainless steel processing, d)
Processed stainless steel sheet……………………………………………..…73
A1.1- Efficiency curve for increasing losses…………………………..…………...83
A1.2- Efficiency curve for increasing length………………………………………83
A1.3- Efficiency curve for increasing losses of designed laser……………..….…..84
A1.4 - Figure of merit for thermal expansion…………………………………..….86
A1.5- Figure of merit for combined thermal expansion and refractive gradient
change effect………………………………………………………………...86
A1.6- Figure of merit for thermal expansion for total reflection mirrors……….....87
A2.1- Realized design: a) General view, b) Cathode side, c) Anode side ………....88
A2.2- General top view of the design ………………………………….…………..89
A2.3- Cathode side view of the design …………………………………..…………90
A2.4- Anode side view of the design ……………….………....................................91
A2.5- Beam bending mirrors assemble view of the design ………..........................92
A3.1- 3D surface plot of beam intensity profiler…………………………………...95
A3.2- Contour plot of beam intensity profiles……………………………………...95
A4.1- Percentage power as a function of distance traveled………………………...97
A4.2- Plot of experimental data, fitted curve and 15.9% and 84.1 % lines as a
function of distance…………………………………………………………..98
A5.1- Experimental data with fitted curves along the propagation direction……..103
A6.1 - Beam delivery unit…………………………………………………………105
A6.2 - Laser beam focusing head and nozzle…………………………………......106
1
CHAPTER 1
INTRODUCTION
In the world of high speed technological development, lasers as general take
their place on top of the subjects studied in the last 40 years. The needs of industry
force the research and development of lasers. Lasers especially that with high power
have wide application areas in manufacturing industry. Their main application areas
are cutting (about %80), welding, heat treatment, alloying and cladding. In these
manufacturing areas the use of CO
2 and Nd:YAG lasers is dominated. Since the
developments in solid state lasers in the last ten years increase their use in
manufacturing market, but still CO
2 lasers are widely used and technological
developments in CO
2 laser fixes their application areas in the market. CO2 lasers are
still cheap and have better beam properties than solid state lasers and this makes
them worth to work on.
In our country very little work on CO
2 lasers is done and needs of our market
for cheap and powerful lasers was the main reason of this work. The lack of
experience in this area force us to start with the basic laser design structures. Also
need of practical methods for determining the beam properties of a powerful laser
beam oriented us to use basic methods such as pinhole and knife edge techniques.
The manuscript text of the work was prepared in such an order first to give basics of
CO
2 lasers than their theory and than to pass to basic design needs according to
which design calculations are made. After the design calculations the realization of
the design was described and experiments carried out were discussed.
2
In Chapter 2, main design criteria of slow flow axial laser is given and
background covering the engineering and technical aspects of laser design was tried
to be established.
In Chapter 3, having in hand some experimental results and technical
considerations obtained from Chapter 2, theoretical background of CO
2 lasers is
given and the physics of CO
2 lasers is established in connection with macroscopic
phenomena taking place in lasers.
Chapter 4 introduces beam properties of lasers and connects these properties to
the mechanical design of laser. Also focusing of a laser beam and some results that
may effect the processing are described.
In Chapter 5, design method and calculations are extensively described and
results that were considered in the mechanical realization of the design are given.
In Chapter 6 experiments carried out on the realized design are described and
obtained results are discussed. In this chapter gas concentration and pressure effect
on lasing, beam profiling, beam propagation factor estimations and possible material
processing applications are given.
Calculations carried out during the work, some technical drawings, photos and
description of used techniques are left for Appendices.
3
CHAPTER 2
FUNDAMENTALS OF CO 2 LASER
2.1-History
Infrared radiation emission from CO
2 was first reported by Patel [1] in 1964 in
pulsed discharge through pure CO
2. Soon after that it had been realized that a much
more efficient system involving the vibrational energy transfer from N
2 to CO2 was
possible. Such a laser had been build by Patel [2] also in the same year. This laser
incorporates the RF excitation of N
2 molecules, which are then injected to the CO2
gas.
In these studies output powers of continuous wave (cw) operating CO
2 lasers
were between 0.1 mW to 200 mW. The addition of N
2 increased the efficiency of
the CO
2 laser from 10
–6
to 10
-3
. Direct excitation of a flowing N2 and CO2 gas
mixture using a dc discharge was found to yield cw power of 11.91 W with
efficiency of ≅ 3 % by Patel [3].
Another major advance occurred when the addition of He was found to
increase the cw power obtainable from a flowing N
2, CO2 gas mixture to 106 W [4].
The efficiency of this system was less than 6%. The theoretical efficiency of
the CO
2 laser operating at a wavelength of 10.6 µm was predicted to be ≅ 40 % and
at these years the efficiencies obtained were far away the theoretical limit. In 1967 a
4
162-ft long cw CO
2 laser had been built and operated at a power output of 2.3 kW
[5]. Subsequently, the development of electric discharge convection and gas dynamic
lasers has resulted in generation of cw laser powers in excess of 100 kW. But the size
of these lasers was not convenient for industrial purposes.
The year 1969 marked a turning point in the development of high-power cw
and pulsed CO
2 lasers. Late in 1969, Beaulieu [6] reported that CO2 laser emission
could be obtained at atmospheric pressure and above by exciting the gas transversely
so that the discharge passed perpendicular to the optical axis. This has led directly to
devices that rely on the creation of high densities of electronic charge using electron
beam excitation [7] or volumetric photoionization [8] independent of the discharge
that is used to excite laser emission. As a result Q-switched CO
2 lasers were
developed.
In the same year development of lasers based on convective cooling [9], [10]
were realized and a report was published describing the operation of a compact
closed system cw laser using rapid transverse gas flow capable of generating powers
of 1 kW [11].
Theoretical development of CO
2 laser was completed in 10 years after its
invention. Through the proceeding years works on these lasers were concentrated on
development of new excitation mechanisms, new resonator types and more efficient
laser optics.
2.2- Slow Axial Flow CO2 Laser
Slow axial flow lasers are no more ‘state of art’ lasers. Main difference from
the other types of lasers is that; cooling mechanism is diffusion and gas flow rate is
maximum 50 cm
3
/min.
5
2.2.1 – General Design
Figure 2.1- Schematic diagram of a conventional CO2 Laser
A laser primarily consists of the following parts; two mirrors one total reflector
and the other partial reflector, a tube used as chamber and anode-cathode pair. Gas
mixture flows in the tube between anode and cathode. During this flow the gas
mixture is excited and as a result of this excitation photons are obtained. Photons
travel between the two mirrors by exciting the gas again and increasing the number
of photons in the cavity. Through the output mirror, which allows photons to escape
out of the cavity, a laser beam is obtained. Fig. 2.1 shows the general sketch on
conventional CO
2 laser.
The laser tube is enclosed by a coolant (water, oil etc.) jacket, by which the
excess heat in the cavity is removed. Cooling of the cavity is of great importance to
the operation of the laser as going to be discussed below.
6
2.2.2- Gas composition and Gas Flow
First lasing from CO
2 laser was obtained from pure CO2 gas. But the output
power obtained was very low. Subsequent addition of N
2 and He increased output
power very much. Fig. 2.2 shows the effect of different gas compositions on laser
gain.
Figure 2.2- Gain versus total mixture pressure [12]
As it can be seen from the figure, CO
2 alone has very low power gain. This is
due to the inefficient electrical excitation made directly to CO
2 gas. Addition of N2
increased the laser gain by factor of 10. This is a result of the efficient excitation of
N
2 and that nitrogen has a very long-lived first-excited vibrational state that almost
exactly matches the upper level of CO
2. The energy transfer between the N2 and CO2
molecule is very efficient as a consequence of the close energy levels. The graph
7
showing the vibrational energy levels of CO
2 and N2 and the excitation mechanism
of CO
2 laser are shown in Fig. 2.3.
Figure 2.3- Excitation Mechanism of CO2 Laser [13]
From Fig. 2.3 we can see that He has no effect in the excitation of CO
2 but
addition of it to pure CO
2 or to CO2-N2 mixture increases power gain very much.
This can be explained by the high heat transfer coefficient of He gas compared with
CO
2 and N2.
Gain in CO
2 laser is not only a function of gas composition but also it is a
function of gas flow rate. In Fig. 2.4 it is shown that with an increase in flow rate
gain of a given laser increases first and then saturates after a given flow rate
depending on tube diameter. But the saturation of gain does not mean an obtainable
constant power. Power can increase after that point but very slowly. The primary
effect of flowing gas on gas discharge is a reduction of gas temperature by
convection, where as in slow flowing gases the heat transfer is obtained by diffusion.
8
Figure 2.4- Gain versus CO2 flow rate for CO2:N2: He mixture at near optimum pressure
and mixture ratio [14]
2.2.3- Tube Diameter
Gain of a laser is a function of tube diameter also. Fig. 2.5 depicts a usual
dependence of gain on tube diameter. Gain decreases with increasing tube diameter.
Although this situation cannot be explained very simply, there are three effects which
are produced with the decrease of tube diameter. First, by decrease in tube diameter
an increase in longitudinal field is observed. This increase in the field leads to
increase in electron temperatures that imply an increase in electron energy. This
leads to increase in pumping rate and gain coefficient. Second, if the volume pump
rate is constant decrease in tube diameter results in increase in the gas velocity that
causes a faster gas exchange. Therefore, during the passage of the gas through the
9
tube, decomposition of CO
2 decreases and gain increases. Third, with the decrease in
tube diameter the temperature difference between the gas center and the tube walls
decreases, That is, heat removal from the gas is enhanced and the gain is increased.
Figure 2.5- Gain versus tube diameter for optimum current density [12]
2.2.4- Pressure and Current
As it is clear from Fig. 2.2 gain is also a function of pressure. For a given gas
mixture gain increases with increasing pressure. This is an expected result since
density of CO
2 and N2 molecules increases giving rise to gain. However, with
increasing pressure, field intensity is growing as a result of which input energy in the
gas and gas temperature are increased resulting in a decrease in the gain. So, for each
gas mixture there exists an optimum pressure for which maximum gain is obtained.
10
In condition of optimum gas mixture and constant flow rate the dependence of
gain on current is depicted in Fig. 2.6. Here increase in current means increase in
input power which is expected to increase gain of the amplifier. But increase in input
power results in excess power stored in gas medium which in consequence increases
gas temperature. Increase in gas temperature as mentioned above means decrease in
amplifier gain. Also from Fig. 2.7 it can be seen that optimum current is also
dependent on tube diameter.
Figure 2.6- Gain versus discharge current for various flowing gas media at optimum
mixture ratios and a constant flow rate in 12 and 37 mm bore amplifier tubes [14]
2.2.5- Discharge Energy and Gas Temperature
Gas temperature plays a key role in gas laser excitation. It depends on input
energy density having a scaled unit of Joule/(Volume x Pressure). In the discharge
tube gas temperature has a radial gradient. That is temperature is high at the center
of. the tube and decreases gradually to the tube wall.
11
Figure 2.7- Optimum current and total pressures for maximum cw oscillator output power
as a function of tube diameter [15]
Fig. 2.8 depicts the behavior of gas temperature and small signal gain in
discharge tube for two different input energies From Fig. 2.8 we can conclude that at
small input energies the gain distribution reflects the input energy distribution.
However as the input energy increases after a certain energy input value the gain
distribution does not continue to reflect the spatial energy input distribution. The
temperature of the gas at the center of the tube increases and the gain starts to
decrease.
The spatial energy distribution posses the same profile as shown in Fig. 2.9 but
gain profile is changing in not preferred way. The decrease of gain is not a direct
function of temperature increase. Increased temperature increases the dissociation of
CO
2 and the increase in content of CO decreases gain of the medium. Carbon-
monoxide production rate as a function of input energy distribution is shown in Fig.
2.10. The CO production is directly proportional to the discharge energy.
In Fig. 2.11 the dependence of gain and gas temperature on input energy is
shown. Temperature increase is a direct function of energy input. But gain has a
maximum at certain temperature or input energy and after that point it starts to
decrease.
12
Figure 2.8- Spatial variation of gain for a 3% CO2: 10%N2: 87% He mixture at 100 torr.
Two gain distributions are shown; the one on the left corresponds to an energy input of
120 J/l atm. The other gain distribution corresponds to an input energy of 350 J/l atm.
Representative gas temperatures are also shown [16]
Figure 2.9- Spatial variation of gain for a 10%CO2: 90% He mixture at 150 Torr [16]
13
Figure 2.10- Carbon-monoxide production in a 10%CO2: 90% He mixture at 150 Torr
[16]
Figure 2.11- Small-signal gain and gas temperature as a function of energy [16]
14
2.2.6- Resonator Configuration and Output Coupling
Generally a resonator is consisting of a discharge medium and two mirrors.
Mirror configurations may vary in regard to their curvature and resonator length.
Possible resonator configurations are given in Fig. 2.12. Selection of resonator is a
question of application. Usually the end mirror is fully reflective. Materials for end
mirror are Au, Ag coated Cu, Mo and Si. Curvature of the mirror is determined by
the desired output beam characteristics and stability criteria of the resonator.
The most important issue in laser design is determination of the reflectivity or
transmission of the output mirror. Less transmission of the output mirror means
obtaining less power from that a resonator can give. However, large transmissions
decrease photons that sustain the laser action in the resonator cavity so obtained
power also decrease. The effect of transmission of the output mirror on the obtained
laser power is given in Fig. 2.13. Here the effect of output coupling mentioned above
can be seen clearly. So an optimum value for the transmission of output mirror
should be chosen. In his report Tyte [15] gives a graph from which an optimum
reflectivity for the output mirror can be chosen with respect to resonator
length/discharge tube diameter. This graph is given in Fig. 2.14. The graph is based
on experimental values and it is difficult to obtain desired results using it.
15
Figure 2.12- Resonator Configurations
16
Figure 2.13- Graph of output power versus window reflectivity [15]
Figure 2.14- Optimum reflectivity as a function of the ratio of laser length to diameter [15]
17
Since each amplifying medium is defined by its saturation intensity and small
signal gain a relation to obtain optimum output transmission should include these
parameters beside the resonator length and diameter. Such a relation is given by
Rigrod [17]. This relation includes reflectivities of both mirrors, resonator length,
discharge cross section, small signal gain and saturation intensity. The formula is:
()
1/ 2
1 12
1/ 2
1/ 2 1/ 2
12 12
ln( )
21
os
o
rtgIS rr
PL
grr rr
=+
+−
(2.1)
where P is the output power, r
1 and r 2 are the reflectivities of end and output mirror
respectively, t is transmission of output mirror, S is cross section area, I
s is saturation
intensity, g
o is small signal gain and L is discharge length. From this formula by
taking out P
av=goIsV which is the available power in the resonator we obtain a
formula for efficiency like
()
1/ 2
112
1/ 2
1/ 2 1/ 2
12 12
ln( )
1
21
av o
rt rrP
PgLrr rr
η
== +
+−
(2.2)
Using this equation we can obtain a plot of efficiency versus transmission for
the output mirror. From the plot transmission value giving the highest efficiency is
selected to be the transmission of the output mirror. In this formula also there is an
unclear part; how can g
o be determined. There are two ways to found go: one by
measuring it, which is the long way and second by taking it from graphs such as Fig.
2.5. From equation (2.2) it is clear that output transmission is a function of mirror
reflectivities and g
oL product. Only one parameter is experimental the others are
given.
18
CHAPTER 3
THEORY OF CO 2 LASER
3.1 – Carbon Dioxide Molecule
Carbon dioxide is a linear, symmetric, triatomic molecule. This molecule can
vibrate in the three independent modes of vibration shown in Fig. 3.1. They are: the
longitudinal symmetric mode in which the carbon atom remains stationary and the
oxygen atoms move in opposite directions along the line of symmetry; the bending
mode in which the atoms all move in a plane perpendicular to the line of symmetry,
the carbon going one way while the two oxygen go the other; and the asymmetric
mode in which the atoms all move along the line of symmetry, the carbon moving in
the opposite direction to the two oxygen atoms. Each of these modes is characterized
by a definite frequency of vibration. According to basic quantum mechanics these
vibrational degrees of freedom are quantized, that is when a molecule vibrates in any
of the modes it can have only a discrete set of energies. Thus if we call ν
1 the
frequency corresponding to the symmetric mode then the molecule can have energies
of only
1
1
2
En h ν
=+
, n= 0,1,2,... (3.1)
when it vibrates in the symmetric stretch mode. Thus the degree of excitation is
19
characterized by the integer n when the carbon dioxide molecule vibrates in the
symmetric stretch mode. In general, since the carbon dioxide molecule can vibrate in
a combination of the three modes the state of vibration can be described by three
integers (nmq); the three integers correspond respectively to the degree of excitation
in the symmetric, bending and asymmetric mode.
Fig. 3 also shows the different
vibrational energy levels taking part in the laser transition. The laser transition at
10.6 µm occurs between the (001) and (100) levels of carbon dioxide.
The CO
2 energy levels, which are of greatest importance in laser action, are
situated 2349.3 cm
-1
, 1388.3 cm
-1
, 1285.5 cm
-1
and 667.3 cm
-1
above the ground
state.
Figure 3.1 - The fundamental modes of vibration of the CO2 molecule
20
3.2- Excitation of Carbon Dioxide
The excitation of the carbon dioxide molecules to the long lived level (001)
occurs both through collisional energy transfer from nearly resonant excited nitrogen
molecules and also from the cascading down of carbon dioxide molecules from
higher energy levels.
The first vibrationally excited level of N
2 lies 2329.66 cm
-1
above the
unexcited ground state. The (001) level of CO
2 is 20 cm
–1
above the first excited
level of N
2. Thus energy can be transferred between N2 and CO2 with high efficiency
when N
2 collides with CO2. The transfer equation is given by
N
2 (v``=1) + CO2 (000) ↔ N 2 (v``=0) + CO2 (001) – 18 cm
-1
(3.2)
and the rate of energy exchange is k=1.9x10
4
Torr/sec at 300
o
K [18].
3.3- Relaxation Process
Early in the development of CO
2 lasers, it was realized that adding He in
quantity to the laser gas resulted in substantially increased output powers. It is now
established that the primary effect of He is to relax the lower laser level while
essentially unaffecting the population in the (001) state. Furthermore, He is also
effective in depopulating the (010) level and may also lower the gas kinetic
temperature. All these effects are beneficial to high power laser operation, since a
radiative relaxation rates are orders of magnitude smaller than those due to collisions
with He atoms and other gaseous components.
Quenching process involving the (010) level include the vibrational-
translational energy transfer
21
CO
2 (01
1
0) + M → CO 2 (00
0
0) + M + 667 cm
-1
(3.3)
The relaxation of the two lower laser levels (10
0
0,02
0
0)I and (10
0
0,02
0
0)II,
these levels correspond to levels with energies of 1388.3 cm
-1
and 1285.5 cm
-1
, may
occur through
CO
2 (10
0
0,02
0
0)I +CO2(00
0
0) ↔ 2CO 2(01
1
0) + 64 cm
-1
(3.4)
CO
2 (10
0
0,02
0
0)II +CO2(00
0
0) ↔ 2CO 2(01
1
0) - 40 cm
-1
(3.5)
or (1000,0200)
I may relax via the process (Tyte [15] )
CO
2 (10
0
0,02
0
0)I +CO2(00
0
0) ↔ CO 2(02
2
0) +CO2 (00
0
0) + 52.7 cm
-1
(3.6)
CO
2 (02
2
0)I +CO2(00
0
0) ↔ 2CO 2(01
1
0) + 1 cm
-1
(3.7)
Rate constants associated with these processes depend on temperature,
pressure, and type of quencher. Lifetimes associated with the states involved in
excitation and relaxation is shown in Fig. 3.2.
3.4- General Energy Transfer Processes in CO2 Laser
Of the many energy-transfer processes, which occur in CO
2-N2-He laser
mixture, the following equations (which also include the ones given above) are found
to be the most important and are given by Fowler [19];
i) Electron excitation and deexcitation of the CO
2 asymmetric stretch mode,
e + CO
2 [l,m,(n-1)] ↔ e + CO 2(l,m,n), (3.8)
22
Figure 3.2- Schematic diagram of the CO2 laser mechanism showing relaxation times for
typical gas mixture (1:1:8 of CO
2:N2:He) at a total pressure of 15 torr and a gas-kinetic
temperature of 420
o
K. Level populations shown are equilibrium vibrational populations
in the absence of laser action [15].
ii) Vibrational energy exchange between N
2 and the asymmetric stretch
mode of CO
2,
N
2(v) +CO2 [l,m,(n-1)] ↔ N 2(v-1) + CO 2 (l,m,n), (3.9)
iii) Vibrational energy exchange between the asymmetric stretch and the
coupled symmetric stretch and bending modes of CO
2,
M+ CO
2 [l,m,(n-1)] ↔ M + CO 2 [(l-1), (m-1), n], (3.10)
iv) Electron excitation and deexcitation of N
2,
e + N
2(v) ↔ e + N 2 (v’) , (3.11)
23
v) Excitation and deexcitation by both electrons and heavy particles of the
bending mode of CO
2,
M+ CO
2 [l,(m-1),n] ↔ M + CO 2 [l, m, n], (3.12)
vi) Absorption and spontaneous emission of 4.3 µm radiation by CO2,
CO
2 [l,m,(n-1)]+photon (4.3 µm) ↔ CO 2 (l, m, n), (3.13)
vii) Absorption and stimulated emission of 10.6 µm radiation by CO
2 ,
CO
2 (100) + 2 photons (10.6 µm) ↔ CO 2 (001) + photon (10.6 µm). (3.14)
These energy-transfer processes are illustrated schematically in Fig. 3.3, which
also shows the energy levels of some of the CO
2 and N2 vibrational states.
Equations (3.8) and (3.12) state that CO
2 molecule can be not excited only by
collisions with excited N
2 molecules but also by electron and other particles
collision. But by excitation mechanism of equation (3.12) a laser radiation cannot be
obtained as seen from Fig. 3.3. From excitation of CO
2 by electron bombardment
lasing action can be obtained but as seen in Fig. 2.2 excitation of CO
2 alone is not
efficient as it is when excited with N
2-He gas mixture. Equation (3.14) is the
process equation of lasing in CO
2-N2-He gas mixtures. From this equation light
amplification is clearly seen. When a 10.6 µm photon collides with a CO
2 molecule
in (001) state two photons of the same wavelength are obtained.
24
Figure 3.3- Diagram of some of the CO2 and N2 vibrational levels involved in the analysis.
25
CHAPTER 4
LASER BEAMS AND THEIR PROPERTIES
4.1- Transverse Modes of Laser Resonators
4.1.1- Intensity Distribution of Transverse Modes
In optical resonator electromagnetic fields can exist whose distribution of
amplitudes and phases reproduce themselves upon repeated reflections between the
mirrors. These particular field configurations compromise the transverse
electromagnetic modes of a passive resonator.
Transverse modes are defined by the designation TEM
mn for Cartesian
coordinates. The integers m and n represent the number of nodes or zeros of intensity
transverse to the beam axis in the vertical and horizontal directions. In cylindrical
coordinates the modes labeled TEM
pl and are characterized by the number of radial
nodes p and angular nodes l. The higher the values of m, n, p, l, the higher the mode
order. The lowest-order mode is TEM
00 mode, which has a Gaussian-like intensity
profile with its maximum on the beam axis. For modes with subscripts of 1 or more ,
intensity maxima occur that are off-axis in a symmetrical pattern. To determine the
location and amplitudes of the peaks and nodes of the oscillation modes, it is
26
necessary to employ higher-order equations, which either involve Hermite or
Laguerre polynomials. The Hermite polynomials are used when working with
rectangular coordinates, while Laguerre polynomials are more convenient when
working with cylindrical coordinates.
In cylindrical coordinates, the radial intensity distribution of allowable circular
symmetric TEM
pl modes is given by the expression
() () ()
2
2
0
,, cos exp
l
pl l p
Irz I L lφρρ φρ=−
(4.1)
with
()
22
2/()rzwzρ= , where z is the propagation direction of the beam, and r, φ
are the polar coordinates in a plane transverse to the beam direction. The radial
intensity distribution are normalized to the spot size of a Gaussian profile; that is
w(z) is the spot size of the Gaussian beam, defined as the radius at which the
intensity of the TEM
00 mode is 1/e
2
of its peak value on the axis. L p is the
generalized Laguerre polynomial of order p and index l. Fig. 4.1 shows a 3D view of
some of the TEM
pl modes.
Figure 4.1- Intensity profile of higher transversal modes of stable laser resonators with
circular symmetry [20]
27
4.1.2- Characteristics of Gaussian Beam
A light beam emitted from a laser with a Gaussian intensity profile is called the
“Fundamental mode” or TEM
00 mode. As a Gaussian beam propagates, its intensity
distribution is Gaussian in every beam cross-section but the width of the intensity
profile changes along the axis. The Gaussian beam contracts to a minimum diameter
2w
0 at the beam waist where the phase front is planar. If one measures z from this
waist, the expansion laws for the beam assume a simple form. The spot size, a
distance z from the beam waist expands as a hyperbola, which has the form
1/ 2
2
0 2
0
() 1
z
wz w
w
λ
π
=+
. (4.2)
Its asymptote is inclined at an angle θ/2 with the axis, as shown in Fig. 4.2, and
defines the far-field divergence angle of the emerging beam. The full divergence
angle for the fundamental mode is given by
θ
∞ z
2 w z ( )
z
lim
→
2 λ
πw
0
. (4.3)
From these considerations it follows that at large distances, the spot size
increases linearly with z, and the beam diverges at a constant cone angle θ. From
equation (4.3) it is clearly seen that the beams with larger diameters have smaller
divergence.
At sufficiently large distances from the beam waist the wave has a spherical
wavefront appearing to emanate from a point on the axis at the waist. If R(z) is the
radius of curvature of the wavefront that intersects the axis at z, then
2
2
0
() 1
w
Rz z
z
π
λ
=+
. (4.4)
28
Figure 4. 2- Contour of a Gaussian beam
Also the wavefront of a Gaussian beam has the same phase across its entire
surface. Sometimes the properties of a TEM
00 mode beam are described by
specifying a confocal parameter
2
o
w
b
π
λ
= , (4.5)
where b is the distance between the points at each side of the beam waist for which
w(z)=(2)
1/2
wo .
29
4.1.3- Laser Resonator
Mode structure of a laser beam is directly related with the resonator
configuration. Mode parameters of a resonator with unequal curvature mirrors are
derived by Kogelnik and Li [21]. The geometry of such a resonator, where the radii
of curvature of the mirrors are R
1 and R2, is shown in Fig. 4.3.
Figure 4.3 - Mode parameters of interest for a resonator with mirrors of unequal
curvature
The diameters of the beam at the mirrors of a stable resonator, 2w
1 and 2w2, are
given by
30
2
4 12
1
112
RRL L
w
RLRRL
λ
π
−
=
−+−
(4.6)
2
4 21
2
212
RRL L
w
RLRRL
λ
π
−
=
−+−
. (4.7)
The diameter of the beam waist 2w
o, which is formed either inside or outside the
resonator, is given by
()()( )
()
2
12124
0 2
12
2
LR LR LR R L
w
RR L
λ
π
−−+−
=
+−
. (4.8)
The distances t
1 and t2 between the waist and the mirrors, measured positive as
shown in Fig. 4.3, are
()
2
1
12
2
LR L
t
RR L
−
=
+−
(4.9)
()
1
2
12
2
LR L
t
RR L
−
=
+−
. (4.10)
These equations treat the most general case of a resonator but they can be simplified
for most of resonator configurations.
4.1.4- Stability of a Resonator
Resonators with regard to their stability are classified as stable and unstable
resonators. In stable resonators rays are continuously refocused and ray content of
the resonator is constant. However, in unstable resonators rays become more and
more dispersed and they escape out of the resonator. But the “stability” referred to in
31
these resonator classifications is that of geometrical ray bouncing forth and back in
the cavity designs mentioned above, and has nothing related with the stability and
instability of laser oscillation in transverse modes.
Stability condition for a stable resonator is given as follows
12
01 1 1
LL
RR
<− − <
. (4.11)
This condition is derived from a paraxial ray tracing in a periodic convergent
lens sequence. This periodic lens sequence is dual to spherical-mirror resonator.
To show graphically which type of resonator is stable and which is unstable, it
is useful to plot a stability diagram on which each resonator type is represented by a
point. This is shown in Fig. 4.4 where the parameters L/R
1 and L/R2 are drawn as the
coordinate axes; unstable systems are represented by points in the shaded areas.
Various resonators types, as characterized by the relative positions of the
centers of curvature of the mirrors, are indicated in the appropriate regions of the
diagram. Also parameters g
1 and g2 are shown in the figure.
32
Figure 4. 4- Stability diagram. Unstable resonator systems lie in the shaded region[21]
4.2- Beam Propagation Factor
For applications of laser beams the characterization of their transversal mode
structure is necessary. Both the beam diameter at the waist 2w
o and the beam
divergence θ has to be determined for this purpose. The quality of laser beams can be
described by the beam parameter product BP:
o
BPωθ.= (4.12)
BP is minimum for diffraction-limited beams.
The values of θ and w
o have to be obtained experimentally. One simple method
is the measurement of the spot size diameter d
focus of the beam at the focal length of a
lens as shown in Fig. 4.5.
33
Figure 4.5- Divergence determined by measuring the waist size behind a focusing lens
The divergence angle can be determined from the focal length f and the
diameter d
focus in the focal distance of the lens by:
f
d
focus
2
=θ . (4.13)
The beam parameter product can be calculated from
f
dD
BP
focuslens
4
.
= (4.14)
with the beam diameter D
lens at the position of the lens and the focal length f.
The quality of a measured beam is described by the ratio of the beam parameter
products of this beam BP
beam relative to the best possible value BPGauss of a Gaussian
beam of the same wavelength λ in the same material with refractive index n. This
ratio is defined as beam quality or beam propagation factor M
2
:
()
λ
π
ωθ
n
BP
BP
M
waist
Gauss
beam
.
2
== . (4.15)
34
Thus M
2
=1.5 means the beam parameter is 1.5 times worse than the best
possible value for the wavelength and thus the focus for a given lens would show 1.5
times larger diameter as for a perfect beam.
4.3- Focusing of a Laser Beam
Focusing a laser beam is of great importance in material processing.
Positioning of a beam in specified location with specified spot size is the main
purpose of focusing. In all focusing methods the laws of geometric optics are
sufficient but to calculate the precise spot size and focal depth one needs to refer to
Gaussian optics and diffraction theory.
4.3.1- Diffraction Limited Spot Size
A beam of finite diameter is focused by a lens onto a plane as shown in Fig.
4.6. The individual parts of the beam striking the lens can be imagined to be point
radiators of a new wave front. The lens will draw the rays together at the focal plane
and thus constructive and destructive interference will take place. There will be
observed a Fraunhofer Diffraction Pattern. The central maximum will contain
approximately 86% of all the power in the beam. The diameter of this central
maximum will be the focused beam diameter, usually measured between the points
where the intensity has fallen to 1/e
2
of the central value.
35
Figure 4. 6- Diagram illustrating the diffraction limited spot size
Equation for the spot diameter of the fundamental mode of a circular beam is given
by
D
f
d
λ
⋅=44.2
min
(4.16)
where D is the beam diameter on the lens and f is the focal length of the lens. The
diameter of diffraction limited spots for other modes (TEM
pl) of a laser beam is
given by
()1244.2
min
++
⋅
⋅= lp
D
f
d
λ. (4.17)
36
4.3.2- Effects of M
2
on Focusing
Beam quality is a property of a beam and should have influence on the beam
size obtained after focusing. In focusing of Gaussian beam the diffraction limited
Gaussian spot size is:
D
f
d
π
λ4
min
= . (4.18)
But not every beam is Gaussian so a correction by M
2
should be applied. As it
was previously mentioned that the spot size of a focused beam is M
2
times that for a
perfect beam. So the equation is given by
2
min4
M
D
f
d ⋅=
π
λ
. (4.19)
4.3.4- Spherical Aberration
There are two reasons why a lens will not focus to a theoretical point; one is
the diffraction-limited problem discussed above and the other is that a spherical lens
is not a perfect shape. Most lenses are made with a spherical shape since this can be
accurately manufactured without too much cost and the alignment of the beam is not
so critical as with a perfect aspheric shape. The net result is that the outer ray
entering the lens is brought to a shorter axial focal point than the rays nearer the
center of the lens, as shown in Fig. 4.7. This leaves a blur in the focal point location.
The plane of best geometric focus (the minimum spot size) is a little short of the
plane of the plane wavefront, the paraxial point.
37
Figure 4.7- Spherical aberration of a single lens focusing a parallel beam
Focusing of a collimated beam produces a transverse spherical aberration
(TSA) and a longitudinal spherical aberration (LSA) shifts as shown in Fig. 4.7. To a
first approximation the focus diameter of the part due to the aberration of lens f
3:
2
3
3
),(
f
D
qnKd
a
= , (4.20)
where
()
()
()
()( )()
−
+−+++−
−
+
−
=
1
12314
1
2
1
1
32
1
),(
3
2
n
n
nnqnq
n
n
nn
qnK (4.21)
with
12
12
RR
RR
q
+
−
= (4.22)
R
1- lens entrance radius
38
R
2- lens exit radius
n- lens refractive index.
In spherical aberration shape parameter q determines the degree of the
aberration. Fig. 4.8 shows the axial difference in focal length (LSA) with respect to
parameter q. Here the focus length for different shapes is constant. Here it is clearly
seen that spherical aberration decreases for a plano-convex lens mounted in one way
than the other.
Figure 4.8 - A graph of spherical aberration for lenses of different shape but the same
focal length [22]
39
4.3.5- Depth of Focus
The depth of focus is the distance over which the focused beam has
approximately the same intensity. It is defined as the distance over which the focal
spot size changes by ± 5%. It is given by
()
() λ
ρπ
12
1
2
2
++
−±=
lp
r
z
o
f
, (4.23)
where ρ is the fractional incrase in beam diameter and p,l show the TEM supscripts.
40
CHAPTER 5
DESIGN OF SLOW AXIAL FLOW CO 2 LASER
A laser is designed for use in a specified application, so design parameters
should be defined accordingly. Starting with the application needs one should derive
the properties of the resonator. In Fig. 4.3 the basic resonator specifications,
resonator length, place of waist, spot sizes on mirrors etc., are shown and these
should be determined. Calculations are given in Appendix I.
5.1- Laser Applications and Specification of Laser Output
Beam Parameters
Applications that are considered to be done in this thesis work are cutting and
welding of metals, polymers and ceramics. For applications related to cutting it is
desirable to have a laser with TEM
00 mode of operation. Transverse electromagnetic
mode of operation of lasers used in welding applications is usually TEM
00 + TEM01
*.
Also it is good to have a beam with small angle of divergence in order of mrad’s.
Desired cutting thickness for ordinary carbon steel is 2 mm, and for welding it is 0.5
mm. Power needed for such applications is around 200 Watts of 10.6 µm laser
41
radiation. As a result of these application needs the laser should have the following
output beam and power properties:
- Laser Power : 200 W
- Beam Mode : TEM
00 +TEM01*
- Beam Divergence: mrad.
5.2- Determination of Laser Resonator Parameters
Starting with the application needs parameters, resonator type, resonator
length, tube bore diameter, size and types of mirrors used, transmission of output
coupler is the most important parameters to be determined.
5.2.1- Resonator Type
Some possible resonator configurations are given in Fig. 2.12. For the design
hemispherical resonator configuration is selected. This type of resonator has one
plain output coupler and a spherical end mirror. The advantage of this type is to
know exactly the place of the beam waist. Place of the waist and its size are the key
parameters that are needed to estimate beam diameter a distance far away from the
resonator. This also helps to calculate the spot size of the focused beam.
To save space it is decided to have U-bend resonator. By using such a
resonator without changing the discharge length overall length of the laser is lowered
to half of the resonator length. Also having such a resonator guarantees to have a
laser beam with linear polarization [26].
42
5.2.2- Discharge and Resonator Lengths
As a general rule lasers operated in slow flow regime have output powers of 50
W/m for TEM
00 and 75 W/m for TEM00+TEM01* mode of operation. In length
calculations 75 W/m value will be considered since radiation of the fundamental
mode is also present there. For power of 225 Watts we should have a discharge
length of 3 m. For usual resonators, resonator length is about 25% larger from the
discharge length and resonator length of 3.75m is obtained for the design. Mirrors of
the resonator should be as far as possible from the electrodes to avoid any
contamination from sputtering of electrodes, electrode material vapor deposition and
from any gas impurities.
5.2.3- Determination of End Mirror Curvature and Stability Check
To determine the curvature of the end mirror first of all the beam divergence
should be exactly specified. Then by using equation (4.3) beam waist w
o (radius of
the beam on output coupler) should be calculated. Inserting that value in equation
(4.8) radius of curvature of the end mirror R
2 can be calculated.
In this work a mirror with R
2=8 m was used. Radius of the beam waist was
calculated as w
o= 3.67mm and divergence for the fundamental mode was calculated
as 1.839 mrad. This value of divergence is sufficient for the thesis applications but
can be lowered by changing the curvature of the end mirror.
Having determined both mirror curvatures, stability of the resonator should be
checked by using inequality (4.11). Product g
1g2=0.531 and resonator is in the stable
region.
43
5.2.4- Determination of Laser Tube Bore Diameter
Having in hand values for R
1= ∞ and R 2= 8 m it is possible to calculate the
beam spot size on both mirrors. Since it was previously specified that the laser will
operate to its TEM
01* mode, in determination of tube bore diameter, spot size of the
second mode should be used. In calculation of beam diameters on the mirrors
equations (4.6) and (4.7) will be used. Beam diameter on the output coupler is small
than the beam diameter on the end mirror and for that reason in determination of tube
bore diameter only the beam spot size on the end mirror will be considered. Beam
diameter on the end mirror for the second mode is 15.1 mm and the bore diameter
was selected to be 16.4 mm, which is the closest to our calculated value in the
market. Also the beam on the output mirror will have a diameter of 11 mm.
5.2.5 – Determination of Output Coupler Transmittance
Most important parameter that should be determined in design of laser is the
transmittance of output coupler. To determine this parameter equation (2.2) should
be used. By plotting efficiency versus transmittance curve, the transmittance giving
maximum efficiency is the optimum transmittance of the output coupler. To plot this
curve we need the reflection of end mirror, cavity losses and small signal gain of the
active medium.
Cavity losses are generally the absorptions of total reflector mirrors used in the
cavity and diffraction losses due to diameter changes inside the cavity along the
beam path. Losses of cavity are increasing during the life period of resonator mainly
due to contaminations on the cavity mirrors. These contaminations arise from vapor
of electrodes used, sputtering and dust contained in gas mixture. Because of that
increase in cavity losses, efficiency curves are plot for different loss factors.
Small signal gain and saturation intensity characterize the discharge medium
44
and should be used in every calculation related with the output parameters of the
resonator. Since from Rigrod’s equation saturation intensity is canceled to give
efficiency of coupling there is no need to have that parameter. But in any case small
signal gain should be known. This parameter can be determined experimentally. Also
there are computational methods to derive this parameter but these methods are not
in the scope of this study. To specify this parameter we should refer to previously
done experimental works. Using Fig. 2.5 small signal gain of 0.5 %/cm was obtained
and used in calculations.
Plot of efficiency versus transmittance curve is given in Fig. 5.1. In plot,
curves for different loss values are given and transmittance of 20% for the output
coupler was selected.
Figure 5.1 - Output coupling efficiency versus transmittance plot
45
5.2.6 – Selection of Mirror Material
In previous section it was stated that in time of operation contamination of
optics surfaces increases and that increase causes absorption of radiation which
results in increase in optics temperature. Temperature increase results in thermal
expansion, which causes change in refractive index of material. Output coupler
subjected to temperature change behaves like a lens (diverging or converging),
changing the properties of output beam. For curved total or partial reflectors increase
in temperature results in dimension changes from which curvature change is the most
critical one. Change in curvature causes change in the previously predicted beam
properties. Also since the radiation possesses Gaussian intensity profile and not all
the surface is irradiated, curvature change is different for any point on a particular
surface.
As a result, in selection of mirror materials thermal and optical properties of
different materials used in CO
2 laser resonators will be studied. A formula including
thermal expansion coefficient, thermal conductivity, refractive index gradient and
percentage absorption for particular material will be used to obtain relative figures of
merit. Formulas [28] are:
1,2
K
F
AX
= (4.1)
where
K: thermal conductivity
A: total absorption
1
dL dn
X
dT dT
=+ : thermal expansion coefficient plus refractive index gradient
or
46
2
dL
X
dT
= : thermal gradient.
X
1 is used for lenses, windows and output couplers where effects of both
focusing and surface distortion are important. X
2 is used for total reflectors where
effect of only surface distortion is important. Also X
2 is used for output couplers
when the effects of focusing can be ignored.
Material properties of materials studied are given in Appendix I. Plots for
figure of merit versus absorption are given below.
5.2.6.1- Material selection for Output coupler
Output coupler material was selected between four materials: GaAs, Ge, KCl
and ZnSe. Plots for the two cases: (1) focusing and surface distortion and (2) surface
distortion only were examined.
From the plots of Fig. 5.2 it is observed that KCl is best when focusing is of
great importance, which is for high power lasers, but when surface distortion only is
a concern performance of KCl is worst among the others. KCl is very good for high
power lasers but needs extra conditioning facilities since it is hydroscopic in nature
and for low power laser where focusing is not problem its performance is worst so it
is not applicable in our design.
GaAs is best when only surface distortion is considered and performs close to
ZnSe and Ge when both focusing and surface distortions are important. GaAs cannot
be easily found in the market and also it is not transparent in visible region
(transparency in visible region is advantage in alignment facilities) so it also is not
applicable.
Ge, in performance sequence comes after GaAs but thermal runaway is near
100
o
C and is not applicable to our design.
47
0 0.02 0.04 0.06 0.08 0.1
0
2
4
KCl
GaAs
ZnSe
Ge
FM plot for Thermal Expansion
Surface Absorbtion (%)
Figure of Merit
a) for X= X2
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
KCl
GaAs
ZnSe
Ge
FM for TE plus Thermooptic Coefficient
Surface Absorbtion ( % )
Figure of Merit
(b) for X=X1
Figure 5.2 - Optical distortion figures of merit compare substrate materials for CO
2 laser
output coupler
48
ZnSe is average in performance but runaway temperature is about 300
o
C, can
be easily found in the market, and is transparent in visible region. Also it is cheap in
comparison to Ge and GaAs. As a result ZnSe is selected as material for the output
coupler.
5.2.6.2- Material selection for full reflector mirrors
Full reflector mirrors substrate material was selected between four materials:
Cu, Mo, Si and Ge. Only the case of surface distortion was examined.
0 0.04 0.08 0.12 0.16 0.2
0
2
4
Cu
Mo
Si
Ge
Surface Absorbtion (%)
Figure of Merit
Figure 5.3 - Optical distortion figures of merit compare substrate materials for CO2 laser
full reflector mirrors
49
From the plot in Fig. 5.3 it is seen that silicon substrates perform best but
performance of coatings on silicon is less from that on copper. Molybdenum is
second in performance and is widely used in not very clean operating conditions.
Germanium having a very low runaway temperature is not good for lasers with high
power. Copper on the other hand is in third position but metallic coatings perform
best on it. Cooling of copper is very easy compared to other materials. For the design
copper substrate mirrors were selected, end mirror with 99% reflection and bending
mirrors with 99.9 % reflection were selected.
50
CHAPTER 6
EXPERIMENTS CARRIED OUT ON THE DESIGNED
LASER
6.1- Experiments on Laser Gas and Electrical Feeding
In the previous chapter laser design parameters were determined and
considering these parameters a resonator was designed and constructed. Technical
drawings and a picture of the final construction of resonator are given in Appendix
II. In this part laser output beam characteristics, input power, current, gas pressure
and gas composition will be measured and relations between these parameters will be
determined.
6.1.1- Laser System Set up
Designed laser is driven by 25 kV power supply which delivers 50 mA current.
Voltmeter shown in Fig. 6.1 directly measures the voltage on the capacitor of supply.
Ammeter is connected to the supply end and total current delivered to
51
the system is measured. SCR`s group is consisted of trystors and resistors which
ensure the ignition of both tubes at the same time. Also resistor R
2 is a part of the
ignition system. Resistors R
1 are the ballast resistors which sustain current delivery
to the discharge after ignition.
Gas mixture of CO
2:N2: He is delivered to the system by three tubes which
separately contain the three gases. Mixture composition is arranged by flow meters
connected to each tube. Gas inlet is at the anode side of the resonator. Exhaust gas is
pumped out by vacuum pump and the pressure inside the resonator is measured by a
vacuum gage.
Laser radiation is measured by a power meter having a range of 0-250 W for
CO
2 lasers.
R1
130K
R1
130K
R2
300M
Vacuum
gage
Power
Meter
Vacuum
Pump
Figure 6.1- Experimental setup
52
6.1.2- Current - Partial Pressure Characteristics of Laser Discharge
Having three different gases, keeping the pressure and flow rate of two of them
constant and changing the pressure of the third gas the behavior of individual gases
in the discharge is obtained. Current-Partial pressure characteristics were obtained by
maximizing the output power of the laser and at that instance current, operating
potential and total gas pressure were recorded. The plots of current versus partial
pressure of one constituent of the mixture are given in Fig. 6.2.
Figure 6.2 - Current -Partial Pressure behavior of laser gas constituents
53
Each gas has different effect on the optimum current that gives maximum
power. Keeping constant pressures of CO
2 and He, current decreases with increasing
partial pressure of N
2. Also keeping constant pressures of N2 and He and increasing
the pressure of CO
2 current behavior is similar to N2 gas instead of that when partial
pressure of CO
2 is less than the partial pressure N2 current is increasing. But current
characteristics of He is completely different from the other two constituents,
increasing the partial pressure of He allows the increase of current to obtain
maximum output laser power.
6.1.3- Effects Partial Gas Pressures on Output Laser Power
Keeping constant pressure of the other two gases and changing the pressure of
the third gas output power was measured for the three gases. Fig. 6.3 gives the
measured results of partial gas pressure and corresponding output power for optimum
current. As it is clear from the figures below CO
2 and N2 have similar effects, both
increase the output power to some maximum value and then with increase of their
partial pressure output power decreases. In contrast to CO
2 and N2 partial pressure
effects, increase in partial pressure of He results in increase in output laser power.
During the experiments saturation for He pressure was not found since the power
delivered from the supply has reached its limit but it can be observed from the Fig.
6.3 that rate of power increase decreases with the increase of partial pressure of He.
54
Figure 6.3 - Effect of Partial pressures on output power
6.1.4- Effects of Gas Composition on Output Power
It should be also noted that the output power strongly depends on gas
composition. So it is of great importance to study the composition rates of all the
three gases at the same time. To make such comparison partial pressures of
individual gases were divided by total pressure so that the sum of the ratios results in
unity. Plotting contour graphs for two of the gas ratios and having as a third variable
the output power the effects of the constituent gases on the output power is obtained.
55
From the plots the third gas ratio is found by subtracting the sum of the two gas
ratios from one. Results are present in Fig. 6.4.
Figure 6.4 - Gas composition effects on Output Power
From the results illustrated in Fig. 6.4 we can conclude that the ratio between the
composition rate of CO
2 and N2 is 1to 1.5 respectively. Rate of He is very much
56
than the proportions of CO
2 or N2. In gas content optimization it is more reasonable
first to fix the values of CO
2 and N2 and then to increase the content of He until
maximum power is obtained.
6.1.5 – Laser Discharge with Constant Gas Composition
Up to now effects of individual laser gas constituents were studied and an
optimum gas composition at partial pressure ratios of (CO
2:N2:He-1:1:6) was
selected. Total pressure was 23 torr and current was changed so that to obtain a graph
indicating effects of power input in the discharge. As in previous experiments
current, potential and output laser power were recorded.
Fig. 6.5 shows effect of increasing current on laser output power. As current
increases laser power also increases up to a maximum power and starts to decrease
with subsequent increase in current. Here the behavior of current is identical to the
behavior of input power and the graph also indicates that much power does not mean
much laser output power since the lasing medium is heated up and this decreases
lasing effect of the discharge.
Figure 6.5- Effect of current on laser power
57
Fig. 6.6 depicts the effect of current on the electric to optical efficiency of the
laser. Increase in current decreases the efficiency of the system and for efficient
operation its better to have less current. Having less current means less input power
but it does not mean maximum output laser power as can be observed from Fig. 6.7.
If Fig. 6.5 and Fig. 6.7 are examined at the same time it is clear that the plot in Fig.
6.7 is the mirror image of the plot in Fig. 6.5. This indicates that as the input power
increases the conversion of electrical power to radiation power decreases. And this
decrease in efficiency is related to the increase in temperature of the discharge and to
the increase in power dissipation in the ballast resistors.
Figure 6.6 - Efficiency versus Current graph
To study the effects of pressure rise on output laser power a constant mixture
of volumetric composition CO
2:N2:He-8.9:13.4:77.7 was used. During the
experiment flowing gas pressure, output laser power, potential and optimum current
that maximizes the output power were recorded. Effects of input power and pressure
on the output power are plotted in Fig. 6.8 and Fig. 6.9 respectively. According to
Fig. 6.8 laser power is directly proportional to input power but it saturates after some
pressure limit. This implies that after some pressure limit it is not possible to increase
58
the laser power further. This result can be seen more clearly from the contour plot in
Fig. 6.10. In that figure red parts indicate the region of maximum laser power and it
is reached for an input power of 140 W and total pressure of 100 torr. And any
further increase in pressure or input power has very little effect on laser power.
Figure 6.7 - Efficiency versus Laser Power graph
Figure 6.8 - Effects of Input Power on Output Laser Power
59
Figure 6.9 - Effects of Pressure on Output Laser Power
Figure 6.10 - Contour Plot of Combined effects of Pressure and Input Power on Output
Laser Power
60
6.1.6- Maximum Power Obtained
During the experiments carried out above a general relation between the gas
composition and electrical feeding of the system was investigated. So these results
should be evaluated qualitatively since quantitative results do not reveal the
maximum obtainable power of the design. Maximum power was obtained under
pressure which was not detectable by our instruments but it was near the
atmospheric pressure. That maximum power was also realized by the constant
mixture of volumetric composition CO
2:N2:He-8.9:13.4:77.7 and current of 20 mA.
Power obtained was 32 W and a photo showing the power meter digital display is
given in Fig. 6.11.
Figure 6.11 - Maximum Power Obtained for Single Tube Operation
61
6.2- Experiments on Laser beam Properties
6.2.1- Laser Beam Profiling
Intensity distribution measurement on the cross section of a beam was
performed. A pinhole with diameter much smaller than the beam diameter was
moved in a plane cutting the beam perpendicular to its path of propagation. A
pinhole of 0.8 mm in diameter was used to profile the intensity distribution of a beam
with 9 mm in diameter. Also the pinhole plate was coated with graphite to prevent
back reflections of radiation into the resonator. Fig. 6.12 depicts the method of
measurement. Circle with dashed lines shows the laser beam. Moving the pinhole in
x and y-directions a two dimensional array of power data was obtained. Beam power
passing through the pinhole was measured by a power meter placed behind the
pinhole plate. Increments of displacement ∆X and ∆Y in x and y-directions
respectively were kept constant to eliminate additional data analysis. During the
measurements after completing one row of power sampling the pinhole plate was
removed and total power of the laser beam was checked to ensure that the beam
power was constant in the period of sampling. Since the pinhole plate was moved by
an X-Y stage there was no considerable errors in placing the pinhole at place where it
was desired to be. It should be noted that the profile is also strongly dependent on
optics alignment and to assure constant alignment mirror adjusting screws were kept
at the same position during all measurements on beam profiling. Power changes were
compensated by pressure adjustments and keeping the current of operation constant.
To eliminate effects of composition changes in operating gas a constant mixture gas
was used. The diffraction of the sample beam passing through the pinhole was not
considered since the aim is not to obtain absolute powers but relative power
distributions over the beam cross-section. This will not affect the beam profile results
but a precaution was taken by placing the power meter just behind and very close to
pinhole plate such that to absorb all the radiation passing the pinhole.
62
Figure 6.12 - Beam Profiling Method of Laser Beam
Obtained data were collected as square matrices and 3-D plots giving directly
the beam intensity distribution were obtained. X and y-axes are the same as they
were in measurements and the z-axis is the power, which are matrix elements. Plot of
raw data does not look very nice and also it represents the values of the data points
only and surrounding points remain under the lines joining data points. It is obvious
that there is no any sharp edge in the intensity distribution of a beam as it is observed
in the plot of raw data. To eliminate such results either a very small pinhole should
be used or data analysis of the measurements should be done. Plot of raw data is
given in Appendix III.
In data analysis instead of joining data points with lines a curve was fitted over
all data points. Method used is known as cubic spline fitting and fits a third order
polynomial to data. Such an analysis results in approximation of points which were
not sampled and gives more realistic plots. In data analysis a package program
MathCAD
®
was used and a sample calculation sheet is given in Appendix III for 10
Watt beam.
Beam profiling was done for three beams with different power so that it will be
63
possible to investigate the effects of beam power on the intensity distribution of the
beam. Fig. 6.13 gives the asymmetric and top vies of the 3-D plots for 10, 20 and 30
watt laser beams.
As it is observed from the plots in Fig. 6.13 asymmetric views give the
intensity profile of a beam but do not give clear information about the shape of the
beam cross-section. To obtain beam cross-section shape a top view of the profiling
plot is needed and this plot is given also in Fig. 6.13. From the plots it is observed
that the beam is more intense in its center and intensity decreases going away from
the center. And also it is clear that the beam shape of the obtained laser beam is not
an exact circle as it was desired to be obtained. The shape of the beam changes to
elliptical one as the power of the beam increases. This change in beam shape can not
be observed from beam burns obtained on wood (Fig. 6.14). Although burn patters
look like a circle the intensity profiles of the beams show that they are not. This
difference is mainly the result of thermal conductivity of material on which burns are
obtained. May be with materials, having lower thermal conductivity and high
absorption of the laser radiation, burns close in shape to the profiles could be
obtained.
The change of beam profiles with increasing powers may be a result of
increasing power in the discharge and increasing gas pressure. The increased
discharge power has destructive effects on the distribution of small signal gain in the
active medium and this will result in change in the beam profile. To eliminate such
drawbacks of increased power, cooling rate of active medium should be increased
either by decreasing the temperature of cooling water or increasing water flow rate or
by increasing the gas flow rate. None of these methods were applicable during the
experiments.
The effect of increased pressure on the beam profile is in connection with the
change in alignment of cavity optics. Since the optics are placed on O-rings to ensure
a good vacuum, easy replacement and easy cleaning there are finite movements in
any pressure change. As a consequence of these finite displacements output power of
the laser will not change by distinguishable amount but beam profile will change
destructively.
64
a)
b)
c)
Figure 6.13 - Beam Intensity profiles obtained from different Laser Powers, a) 10 W, b) 20
W, and c) 30 W
65
Figure 6.14- Burn patterns of laser beam on wood for 10W, 20 W and 30 W
From beam profiling it is possible to conclude for the transverse
electromagnetic mode content of laser beam. In the design part it was desired to
operate in TEM
00+TEM01 and this mode should be satisfied by the profiles obtained.
Theoretical intensity distribution is given by equation (4.1) and Fig. 4.1 depicts the
3-D plots for different modes separately. But in time of operation it is not possible to
separate these modes and the result is their sum. In Fig. 6.15 two dimensional plots
of intensity distributions of TEM
00, TEM01 and their sum is given. As it is seen form
the graph it is difficult to distinguish from a single profile the mode of operation
since the intensity distributions of TEM
00 and the sum TEM00+TEM01 are nearly the
same. Only the top of the sum is flat which means that in the profile we should have
a larger red spot in comparison to a TEM
00 beam. Any measurement on TEM00 beam
was not carried out and we can not conclude on the mode content of the beam with
experimental results. But our theoretical calculation results indicate that considering
curvature of the end mirror, resonator length and laser tube diameter we should have
obtained a beam with mode content which is the sum of first two transverse
electromagnetic modes.
The intensity distribution of a beam is of great importance in material
processing. Since the intensity profile of the beam is conserved in the focus point it is
important to locate the point of maximum intensity on the axis of propagation that is
in the center of the beam. In material processing the region or line that is desired to
cut, weld or heat treated is placed on the axis of beam propagation.
66
Figure 6.15 -Radial Intensity distribution for TEM00, TEM01 and sum TEM00+TEM01
Also the profile should posses a distribution such that points having equal
magnitude of intensity should be located in equal distance around the point having
maximum intensity, i.e. beam should have a circular cross-section. If this condition is
not satisfied the path of processing could not be located properly.
6.2.2- Beam Propagation Parameter and Divergence
The two most important parameters that characterize the laser beam are its
beam propagation factor, i.e. beam quality, and divergence. To determine these
parameters a lens of known focal length is used to focus the out coming laser beam
and by knife edge method beam cuts are made on both sides of focus point to
determine the place and size of the beam waist. Setup of the experiment is given in
Fig. 6.16. At each cut point diameters are measured in two directions. This allows to
observe the difference in beam propagation along the two transversal axis, x and y.
67
For a beam with acceptable characteristics it is expected to have the same properties
in both directions.
To determine the desired properties at twelve points on the propagation path of
the focused beam knife edge cuts were made and diameter of the beam at each cut
was determined. In Appendix IV use of knife edge cut method is described and a
sample calculation is given. Calculated beam diameters are plotted with respect to z-
axis in Fig. 6.17.
Figure 6.16 - Experimental Setup for knife edge method
A laser beam posses the form of propagation of the equation (4.2). Diameters
determined by knife edge method should be fitted to equation (4.2). Since our beam
is not purely Gaussian beam it is expected the beam to deviate by a factor equal to
the beam quality of the beam. So equation (4.2) is modified such that include the
beam propagation factor as well. The modified equation is given below
68
()
1
2
22
2
4
2()2 1
(2 )
o
o
o
Mzz
Wz W
W
λ
π
−
=+
(6.1)
Since parameters z
o, 2Wo and M
2
are not known exactly a trial and error
approach was applied in order to fit the equation. For the three unknown parameters
a range was specified and equation (6.1) was calculated at data points. The results are
determined by estimating the minimum difference between the calculated and
measured diameters at data points. At minimum total error the beam waist, its
location and beam quality were determined. A plot of the fitted curve for 10 W beam
is given in Fig. 6.17.
Curve fitting process was applied for three beams with power of 10W, 20 W
and 30 W for two directions and totally six waist diameters and beam quality
parameters were determined. From the curves approximating the propagation of the
beam, which beam diameters at twelve points along the propagation path were
obtained, Rayleigh range for each beam was estimated. By using the beam waist
diameter and the Rayleigh range for that beam the divergence for each beam were
also calculated. By using the previously measured beam diameters on the focusing
lens and calculated beam propagation factors theoretical beam diameters at focal
point of the lens were obtained. Also beam focal depth for each beam was calculated.
Results of Laser beam properties are given in Table 6.1. A sample calculation for
curve fitting and calculation of other beam parameters is given in Appendix-V.
In previous section intensity profiles of beams were determined and it was
observed that beams are elliptical. This result was confirmed by the diameter
measurements of beams on the lens. This beam ellipticity was conserved through all
path of propagation after beam passes focusing lens. From Fig. 6.17 this result can
be easily observed. As a result of beam ellipticity different results of beam properties
for the different axis of each beam were observed. Due to this observed beam shape
evaluation of the results will be done with respect to a given axis at all three power
levels.
As general result we can conclude that our beams are of TEM
00 mode with a small
contribution of second TEM
01 mode since average M2≅1.07. This result is consistent
with the plot in Fig. 6.15, that is the area below the curve of TEM
01 is much less than
the area below the curve of TEM
00 mode.
69
Figure 6.17 - Curve fitted to data of 10 W laser Beam
70
Table 6.1- Results on Laser Beam Properties
71
Comparing the beam diameters determined from the curve fitting at focal point
and calculated theoretical diameters they are consistent and calculated diameters are
slightly lager (+ 0.01 mm ) than that determined by curve fitting.
Waist position compared for each beam is different both axes. From this result
we can conclude that the beam posses a slight astigmatism.
Comparing the Rayleigh range of beams in one direction(x or y) its observed
that it decreases with increasing power. But beam divergence on the other hand
increases with increasing power. Behavior of beam propagation factor is the same as
beam divergence it increases with increasing power. Focal depth in contrast to beam
divergence and beam propagation factor decreases with increasing power. The
increase in beam divergence and beam propagation factor and the decrease in
Rayleigh range and focal depth indicate the same result, beam is getting worst with
increasing power.
The results of different beam properties on different axis of evaluation are the
result of worst alignment but differences are less than 10 % and that difference is
acceptable.
6.3 – Experiments on Material Processing
The possibility of material processing with the designed laser was investigated.
Since the laser output power is low (only 30 W) it is difficult to find a wide
application area in material processing. So during the thesis work concentration was
made on materials that absorb 10.6 µm wavelength radiation. These materials are
Teflon, Plexiglas, pyrex glass and packing paper. Also an attempt was done to cut
stainless steel thin sheet of 0.05mm. Primary processing considered was cutting and
any work on process optimization was not in the scope of this work.
Laser beam delivery system and nozzle were designed by the author and used
in the applications. A photo of the beam delivery unit and a technical drawing of the
nozzle are given in Appendix VI.
72
Cutting of three non metal materials was successful these materials are teflon,
pleksiglas and paper.
Thickness of teflon used in experiment was of 4 mm and cuts of 2 mm in depth
were realized. During the cutting of teflon there was a little flame and exhaust was
not melt but snow like powder. In processing of teflon the cutting mechanism may be
chemical bond breaking primarily and very little evaporation which results in flame
during the processing. In Fig. 6.18a a photo taken during the cutting of teflon is
shown and two cuts on the sample are also given in Fig. 6.18b.
Attempt to cut a hard 3 mm thick paper was done. It was successful but the
speed of cutting was very low 9 cm/min. In paper cutting mechanism was burning
there was no melting no vaporization. A performed cut is given in Fig. 6.18c.
5 mm thick pleksiglas was cutted very successfully. The cutting mechanism
was vaporization and there was not any melt on the plate. The cut width was
measured to lager than 0.1mm but less than 0.15 mm. A photo of the cut is given in
Fig. 6.18d.
Also cutting of 1 mm pyrex glass was performed but it was unsuccessful.
During the cutting operation glass was broken into pieces in one trial but in the
other cut was performed but glass melts were solidified at the edge of the cut. This
indicates that the process is not optimized and there is a possibility to cut this
material. Cutting mechanism in glass cutting is crack propagation and melting of the
material is not allowed. With good cooling during the processing, glass can be
successfully cutted.
Possibility of cutting highly reflective materials such as stainless steel was also
investigated and with a proper gas selection and processing speed optimization
cutting of a 0.05 mm thick material can be performed. In Fig. 6.18e photo taken
during the processing of steel is given. The flame is bright and likes the one in
welding applications and this flame indicates the melting of material. In Fig. 6.18f
the cut performed on the stainless sheet is given. As it is observed very clearly heat
affected zone around the cut is nearly two times large than the cut width. This result
shows that cooling of the sample during the processing was not enough. During
processing N
2 gas was used but by using a gas with high heat conduction constant or
by use of reactive gas such as oxygen better cuts can be obtained.
73
a) b)
c) d)
e) f)
Figure 6.18 - Processed materials; a) Teflon processing, b) Teflon processed, c) Paper
processed, d) Cut performed in plexiglas, e) Stainless steel processing, d) Processed
stainless steel sheet.
74
CHAPTER 7
DISCUSSION and CONCLUSION
The main aim of this work was to get experience with the basic structure of
CO
2 laser resonators and to get basic understanding of lasing processes. Beside these
also laser beam characterization, laser beam guiding and laser beam material
processing were the secondary subjects worked on.
The two basic problems of laser resonators are their mechanical and electrical
stability. The design realized was faced with both of the problems. Realization of
discharge at both tubes was not possible either by using single or two separate power
supplies. By using single supply connected to both tubes results in discharge in a
single tube completely randomly. By using two separate supplies also results in
discharge in a single tube. Second discharge is obtained between the anodes of the
resonator and not between the anode and cathode of the second tube.
Although some mechanical solutions for the problem exist during the work
electrical solutions were considered. Using of SCR`s group was the solution worked
on but it was successful only at pressures below 3 torr. The problem needs further
investigations and during the experimental part only one tube was operated.
The mechanical stability of the resonator was another problem faced with. Due
to frequent pressure changes fasteners of the mirrors are loosen and this destroys
mirror alignment. During the work bending mirrors were stabilized by locking of
fasteners but end mirror and output mirror were not stabilized since adjusting
micrometer screws were used.
75
The output power obtained was maximum 32 watts and this result is very far
from the design power of 225 watts even if two tubes were in operation. Main reason
of that result was the length between the anode and cathode. It is known that small
signal gain of a discharge decreases with the increase of length between anode and
cathode [27]. Keeping discharge length constant and increasing the anode-cathode
pairs is a way to increase the output power of the laser.
Operational experiments of the design were performed considering gas
composition, gas pressure and electrical feeding. Results consistent with general
relations obtained in previous works [12] [14] were obtained. Since all of such
experiments are not performed in maximum operation power or resulting in that
power, these results should be evaluated in qualitative manner rather than in
quantitative.
Experiments related with beam properties are performed by basic methods
although there are available instruments performing such measurements. Beam
profiling was performed by pinhole measurements and the size of the pinhole may be
critical in evaluation of the results. To minimize the effect of pinhole size on the
results 2D curve fitting was applied. This method was used to interpolate the values
of points that are not measured between the points that are measured. From obtained
profiles it was observed that the beam possess shape of ellipse rather than a circle
and this result is a consequence of bad alignment. Beam profiling was performed for
three different operational powers and the ellipticity was observed to increase with
increase in beam power. This increase may refer to either bad alignment or to the
destruction of discharge uniformity with increasing power. Further investigations of
the results should be performed.
Beam propagation factor and beam divergence were obtained by curve fitting
to data obtained from beam diameter measurements at both sides of the focus point
of a focused beam. Twelve diameters were measured for a beam and modified
Gaussian beam diameter function with respect to distance traveled were fitted to
these data. Curve fitting was done in regard to minimize the error of diameters at
data points between the calculated diameters from fitted curve and measured
diameters. From the fitted curve beam propagation factor, beam divergence,
Rayleigh range for the focused beam, focused beam diameter and focal depth of the
76
laser beam were obtained. Beam diameters at focal point of the lens were compared
to the theoretically calculated beam diameters. Diameter measurements were
performed for three beams with different power and for two axes in the cross-section
of a cut. So six different sets of beam parameters were obtained and compared.
Ellipticity of beams was observed also through diameter measurements and this
beam cross-section shape shows his effects on the beam properties. These effects
were different divergence and beam propagation factor for each axis of a given
power beam.
Possibility of the designed laser to be used in material processing was also
investigated at end of the work. Primarily materials that absorb 10.6 mm wavelength
radiation were considered. Fine cuts were obtained in paper, teflon and plexiglas.
Also it is possible to cut stainless steel if process is optimized. Range of materials
that can be processed with 30 W laser can be extended by further investigations.
During the work only cutting process was studied and any process optimization work
was not carried out. To settle down the application area of such an instrument
process optimization and economical effectiveness of the process need to work on.
77
REFERENCES
1. Patel C.K.N. ,1964, Physical Review Letters, V12, pp. 588
2. Patel C.K.N. 1964, Physical Review Letters, V13, pp. 617
3. Patel C.K.N. 1964, Physical Reviews, V136, A1187
4. Patel C.K.N. 1965, Physical Review Letters, V17, pp. 15
5. Roberts T.G., Hutcheson G. J., Ehrlich J.J., Hales W.L., Barr T.A., 1967,
IEEE Journal of Quantum Electronics, V3, pp 605-609
6. Bealieu A.U. 1970, Applied Physics Letters, V16, pp. 504
7. Festermacher C.A., Nutter, M.J., Rink, J.P., Boyer, K. 1971, Bulletin of
American Physical Society, V16, pp.42
8. Seguin H.J., Tulip, J., 1972, Applied Physics Letters, V21, pp.414
9. Deutch T.F., Horrigan, F.A., Rudko, R.I., 1969, Applied Physic Letters,
V15, pp.88
10. Cool T.A., Shirley, J.A., 1969, Applied Physic Letters, V14, pp70
11. Tiffany W.B., Targ, R., Foster, J.D., 1969, Applied Physic Letters, V15,
pp.91
12. Antropov E.T., Silin-Bekchurin, I.A., Sobolev, N.N., Sokovikov, V.V.,
1968, IEEE journal of Quantum Electronics, V4, pp.790
13. Matrhews S., 2001, Laser Focus World, V37, pp.185
14. Cheo P.K.,1967, IEEE Journal of Quantum Electronics,V3, pp.683
15. Tyte D. C.,1970, In `Advances in Quantum Electronics` (D.W. Goodwin,
ed.), Vol. 1, Academic Press, New York
16. Ballik E.A., Garside, B.K., Reid, J., Tricker, T.,1975, Journal of Applied
Physics, Vol. 46, pp. 1322
17. Rigrod W. W.,1965, Journal of Applied Physics, V34, pp. 2487
18. Taylor R.L., Bitterman, S.,1969, Review of Modern Physics, V41, pp.26
78
19. Fowler M.C., 1972, Journal of Applied Physics, V43, pp. 3480
20. Menzel Ralf, 2001, Photonics- Linear and Nonlinear Interactions of Laser
Light and Matter, Springer-Verlag, Berlin,
21. Kogelnik H. , Li, T., 1966, Proceedings of the IEEE, pp. 97
22. Jenkins F.A., 1976, Fundamentals of Optics, 4th Ed., McGraw-Hill, Tokyo,
23. Koechner W., 1999, Solid State Laser Engineering, 5
th
Ed., Springer-Verlag,
Berlin,
24. Weber M. J., 2002,
Handbook of Optical Materials, 1
st
ed., CRC Press, New
York
25. Callister W.D., 1997, Materials Science and Engineering: an Introduction, 4
th
ed., John Willey & Sons, New York
26. Welch M., Lasers&Applications , 1986, pp.67-71
27. Bogaerts A., Grozeva, M., 2002, Applied Physics B, V75, pp.731-738
28. Sherman Glen F., Danielewicz, Edward J., Rudisill, J.Earl, Lasers &
Optronics, September 1988
79
APPENDIX I
LASER DESIGN CALCULATIONS
A1.1 - Laser Output Parameters
P
out
225
W:=
θ210
3−
⋅rad:=
TEM
00
+ TEM
01
* mode of operation
A1.2 - Slow flow laser power constants
p
00
50
W
m
:= for laser operation in TEM
00
mode
p
01
75
W
m
:= for laser operation in TEM
00
+ TEM
01
*mode
A1.3 - Calculation of Discharge and Resonator Lengths
A1.3.1- Discharge Length
L
d
P
out
p
01
:= L
d
3m=
80
A1.3.2- Optical Length
L 1.25 L
d
⋅:=
L 3.75m=
A1.4 - Calculation of mirror radii
A1.4.1-Curvature of End Mirror
λ10.6µ
m:=
w
o
2λ⋅
θπ⋅
:= w
o
3.374mm=
R
2
w
o
2π
λ
⋅
2
1
L
⋅L+:= R
2
6.786m=
A mirror with larger value that is in hand is selected
R
2
8
m:=
A1.4.2 - Curvature of Output Mirror
Resonator type is hemispherical and the output coupler is plane.
R
1
10
20
m:= (i.e. plane)
A1.5- Stability of The Resonator
g
1
1
L
R
1
−:= g
2
1
L
R
2
−:= g
1
g
2
⋅0.531= 0 < g
1
.g
2
<1
A1.6- Beam diameters on mirrors
A1.6.1- Output Beam Diameter for Fundamental Mode
w
o
λ
π
LR
2
L−()⋅⋅
1
2
:= D
o
2w
o
⋅:= D
o
7.3mm=
81
A1.6.2-Beam Diameter on the End Mirror for Fundamental Mode
w
e
λR
2
⋅
π
2
R
1
L−
R
2
L−
⋅
L
R
1
R
2
+ L−
⋅
1
4
:= D
e
2w
e
⋅:= D
e
10.1mm=
A1.6.3 - Mode Content of Resonator
C
1
1.9
2.42
1.5
2.15
2.63
:= Mode Coefficients [23]
Laser TEM mode abbreviations
TEM
00
10
20
01
11
21
:=
A1.6.3.1 - On output mirror
D
o
2C⋅w
o
⋅:= D
o
7.3
13.9
17.8
11
15.8
19.3
mm=
A1.6.3.2 - On end Mirror
D
e
D
o
g
1
g
2
⋅:= D
e
10.1
19.1
24.4
15.1
21.7
26.5
mm=
A1.6.4 - Selection of tube diameter
TEM 01 diameter on the output mirror is
D
o
01,
11m
m= and on the end mirror is
D
e
01,
15.1m
m=
Selected Diameter :
D
tube
16.4m
m:=
82
A1.7- Beam Divergence of the Fundamental Mode
θ
2λ⋅
πw
o
⋅
:= θ1.839 10
3−
× rad=
A1.8 - Transmittance of Output Coupler
A1.8.1 - Mirror Losses
a
1
0.0
1:=
a
2
0.005:= a
3
0.005:= a
4
γ()γ:=
A1.8.2 - Mirror Reflectances
r
1
1a
1
a
2
+ a
3
+
( )−:=
All total reflectors are considered as one
r
2
γt,() 1t−
γ−:=
Reflectance of output mirror
A1.8.3 - Calculation of transmittance by Rigrod's method
ηtL,g
o
,γ,()
r
1
0.5
t⋅
r
1
0.5
r
2
γt,()()
0.5
+
1r
1
r
2
γt, ()⋅()
0.5
−
⋅
1
ln r
1
r
2
γt,
()⋅()
2g
o
⋅L⋅+
⋅:=
83
A1.8.3.1 - Behavior of Rigrods formula for different conditions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11 % loss curve
10 % loss curve
30 % loss curve
50 % loss curve
80 % loss curve
Efficiency Curve for increasing losses
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.1 - Efficiency Curve for increasing losses
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 m length curve
2 m length curve
4 m length curve
6 m length curve
Efficiency Curve for increasing length
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.2- Efficiency Curve for increasing length
84
A1.8.3.2 - Selection of go for transmission calculations
From Fig 5 for a tube with diameter of 16.4 mm and pressure of 4 torr value of 3
was selected.
α3d
B:= α10−log g
o ()⋅
g
o
10
α−
101
m
⋅:= g
o
0.501
1
m
=
A1.8.3.3 - Efficiency plot and Transmission determination for design
L
d
3m= g
o
0.501
1
m
=
γincreasing
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1% loss curve
2% loss curve
3% loss curve
4% loss curve
5% loss curve
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.3 - Efficiency Curve for increasing losses of designed laser
T20%:= is selected for the output transmission
85
A1.9 - Selection of Mirror Materials
A1.9.1- Output Coupler Material
Table A1.1- Material Properties of possible Output Coupler Material s [24]
Function for figure of merit:
F
K
AX⋅
K - Thermal Conductivity ; A - Total Absorption
X
1
T
L
d
dT
n
d
d
+ Thermal Expansion coefficient + Refractive index gradient
or
X
2
T
L
d
d
Thermal Expansion
FψA,∆L,()
ψ
mK⋅
W
⋅
A∆L
()⋅ K⋅10
8
⋅
:=
Figure of Merit Function for thermal Expansion
FψA,∆L,∆n,()
ψ
mK⋅
W
⋅
A∆L∆n+
()⋅ K⋅10
8
⋅
:=
Figure of Merit Function for Thermal
Expansion coefficient plus Refractive index gradient
Material
Thermal
Expansion
Coefficient
(10
-6
/K)
Thermal
Conductivity
(W/m.K)
Thermo
Optic
Coefficient
(10
-6
/K)
GaAs 5 65 200
Ge 5.7 59.9 401
KCl 36.5 6.7 -34.8
ZnSe 7.1 13 61
86
0 0.02 0.04 0.06 0.08 0.1
0
2
4
KCl
GaAs
ZnSe
Ge
FM plot for Thermal Expansion
Surface Absorbtion (%)
Figure of Merit
Figure A1.4 - Figure of Merit for Thermal Expansion
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
KCl
GaAs
ZnSe
Ge
FM for TE plus Thermooptic Coefficient
Surface Absorbtion ( % )
Figure of Merit
Figure A1.5 - Figure of Merit for Combined Thermal Expansion and Refractive gradient
change effect
87
A1.9.2 - Material Selection for Full Reflector Mirrors
Table A1.2 – Material Properties of Possible Materials for Total reflector mirrors [24]
Material
Thermal
Expansion
Coefficient
(10
-6
/K)
Thermal
Conductivity
(W/m.K)
Cu 17 398
Si 4.9 142
Mo 2.5 141
FψA,∆L,()
ψ
mK⋅
W
⋅
A∆L
()⋅ K⋅10
9
⋅
:=
Figure of Merit Function for thermal Expansion
0 0.04 0.08 0.12 0.16 0.2
0
2
4
Cu
Mo
Si
Ge
Surface Absorbtion (%)
Figure of Merit
Figure A1.6 - Figure of Merit for Thermal Expansion for total reflection mirrors
88
APPENDIX II
REALIZED DESIGN AND TECHNICAL DRAWINGS
a)
b)
c)
Figure A2.1- Realized design: a) General view, b) Cathode side, c) Anode side
89
Figure A2.2- General Top view of the Design
90
Figure A2.3- Cathode side view of the Design
91
FigureA2. 4- Anode side view of the Design
92
FigureA2. 5- Beam Bending Mirrors Assemble view of the Design
93
APPENDIX III
BEAM PROFILING
A3.1 - Obtained Data from pinhole power measurements
Data
12345678910
1
2
3
4
5
6
7
8
9
10
0.06 0.07 0.07 0.07 0.07 0.06 0.07 0.06 0.05 0.05
0.05 0.06 0.06 0.07 0.08 0.07 0.07 0.05 0.04 0.03
0.06 0.07 0.07 0.07 0.09 0.1 0.1 0.08 0.05 0.04
0.06 0.06 0.07 0.1 0.14 0.15 0.17 0.1 0.06 0.04
0.05 0.05 0.06 0.08 0.15 0.2 0.15 0.12 0.07 0.04
0.05 0.05 0.05 0.08 0.15 0.23 0.16 0.12 0.08 0.06
0.05 0.05 0.06 0.08 0.13 0.17 0.14 0.09 0.07 0.05
0.05 0.05 0.06 0.07 0.11 0.12 0.1 0.07 0.07 0.05
0.05 0.05 0.05 0.06 0.07 0.08 0.07 0.06 0.06 0.04
0.05 0.05 0.03 0.05 0.06 0.05 0.05 0.06 0.06 0.05
W=
A3.2 - Data Normalization
Data are normalized to the maximum power measured.
M
Data
max Data()
:=
94
M
12345678910
1
2
3
4
5
6
7
8
9
10
0.261 0.304 0.304 0.304 0.304 0.261 0.304 0.261 0.217 0.217
0.217 0.261 0.261 0.304 0.348 0.304 0.304 0.217 0.174 0.13
0.261 0.304 0.304 0.304 0.391 0.435 0.435 0.348 0.217 0.174
0.261 0.261 0.304 0.435 0.609 0.652 0.739 0.435 0.261 0.174
0.217 0.217 0.261 0.348 0.652 0.87 0.652 0.522 0.304 0.174
0.217 0.217 0.217 0.348 0.652 1 0.696 0.522 0.348 0.261
0.217 0.217 0.261 0.348 0.565 0.739 0.609 0.391 0.304 0.217
0.217 0.217 0.261 0.304 0.478 0.522 0.435 0.304 0.304 0.217
0.217 0.217 0.217 0.261 0.304 0.348 0.304 0.261 0.261 0.174
0.217 0.217 0.13 0.217 0.261 0.217 0.217 0.261 0.261 0.217
=
A3.3 - 2D Curve fitting
Since obtained data correspond to specified points on the beam cross section, value of
surrounding and not measured points can be found by interpolation. For our case 2D cubic spline
will be used for curve fitting interpolation function.
rows M() 10= cols M() 10= n rows M():=
X and Y n-vectors that determine the mesh for the matrix:
X
0
1
2
3
4
5
6
7
8
9
:= Y
0
1
2
3
4
5
6
7
8
9
:=
Mxy augment sort X( ) sort Y(),():=
rows Mxy()10=
Computed spline coefficients:
S cspline Mxy M,():=
Fitting function for surface:
fit x y,( ) interp S Mxy, M,
x
y
,
:= xlow Mxy
11,
:= xhigh Mxy
n1,
:=
95
ylow Mxy
1
2,
:= yhigh Mxy
n2,
:=
Density of mesh for interpolation:
xn 10
n⋅:= yn 10 n⋅:= i1x n..:=
j1y n..:=
xind
i
xlow i
xhigh xlow−
xn
⋅+:= yind
j
ylow j
yhigh ylow−
yn
⋅+:=
Curve Fitting Function:
FIT
ij,
fit xind
i
yind
j
,
( )
:=
Surface Plot of 2D Spline Interpolated Surface
Original Surface
Figure A3.1- 3D surface plot of beam intensity profiler
Contour Plot of 2D Spline Interpolated Surface
Contour Plot of Original Surface
Figure A3.2- Contour plot of beam intensity profiles
96
APPENDIX IV
KNIFE EDGE METHOD FOR LASER BEAM DIAMETER
MEASUREMENTS
A4.1 - Description of Method
A plate with very thin edge (Knife Edge) is placed perpendicularly on the path of
propagation of the beam, which diameter is desired to be measured, such that to prevent the beam
to pass across the knife edge plate. After that the knife edge plate is pulled out by recording
power passing across the edge with respect to distance traveled. Recorded powers are divided by
the maximum power recorded and multiplied by one hundred in order to obtain percentage
values with respect to the beam total power. Percentage power changes are plotted with respect
to traveled distance and distance between %15.9 and % 84.1 is obtained graphically. Two times
that distance gives the diameter of the beam. This diameter is equal to the (1-1/e
2
) of the
diameter of a Gaussian beam.
97
A4.2 - Obtained Data
a 0.0254mm:= Distance traveled by one step of micrometer screw.
Distance Traveled: Power corresponding to traveled distance:
X
0
25
75
100
125
150
175
200
225
250
275
300
350
400
a⋅:=
P
0.77
1.18
2.41
3.45
4.76
6.33
8.59
10.99
13.22
15.35
17.02
18.19
19.59
20.58
W:=
A4.3 - Data Analysis
0 5 10 15
0
50
100
Travelled distance (mm)
Percentage Power (%)
Figure A4.1 - Percentage power as a function of distance traveled.
P
c
M
i
P
i
max P()
100⋅←
i 1 rows P()..∈for
M
:=
The Loop on the left estimates percentage of transmitted power with
respect to total beam power.
98
A4.3.1 - Fitting Cubic Spline to approximate obtained data
Algorithm a Cubic Spline fit in MathCAD®:
data
M
i1,
X
i
mm
←
M
i2,
P
c
i
←
i 1 rows P()..∈for
M
:=
data csort data 1,():= X data
1〈〉
:=
Y data
2〈〉
:=
Spline coefficients:
S cspline X Y,():=
Fitting function:
fit x( ) interp S X,Y,x,():=
Figure A4.2- Plot of experimental data, Fitted curve and 15.9% and 84.1 % lines as a
function of distance.
Beam Diameter:
D root fit x( ) 15.9− x,0,7,( ) root fit x ( ) 84.1− x,5,10,()−
( )2⋅:= D 9.353=
99
APPENDIX V
DETERMINATION OF BEAM PROPERTIES
A5.1- Experimental Data
Distance from the reference
Point
Measured diameters for 10W, 20W, 30 W
laser beams in x and y-directions. First, third
and fifth columns are diameters measured for
x –axis and the others are for y-axis
respective to power increasing order
Z
20
50
70
85
90
95
100
105
110
120
130
150
:=
W
7.246
4.178
2.111
1.09
0.531
0.137
0.281
0.579
1.039
2.097
2.865
4.411
7.377
4.317
2.451
1.083
0.486
0.146
0.227
0.577
1.219
2.009
2.85
4.628
8.255
4.386
2.63
0.87
0.433
0.067
0.256
0.537
1.161
2.033
2.993
4.602
7.897
4.686
2.639
1.051
0.812
0.255
0.245
0.91
1.263
2.303
3.049
5.136
7.403
4.231
2.248
0.94
0.331
0.175
0.147
0.603
1.215
2.218
3.119
4.929
8.44
4.847
2.64
1.186
0.4
0.113
0.279
0.454
1.281
2.131
3.424
5.286
=
A5.2 – Derivation of Curve fit Equation
The spot size of a Gaussian beam a distance z from the beam waist expands as a hyperbola,
which has the form
ωz()ω o1
λz
πω
o()
2
2
+
1
2
(A5-1)
100
Divergence of Gaussian Beam at far field:
θ
2λ
πω
o
⋅
(A5-2)
By inserting (A5-2) in (A5-1):
ωz()ω o1
θz
2ω
o⋅
2
+
1
2
(A5-3)
Divergence of a real beam:
ΘMθ⋅
(A5-4)
Diameter of a real beam at waist:
W
o
Mω
o
⋅ (A5-5)
Beam Product of a real beam:
W
o
Θ⋅M
2
θ⋅ω
o
⋅
M
22λ⋅
πω
o
⋅
⋅ω
o
⋅ (A5-6)
Beam Propagation Factor:
M
2
πΘ2W
o()
4λ
(A5-7)
By inserting (A5-7) in (A5-3):
Wz() W
o
1
M
2
λ⋅z⋅
πW
o()
2
2
+
1
2
(A5-8)
Replacing
Wz()
by 2W(z) and
W
o
by
2W
o
⋅
in equation (A5-8)
2W z() 2W
o
1
4M
2
λ⋅z⋅
π2W
o()
2
2
+
1
2
(A5-9)
Replacing 2W(z) by W(z) and 2W
o by Wo in equation (A5-9) we obtain a function giving beam
diameters
Wz() W
o
1
4M
2
λ⋅z⋅
πW
o()
2
2
+
1
2
(A5-10)
and this equation will be fitted to data obtained.
101
A5.3 - Curve Fitting
Since from obtained data we cannot conclude on waist diameter, location of waist and
beam quality factor a simple approach is to define a range for these constants and to calculate
equation (A5-10) over that range at data points and to minimize the error between calculated and
obtained beam diameters. The constants for which the errors between calculated and measured
results are minimum are the parameters of our beam.
Range for
z
o
: 90 110..
Range for
W
o
:
1
2
min W
column_number
〈〉
()⋅ 4 min W
column_number
〈〉
()⋅..
Range for
M
:
12.5..
Curve fitting Function:
ωxw
o
,z
o
,M,() w
o
1
4M
2
⋅λ xz
o
− ()⋅
πw
o
()
2
⋅
2
+
1
2
⋅:= (A5-11)
Minimum Error Loop:
I
err 10←
ror
1
rows Z()
i
W
j〈〉()
i
ωZ
i
φ,υ,τ,()
−
∑
=
←
U
1j,
φ←
U
2j,
υ←
U
3j,
τ←
err ror←
err
err ror>if
err
τ1 1.01, 5..∈for
υ90 90.25, 110..∈for
φ0.5 min W
j〈〉()
⋅ 0.5 min W
j〈〉()
⋅ 0.01+()
, 4 min W
j〈〉()
⋅..∈forj 1 cols W()..∈for
U
:=
102
Loop Results:
I
0.168
97.25
1.05
0.153
97.75
1.01
0.164
97.75
1.06
0.143
97.5
1.02
0.153
96
1.02
0.147
97.5
1.05
=
A5.4 - Beam Constants
A5.4.1- Waist Location
z
o
j
I
2
j,
:=
z
o
97.25
97.75
97.75
97.5
96
97.5
=
A5.4.2- Rayleigh Range for each beam
Z
R
j
I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
2
I
1j,
⋅− x,I
2j,
,160,( )− I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
2I
1j,
⋅− x,50,I
2j,
,( )−+
2
:=
Z
R
1.908
1.7
1.763
1.446
1.678
1.442
=
A5.4.3 - Beam Propagation Factor
M
j
I
3j,
()
:=
M
j()
2
1.103
1.02
1.124
1.04
1.04
1.103
=
A5.4.4 - Beam Divergence
Θj
I
1j,
Z
R
j
:=
Θj
88.3
90
92.7
98.5
91.5
101.6
mrad=
103
0 16 32 48 64 80 96 112 128 144 160
0
1
2
3
4
5
6
7
8
9
10
10 W x-axis
10 W y-axis
20 W x-axis
10 W y-axis
30 W x-axis
30 W y-axis
Curve of 10 W x-axis
Curve of 10 W y-axis
Curve of 20 W x-axis
Curve of 20 W y-axis
Curve of 30 W x-axis
Curve of 30 W y-axis
Distance Z (mm)
Diameter D (mm)
Figure A5.1 - Experimental data with fitted curves along the propagation direction
104
A5.4.5 - Theoretical Calculation of Beam Diameters at focal point of the lens
f 100m
m:= Lens focal length
D
9.026
8.698
8.574
9.353
8.691
9.937
mm:= Beam Diameters on the lens
d
j
4
π
λf⋅
D
j
⋅ M
j()
2
⋅
0.0286
D
j()
3
f
2
+:= d
0.167
0.16
0.179
0.152
0.163
0.153
mm=
difference
j
d
j
mm
I
1j,
−
d
j
mm
100⋅:= difference
j
0.9
4.5
8.5
6.5
6.1
4
=
A5.4.6 - Focal Depth
z
f
j
I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
1.05
I
1j,
⋅− x,I
2j,
,160,( )− I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
1.05I
1j,
⋅− x,50,I
2j,
,( )−+( )mm:=
z
f
0.85
0.76
0.79
0.65
0.75
0.65
mm=
105
APPENDIX VI
BEAM DELIVERY UNIT and NOZLE DRAWING
Figure A6.1 – Beam delivery unit
106
Figure A6.2 - Laser beam focusing head and nozzle
FOLDED CARBON DIOXIDE LASER
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
THE MIDDLE EAST TECHNICAL UNIVERSITY
BY
NECMETTİN KENAR
IN PARTIAL FULFILLEMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
THE DEPARTMENT OF PHYSICS
AUGUST 2003
ii
Approval of the Graduate School of Natural and Applied Sciences
______________________________
Prof. Dr. Canan ÖZGEN
Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of
Master of Science.
_____________________________
Prof. Dr. Sinan BİLİKMEN
Head of Department
This is to certify that we have read this thesis and that in our opinion it is fully
adequate, in scope and quality, as a thesis for the degree of Master of Science
___________________________
Assoc. Prof. Dr. Gülay ÖKE
Superviser
Examining Committee Members
Prof. Dr. Sinan BİLİKMEN ___________________________
Assoc. Prof. Dr. Serhat ÇAKIR _________________________
Assoc. Prof. Dr. Akif ESENDEMİR ___________________________
Dr. Ali ALAÇAKIR ____________________________
Assoc. Prof. Dr. Gülay ÖKE ____________________________
iii
ABSTRACT
DESIGN AND CONSTRUCTION OF AXIAL SLOW FLOW CONTINUOUS
WAVE FOLDED CARBON DIOXIDE LASER
KENAR, Necmettin
M.S., Department of Physics
Supervisor: Assoc. Prof. Dr. Gülay ÖKE
August 2003, 106 pages
Design and realization of a conventional carbon dioxide laser was performed.
Gas composition and gas pressure effects on laser output power were studied. Effects
of input electrical power and current on laser power were also investigated. Beam
profiling of the laser beam was performed by pinhole method. Laser beam
parameters like beam divergence, beam propagation factor were measured. These
properties were extracted from focusing a laser beam in near field and performing a
number of cuts across the beam cross-section and measuring the beam diameter at
these points. Diameter measurements were obtained by knife edge method. Laser
beam parameters were obtained for three different power laser beams in two axes
across the beam. Found parameters were compared with regard to beam power and
beam cross-section axis. Also possibility of using the obtained laser beam in material
processing was investigated.
Keywords: Carbon dioxide laser, beam profiling, laser beam propagation
factor.
iv
ÖZ
YAVAŞ EKSENEL AKIŞLI SUREKLİ DALGA BÜKÜK KARBON D İOKSİT
LASERİ TASARIM VE YAPIMI
KENAR, Necmettin
Yüksek Lisans, Fizik Bölümü
Tez Yöneticisi: Doç. Dr. Gülay ÖKE
Ağustos 2003, 106 sayfa
Geleneksel karbon dioksit laseri tasarımı ve gerçekleştirilmesi çalışması
yapılmıştır. Gaz karışım oranlarının ve gaz basıncının laserin çıkış gücü üzerine
etkileri çalışılmıştır. Elektrik giriş gücü ve akımının laserin gücü üzerine etkileride
ayrıca incelenmiştir. Laser demeti kesitinde üç boyutlu güç şiddeti profili iğne deliği
yöntemiyle çıkartılmıştır. Laser demeti parametreleri , demet genişleme ve demet
yayılma katsayısı ölçülmüştür. Bu özellikler yakın mesafede laser demeti
odaklanarak, demetten çok sayıda kesit alınarak ölçülen demet çaplarından
çıkartılmıştır. Çap ölçümleri bıçak kenarı yöntemi ile yapılmıştır. Laser demet
parametreleri üç ayrı güçte laser demetinin her iki kesit ekseni için elde edilmiştir.
Bulunan parametreler demet gücü ve eksen açısından değerlendirilmiştir. Ayrıca elde
edilen laser demetinin malzeme işlemede kullanılma imkanları araştırılmıştır.
Anahtar Kelimeler: Karbon dioksit laseri, laser demeti güç şiddeti profili,
laser demeti yayılma katsayısı.
v
In Memory of Dr. Iulian GUTU
vi
ACKNOWLEDGEMENTS
I would like to express my sincere thanks to my advisor Assoc. Prof. Dr. Gülay
ÖKE for her support and careful reading of manuscript.
I would like to express my sincere thanks to Prof. Dr. Sinan BİLİKMEN for
the opportunity to study on CO
2 lasers as a member of Laser Laboratory.
I am very grateful to late Dr. Iulian GUTU for the discussions on CO
2 lasers
and laser material processing. He was the person who established my basics on the
subject.
I am grateful to Dr. Ali Alacakır for tolerating my faults and his help during
experiments.
I am also grateful to Mr. Oğuz PERVAN for his encouragement and help
during experiments.
I thank to Dr. Hilal GÖKTAŞ for allowing me to use his instruments.
Special thanks go to İlker YILDIZ and Ertan ERYILMAZ for their friendship
and help during my work.
I express my thanks to our technicians İsmail DOĞRU, Muharrem KUZU and
Murat AYDIN for their help.
I would like to thank to Mr. Semih ÖZEL from Genel Makina Tasarım Ltd. Şti.
for manufacturing of beam delivery unit and 3-axis stage, without his kind help this
work could not be completed.
I would like to thank to Mr. Hikmet ÖZGÜR from Makina Dizayn Ltd. Şti. for
spending his valuable time in supplying material that I could not find,
Also I thank to Mr. İbrahim DİNÇER from Dinçer Medikal Ltd. Şti. for his
valuable discussions and encouragement.
At last but not least I would like to express my sincere thanks to my family for
their patient and support during my work.
vii
TABLE OF CONTENTS
ABSTRACT……………………… ………………………………………………iii
ÖZ…………………………………………… ……… ……………….……………iv
ACKNOWLEDGMENT …………………………… ………………...…...…….vi
TABLE OF CONTENTS……… ………… ……………………………...……vii
LIST OF TABLES……………………………………………………………………………. .x
LIST OF FIGURES…………………………………… … … ………………….… …….….xi
CHAPTER
1.
INTRODUCTION………………………………………….…….………1
2.
FUNDAMENTALS OF CO 2 LASER……………….………….….....3
2.1-
History……………………………………………………….…...……3
2.2-
Slow Axial Flow CO2 Laser……………………….………..4
2.2.1- General Design………………………………..……….…………5
2.2.2- Gas Composition and Gas Flow……………………..……...……6
2.2.3- Tube Diameter…………………………………………...……….8
2.2.4- Pressure and Current…………………………...……….…..….…9
2.2.5- Discharge Energy and Gas Temperature……………….……….10
2.2.6- Resonator Configuration and Output Coupling………….…...…14
3.
THEORY OF CO2 LASER . . . . . . . . . . . . . . . . . . . . . . . . . . ……. . . . 18
3.1
- Carbon Dioxide Molecule . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 18
viii
3.2-
Excitation of Carbon Dioxide . . . . . . . . . . . . . . . .. . . . . ………. . . . . . . 20
3.3-
Relaxation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4
- General Energy Transfer Processes in CO2 Laser . . . . . . . . . 21
4.
LASER BEAMS AND THEIR PROPERTIES . . . . . . ……….. . . 25
4.1-
Transverse Modes of Laser Resonators . . . . . . . . . . . . . . .. . . . 25
4.1.1- Intensity Distribution of Transverse Modes . . . . . . . . . . . ... . . . 25
4.1.2- Characteristics of Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.3- Laser Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.4- Stability of a Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
4.2
- Beam Propagation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3-
Focusing of a Laser Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1- Diffraction Limited Spot Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2- Effects of M
2
on Focusing. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .36
4.3.4- Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.5- Depth of Focus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 39
5.
DESIGN OF SLOW AXIAL FLOW CO 2 LASER. ……….... . . . .40
5.1-
Laser Applications and Specification of Laser Output Beam
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2-
Determination of Laser Resonator Parameters ……………….41
5.2.1- Resonator Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
5.2.2- Discharge and Resonator Lengths . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.3- Determination of End Mirror Curvature and Stability Check . . 42
5.2.4- Determination of Laser Tube Bore Diameter. . . . . . . . . . . . . . . .43
5.2.5–Determination of Output Coupler Transmittance . . . . . . . .. . . . . 43
5.2.6–Selection of Mirror Material. . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
5.2.6.1- Material Selection for Output Coupler . ........................ 46
5.2.6.2- Material Selection for Full Reflector Mirrors. . . . . . . . 48
6
. EXPERIMENTS CARRIED OUT ON THE DESIGNED
LASER
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1-
Experiments on Laser Gas and Electrical Feeding. . . . . . . . . 50
ix
6.1.1- Laser System Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1.2- Current - Partial Pressure Characteristics of Laser Discharge .. 52
6.1.3- Effects Partial Gas Pressures on Output Laser Power. . . . . . .. . .53
6.1.4- Effects of Gas Composition on Output Power. . . . . . . . . . . . . . .54
6.1.5-Laser Discharge with Constant Gas Composition . . . . . . . .. . . . . 56
6.1.6- Maximum Power Obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
6.2-
Experiments on Laser beam Properties . . . . . . . . . . . . . . . . . . 61
6.2.1- Laser Beam Profiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
6.2.2- Beam Propagation Parameter and Divergence . . . . .. . . . . . . . . 66
6.3-
Experiments on Material Processing . . . . . . . . . . . . . . . . . . . . 71
7.
DISCUSSION and CONCLUSION ……………….….……………...74
REFERENCES…………………………………………………….……….…….77
APPENDICES
I. LASER DESIGN CALCULATIONS . . . . . . . . . . . . . . . . ... . . . . . . ……..79
II.
REALIZED DESIGN AND TECHNICAL DRAWINGS . ... …………88
III.
BEAM PROFILING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . ................93
IV.
KNIFE EDGE METHOD FOR LASER BEAM DIAMETER
MEASUREMENTS
. ……………….... . . . . . . . . . . . . . .. . . . . . . . . . . ……96
V. DETERMINATION OF BEAM PROPERTIES . . . . . . . . . . . …..……..99
VI
. BEAM DELIVERY UNIT AND NOZLE DRAWING . . . …………105
x
LIST OF TABLES
TABLE
6.1- Results on Laser Beam Properties
………………………………………...… 70
A1.1- Material Properties of possible Output Coupler Materials………...…85
A1.2- Material Properties of Possible Materials for Total Reflector
Mirrors………………………………………………………...………87
xi
LIST OF FIGURES
FIGURES
2.1- Schematic diagram of a conventional CO
2 Laser………………………………..5
2.2- Gain versus total mixture pressure……………………………………………....6
2.3- Excitation Mechanism of CO
2 Laser………………………………………….....7
2.4- Gain versus CO
2 flow rate for CO2:N2: He mixture at near optimum pressure
and mixture ratio………………………………………………………………..8
2.5- Gain versus tube diameter for optimum current density………………………...9
2.6- Gain versus discharge current for various flowing gas media at optimum
mixture ratios and a constant flow rate in 12 and 37 mm bore amplifier tubes...10
2.7- Optimum current and total pressures for maximum cw oscillator output power
as a function of tube diameter…………………………………………………11
2.8- Spatial variation of gain for a 3% CO
2: 10%N2: 87% He mixture at 100 torr.
Two gain distributions are shown; the one on the left corresponds to energy
input of 120 J/l atm. The other gain distribution corresponds to an input
energy of 350 J/l atm. Representative gas temperatures are also shown……...12
2.9- Spatial variation of gain for a 10%CO
2: 90% He mixture at 150 Torr…...…...12
2.10- Carbon-monoxide production in a 10%CO
2: 90% He mixture at 150 Torr.…13
2.11- Small-signal gain and gas temperature as a function of energy……………...13
2.12- Resonator Configurations…………………………………………………….15
2.13- Graph of output power versus window reflectivity…………………………..16
2.14- Optimum reflectivity as a function of the ratio of laser length to diameter….16
3.1- The fundamental modes of vibration of the CO
2 molecule…………………....19
3.2- Schematic diagram of the CO
2 laser mechanism showing relaxation times for
typical gas mixture (1:1:8 of CO
2:N2:He) at a total pressure of 15 torr and a
xii
gas-kinetic temperature of 420
o
K. Level populations shown are equilibrium
vibrational populations in the absence of laser action……………………..…..22
3.3- Diagram of some of the CO
2 and N2 vibrational levels involved in the
analysis…………………………………………………………………………24
4.1- Intensity profile of higher transversal modes of stable laser resonators with
circular symmetry…………………………………………………………….26
4.2- Contour of a Gaussian beam……………………………………………….…...28
4.3- Mode parameters of interest for a resonator with mirrors of unequal
curvature………………………………………………………………………..29
4.4- Stability diagram. Unstable resonator systems lie in the shaded regions……....32
4.5- Divergence determined by measuring the waist size behind a focusing lens…33
4.6- Diagram illustrating the diffraction limited spot size……………………….….35
4.7- Spherical aberration of a single lens focusing a parallel beam…………………37
4.8- A graph of spherical aberration for lenses of different shape but the same
focal length………………………………………...…………………………...38
5.1- Output Coupling Efficiency versus transmittance plot………………………...44
5.2- Optical distortion figures of merit compare substrate materials for CO
2 laser
output coupler…………........…………………………………………………47
5.3- Optical distortion figures of merit compare substrate materials for CO
2 laser
full reflector mirrors…………………………………………………….…….48
6.1- Experimental setup…………………………………………………………….51
6.2- Current -partial pressure behavior of laser gas constituents…………….……..52
6.3- Effect of partial pressures on output power……………………………………54
6.4- Gas composition effects on output power……………………………….…….55
6.5- Effect of current on laser power ……………………………………………….56
6.6- Efficiency versus current graph ……………………………………………….57
6.7- Efficiency versus laser power graph……………………………………..…….58
6.8- Effects of input power on output laser power……………………………...…..58
6.9- Effects of pressure on output laser power…………………………………..…59
6.10- Contour plot of combined effects of pressure and input power on output
laser power…………………………………………………………………...59
6.11- Maximum power obtained for single tube operation…………………………60
xiii
6.12- Beam brofiling method of laser beam…………………………..……………62
6.13- Beam intensity profiles obtained from different laser powers, a) 10 W, b) 20
W, and c) 30 W……………………………………………………………….64
6.14- Burn patterns of laser beam on wood for 10W, 20 W and 30 W…………….65
6.15-Radial intensity distribution for TEM
00, TEM01 and sum TEM00+TEM01…....66
6.16- Experimental setup for knife edge method………………………….………..67
6.17- Curve fitted to data of 10 W laser beam……………………………….……..69
6.18- Processed materials; a) Teflon processing, b) Teflon processed, c) Paper
processed, d) Cut performed in plexiglas, e) Stainless steel processing, d)
Processed stainless steel sheet……………………………………………..…73
A1.1- Efficiency curve for increasing losses…………………………..…………...83
A1.2- Efficiency curve for increasing length………………………………………83
A1.3- Efficiency curve for increasing losses of designed laser……………..….…..84
A1.4 - Figure of merit for thermal expansion…………………………………..….86
A1.5- Figure of merit for combined thermal expansion and refractive gradient
change effect………………………………………………………………...86
A1.6- Figure of merit for thermal expansion for total reflection mirrors……….....87
A2.1- Realized design: a) General view, b) Cathode side, c) Anode side ………....88
A2.2- General top view of the design ………………………………….…………..89
A2.3- Cathode side view of the design …………………………………..…………90
A2.4- Anode side view of the design ……………….………....................................91
A2.5- Beam bending mirrors assemble view of the design ………..........................92
A3.1- 3D surface plot of beam intensity profiler…………………………………...95
A3.2- Contour plot of beam intensity profiles……………………………………...95
A4.1- Percentage power as a function of distance traveled………………………...97
A4.2- Plot of experimental data, fitted curve and 15.9% and 84.1 % lines as a
function of distance…………………………………………………………..98
A5.1- Experimental data with fitted curves along the propagation direction……..103
A6.1 - Beam delivery unit…………………………………………………………105
A6.2 - Laser beam focusing head and nozzle…………………………………......106
1
CHAPTER 1
INTRODUCTION
In the world of high speed technological development, lasers as general take
their place on top of the subjects studied in the last 40 years. The needs of industry
force the research and development of lasers. Lasers especially that with high power
have wide application areas in manufacturing industry. Their main application areas
are cutting (about %80), welding, heat treatment, alloying and cladding. In these
manufacturing areas the use of CO
2 and Nd:YAG lasers is dominated. Since the
developments in solid state lasers in the last ten years increase their use in
manufacturing market, but still CO
2 lasers are widely used and technological
developments in CO
2 laser fixes their application areas in the market. CO2 lasers are
still cheap and have better beam properties than solid state lasers and this makes
them worth to work on.
In our country very little work on CO
2 lasers is done and needs of our market
for cheap and powerful lasers was the main reason of this work. The lack of
experience in this area force us to start with the basic laser design structures. Also
need of practical methods for determining the beam properties of a powerful laser
beam oriented us to use basic methods such as pinhole and knife edge techniques.
The manuscript text of the work was prepared in such an order first to give basics of
CO
2 lasers than their theory and than to pass to basic design needs according to
which design calculations are made. After the design calculations the realization of
the design was described and experiments carried out were discussed.
2
In Chapter 2, main design criteria of slow flow axial laser is given and
background covering the engineering and technical aspects of laser design was tried
to be established.
In Chapter 3, having in hand some experimental results and technical
considerations obtained from Chapter 2, theoretical background of CO
2 lasers is
given and the physics of CO
2 lasers is established in connection with macroscopic
phenomena taking place in lasers.
Chapter 4 introduces beam properties of lasers and connects these properties to
the mechanical design of laser. Also focusing of a laser beam and some results that
may effect the processing are described.
In Chapter 5, design method and calculations are extensively described and
results that were considered in the mechanical realization of the design are given.
In Chapter 6 experiments carried out on the realized design are described and
obtained results are discussed. In this chapter gas concentration and pressure effect
on lasing, beam profiling, beam propagation factor estimations and possible material
processing applications are given.
Calculations carried out during the work, some technical drawings, photos and
description of used techniques are left for Appendices.
3
CHAPTER 2
FUNDAMENTALS OF CO 2 LASER
2.1-History
Infrared radiation emission from CO
2 was first reported by Patel [1] in 1964 in
pulsed discharge through pure CO
2. Soon after that it had been realized that a much
more efficient system involving the vibrational energy transfer from N
2 to CO2 was
possible. Such a laser had been build by Patel [2] also in the same year. This laser
incorporates the RF excitation of N
2 molecules, which are then injected to the CO2
gas.
In these studies output powers of continuous wave (cw) operating CO
2 lasers
were between 0.1 mW to 200 mW. The addition of N
2 increased the efficiency of
the CO
2 laser from 10
–6
to 10
-3
. Direct excitation of a flowing N2 and CO2 gas
mixture using a dc discharge was found to yield cw power of 11.91 W with
efficiency of ≅ 3 % by Patel [3].
Another major advance occurred when the addition of He was found to
increase the cw power obtainable from a flowing N
2, CO2 gas mixture to 106 W [4].
The efficiency of this system was less than 6%. The theoretical efficiency of
the CO
2 laser operating at a wavelength of 10.6 µm was predicted to be ≅ 40 % and
at these years the efficiencies obtained were far away the theoretical limit. In 1967 a
4
162-ft long cw CO
2 laser had been built and operated at a power output of 2.3 kW
[5]. Subsequently, the development of electric discharge convection and gas dynamic
lasers has resulted in generation of cw laser powers in excess of 100 kW. But the size
of these lasers was not convenient for industrial purposes.
The year 1969 marked a turning point in the development of high-power cw
and pulsed CO
2 lasers. Late in 1969, Beaulieu [6] reported that CO2 laser emission
could be obtained at atmospheric pressure and above by exciting the gas transversely
so that the discharge passed perpendicular to the optical axis. This has led directly to
devices that rely on the creation of high densities of electronic charge using electron
beam excitation [7] or volumetric photoionization [8] independent of the discharge
that is used to excite laser emission. As a result Q-switched CO
2 lasers were
developed.
In the same year development of lasers based on convective cooling [9], [10]
were realized and a report was published describing the operation of a compact
closed system cw laser using rapid transverse gas flow capable of generating powers
of 1 kW [11].
Theoretical development of CO
2 laser was completed in 10 years after its
invention. Through the proceeding years works on these lasers were concentrated on
development of new excitation mechanisms, new resonator types and more efficient
laser optics.
2.2- Slow Axial Flow CO2 Laser
Slow axial flow lasers are no more ‘state of art’ lasers. Main difference from
the other types of lasers is that; cooling mechanism is diffusion and gas flow rate is
maximum 50 cm
3
/min.
5
2.2.1 – General Design
Figure 2.1- Schematic diagram of a conventional CO2 Laser
A laser primarily consists of the following parts; two mirrors one total reflector
and the other partial reflector, a tube used as chamber and anode-cathode pair. Gas
mixture flows in the tube between anode and cathode. During this flow the gas
mixture is excited and as a result of this excitation photons are obtained. Photons
travel between the two mirrors by exciting the gas again and increasing the number
of photons in the cavity. Through the output mirror, which allows photons to escape
out of the cavity, a laser beam is obtained. Fig. 2.1 shows the general sketch on
conventional CO
2 laser.
The laser tube is enclosed by a coolant (water, oil etc.) jacket, by which the
excess heat in the cavity is removed. Cooling of the cavity is of great importance to
the operation of the laser as going to be discussed below.
6
2.2.2- Gas composition and Gas Flow
First lasing from CO
2 laser was obtained from pure CO2 gas. But the output
power obtained was very low. Subsequent addition of N
2 and He increased output
power very much. Fig. 2.2 shows the effect of different gas compositions on laser
gain.
Figure 2.2- Gain versus total mixture pressure [12]
As it can be seen from the figure, CO
2 alone has very low power gain. This is
due to the inefficient electrical excitation made directly to CO
2 gas. Addition of N2
increased the laser gain by factor of 10. This is a result of the efficient excitation of
N
2 and that nitrogen has a very long-lived first-excited vibrational state that almost
exactly matches the upper level of CO
2. The energy transfer between the N2 and CO2
molecule is very efficient as a consequence of the close energy levels. The graph
7
showing the vibrational energy levels of CO
2 and N2 and the excitation mechanism
of CO
2 laser are shown in Fig. 2.3.
Figure 2.3- Excitation Mechanism of CO2 Laser [13]
From Fig. 2.3 we can see that He has no effect in the excitation of CO
2 but
addition of it to pure CO
2 or to CO2-N2 mixture increases power gain very much.
This can be explained by the high heat transfer coefficient of He gas compared with
CO
2 and N2.
Gain in CO
2 laser is not only a function of gas composition but also it is a
function of gas flow rate. In Fig. 2.4 it is shown that with an increase in flow rate
gain of a given laser increases first and then saturates after a given flow rate
depending on tube diameter. But the saturation of gain does not mean an obtainable
constant power. Power can increase after that point but very slowly. The primary
effect of flowing gas on gas discharge is a reduction of gas temperature by
convection, where as in slow flowing gases the heat transfer is obtained by diffusion.
8
Figure 2.4- Gain versus CO2 flow rate for CO2:N2: He mixture at near optimum pressure
and mixture ratio [14]
2.2.3- Tube Diameter
Gain of a laser is a function of tube diameter also. Fig. 2.5 depicts a usual
dependence of gain on tube diameter. Gain decreases with increasing tube diameter.
Although this situation cannot be explained very simply, there are three effects which
are produced with the decrease of tube diameter. First, by decrease in tube diameter
an increase in longitudinal field is observed. This increase in the field leads to
increase in electron temperatures that imply an increase in electron energy. This
leads to increase in pumping rate and gain coefficient. Second, if the volume pump
rate is constant decrease in tube diameter results in increase in the gas velocity that
causes a faster gas exchange. Therefore, during the passage of the gas through the
9
tube, decomposition of CO
2 decreases and gain increases. Third, with the decrease in
tube diameter the temperature difference between the gas center and the tube walls
decreases, That is, heat removal from the gas is enhanced and the gain is increased.
Figure 2.5- Gain versus tube diameter for optimum current density [12]
2.2.4- Pressure and Current
As it is clear from Fig. 2.2 gain is also a function of pressure. For a given gas
mixture gain increases with increasing pressure. This is an expected result since
density of CO
2 and N2 molecules increases giving rise to gain. However, with
increasing pressure, field intensity is growing as a result of which input energy in the
gas and gas temperature are increased resulting in a decrease in the gain. So, for each
gas mixture there exists an optimum pressure for which maximum gain is obtained.
10
In condition of optimum gas mixture and constant flow rate the dependence of
gain on current is depicted in Fig. 2.6. Here increase in current means increase in
input power which is expected to increase gain of the amplifier. But increase in input
power results in excess power stored in gas medium which in consequence increases
gas temperature. Increase in gas temperature as mentioned above means decrease in
amplifier gain. Also from Fig. 2.7 it can be seen that optimum current is also
dependent on tube diameter.
Figure 2.6- Gain versus discharge current for various flowing gas media at optimum
mixture ratios and a constant flow rate in 12 and 37 mm bore amplifier tubes [14]
2.2.5- Discharge Energy and Gas Temperature
Gas temperature plays a key role in gas laser excitation. It depends on input
energy density having a scaled unit of Joule/(Volume x Pressure). In the discharge
tube gas temperature has a radial gradient. That is temperature is high at the center
of. the tube and decreases gradually to the tube wall.
11
Figure 2.7- Optimum current and total pressures for maximum cw oscillator output power
as a function of tube diameter [15]
Fig. 2.8 depicts the behavior of gas temperature and small signal gain in
discharge tube for two different input energies From Fig. 2.8 we can conclude that at
small input energies the gain distribution reflects the input energy distribution.
However as the input energy increases after a certain energy input value the gain
distribution does not continue to reflect the spatial energy input distribution. The
temperature of the gas at the center of the tube increases and the gain starts to
decrease.
The spatial energy distribution posses the same profile as shown in Fig. 2.9 but
gain profile is changing in not preferred way. The decrease of gain is not a direct
function of temperature increase. Increased temperature increases the dissociation of
CO
2 and the increase in content of CO decreases gain of the medium. Carbon-
monoxide production rate as a function of input energy distribution is shown in Fig.
2.10. The CO production is directly proportional to the discharge energy.
In Fig. 2.11 the dependence of gain and gas temperature on input energy is
shown. Temperature increase is a direct function of energy input. But gain has a
maximum at certain temperature or input energy and after that point it starts to
decrease.
12
Figure 2.8- Spatial variation of gain for a 3% CO2: 10%N2: 87% He mixture at 100 torr.
Two gain distributions are shown; the one on the left corresponds to an energy input of
120 J/l atm. The other gain distribution corresponds to an input energy of 350 J/l atm.
Representative gas temperatures are also shown [16]
Figure 2.9- Spatial variation of gain for a 10%CO2: 90% He mixture at 150 Torr [16]
13
Figure 2.10- Carbon-monoxide production in a 10%CO2: 90% He mixture at 150 Torr
[16]
Figure 2.11- Small-signal gain and gas temperature as a function of energy [16]
14
2.2.6- Resonator Configuration and Output Coupling
Generally a resonator is consisting of a discharge medium and two mirrors.
Mirror configurations may vary in regard to their curvature and resonator length.
Possible resonator configurations are given in Fig. 2.12. Selection of resonator is a
question of application. Usually the end mirror is fully reflective. Materials for end
mirror are Au, Ag coated Cu, Mo and Si. Curvature of the mirror is determined by
the desired output beam characteristics and stability criteria of the resonator.
The most important issue in laser design is determination of the reflectivity or
transmission of the output mirror. Less transmission of the output mirror means
obtaining less power from that a resonator can give. However, large transmissions
decrease photons that sustain the laser action in the resonator cavity so obtained
power also decrease. The effect of transmission of the output mirror on the obtained
laser power is given in Fig. 2.13. Here the effect of output coupling mentioned above
can be seen clearly. So an optimum value for the transmission of output mirror
should be chosen. In his report Tyte [15] gives a graph from which an optimum
reflectivity for the output mirror can be chosen with respect to resonator
length/discharge tube diameter. This graph is given in Fig. 2.14. The graph is based
on experimental values and it is difficult to obtain desired results using it.
15
Figure 2.12- Resonator Configurations
16
Figure 2.13- Graph of output power versus window reflectivity [15]
Figure 2.14- Optimum reflectivity as a function of the ratio of laser length to diameter [15]
17
Since each amplifying medium is defined by its saturation intensity and small
signal gain a relation to obtain optimum output transmission should include these
parameters beside the resonator length and diameter. Such a relation is given by
Rigrod [17]. This relation includes reflectivities of both mirrors, resonator length,
discharge cross section, small signal gain and saturation intensity. The formula is:
()
1/ 2
1 12
1/ 2
1/ 2 1/ 2
12 12
ln( )
21
os
o
rtgIS rr
PL
grr rr
=+
+−
(2.1)
where P is the output power, r
1 and r 2 are the reflectivities of end and output mirror
respectively, t is transmission of output mirror, S is cross section area, I
s is saturation
intensity, g
o is small signal gain and L is discharge length. From this formula by
taking out P
av=goIsV which is the available power in the resonator we obtain a
formula for efficiency like
()
1/ 2
112
1/ 2
1/ 2 1/ 2
12 12
ln( )
1
21
av o
rt rrP
PgLrr rr
η
== +
+−
(2.2)
Using this equation we can obtain a plot of efficiency versus transmission for
the output mirror. From the plot transmission value giving the highest efficiency is
selected to be the transmission of the output mirror. In this formula also there is an
unclear part; how can g
o be determined. There are two ways to found go: one by
measuring it, which is the long way and second by taking it from graphs such as Fig.
2.5. From equation (2.2) it is clear that output transmission is a function of mirror
reflectivities and g
oL product. Only one parameter is experimental the others are
given.
18
CHAPTER 3
THEORY OF CO 2 LASER
3.1 – Carbon Dioxide Molecule
Carbon dioxide is a linear, symmetric, triatomic molecule. This molecule can
vibrate in the three independent modes of vibration shown in Fig. 3.1. They are: the
longitudinal symmetric mode in which the carbon atom remains stationary and the
oxygen atoms move in opposite directions along the line of symmetry; the bending
mode in which the atoms all move in a plane perpendicular to the line of symmetry,
the carbon going one way while the two oxygen go the other; and the asymmetric
mode in which the atoms all move along the line of symmetry, the carbon moving in
the opposite direction to the two oxygen atoms. Each of these modes is characterized
by a definite frequency of vibration. According to basic quantum mechanics these
vibrational degrees of freedom are quantized, that is when a molecule vibrates in any
of the modes it can have only a discrete set of energies. Thus if we call ν
1 the
frequency corresponding to the symmetric mode then the molecule can have energies
of only
1
1
2
En h ν
=+
, n= 0,1,2,... (3.1)
when it vibrates in the symmetric stretch mode. Thus the degree of excitation is
19
characterized by the integer n when the carbon dioxide molecule vibrates in the
symmetric stretch mode. In general, since the carbon dioxide molecule can vibrate in
a combination of the three modes the state of vibration can be described by three
integers (nmq); the three integers correspond respectively to the degree of excitation
in the symmetric, bending and asymmetric mode.
Fig. 3 also shows the different
vibrational energy levels taking part in the laser transition. The laser transition at
10.6 µm occurs between the (001) and (100) levels of carbon dioxide.
The CO
2 energy levels, which are of greatest importance in laser action, are
situated 2349.3 cm
-1
, 1388.3 cm
-1
, 1285.5 cm
-1
and 667.3 cm
-1
above the ground
state.
Figure 3.1 - The fundamental modes of vibration of the CO2 molecule
20
3.2- Excitation of Carbon Dioxide
The excitation of the carbon dioxide molecules to the long lived level (001)
occurs both through collisional energy transfer from nearly resonant excited nitrogen
molecules and also from the cascading down of carbon dioxide molecules from
higher energy levels.
The first vibrationally excited level of N
2 lies 2329.66 cm
-1
above the
unexcited ground state. The (001) level of CO
2 is 20 cm
–1
above the first excited
level of N
2. Thus energy can be transferred between N2 and CO2 with high efficiency
when N
2 collides with CO2. The transfer equation is given by
N
2 (v``=1) + CO2 (000) ↔ N 2 (v``=0) + CO2 (001) – 18 cm
-1
(3.2)
and the rate of energy exchange is k=1.9x10
4
Torr/sec at 300
o
K [18].
3.3- Relaxation Process
Early in the development of CO
2 lasers, it was realized that adding He in
quantity to the laser gas resulted in substantially increased output powers. It is now
established that the primary effect of He is to relax the lower laser level while
essentially unaffecting the population in the (001) state. Furthermore, He is also
effective in depopulating the (010) level and may also lower the gas kinetic
temperature. All these effects are beneficial to high power laser operation, since a
radiative relaxation rates are orders of magnitude smaller than those due to collisions
with He atoms and other gaseous components.
Quenching process involving the (010) level include the vibrational-
translational energy transfer
21
CO
2 (01
1
0) + M → CO 2 (00
0
0) + M + 667 cm
-1
(3.3)
The relaxation of the two lower laser levels (10
0
0,02
0
0)I and (10
0
0,02
0
0)II,
these levels correspond to levels with energies of 1388.3 cm
-1
and 1285.5 cm
-1
, may
occur through
CO
2 (10
0
0,02
0
0)I +CO2(00
0
0) ↔ 2CO 2(01
1
0) + 64 cm
-1
(3.4)
CO
2 (10
0
0,02
0
0)II +CO2(00
0
0) ↔ 2CO 2(01
1
0) - 40 cm
-1
(3.5)
or (1000,0200)
I may relax via the process (Tyte [15] )
CO
2 (10
0
0,02
0
0)I +CO2(00
0
0) ↔ CO 2(02
2
0) +CO2 (00
0
0) + 52.7 cm
-1
(3.6)
CO
2 (02
2
0)I +CO2(00
0
0) ↔ 2CO 2(01
1
0) + 1 cm
-1
(3.7)
Rate constants associated with these processes depend on temperature,
pressure, and type of quencher. Lifetimes associated with the states involved in
excitation and relaxation is shown in Fig. 3.2.
3.4- General Energy Transfer Processes in CO2 Laser
Of the many energy-transfer processes, which occur in CO
2-N2-He laser
mixture, the following equations (which also include the ones given above) are found
to be the most important and are given by Fowler [19];
i) Electron excitation and deexcitation of the CO
2 asymmetric stretch mode,
e + CO
2 [l,m,(n-1)] ↔ e + CO 2(l,m,n), (3.8)
22
Figure 3.2- Schematic diagram of the CO2 laser mechanism showing relaxation times for
typical gas mixture (1:1:8 of CO
2:N2:He) at a total pressure of 15 torr and a gas-kinetic
temperature of 420
o
K. Level populations shown are equilibrium vibrational populations
in the absence of laser action [15].
ii) Vibrational energy exchange between N
2 and the asymmetric stretch
mode of CO
2,
N
2(v) +CO2 [l,m,(n-1)] ↔ N 2(v-1) + CO 2 (l,m,n), (3.9)
iii) Vibrational energy exchange between the asymmetric stretch and the
coupled symmetric stretch and bending modes of CO
2,
M+ CO
2 [l,m,(n-1)] ↔ M + CO 2 [(l-1), (m-1), n], (3.10)
iv) Electron excitation and deexcitation of N
2,
e + N
2(v) ↔ e + N 2 (v’) , (3.11)
23
v) Excitation and deexcitation by both electrons and heavy particles of the
bending mode of CO
2,
M+ CO
2 [l,(m-1),n] ↔ M + CO 2 [l, m, n], (3.12)
vi) Absorption and spontaneous emission of 4.3 µm radiation by CO2,
CO
2 [l,m,(n-1)]+photon (4.3 µm) ↔ CO 2 (l, m, n), (3.13)
vii) Absorption and stimulated emission of 10.6 µm radiation by CO
2 ,
CO
2 (100) + 2 photons (10.6 µm) ↔ CO 2 (001) + photon (10.6 µm). (3.14)
These energy-transfer processes are illustrated schematically in Fig. 3.3, which
also shows the energy levels of some of the CO
2 and N2 vibrational states.
Equations (3.8) and (3.12) state that CO
2 molecule can be not excited only by
collisions with excited N
2 molecules but also by electron and other particles
collision. But by excitation mechanism of equation (3.12) a laser radiation cannot be
obtained as seen from Fig. 3.3. From excitation of CO
2 by electron bombardment
lasing action can be obtained but as seen in Fig. 2.2 excitation of CO
2 alone is not
efficient as it is when excited with N
2-He gas mixture. Equation (3.14) is the
process equation of lasing in CO
2-N2-He gas mixtures. From this equation light
amplification is clearly seen. When a 10.6 µm photon collides with a CO
2 molecule
in (001) state two photons of the same wavelength are obtained.
24
Figure 3.3- Diagram of some of the CO2 and N2 vibrational levels involved in the analysis.
25
CHAPTER 4
LASER BEAMS AND THEIR PROPERTIES
4.1- Transverse Modes of Laser Resonators
4.1.1- Intensity Distribution of Transverse Modes
In optical resonator electromagnetic fields can exist whose distribution of
amplitudes and phases reproduce themselves upon repeated reflections between the
mirrors. These particular field configurations compromise the transverse
electromagnetic modes of a passive resonator.
Transverse modes are defined by the designation TEM
mn for Cartesian
coordinates. The integers m and n represent the number of nodes or zeros of intensity
transverse to the beam axis in the vertical and horizontal directions. In cylindrical
coordinates the modes labeled TEM
pl and are characterized by the number of radial
nodes p and angular nodes l. The higher the values of m, n, p, l, the higher the mode
order. The lowest-order mode is TEM
00 mode, which has a Gaussian-like intensity
profile with its maximum on the beam axis. For modes with subscripts of 1 or more ,
intensity maxima occur that are off-axis in a symmetrical pattern. To determine the
location and amplitudes of the peaks and nodes of the oscillation modes, it is
26
necessary to employ higher-order equations, which either involve Hermite or
Laguerre polynomials. The Hermite polynomials are used when working with
rectangular coordinates, while Laguerre polynomials are more convenient when
working with cylindrical coordinates.
In cylindrical coordinates, the radial intensity distribution of allowable circular
symmetric TEM
pl modes is given by the expression
() () ()
2
2
0
,, cos exp
l
pl l p
Irz I L lφρρ φρ=−
(4.1)
with
()
22
2/()rzwzρ= , where z is the propagation direction of the beam, and r, φ
are the polar coordinates in a plane transverse to the beam direction. The radial
intensity distribution are normalized to the spot size of a Gaussian profile; that is
w(z) is the spot size of the Gaussian beam, defined as the radius at which the
intensity of the TEM
00 mode is 1/e
2
of its peak value on the axis. L p is the
generalized Laguerre polynomial of order p and index l. Fig. 4.1 shows a 3D view of
some of the TEM
pl modes.
Figure 4.1- Intensity profile of higher transversal modes of stable laser resonators with
circular symmetry [20]
27
4.1.2- Characteristics of Gaussian Beam
A light beam emitted from a laser with a Gaussian intensity profile is called the
“Fundamental mode” or TEM
00 mode. As a Gaussian beam propagates, its intensity
distribution is Gaussian in every beam cross-section but the width of the intensity
profile changes along the axis. The Gaussian beam contracts to a minimum diameter
2w
0 at the beam waist where the phase front is planar. If one measures z from this
waist, the expansion laws for the beam assume a simple form. The spot size, a
distance z from the beam waist expands as a hyperbola, which has the form
1/ 2
2
0 2
0
() 1
z
wz w
w
λ
π
=+
. (4.2)
Its asymptote is inclined at an angle θ/2 with the axis, as shown in Fig. 4.2, and
defines the far-field divergence angle of the emerging beam. The full divergence
angle for the fundamental mode is given by
θ
∞ z
2 w z ( )
z
lim
→
2 λ
πw
0
. (4.3)
From these considerations it follows that at large distances, the spot size
increases linearly with z, and the beam diverges at a constant cone angle θ. From
equation (4.3) it is clearly seen that the beams with larger diameters have smaller
divergence.
At sufficiently large distances from the beam waist the wave has a spherical
wavefront appearing to emanate from a point on the axis at the waist. If R(z) is the
radius of curvature of the wavefront that intersects the axis at z, then
2
2
0
() 1
w
Rz z
z
π
λ
=+
. (4.4)
28
Figure 4. 2- Contour of a Gaussian beam
Also the wavefront of a Gaussian beam has the same phase across its entire
surface. Sometimes the properties of a TEM
00 mode beam are described by
specifying a confocal parameter
2
o
w
b
π
λ
= , (4.5)
where b is the distance between the points at each side of the beam waist for which
w(z)=(2)
1/2
wo .
29
4.1.3- Laser Resonator
Mode structure of a laser beam is directly related with the resonator
configuration. Mode parameters of a resonator with unequal curvature mirrors are
derived by Kogelnik and Li [21]. The geometry of such a resonator, where the radii
of curvature of the mirrors are R
1 and R2, is shown in Fig. 4.3.
Figure 4.3 - Mode parameters of interest for a resonator with mirrors of unequal
curvature
The diameters of the beam at the mirrors of a stable resonator, 2w
1 and 2w2, are
given by
30
2
4 12
1
112
RRL L
w
RLRRL
λ
π
−
=
−+−
(4.6)
2
4 21
2
212
RRL L
w
RLRRL
λ
π
−
=
−+−
. (4.7)
The diameter of the beam waist 2w
o, which is formed either inside or outside the
resonator, is given by
()()( )
()
2
12124
0 2
12
2
LR LR LR R L
w
RR L
λ
π
−−+−
=
+−
. (4.8)
The distances t
1 and t2 between the waist and the mirrors, measured positive as
shown in Fig. 4.3, are
()
2
1
12
2
LR L
t
RR L
−
=
+−
(4.9)
()
1
2
12
2
LR L
t
RR L
−
=
+−
. (4.10)
These equations treat the most general case of a resonator but they can be simplified
for most of resonator configurations.
4.1.4- Stability of a Resonator
Resonators with regard to their stability are classified as stable and unstable
resonators. In stable resonators rays are continuously refocused and ray content of
the resonator is constant. However, in unstable resonators rays become more and
more dispersed and they escape out of the resonator. But the “stability” referred to in
31
these resonator classifications is that of geometrical ray bouncing forth and back in
the cavity designs mentioned above, and has nothing related with the stability and
instability of laser oscillation in transverse modes.
Stability condition for a stable resonator is given as follows
12
01 1 1
LL
RR
<− − <
. (4.11)
This condition is derived from a paraxial ray tracing in a periodic convergent
lens sequence. This periodic lens sequence is dual to spherical-mirror resonator.
To show graphically which type of resonator is stable and which is unstable, it
is useful to plot a stability diagram on which each resonator type is represented by a
point. This is shown in Fig. 4.4 where the parameters L/R
1 and L/R2 are drawn as the
coordinate axes; unstable systems are represented by points in the shaded areas.
Various resonators types, as characterized by the relative positions of the
centers of curvature of the mirrors, are indicated in the appropriate regions of the
diagram. Also parameters g
1 and g2 are shown in the figure.
32
Figure 4. 4- Stability diagram. Unstable resonator systems lie in the shaded region[21]
4.2- Beam Propagation Factor
For applications of laser beams the characterization of their transversal mode
structure is necessary. Both the beam diameter at the waist 2w
o and the beam
divergence θ has to be determined for this purpose. The quality of laser beams can be
described by the beam parameter product BP:
o
BPωθ.= (4.12)
BP is minimum for diffraction-limited beams.
The values of θ and w
o have to be obtained experimentally. One simple method
is the measurement of the spot size diameter d
focus of the beam at the focal length of a
lens as shown in Fig. 4.5.
33
Figure 4.5- Divergence determined by measuring the waist size behind a focusing lens
The divergence angle can be determined from the focal length f and the
diameter d
focus in the focal distance of the lens by:
f
d
focus
2
=θ . (4.13)
The beam parameter product can be calculated from
f
dD
BP
focuslens
4
.
= (4.14)
with the beam diameter D
lens at the position of the lens and the focal length f.
The quality of a measured beam is described by the ratio of the beam parameter
products of this beam BP
beam relative to the best possible value BPGauss of a Gaussian
beam of the same wavelength λ in the same material with refractive index n. This
ratio is defined as beam quality or beam propagation factor M
2
:
()
λ
π
ωθ
n
BP
BP
M
waist
Gauss
beam
.
2
== . (4.15)
34
Thus M
2
=1.5 means the beam parameter is 1.5 times worse than the best
possible value for the wavelength and thus the focus for a given lens would show 1.5
times larger diameter as for a perfect beam.
4.3- Focusing of a Laser Beam
Focusing a laser beam is of great importance in material processing.
Positioning of a beam in specified location with specified spot size is the main
purpose of focusing. In all focusing methods the laws of geometric optics are
sufficient but to calculate the precise spot size and focal depth one needs to refer to
Gaussian optics and diffraction theory.
4.3.1- Diffraction Limited Spot Size
A beam of finite diameter is focused by a lens onto a plane as shown in Fig.
4.6. The individual parts of the beam striking the lens can be imagined to be point
radiators of a new wave front. The lens will draw the rays together at the focal plane
and thus constructive and destructive interference will take place. There will be
observed a Fraunhofer Diffraction Pattern. The central maximum will contain
approximately 86% of all the power in the beam. The diameter of this central
maximum will be the focused beam diameter, usually measured between the points
where the intensity has fallen to 1/e
2
of the central value.
35
Figure 4. 6- Diagram illustrating the diffraction limited spot size
Equation for the spot diameter of the fundamental mode of a circular beam is given
by
D
f
d
λ
⋅=44.2
min
(4.16)
where D is the beam diameter on the lens and f is the focal length of the lens. The
diameter of diffraction limited spots for other modes (TEM
pl) of a laser beam is
given by
()1244.2
min
++
⋅
⋅= lp
D
f
d
λ. (4.17)
36
4.3.2- Effects of M
2
on Focusing
Beam quality is a property of a beam and should have influence on the beam
size obtained after focusing. In focusing of Gaussian beam the diffraction limited
Gaussian spot size is:
D
f
d
π
λ4
min
= . (4.18)
But not every beam is Gaussian so a correction by M
2
should be applied. As it
was previously mentioned that the spot size of a focused beam is M
2
times that for a
perfect beam. So the equation is given by
2
min4
M
D
f
d ⋅=
π
λ
. (4.19)
4.3.4- Spherical Aberration
There are two reasons why a lens will not focus to a theoretical point; one is
the diffraction-limited problem discussed above and the other is that a spherical lens
is not a perfect shape. Most lenses are made with a spherical shape since this can be
accurately manufactured without too much cost and the alignment of the beam is not
so critical as with a perfect aspheric shape. The net result is that the outer ray
entering the lens is brought to a shorter axial focal point than the rays nearer the
center of the lens, as shown in Fig. 4.7. This leaves a blur in the focal point location.
The plane of best geometric focus (the minimum spot size) is a little short of the
plane of the plane wavefront, the paraxial point.
37
Figure 4.7- Spherical aberration of a single lens focusing a parallel beam
Focusing of a collimated beam produces a transverse spherical aberration
(TSA) and a longitudinal spherical aberration (LSA) shifts as shown in Fig. 4.7. To a
first approximation the focus diameter of the part due to the aberration of lens f
3:
2
3
3
),(
f
D
qnKd
a
= , (4.20)
where
()
()
()
()( )()
−
+−+++−
−
+
−
=
1
12314
1
2
1
1
32
1
),(
3
2
n
n
nnqnq
n
n
nn
qnK (4.21)
with
12
12
RR
RR
q
+
−
= (4.22)
R
1- lens entrance radius
38
R
2- lens exit radius
n- lens refractive index.
In spherical aberration shape parameter q determines the degree of the
aberration. Fig. 4.8 shows the axial difference in focal length (LSA) with respect to
parameter q. Here the focus length for different shapes is constant. Here it is clearly
seen that spherical aberration decreases for a plano-convex lens mounted in one way
than the other.
Figure 4.8 - A graph of spherical aberration for lenses of different shape but the same
focal length [22]
39
4.3.5- Depth of Focus
The depth of focus is the distance over which the focused beam has
approximately the same intensity. It is defined as the distance over which the focal
spot size changes by ± 5%. It is given by
()
() λ
ρπ
12
1
2
2
++
−±=
lp
r
z
o
f
, (4.23)
where ρ is the fractional incrase in beam diameter and p,l show the TEM supscripts.
40
CHAPTER 5
DESIGN OF SLOW AXIAL FLOW CO 2 LASER
A laser is designed for use in a specified application, so design parameters
should be defined accordingly. Starting with the application needs one should derive
the properties of the resonator. In Fig. 4.3 the basic resonator specifications,
resonator length, place of waist, spot sizes on mirrors etc., are shown and these
should be determined. Calculations are given in Appendix I.
5.1- Laser Applications and Specification of Laser Output
Beam Parameters
Applications that are considered to be done in this thesis work are cutting and
welding of metals, polymers and ceramics. For applications related to cutting it is
desirable to have a laser with TEM
00 mode of operation. Transverse electromagnetic
mode of operation of lasers used in welding applications is usually TEM
00 + TEM01
*.
Also it is good to have a beam with small angle of divergence in order of mrad’s.
Desired cutting thickness for ordinary carbon steel is 2 mm, and for welding it is 0.5
mm. Power needed for such applications is around 200 Watts of 10.6 µm laser
41
radiation. As a result of these application needs the laser should have the following
output beam and power properties:
- Laser Power : 200 W
- Beam Mode : TEM
00 +TEM01*
- Beam Divergence: mrad.
5.2- Determination of Laser Resonator Parameters
Starting with the application needs parameters, resonator type, resonator
length, tube bore diameter, size and types of mirrors used, transmission of output
coupler is the most important parameters to be determined.
5.2.1- Resonator Type
Some possible resonator configurations are given in Fig. 2.12. For the design
hemispherical resonator configuration is selected. This type of resonator has one
plain output coupler and a spherical end mirror. The advantage of this type is to
know exactly the place of the beam waist. Place of the waist and its size are the key
parameters that are needed to estimate beam diameter a distance far away from the
resonator. This also helps to calculate the spot size of the focused beam.
To save space it is decided to have U-bend resonator. By using such a
resonator without changing the discharge length overall length of the laser is lowered
to half of the resonator length. Also having such a resonator guarantees to have a
laser beam with linear polarization [26].
42
5.2.2- Discharge and Resonator Lengths
As a general rule lasers operated in slow flow regime have output powers of 50
W/m for TEM
00 and 75 W/m for TEM00+TEM01* mode of operation. In length
calculations 75 W/m value will be considered since radiation of the fundamental
mode is also present there. For power of 225 Watts we should have a discharge
length of 3 m. For usual resonators, resonator length is about 25% larger from the
discharge length and resonator length of 3.75m is obtained for the design. Mirrors of
the resonator should be as far as possible from the electrodes to avoid any
contamination from sputtering of electrodes, electrode material vapor deposition and
from any gas impurities.
5.2.3- Determination of End Mirror Curvature and Stability Check
To determine the curvature of the end mirror first of all the beam divergence
should be exactly specified. Then by using equation (4.3) beam waist w
o (radius of
the beam on output coupler) should be calculated. Inserting that value in equation
(4.8) radius of curvature of the end mirror R
2 can be calculated.
In this work a mirror with R
2=8 m was used. Radius of the beam waist was
calculated as w
o= 3.67mm and divergence for the fundamental mode was calculated
as 1.839 mrad. This value of divergence is sufficient for the thesis applications but
can be lowered by changing the curvature of the end mirror.
Having determined both mirror curvatures, stability of the resonator should be
checked by using inequality (4.11). Product g
1g2=0.531 and resonator is in the stable
region.
43
5.2.4- Determination of Laser Tube Bore Diameter
Having in hand values for R
1= ∞ and R 2= 8 m it is possible to calculate the
beam spot size on both mirrors. Since it was previously specified that the laser will
operate to its TEM
01* mode, in determination of tube bore diameter, spot size of the
second mode should be used. In calculation of beam diameters on the mirrors
equations (4.6) and (4.7) will be used. Beam diameter on the output coupler is small
than the beam diameter on the end mirror and for that reason in determination of tube
bore diameter only the beam spot size on the end mirror will be considered. Beam
diameter on the end mirror for the second mode is 15.1 mm and the bore diameter
was selected to be 16.4 mm, which is the closest to our calculated value in the
market. Also the beam on the output mirror will have a diameter of 11 mm.
5.2.5 – Determination of Output Coupler Transmittance
Most important parameter that should be determined in design of laser is the
transmittance of output coupler. To determine this parameter equation (2.2) should
be used. By plotting efficiency versus transmittance curve, the transmittance giving
maximum efficiency is the optimum transmittance of the output coupler. To plot this
curve we need the reflection of end mirror, cavity losses and small signal gain of the
active medium.
Cavity losses are generally the absorptions of total reflector mirrors used in the
cavity and diffraction losses due to diameter changes inside the cavity along the
beam path. Losses of cavity are increasing during the life period of resonator mainly
due to contaminations on the cavity mirrors. These contaminations arise from vapor
of electrodes used, sputtering and dust contained in gas mixture. Because of that
increase in cavity losses, efficiency curves are plot for different loss factors.
Small signal gain and saturation intensity characterize the discharge medium
44
and should be used in every calculation related with the output parameters of the
resonator. Since from Rigrod’s equation saturation intensity is canceled to give
efficiency of coupling there is no need to have that parameter. But in any case small
signal gain should be known. This parameter can be determined experimentally. Also
there are computational methods to derive this parameter but these methods are not
in the scope of this study. To specify this parameter we should refer to previously
done experimental works. Using Fig. 2.5 small signal gain of 0.5 %/cm was obtained
and used in calculations.
Plot of efficiency versus transmittance curve is given in Fig. 5.1. In plot,
curves for different loss values are given and transmittance of 20% for the output
coupler was selected.
Figure 5.1 - Output coupling efficiency versus transmittance plot
45
5.2.6 – Selection of Mirror Material
In previous section it was stated that in time of operation contamination of
optics surfaces increases and that increase causes absorption of radiation which
results in increase in optics temperature. Temperature increase results in thermal
expansion, which causes change in refractive index of material. Output coupler
subjected to temperature change behaves like a lens (diverging or converging),
changing the properties of output beam. For curved total or partial reflectors increase
in temperature results in dimension changes from which curvature change is the most
critical one. Change in curvature causes change in the previously predicted beam
properties. Also since the radiation possesses Gaussian intensity profile and not all
the surface is irradiated, curvature change is different for any point on a particular
surface.
As a result, in selection of mirror materials thermal and optical properties of
different materials used in CO
2 laser resonators will be studied. A formula including
thermal expansion coefficient, thermal conductivity, refractive index gradient and
percentage absorption for particular material will be used to obtain relative figures of
merit. Formulas [28] are:
1,2
K
F
AX
= (4.1)
where
K: thermal conductivity
A: total absorption
1
dL dn
X
dT dT
=+ : thermal expansion coefficient plus refractive index gradient
or
46
2
dL
X
dT
= : thermal gradient.
X
1 is used for lenses, windows and output couplers where effects of both
focusing and surface distortion are important. X
2 is used for total reflectors where
effect of only surface distortion is important. Also X
2 is used for output couplers
when the effects of focusing can be ignored.
Material properties of materials studied are given in Appendix I. Plots for
figure of merit versus absorption are given below.
5.2.6.1- Material selection for Output coupler
Output coupler material was selected between four materials: GaAs, Ge, KCl
and ZnSe. Plots for the two cases: (1) focusing and surface distortion and (2) surface
distortion only were examined.
From the plots of Fig. 5.2 it is observed that KCl is best when focusing is of
great importance, which is for high power lasers, but when surface distortion only is
a concern performance of KCl is worst among the others. KCl is very good for high
power lasers but needs extra conditioning facilities since it is hydroscopic in nature
and for low power laser where focusing is not problem its performance is worst so it
is not applicable in our design.
GaAs is best when only surface distortion is considered and performs close to
ZnSe and Ge when both focusing and surface distortions are important. GaAs cannot
be easily found in the market and also it is not transparent in visible region
(transparency in visible region is advantage in alignment facilities) so it also is not
applicable.
Ge, in performance sequence comes after GaAs but thermal runaway is near
100
o
C and is not applicable to our design.
47
0 0.02 0.04 0.06 0.08 0.1
0
2
4
KCl
GaAs
ZnSe
Ge
FM plot for Thermal Expansion
Surface Absorbtion (%)
Figure of Merit
a) for X= X2
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
KCl
GaAs
ZnSe
Ge
FM for TE plus Thermooptic Coefficient
Surface Absorbtion ( % )
Figure of Merit
(b) for X=X1
Figure 5.2 - Optical distortion figures of merit compare substrate materials for CO
2 laser
output coupler
48
ZnSe is average in performance but runaway temperature is about 300
o
C, can
be easily found in the market, and is transparent in visible region. Also it is cheap in
comparison to Ge and GaAs. As a result ZnSe is selected as material for the output
coupler.
5.2.6.2- Material selection for full reflector mirrors
Full reflector mirrors substrate material was selected between four materials:
Cu, Mo, Si and Ge. Only the case of surface distortion was examined.
0 0.04 0.08 0.12 0.16 0.2
0
2
4
Cu
Mo
Si
Ge
Surface Absorbtion (%)
Figure of Merit
Figure 5.3 - Optical distortion figures of merit compare substrate materials for CO2 laser
full reflector mirrors
49
From the plot in Fig. 5.3 it is seen that silicon substrates perform best but
performance of coatings on silicon is less from that on copper. Molybdenum is
second in performance and is widely used in not very clean operating conditions.
Germanium having a very low runaway temperature is not good for lasers with high
power. Copper on the other hand is in third position but metallic coatings perform
best on it. Cooling of copper is very easy compared to other materials. For the design
copper substrate mirrors were selected, end mirror with 99% reflection and bending
mirrors with 99.9 % reflection were selected.
50
CHAPTER 6
EXPERIMENTS CARRIED OUT ON THE DESIGNED
LASER
6.1- Experiments on Laser Gas and Electrical Feeding
In the previous chapter laser design parameters were determined and
considering these parameters a resonator was designed and constructed. Technical
drawings and a picture of the final construction of resonator are given in Appendix
II. In this part laser output beam characteristics, input power, current, gas pressure
and gas composition will be measured and relations between these parameters will be
determined.
6.1.1- Laser System Set up
Designed laser is driven by 25 kV power supply which delivers 50 mA current.
Voltmeter shown in Fig. 6.1 directly measures the voltage on the capacitor of supply.
Ammeter is connected to the supply end and total current delivered to
51
the system is measured. SCR`s group is consisted of trystors and resistors which
ensure the ignition of both tubes at the same time. Also resistor R
2 is a part of the
ignition system. Resistors R
1 are the ballast resistors which sustain current delivery
to the discharge after ignition.
Gas mixture of CO
2:N2: He is delivered to the system by three tubes which
separately contain the three gases. Mixture composition is arranged by flow meters
connected to each tube. Gas inlet is at the anode side of the resonator. Exhaust gas is
pumped out by vacuum pump and the pressure inside the resonator is measured by a
vacuum gage.
Laser radiation is measured by a power meter having a range of 0-250 W for
CO
2 lasers.
R1
130K
R1
130K
R2
300M
Vacuum
gage
Power
Meter
Vacuum
Pump
Figure 6.1- Experimental setup
52
6.1.2- Current - Partial Pressure Characteristics of Laser Discharge
Having three different gases, keeping the pressure and flow rate of two of them
constant and changing the pressure of the third gas the behavior of individual gases
in the discharge is obtained. Current-Partial pressure characteristics were obtained by
maximizing the output power of the laser and at that instance current, operating
potential and total gas pressure were recorded. The plots of current versus partial
pressure of one constituent of the mixture are given in Fig. 6.2.
Figure 6.2 - Current -Partial Pressure behavior of laser gas constituents
53
Each gas has different effect on the optimum current that gives maximum
power. Keeping constant pressures of CO
2 and He, current decreases with increasing
partial pressure of N
2. Also keeping constant pressures of N2 and He and increasing
the pressure of CO
2 current behavior is similar to N2 gas instead of that when partial
pressure of CO
2 is less than the partial pressure N2 current is increasing. But current
characteristics of He is completely different from the other two constituents,
increasing the partial pressure of He allows the increase of current to obtain
maximum output laser power.
6.1.3- Effects Partial Gas Pressures on Output Laser Power
Keeping constant pressure of the other two gases and changing the pressure of
the third gas output power was measured for the three gases. Fig. 6.3 gives the
measured results of partial gas pressure and corresponding output power for optimum
current. As it is clear from the figures below CO
2 and N2 have similar effects, both
increase the output power to some maximum value and then with increase of their
partial pressure output power decreases. In contrast to CO
2 and N2 partial pressure
effects, increase in partial pressure of He results in increase in output laser power.
During the experiments saturation for He pressure was not found since the power
delivered from the supply has reached its limit but it can be observed from the Fig.
6.3 that rate of power increase decreases with the increase of partial pressure of He.
54
Figure 6.3 - Effect of Partial pressures on output power
6.1.4- Effects of Gas Composition on Output Power
It should be also noted that the output power strongly depends on gas
composition. So it is of great importance to study the composition rates of all the
three gases at the same time. To make such comparison partial pressures of
individual gases were divided by total pressure so that the sum of the ratios results in
unity. Plotting contour graphs for two of the gas ratios and having as a third variable
the output power the effects of the constituent gases on the output power is obtained.
55
From the plots the third gas ratio is found by subtracting the sum of the two gas
ratios from one. Results are present in Fig. 6.4.
Figure 6.4 - Gas composition effects on Output Power
From the results illustrated in Fig. 6.4 we can conclude that the ratio between the
composition rate of CO
2 and N2 is 1to 1.5 respectively. Rate of He is very much
56
than the proportions of CO
2 or N2. In gas content optimization it is more reasonable
first to fix the values of CO
2 and N2 and then to increase the content of He until
maximum power is obtained.
6.1.5 – Laser Discharge with Constant Gas Composition
Up to now effects of individual laser gas constituents were studied and an
optimum gas composition at partial pressure ratios of (CO
2:N2:He-1:1:6) was
selected. Total pressure was 23 torr and current was changed so that to obtain a graph
indicating effects of power input in the discharge. As in previous experiments
current, potential and output laser power were recorded.
Fig. 6.5 shows effect of increasing current on laser output power. As current
increases laser power also increases up to a maximum power and starts to decrease
with subsequent increase in current. Here the behavior of current is identical to the
behavior of input power and the graph also indicates that much power does not mean
much laser output power since the lasing medium is heated up and this decreases
lasing effect of the discharge.
Figure 6.5- Effect of current on laser power
57
Fig. 6.6 depicts the effect of current on the electric to optical efficiency of the
laser. Increase in current decreases the efficiency of the system and for efficient
operation its better to have less current. Having less current means less input power
but it does not mean maximum output laser power as can be observed from Fig. 6.7.
If Fig. 6.5 and Fig. 6.7 are examined at the same time it is clear that the plot in Fig.
6.7 is the mirror image of the plot in Fig. 6.5. This indicates that as the input power
increases the conversion of electrical power to radiation power decreases. And this
decrease in efficiency is related to the increase in temperature of the discharge and to
the increase in power dissipation in the ballast resistors.
Figure 6.6 - Efficiency versus Current graph
To study the effects of pressure rise on output laser power a constant mixture
of volumetric composition CO
2:N2:He-8.9:13.4:77.7 was used. During the
experiment flowing gas pressure, output laser power, potential and optimum current
that maximizes the output power were recorded. Effects of input power and pressure
on the output power are plotted in Fig. 6.8 and Fig. 6.9 respectively. According to
Fig. 6.8 laser power is directly proportional to input power but it saturates after some
pressure limit. This implies that after some pressure limit it is not possible to increase
58
the laser power further. This result can be seen more clearly from the contour plot in
Fig. 6.10. In that figure red parts indicate the region of maximum laser power and it
is reached for an input power of 140 W and total pressure of 100 torr. And any
further increase in pressure or input power has very little effect on laser power.
Figure 6.7 - Efficiency versus Laser Power graph
Figure 6.8 - Effects of Input Power on Output Laser Power
59
Figure 6.9 - Effects of Pressure on Output Laser Power
Figure 6.10 - Contour Plot of Combined effects of Pressure and Input Power on Output
Laser Power
60
6.1.6- Maximum Power Obtained
During the experiments carried out above a general relation between the gas
composition and electrical feeding of the system was investigated. So these results
should be evaluated qualitatively since quantitative results do not reveal the
maximum obtainable power of the design. Maximum power was obtained under
pressure which was not detectable by our instruments but it was near the
atmospheric pressure. That maximum power was also realized by the constant
mixture of volumetric composition CO
2:N2:He-8.9:13.4:77.7 and current of 20 mA.
Power obtained was 32 W and a photo showing the power meter digital display is
given in Fig. 6.11.
Figure 6.11 - Maximum Power Obtained for Single Tube Operation
61
6.2- Experiments on Laser beam Properties
6.2.1- Laser Beam Profiling
Intensity distribution measurement on the cross section of a beam was
performed. A pinhole with diameter much smaller than the beam diameter was
moved in a plane cutting the beam perpendicular to its path of propagation. A
pinhole of 0.8 mm in diameter was used to profile the intensity distribution of a beam
with 9 mm in diameter. Also the pinhole plate was coated with graphite to prevent
back reflections of radiation into the resonator. Fig. 6.12 depicts the method of
measurement. Circle with dashed lines shows the laser beam. Moving the pinhole in
x and y-directions a two dimensional array of power data was obtained. Beam power
passing through the pinhole was measured by a power meter placed behind the
pinhole plate. Increments of displacement ∆X and ∆Y in x and y-directions
respectively were kept constant to eliminate additional data analysis. During the
measurements after completing one row of power sampling the pinhole plate was
removed and total power of the laser beam was checked to ensure that the beam
power was constant in the period of sampling. Since the pinhole plate was moved by
an X-Y stage there was no considerable errors in placing the pinhole at place where it
was desired to be. It should be noted that the profile is also strongly dependent on
optics alignment and to assure constant alignment mirror adjusting screws were kept
at the same position during all measurements on beam profiling. Power changes were
compensated by pressure adjustments and keeping the current of operation constant.
To eliminate effects of composition changes in operating gas a constant mixture gas
was used. The diffraction of the sample beam passing through the pinhole was not
considered since the aim is not to obtain absolute powers but relative power
distributions over the beam cross-section. This will not affect the beam profile results
but a precaution was taken by placing the power meter just behind and very close to
pinhole plate such that to absorb all the radiation passing the pinhole.
62
Figure 6.12 - Beam Profiling Method of Laser Beam
Obtained data were collected as square matrices and 3-D plots giving directly
the beam intensity distribution were obtained. X and y-axes are the same as they
were in measurements and the z-axis is the power, which are matrix elements. Plot of
raw data does not look very nice and also it represents the values of the data points
only and surrounding points remain under the lines joining data points. It is obvious
that there is no any sharp edge in the intensity distribution of a beam as it is observed
in the plot of raw data. To eliminate such results either a very small pinhole should
be used or data analysis of the measurements should be done. Plot of raw data is
given in Appendix III.
In data analysis instead of joining data points with lines a curve was fitted over
all data points. Method used is known as cubic spline fitting and fits a third order
polynomial to data. Such an analysis results in approximation of points which were
not sampled and gives more realistic plots. In data analysis a package program
MathCAD
®
was used and a sample calculation sheet is given in Appendix III for 10
Watt beam.
Beam profiling was done for three beams with different power so that it will be
63
possible to investigate the effects of beam power on the intensity distribution of the
beam. Fig. 6.13 gives the asymmetric and top vies of the 3-D plots for 10, 20 and 30
watt laser beams.
As it is observed from the plots in Fig. 6.13 asymmetric views give the
intensity profile of a beam but do not give clear information about the shape of the
beam cross-section. To obtain beam cross-section shape a top view of the profiling
plot is needed and this plot is given also in Fig. 6.13. From the plots it is observed
that the beam is more intense in its center and intensity decreases going away from
the center. And also it is clear that the beam shape of the obtained laser beam is not
an exact circle as it was desired to be obtained. The shape of the beam changes to
elliptical one as the power of the beam increases. This change in beam shape can not
be observed from beam burns obtained on wood (Fig. 6.14). Although burn patters
look like a circle the intensity profiles of the beams show that they are not. This
difference is mainly the result of thermal conductivity of material on which burns are
obtained. May be with materials, having lower thermal conductivity and high
absorption of the laser radiation, burns close in shape to the profiles could be
obtained.
The change of beam profiles with increasing powers may be a result of
increasing power in the discharge and increasing gas pressure. The increased
discharge power has destructive effects on the distribution of small signal gain in the
active medium and this will result in change in the beam profile. To eliminate such
drawbacks of increased power, cooling rate of active medium should be increased
either by decreasing the temperature of cooling water or increasing water flow rate or
by increasing the gas flow rate. None of these methods were applicable during the
experiments.
The effect of increased pressure on the beam profile is in connection with the
change in alignment of cavity optics. Since the optics are placed on O-rings to ensure
a good vacuum, easy replacement and easy cleaning there are finite movements in
any pressure change. As a consequence of these finite displacements output power of
the laser will not change by distinguishable amount but beam profile will change
destructively.
64
a)
b)
c)
Figure 6.13 - Beam Intensity profiles obtained from different Laser Powers, a) 10 W, b) 20
W, and c) 30 W
65
Figure 6.14- Burn patterns of laser beam on wood for 10W, 20 W and 30 W
From beam profiling it is possible to conclude for the transverse
electromagnetic mode content of laser beam. In the design part it was desired to
operate in TEM
00+TEM01 and this mode should be satisfied by the profiles obtained.
Theoretical intensity distribution is given by equation (4.1) and Fig. 4.1 depicts the
3-D plots for different modes separately. But in time of operation it is not possible to
separate these modes and the result is their sum. In Fig. 6.15 two dimensional plots
of intensity distributions of TEM
00, TEM01 and their sum is given. As it is seen form
the graph it is difficult to distinguish from a single profile the mode of operation
since the intensity distributions of TEM
00 and the sum TEM00+TEM01 are nearly the
same. Only the top of the sum is flat which means that in the profile we should have
a larger red spot in comparison to a TEM
00 beam. Any measurement on TEM00 beam
was not carried out and we can not conclude on the mode content of the beam with
experimental results. But our theoretical calculation results indicate that considering
curvature of the end mirror, resonator length and laser tube diameter we should have
obtained a beam with mode content which is the sum of first two transverse
electromagnetic modes.
The intensity distribution of a beam is of great importance in material
processing. Since the intensity profile of the beam is conserved in the focus point it is
important to locate the point of maximum intensity on the axis of propagation that is
in the center of the beam. In material processing the region or line that is desired to
cut, weld or heat treated is placed on the axis of beam propagation.
66
Figure 6.15 -Radial Intensity distribution for TEM00, TEM01 and sum TEM00+TEM01
Also the profile should posses a distribution such that points having equal
magnitude of intensity should be located in equal distance around the point having
maximum intensity, i.e. beam should have a circular cross-section. If this condition is
not satisfied the path of processing could not be located properly.
6.2.2- Beam Propagation Parameter and Divergence
The two most important parameters that characterize the laser beam are its
beam propagation factor, i.e. beam quality, and divergence. To determine these
parameters a lens of known focal length is used to focus the out coming laser beam
and by knife edge method beam cuts are made on both sides of focus point to
determine the place and size of the beam waist. Setup of the experiment is given in
Fig. 6.16. At each cut point diameters are measured in two directions. This allows to
observe the difference in beam propagation along the two transversal axis, x and y.
67
For a beam with acceptable characteristics it is expected to have the same properties
in both directions.
To determine the desired properties at twelve points on the propagation path of
the focused beam knife edge cuts were made and diameter of the beam at each cut
was determined. In Appendix IV use of knife edge cut method is described and a
sample calculation is given. Calculated beam diameters are plotted with respect to z-
axis in Fig. 6.17.
Figure 6.16 - Experimental Setup for knife edge method
A laser beam posses the form of propagation of the equation (4.2). Diameters
determined by knife edge method should be fitted to equation (4.2). Since our beam
is not purely Gaussian beam it is expected the beam to deviate by a factor equal to
the beam quality of the beam. So equation (4.2) is modified such that include the
beam propagation factor as well. The modified equation is given below
68
()
1
2
22
2
4
2()2 1
(2 )
o
o
o
Mzz
Wz W
W
λ
π
−
=+
(6.1)
Since parameters z
o, 2Wo and M
2
are not known exactly a trial and error
approach was applied in order to fit the equation. For the three unknown parameters
a range was specified and equation (6.1) was calculated at data points. The results are
determined by estimating the minimum difference between the calculated and
measured diameters at data points. At minimum total error the beam waist, its
location and beam quality were determined. A plot of the fitted curve for 10 W beam
is given in Fig. 6.17.
Curve fitting process was applied for three beams with power of 10W, 20 W
and 30 W for two directions and totally six waist diameters and beam quality
parameters were determined. From the curves approximating the propagation of the
beam, which beam diameters at twelve points along the propagation path were
obtained, Rayleigh range for each beam was estimated. By using the beam waist
diameter and the Rayleigh range for that beam the divergence for each beam were
also calculated. By using the previously measured beam diameters on the focusing
lens and calculated beam propagation factors theoretical beam diameters at focal
point of the lens were obtained. Also beam focal depth for each beam was calculated.
Results of Laser beam properties are given in Table 6.1. A sample calculation for
curve fitting and calculation of other beam parameters is given in Appendix-V.
In previous section intensity profiles of beams were determined and it was
observed that beams are elliptical. This result was confirmed by the diameter
measurements of beams on the lens. This beam ellipticity was conserved through all
path of propagation after beam passes focusing lens. From Fig. 6.17 this result can
be easily observed. As a result of beam ellipticity different results of beam properties
for the different axis of each beam were observed. Due to this observed beam shape
evaluation of the results will be done with respect to a given axis at all three power
levels.
As general result we can conclude that our beams are of TEM
00 mode with a small
contribution of second TEM
01 mode since average M2≅1.07. This result is consistent
with the plot in Fig. 6.15, that is the area below the curve of TEM
01 is much less than
the area below the curve of TEM
00 mode.
69
Figure 6.17 - Curve fitted to data of 10 W laser Beam
70
Table 6.1- Results on Laser Beam Properties
71
Comparing the beam diameters determined from the curve fitting at focal point
and calculated theoretical diameters they are consistent and calculated diameters are
slightly lager (+ 0.01 mm ) than that determined by curve fitting.
Waist position compared for each beam is different both axes. From this result
we can conclude that the beam posses a slight astigmatism.
Comparing the Rayleigh range of beams in one direction(x or y) its observed
that it decreases with increasing power. But beam divergence on the other hand
increases with increasing power. Behavior of beam propagation factor is the same as
beam divergence it increases with increasing power. Focal depth in contrast to beam
divergence and beam propagation factor decreases with increasing power. The
increase in beam divergence and beam propagation factor and the decrease in
Rayleigh range and focal depth indicate the same result, beam is getting worst with
increasing power.
The results of different beam properties on different axis of evaluation are the
result of worst alignment but differences are less than 10 % and that difference is
acceptable.
6.3 – Experiments on Material Processing
The possibility of material processing with the designed laser was investigated.
Since the laser output power is low (only 30 W) it is difficult to find a wide
application area in material processing. So during the thesis work concentration was
made on materials that absorb 10.6 µm wavelength radiation. These materials are
Teflon, Plexiglas, pyrex glass and packing paper. Also an attempt was done to cut
stainless steel thin sheet of 0.05mm. Primary processing considered was cutting and
any work on process optimization was not in the scope of this work.
Laser beam delivery system and nozzle were designed by the author and used
in the applications. A photo of the beam delivery unit and a technical drawing of the
nozzle are given in Appendix VI.
72
Cutting of three non metal materials was successful these materials are teflon,
pleksiglas and paper.
Thickness of teflon used in experiment was of 4 mm and cuts of 2 mm in depth
were realized. During the cutting of teflon there was a little flame and exhaust was
not melt but snow like powder. In processing of teflon the cutting mechanism may be
chemical bond breaking primarily and very little evaporation which results in flame
during the processing. In Fig. 6.18a a photo taken during the cutting of teflon is
shown and two cuts on the sample are also given in Fig. 6.18b.
Attempt to cut a hard 3 mm thick paper was done. It was successful but the
speed of cutting was very low 9 cm/min. In paper cutting mechanism was burning
there was no melting no vaporization. A performed cut is given in Fig. 6.18c.
5 mm thick pleksiglas was cutted very successfully. The cutting mechanism
was vaporization and there was not any melt on the plate. The cut width was
measured to lager than 0.1mm but less than 0.15 mm. A photo of the cut is given in
Fig. 6.18d.
Also cutting of 1 mm pyrex glass was performed but it was unsuccessful.
During the cutting operation glass was broken into pieces in one trial but in the
other cut was performed but glass melts were solidified at the edge of the cut. This
indicates that the process is not optimized and there is a possibility to cut this
material. Cutting mechanism in glass cutting is crack propagation and melting of the
material is not allowed. With good cooling during the processing, glass can be
successfully cutted.
Possibility of cutting highly reflective materials such as stainless steel was also
investigated and with a proper gas selection and processing speed optimization
cutting of a 0.05 mm thick material can be performed. In Fig. 6.18e photo taken
during the processing of steel is given. The flame is bright and likes the one in
welding applications and this flame indicates the melting of material. In Fig. 6.18f
the cut performed on the stainless sheet is given. As it is observed very clearly heat
affected zone around the cut is nearly two times large than the cut width. This result
shows that cooling of the sample during the processing was not enough. During
processing N
2 gas was used but by using a gas with high heat conduction constant or
by use of reactive gas such as oxygen better cuts can be obtained.
73
a) b)
c) d)
e) f)
Figure 6.18 - Processed materials; a) Teflon processing, b) Teflon processed, c) Paper
processed, d) Cut performed in plexiglas, e) Stainless steel processing, d) Processed
stainless steel sheet.
74
CHAPTER 7
DISCUSSION and CONCLUSION
The main aim of this work was to get experience with the basic structure of
CO
2 laser resonators and to get basic understanding of lasing processes. Beside these
also laser beam characterization, laser beam guiding and laser beam material
processing were the secondary subjects worked on.
The two basic problems of laser resonators are their mechanical and electrical
stability. The design realized was faced with both of the problems. Realization of
discharge at both tubes was not possible either by using single or two separate power
supplies. By using single supply connected to both tubes results in discharge in a
single tube completely randomly. By using two separate supplies also results in
discharge in a single tube. Second discharge is obtained between the anodes of the
resonator and not between the anode and cathode of the second tube.
Although some mechanical solutions for the problem exist during the work
electrical solutions were considered. Using of SCR`s group was the solution worked
on but it was successful only at pressures below 3 torr. The problem needs further
investigations and during the experimental part only one tube was operated.
The mechanical stability of the resonator was another problem faced with. Due
to frequent pressure changes fasteners of the mirrors are loosen and this destroys
mirror alignment. During the work bending mirrors were stabilized by locking of
fasteners but end mirror and output mirror were not stabilized since adjusting
micrometer screws were used.
75
The output power obtained was maximum 32 watts and this result is very far
from the design power of 225 watts even if two tubes were in operation. Main reason
of that result was the length between the anode and cathode. It is known that small
signal gain of a discharge decreases with the increase of length between anode and
cathode [27]. Keeping discharge length constant and increasing the anode-cathode
pairs is a way to increase the output power of the laser.
Operational experiments of the design were performed considering gas
composition, gas pressure and electrical feeding. Results consistent with general
relations obtained in previous works [12] [14] were obtained. Since all of such
experiments are not performed in maximum operation power or resulting in that
power, these results should be evaluated in qualitative manner rather than in
quantitative.
Experiments related with beam properties are performed by basic methods
although there are available instruments performing such measurements. Beam
profiling was performed by pinhole measurements and the size of the pinhole may be
critical in evaluation of the results. To minimize the effect of pinhole size on the
results 2D curve fitting was applied. This method was used to interpolate the values
of points that are not measured between the points that are measured. From obtained
profiles it was observed that the beam possess shape of ellipse rather than a circle
and this result is a consequence of bad alignment. Beam profiling was performed for
three different operational powers and the ellipticity was observed to increase with
increase in beam power. This increase may refer to either bad alignment or to the
destruction of discharge uniformity with increasing power. Further investigations of
the results should be performed.
Beam propagation factor and beam divergence were obtained by curve fitting
to data obtained from beam diameter measurements at both sides of the focus point
of a focused beam. Twelve diameters were measured for a beam and modified
Gaussian beam diameter function with respect to distance traveled were fitted to
these data. Curve fitting was done in regard to minimize the error of diameters at
data points between the calculated diameters from fitted curve and measured
diameters. From the fitted curve beam propagation factor, beam divergence,
Rayleigh range for the focused beam, focused beam diameter and focal depth of the
76
laser beam were obtained. Beam diameters at focal point of the lens were compared
to the theoretically calculated beam diameters. Diameter measurements were
performed for three beams with different power and for two axes in the cross-section
of a cut. So six different sets of beam parameters were obtained and compared.
Ellipticity of beams was observed also through diameter measurements and this
beam cross-section shape shows his effects on the beam properties. These effects
were different divergence and beam propagation factor for each axis of a given
power beam.
Possibility of the designed laser to be used in material processing was also
investigated at end of the work. Primarily materials that absorb 10.6 mm wavelength
radiation were considered. Fine cuts were obtained in paper, teflon and plexiglas.
Also it is possible to cut stainless steel if process is optimized. Range of materials
that can be processed with 30 W laser can be extended by further investigations.
During the work only cutting process was studied and any process optimization work
was not carried out. To settle down the application area of such an instrument
process optimization and economical effectiveness of the process need to work on.
77
REFERENCES
1. Patel C.K.N. ,1964, Physical Review Letters, V12, pp. 588
2. Patel C.K.N. 1964, Physical Review Letters, V13, pp. 617
3. Patel C.K.N. 1964, Physical Reviews, V136, A1187
4. Patel C.K.N. 1965, Physical Review Letters, V17, pp. 15
5. Roberts T.G., Hutcheson G. J., Ehrlich J.J., Hales W.L., Barr T.A., 1967,
IEEE Journal of Quantum Electronics, V3, pp 605-609
6. Bealieu A.U. 1970, Applied Physics Letters, V16, pp. 504
7. Festermacher C.A., Nutter, M.J., Rink, J.P., Boyer, K. 1971, Bulletin of
American Physical Society, V16, pp.42
8. Seguin H.J., Tulip, J., 1972, Applied Physics Letters, V21, pp.414
9. Deutch T.F., Horrigan, F.A., Rudko, R.I., 1969, Applied Physic Letters,
V15, pp.88
10. Cool T.A., Shirley, J.A., 1969, Applied Physic Letters, V14, pp70
11. Tiffany W.B., Targ, R., Foster, J.D., 1969, Applied Physic Letters, V15,
pp.91
12. Antropov E.T., Silin-Bekchurin, I.A., Sobolev, N.N., Sokovikov, V.V.,
1968, IEEE journal of Quantum Electronics, V4, pp.790
13. Matrhews S., 2001, Laser Focus World, V37, pp.185
14. Cheo P.K.,1967, IEEE Journal of Quantum Electronics,V3, pp.683
15. Tyte D. C.,1970, In `Advances in Quantum Electronics` (D.W. Goodwin,
ed.), Vol. 1, Academic Press, New York
16. Ballik E.A., Garside, B.K., Reid, J., Tricker, T.,1975, Journal of Applied
Physics, Vol. 46, pp. 1322
17. Rigrod W. W.,1965, Journal of Applied Physics, V34, pp. 2487
18. Taylor R.L., Bitterman, S.,1969, Review of Modern Physics, V41, pp.26
78
19. Fowler M.C., 1972, Journal of Applied Physics, V43, pp. 3480
20. Menzel Ralf, 2001, Photonics- Linear and Nonlinear Interactions of Laser
Light and Matter, Springer-Verlag, Berlin,
21. Kogelnik H. , Li, T., 1966, Proceedings of the IEEE, pp. 97
22. Jenkins F.A., 1976, Fundamentals of Optics, 4th Ed., McGraw-Hill, Tokyo,
23. Koechner W., 1999, Solid State Laser Engineering, 5
th
Ed., Springer-Verlag,
Berlin,
24. Weber M. J., 2002,
Handbook of Optical Materials, 1
st
ed., CRC Press, New
York
25. Callister W.D., 1997, Materials Science and Engineering: an Introduction, 4
th
ed., John Willey & Sons, New York
26. Welch M., Lasers&Applications , 1986, pp.67-71
27. Bogaerts A., Grozeva, M., 2002, Applied Physics B, V75, pp.731-738
28. Sherman Glen F., Danielewicz, Edward J., Rudisill, J.Earl, Lasers &
Optronics, September 1988
79
APPENDIX I
LASER DESIGN CALCULATIONS
A1.1 - Laser Output Parameters
P
out
225
W:=
θ210
3−
⋅rad:=
TEM
00
+ TEM
01
* mode of operation
A1.2 - Slow flow laser power constants
p
00
50
W
m
:= for laser operation in TEM
00
mode
p
01
75
W
m
:= for laser operation in TEM
00
+ TEM
01
*mode
A1.3 - Calculation of Discharge and Resonator Lengths
A1.3.1- Discharge Length
L
d
P
out
p
01
:= L
d
3m=
80
A1.3.2- Optical Length
L 1.25 L
d
⋅:=
L 3.75m=
A1.4 - Calculation of mirror radii
A1.4.1-Curvature of End Mirror
λ10.6µ
m:=
w
o
2λ⋅
θπ⋅
:= w
o
3.374mm=
R
2
w
o
2π
λ
⋅
2
1
L
⋅L+:= R
2
6.786m=
A mirror with larger value that is in hand is selected
R
2
8
m:=
A1.4.2 - Curvature of Output Mirror
Resonator type is hemispherical and the output coupler is plane.
R
1
10
20
m:= (i.e. plane)
A1.5- Stability of The Resonator
g
1
1
L
R
1
−:= g
2
1
L
R
2
−:= g
1
g
2
⋅0.531= 0 < g
1
.g
2
<1
A1.6- Beam diameters on mirrors
A1.6.1- Output Beam Diameter for Fundamental Mode
w
o
λ
π
LR
2
L−()⋅⋅
1
2
:= D
o
2w
o
⋅:= D
o
7.3mm=
81
A1.6.2-Beam Diameter on the End Mirror for Fundamental Mode
w
e
λR
2
⋅
π
2
R
1
L−
R
2
L−
⋅
L
R
1
R
2
+ L−
⋅
1
4
:= D
e
2w
e
⋅:= D
e
10.1mm=
A1.6.3 - Mode Content of Resonator
C
1
1.9
2.42
1.5
2.15
2.63
:= Mode Coefficients [23]
Laser TEM mode abbreviations
TEM
00
10
20
01
11
21
:=
A1.6.3.1 - On output mirror
D
o
2C⋅w
o
⋅:= D
o
7.3
13.9
17.8
11
15.8
19.3
mm=
A1.6.3.2 - On end Mirror
D
e
D
o
g
1
g
2
⋅:= D
e
10.1
19.1
24.4
15.1
21.7
26.5
mm=
A1.6.4 - Selection of tube diameter
TEM 01 diameter on the output mirror is
D
o
01,
11m
m= and on the end mirror is
D
e
01,
15.1m
m=
Selected Diameter :
D
tube
16.4m
m:=
82
A1.7- Beam Divergence of the Fundamental Mode
θ
2λ⋅
πw
o
⋅
:= θ1.839 10
3−
× rad=
A1.8 - Transmittance of Output Coupler
A1.8.1 - Mirror Losses
a
1
0.0
1:=
a
2
0.005:= a
3
0.005:= a
4
γ()γ:=
A1.8.2 - Mirror Reflectances
r
1
1a
1
a
2
+ a
3
+
( )−:=
All total reflectors are considered as one
r
2
γt,() 1t−
γ−:=
Reflectance of output mirror
A1.8.3 - Calculation of transmittance by Rigrod's method
ηtL,g
o
,γ,()
r
1
0.5
t⋅
r
1
0.5
r
2
γt,()()
0.5
+
1r
1
r
2
γt, ()⋅()
0.5
−
⋅
1
ln r
1
r
2
γt,
()⋅()
2g
o
⋅L⋅+
⋅:=
83
A1.8.3.1 - Behavior of Rigrods formula for different conditions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11 % loss curve
10 % loss curve
30 % loss curve
50 % loss curve
80 % loss curve
Efficiency Curve for increasing losses
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.1 - Efficiency Curve for increasing losses
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 m length curve
2 m length curve
4 m length curve
6 m length curve
Efficiency Curve for increasing length
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.2- Efficiency Curve for increasing length
84
A1.8.3.2 - Selection of go for transmission calculations
From Fig 5 for a tube with diameter of 16.4 mm and pressure of 4 torr value of 3
was selected.
α3d
B:= α10−log g
o ()⋅
g
o
10
α−
101
m
⋅:= g
o
0.501
1
m
=
A1.8.3.3 - Efficiency plot and Transmission determination for design
L
d
3m= g
o
0.501
1
m
=
γincreasing
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1% loss curve
2% loss curve
3% loss curve
4% loss curve
5% loss curve
Output Coupler Transmission (%)
Coupling Efficiency (%)
Figure A1.3 - Efficiency Curve for increasing losses of designed laser
T20%:= is selected for the output transmission
85
A1.9 - Selection of Mirror Materials
A1.9.1- Output Coupler Material
Table A1.1- Material Properties of possible Output Coupler Material s [24]
Function for figure of merit:
F
K
AX⋅
K - Thermal Conductivity ; A - Total Absorption
X
1
T
L
d
dT
n
d
d
+ Thermal Expansion coefficient + Refractive index gradient
or
X
2
T
L
d
d
Thermal Expansion
FψA,∆L,()
ψ
mK⋅
W
⋅
A∆L
()⋅ K⋅10
8
⋅
:=
Figure of Merit Function for thermal Expansion
FψA,∆L,∆n,()
ψ
mK⋅
W
⋅
A∆L∆n+
()⋅ K⋅10
8
⋅
:=
Figure of Merit Function for Thermal
Expansion coefficient plus Refractive index gradient
Material
Thermal
Expansion
Coefficient
(10
-6
/K)
Thermal
Conductivity
(W/m.K)
Thermo
Optic
Coefficient
(10
-6
/K)
GaAs 5 65 200
Ge 5.7 59.9 401
KCl 36.5 6.7 -34.8
ZnSe 7.1 13 61
86
0 0.02 0.04 0.06 0.08 0.1
0
2
4
KCl
GaAs
ZnSe
Ge
FM plot for Thermal Expansion
Surface Absorbtion (%)
Figure of Merit
Figure A1.4 - Figure of Merit for Thermal Expansion
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
KCl
GaAs
ZnSe
Ge
FM for TE plus Thermooptic Coefficient
Surface Absorbtion ( % )
Figure of Merit
Figure A1.5 - Figure of Merit for Combined Thermal Expansion and Refractive gradient
change effect
87
A1.9.2 - Material Selection for Full Reflector Mirrors
Table A1.2 – Material Properties of Possible Materials for Total reflector mirrors [24]
Material
Thermal
Expansion
Coefficient
(10
-6
/K)
Thermal
Conductivity
(W/m.K)
Cu 17 398
Si 4.9 142
Mo 2.5 141
FψA,∆L,()
ψ
mK⋅
W
⋅
A∆L
()⋅ K⋅10
9
⋅
:=
Figure of Merit Function for thermal Expansion
0 0.04 0.08 0.12 0.16 0.2
0
2
4
Cu
Mo
Si
Ge
Surface Absorbtion (%)
Figure of Merit
Figure A1.6 - Figure of Merit for Thermal Expansion for total reflection mirrors
88
APPENDIX II
REALIZED DESIGN AND TECHNICAL DRAWINGS
a)
b)
c)
Figure A2.1- Realized design: a) General view, b) Cathode side, c) Anode side
89
Figure A2.2- General Top view of the Design
90
Figure A2.3- Cathode side view of the Design
91
FigureA2. 4- Anode side view of the Design
92
FigureA2. 5- Beam Bending Mirrors Assemble view of the Design
93
APPENDIX III
BEAM PROFILING
A3.1 - Obtained Data from pinhole power measurements
Data
12345678910
1
2
3
4
5
6
7
8
9
10
0.06 0.07 0.07 0.07 0.07 0.06 0.07 0.06 0.05 0.05
0.05 0.06 0.06 0.07 0.08 0.07 0.07 0.05 0.04 0.03
0.06 0.07 0.07 0.07 0.09 0.1 0.1 0.08 0.05 0.04
0.06 0.06 0.07 0.1 0.14 0.15 0.17 0.1 0.06 0.04
0.05 0.05 0.06 0.08 0.15 0.2 0.15 0.12 0.07 0.04
0.05 0.05 0.05 0.08 0.15 0.23 0.16 0.12 0.08 0.06
0.05 0.05 0.06 0.08 0.13 0.17 0.14 0.09 0.07 0.05
0.05 0.05 0.06 0.07 0.11 0.12 0.1 0.07 0.07 0.05
0.05 0.05 0.05 0.06 0.07 0.08 0.07 0.06 0.06 0.04
0.05 0.05 0.03 0.05 0.06 0.05 0.05 0.06 0.06 0.05
W=
A3.2 - Data Normalization
Data are normalized to the maximum power measured.
M
Data
max Data()
:=
94
M
12345678910
1
2
3
4
5
6
7
8
9
10
0.261 0.304 0.304 0.304 0.304 0.261 0.304 0.261 0.217 0.217
0.217 0.261 0.261 0.304 0.348 0.304 0.304 0.217 0.174 0.13
0.261 0.304 0.304 0.304 0.391 0.435 0.435 0.348 0.217 0.174
0.261 0.261 0.304 0.435 0.609 0.652 0.739 0.435 0.261 0.174
0.217 0.217 0.261 0.348 0.652 0.87 0.652 0.522 0.304 0.174
0.217 0.217 0.217 0.348 0.652 1 0.696 0.522 0.348 0.261
0.217 0.217 0.261 0.348 0.565 0.739 0.609 0.391 0.304 0.217
0.217 0.217 0.261 0.304 0.478 0.522 0.435 0.304 0.304 0.217
0.217 0.217 0.217 0.261 0.304 0.348 0.304 0.261 0.261 0.174
0.217 0.217 0.13 0.217 0.261 0.217 0.217 0.261 0.261 0.217
=
A3.3 - 2D Curve fitting
Since obtained data correspond to specified points on the beam cross section, value of
surrounding and not measured points can be found by interpolation. For our case 2D cubic spline
will be used for curve fitting interpolation function.
rows M() 10= cols M() 10= n rows M():=
X and Y n-vectors that determine the mesh for the matrix:
X
0
1
2
3
4
5
6
7
8
9
:= Y
0
1
2
3
4
5
6
7
8
9
:=
Mxy augment sort X( ) sort Y(),():=
rows Mxy()10=
Computed spline coefficients:
S cspline Mxy M,():=
Fitting function for surface:
fit x y,( ) interp S Mxy, M,
x
y
,
:= xlow Mxy
11,
:= xhigh Mxy
n1,
:=
95
ylow Mxy
1
2,
:= yhigh Mxy
n2,
:=
Density of mesh for interpolation:
xn 10
n⋅:= yn 10 n⋅:= i1x n..:=
j1y n..:=
xind
i
xlow i
xhigh xlow−
xn
⋅+:= yind
j
ylow j
yhigh ylow−
yn
⋅+:=
Curve Fitting Function:
FIT
ij,
fit xind
i
yind
j
,
( )
:=
Surface Plot of 2D Spline Interpolated Surface
Original Surface
Figure A3.1- 3D surface plot of beam intensity profiler
Contour Plot of 2D Spline Interpolated Surface
Contour Plot of Original Surface
Figure A3.2- Contour plot of beam intensity profiles
96
APPENDIX IV
KNIFE EDGE METHOD FOR LASER BEAM DIAMETER
MEASUREMENTS
A4.1 - Description of Method
A plate with very thin edge (Knife Edge) is placed perpendicularly on the path of
propagation of the beam, which diameter is desired to be measured, such that to prevent the beam
to pass across the knife edge plate. After that the knife edge plate is pulled out by recording
power passing across the edge with respect to distance traveled. Recorded powers are divided by
the maximum power recorded and multiplied by one hundred in order to obtain percentage
values with respect to the beam total power. Percentage power changes are plotted with respect
to traveled distance and distance between %15.9 and % 84.1 is obtained graphically. Two times
that distance gives the diameter of the beam. This diameter is equal to the (1-1/e
2
) of the
diameter of a Gaussian beam.
97
A4.2 - Obtained Data
a 0.0254mm:= Distance traveled by one step of micrometer screw.
Distance Traveled: Power corresponding to traveled distance:
X
0
25
75
100
125
150
175
200
225
250
275
300
350
400
a⋅:=
P
0.77
1.18
2.41
3.45
4.76
6.33
8.59
10.99
13.22
15.35
17.02
18.19
19.59
20.58
W:=
A4.3 - Data Analysis
0 5 10 15
0
50
100
Travelled distance (mm)
Percentage Power (%)
Figure A4.1 - Percentage power as a function of distance traveled.
P
c
M
i
P
i
max P()
100⋅←
i 1 rows P()..∈for
M
:=
The Loop on the left estimates percentage of transmitted power with
respect to total beam power.
98
A4.3.1 - Fitting Cubic Spline to approximate obtained data
Algorithm a Cubic Spline fit in MathCAD®:
data
M
i1,
X
i
mm
←
M
i2,
P
c
i
←
i 1 rows P()..∈for
M
:=
data csort data 1,():= X data
1〈〉
:=
Y data
2〈〉
:=
Spline coefficients:
S cspline X Y,():=
Fitting function:
fit x( ) interp S X,Y,x,():=
Figure A4.2- Plot of experimental data, Fitted curve and 15.9% and 84.1 % lines as a
function of distance.
Beam Diameter:
D root fit x( ) 15.9− x,0,7,( ) root fit x ( ) 84.1− x,5,10,()−
( )2⋅:= D 9.353=
99
APPENDIX V
DETERMINATION OF BEAM PROPERTIES
A5.1- Experimental Data
Distance from the reference
Point
Measured diameters for 10W, 20W, 30 W
laser beams in x and y-directions. First, third
and fifth columns are diameters measured for
x –axis and the others are for y-axis
respective to power increasing order
Z
20
50
70
85
90
95
100
105
110
120
130
150
:=
W
7.246
4.178
2.111
1.09
0.531
0.137
0.281
0.579
1.039
2.097
2.865
4.411
7.377
4.317
2.451
1.083
0.486
0.146
0.227
0.577
1.219
2.009
2.85
4.628
8.255
4.386
2.63
0.87
0.433
0.067
0.256
0.537
1.161
2.033
2.993
4.602
7.897
4.686
2.639
1.051
0.812
0.255
0.245
0.91
1.263
2.303
3.049
5.136
7.403
4.231
2.248
0.94
0.331
0.175
0.147
0.603
1.215
2.218
3.119
4.929
8.44
4.847
2.64
1.186
0.4
0.113
0.279
0.454
1.281
2.131
3.424
5.286
=
A5.2 – Derivation of Curve fit Equation
The spot size of a Gaussian beam a distance z from the beam waist expands as a hyperbola,
which has the form
ωz()ω o1
λz
πω
o()
2
2
+
1
2
(A5-1)
100
Divergence of Gaussian Beam at far field:
θ
2λ
πω
o
⋅
(A5-2)
By inserting (A5-2) in (A5-1):
ωz()ω o1
θz
2ω
o⋅
2
+
1
2
(A5-3)
Divergence of a real beam:
ΘMθ⋅
(A5-4)
Diameter of a real beam at waist:
W
o
Mω
o
⋅ (A5-5)
Beam Product of a real beam:
W
o
Θ⋅M
2
θ⋅ω
o
⋅
M
22λ⋅
πω
o
⋅
⋅ω
o
⋅ (A5-6)
Beam Propagation Factor:
M
2
πΘ2W
o()
4λ
(A5-7)
By inserting (A5-7) in (A5-3):
Wz() W
o
1
M
2
λ⋅z⋅
πW
o()
2
2
+
1
2
(A5-8)
Replacing
Wz()
by 2W(z) and
W
o
by
2W
o
⋅
in equation (A5-8)
2W z() 2W
o
1
4M
2
λ⋅z⋅
π2W
o()
2
2
+
1
2
(A5-9)
Replacing 2W(z) by W(z) and 2W
o by Wo in equation (A5-9) we obtain a function giving beam
diameters
Wz() W
o
1
4M
2
λ⋅z⋅
πW
o()
2
2
+
1
2
(A5-10)
and this equation will be fitted to data obtained.
101
A5.3 - Curve Fitting
Since from obtained data we cannot conclude on waist diameter, location of waist and
beam quality factor a simple approach is to define a range for these constants and to calculate
equation (A5-10) over that range at data points and to minimize the error between calculated and
obtained beam diameters. The constants for which the errors between calculated and measured
results are minimum are the parameters of our beam.
Range for
z
o
: 90 110..
Range for
W
o
:
1
2
min W
column_number
〈〉
()⋅ 4 min W
column_number
〈〉
()⋅..
Range for
M
:
12.5..
Curve fitting Function:
ωxw
o
,z
o
,M,() w
o
1
4M
2
⋅λ xz
o
− ()⋅
πw
o
()
2
⋅
2
+
1
2
⋅:= (A5-11)
Minimum Error Loop:
I
err 10←
ror
1
rows Z()
i
W
j〈〉()
i
ωZ
i
φ,υ,τ,()
−
∑
=
←
U
1j,
φ←
U
2j,
υ←
U
3j,
τ←
err ror←
err
err ror>if
err
τ1 1.01, 5..∈for
υ90 90.25, 110..∈for
φ0.5 min W
j〈〉()
⋅ 0.5 min W
j〈〉()
⋅ 0.01+()
, 4 min W
j〈〉()
⋅..∈forj 1 cols W()..∈for
U
:=
102
Loop Results:
I
0.168
97.25
1.05
0.153
97.75
1.01
0.164
97.75
1.06
0.143
97.5
1.02
0.153
96
1.02
0.147
97.5
1.05
=
A5.4 - Beam Constants
A5.4.1- Waist Location
z
o
j
I
2
j,
:=
z
o
97.25
97.75
97.75
97.5
96
97.5
=
A5.4.2- Rayleigh Range for each beam
Z
R
j
I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
2
I
1j,
⋅− x,I
2j,
,160,( )− I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
2I
1j,
⋅− x,50,I
2j,
,( )−+
2
:=
Z
R
1.908
1.7
1.763
1.446
1.678
1.442
=
A5.4.3 - Beam Propagation Factor
M
j
I
3j,
()
:=
M
j()
2
1.103
1.02
1.124
1.04
1.04
1.103
=
A5.4.4 - Beam Divergence
Θj
I
1j,
Z
R
j
:=
Θj
88.3
90
92.7
98.5
91.5
101.6
mrad=
103
0 16 32 48 64 80 96 112 128 144 160
0
1
2
3
4
5
6
7
8
9
10
10 W x-axis
10 W y-axis
20 W x-axis
10 W y-axis
30 W x-axis
30 W y-axis
Curve of 10 W x-axis
Curve of 10 W y-axis
Curve of 20 W x-axis
Curve of 20 W y-axis
Curve of 30 W x-axis
Curve of 30 W y-axis
Distance Z (mm)
Diameter D (mm)
Figure A5.1 - Experimental data with fitted curves along the propagation direction
104
A5.4.5 - Theoretical Calculation of Beam Diameters at focal point of the lens
f 100m
m:= Lens focal length
D
9.026
8.698
8.574
9.353
8.691
9.937
mm:= Beam Diameters on the lens
d
j
4
π
λf⋅
D
j
⋅ M
j()
2
⋅
0.0286
D
j()
3
f
2
+:= d
0.167
0.16
0.179
0.152
0.163
0.153
mm=
difference
j
d
j
mm
I
1j,
−
d
j
mm
100⋅:= difference
j
0.9
4.5
8.5
6.5
6.1
4
=
A5.4.6 - Focal Depth
z
f
j
I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
1.05
I
1j,
⋅− x,I
2j,
,160,( )− I
2j,
rootωxI
1j,
,I
2j,
,I
3j,
,()
1.05I
1j,
⋅− x,50,I
2j,
,( )−+( )mm:=
z
f
0.85
0.76
0.79
0.65
0.75
0.65
mm=
105
APPENDIX VI
BEAM DELIVERY UNIT and NOZLE DRAWING
Figure A6.1 – Beam delivery unit
106
Figure A6.2 - Laser beam focusing head and nozzle