Communications in Physics, Vol. 24, No. 3 (2014), pp. 239-245
DOI:10.15625/0868-3166/24/3/3948
ROLE OF CALCULATION MODEL IN EXPERIMENTALLY DETERMINING
THERMAL ENERGY GENERATED IN LASER ROD OF SOLID LASER
PHAM VU THINH AND LE NGOC ANH
Institute of Technical Physics, Military Institute of Science and Technology
E-mail: ngocanhle74@gmail.com
Received 25 April 2014
Accepted for publication 26 Agust 2014
Abstract. In many solid-state laser pumped by flash lamps, thermal source distributed in laser rod is heterogeneous,
complicated and cannot be described by explicit functions. Pumping process can be considered an adiabatic process.
Thermal gradient generated within the laser rod is the result of the inhomogeneous distribution of absorbed energy.
Calculation model and numerical method can help to determine the relation between the experimental values and
thermal energy generated in laser rod.
Keywords: solid-state laser, lase rod.
I. INTRODUCTION
In a solid-state laser, a significant fraction of absorbed pump radiation is converted into
heat instead of contributing to laser radiation [1]. In flashlamp-pumped system, the broad spectral
distribution of the pump source causes a significant amount of generated heat. For high repetition
rate and high power lasers, thermal effects should be mitigated because these effects may reduce
remarkably laser beam quality, interfere with efficient laser operations. Thermal stresses may
cause some thermal fracture damages on the laser rod. For this reason, one of the most critical
problems in designing, manufacturing a high repetition rate and high power laser system is to
address heat dissipation issue. In order to solve the thermo physics problem of the laser rod, we
need to determine the generated thermal energy and its distribution within pumped laser rod.
II. EXPERIMENTAL CAPABILITIES
Direct measurement of generated thermal energy in laser rod is a complicated problem and
technically unfeasible, especially when the pulse width is within the range from a few dozen to
hundreds of µs, even for flashlamp-pumped system. Therefore, in most cases, its thermal effects
determine the thermal energy indirectly.
There are many techniques to evaluate the generated thermal effects in laser rod. Among
those, the simplest method is the technique using a probe laser beam [2]. The principle of this
method is to measure and determine the variation in the parameters of output probe beam after
c
2014
Vietnam Academy of Science and Technology
240 provides
VU THINH
AND LEmethod.
NGOC ANH
a simple but lack ofPHAM
sufficient
accuracy
Among other techniques,
interferometric technique [3,4] provides results with much higher precision. Based on
it travels
through
pumpedpatterns,
active medium.
Based on caustic
changes,
we can
provided
interference
we can determine
wavefront
distortion
afterdetermine
the beamthe focal
length of the thermal lens. Probe beam technique provides a simple but lack of sufficient accutravels
through
theother
laser techniques,
rod. However,
this methodtechnique
has the disadvantage
ofresults
intricate
racy
method.
Among
interferometric
[3,4] provides
with much
experimental
and interference
instabilitiespatterns,
due to we
mechanical
vibrations.
Thedistortion
higher
precision. configurations
Based on provided
can determine
wavefront
after
the beamsensor
travelsmethod
through(Shack-Hartmann)
the laser rod. However,
this method
disadvantage
of intricate
wavefront
is considered
to behas
ofthe
adequate
accuracy
experimental
configurations
and
instabilities
due
to
mechanical
vibrations.
The
wavefront
compared to interferometric method while it requires a much simpler experimental sensor
method (Shack-Hartmann) is considered to be of adequate accuracy compared to interferometric
configuration
and the aresults
obtainedexperimental
are less sensitive
to mechanical
vibrations
method
while it requires
much simpler
configuration
and the results
obtained are
[5,6,7].
With
commercial
wavefront
sensors,
we
often
acquire
the
accuracy
within
theoften acless sensitive to mechanical vibrations [5–7]. With commercial wavefront sensors, we
quire
the accuracy
withinShack-Hartmann
the range of λ /15wavefront
÷ λ /20. Shack-Hartmann
wavefront
range
of λ/15÷λ/20.
sensors system in
compactsensors
form system
in compact
form
including
specialized
camera
and
compatible
software
helps
to
ease
the measureincluding specialized camera and compatible software helps to ease the measurement
ment of changes of wavefront.
of changes of wavefront.
Fig 1 : Schematic diagram of Nd:YAG laser quantron cross section
Fig. 1. Schematic diagram of Nd:YAG laser quantron cross section. 1. Laser rod; 2.
Flash lamp; 3. Cooling liquid; 4. Cooling liquid tube; 5. Reflector chamber.
1. Laser rod 2. Flash lamp 3. Cooling liquid 4. Cooling liquid tube 5. Reflector chamber
Ourobjective
objectiveisistoto
investigate
quantron
of flashlamp-pumped
Nd:YAG
solid
Our
investigate
thethe
quantron
of flashlamp-pumped
Nd:YAG
solid
laser. Fig. 1
laser.
Figure
1
illustrates
the
cross
section
of
this
quantron.
We
use
a
Nd:YAG
laser
illustrates the cross section of this quantron. We use a Nd:YAG laser rod of 1.0 at.%. The surfaces
of the
rod’s
endsThe
aresurfaces
covered of
with
coating.
The scattering
surrounding surface
rod two
of 1.0
at.%.
theanti-reflective
two rod’s ends
are covered
with anti-reflective
is about 20% to avoid parasitic modes. The length of the laser rod is 50.0 mm. The flash lamp
coating. The scattering surrounding surface is about 20% to avoid parasitic modes. The
used is of type Xe 450 torr. The light bulb is made of fused quartz and the distance between the
of theislaser
50.0 mm.
Theasflash
lamp
usedand
is of
Xe 450
The of pyrex
twolength
electrodes
50.0 rod
mm.isWater
is used
cooling
liquid
thetype
routing
tubetorr.
is made
lightThe
bulb
is made
of fused
quartz
thethe
distance
between
the two by
electrodes
is
glass.
reflector
chamber
is 50.0
mmand
long,
inner surface
is covered
metal coating
with
reflective
index
of
90%.
The
experimental
results
based
on
caustic
parameters
of
the
probe
beam
50.0mm. Water is used as cooling liquid and the routing tube is made of pyrex glass.
areThe
presented
in details
in is
[2].
reflector
chamber
50.0 mm long, the inner surface is covered by metal coating
We use the Shack-Hartmann wavefront sensor produced by Thorlab, including standardized
with reflective
indexcamera,
of 90%.compatible
The experimental
on caustic
microlens
array, CCD
softwareresults
in our based
experiments.
Theparameters
microlens of
array is of
theWFS
probe
beam are
presented
in details is
inof
[2].
type
150-5C,
diameter
of microlens
146.0µm, distance between two adjacent microlens
center is 150.0µm, focal length is 3.7 mm. These microlenses are made of fused quartz. The
Weofuse
Shack-Hartmann
wavefront
sensor
by pixel
Thorlab,
including
resolution
thethe
CCD
camera is 1280×1024
pixel,
the produced
size of each
is 4.65µm
× 4.65µm,
microlens
array,
camera,
compatible
in focus
our experiments.
thestandardized
size of the effective
sensor
is CCD
5.95 mm
× 4.76
mm. The software
number of
points on microlens
array is 29×29 when the size of region of interest is 4 mm×4 mm. The precision range of the
2
sensor is λ /15. CCD camera can be triggered synchronically with pumping pulses.
The microlens array is of type WFS 150-5C, diameter of microlens is of 146.0µm,
1
3
distance between two adjacent microlens center is 150.0µm, focal length is 3.7mm.
These microlenses are made of fused quartz. The resolution of the CCD camera is
6
2
1280×1024
pixel, the size
4 of each pixel is74.65µm×4.65µm, the size of the effective
sensor is 5.95mm×4.76mm. 5The number of focus points on microlens array is 29×29
8
when the MODEL
size of IN
region
of interest isDETERMINING
4mm×4mm.THERMAL
The precision
range
ROLE OF CALCULATION
EXPERIMENTALLY
ENERGY
...
of the
241 sensor is
Fig 2: Experimental diagram
of camera
a system
Shack-Hartmann
wavefront
sensor
for
λ/15. CCD
canusing
be triggered
synchronically
with pumping
pulses.
measurement
of thermal
Because
the diameter
of thelens effect in solid-state laser rod
laser rod is 4.0 mm, it does not re1. He-Ne laser
λ = 0,6328µm
2. Telescope optical
system
3. Laser
rod
4. Flashlamp
1
3
quire additional optical system to re5. Reflective
chamber
6. Microlens
array
7. CCD Camera
8. Computer
build images
on the sensors.
A He6
2
Ne laser source with suitable optical
7
4
Because
the provides
diameter
of the beam.
laser rod is 4.0mm, it doesn’t
require additional
system
collimated
5
8
Thorlab’s
sensors can
set anyon
waveFig.A2.He-Ne
Experimental
diagram
of awith
system
optical system
to rebuild
images
the
sensors.
laser
source
suitable
Fig 2: Experimentalusing
diagram
of a system usingwavefront
Shack-Hartmann
sensor for
front to be the benchmark to evaluate
Shack-Hartmann
sensorwavefront
for
measurement
of
thermal
lens
effect
in
solid-state
laser
rod
variation
which may
occur. We
set
optical system
provides
collimated
beam.
Thorlab’s
sensors
can set
to
measurement
of thermal
lens any
effect wavefront
in solid-
laser
rod. 1. He-Ne
λ = 0,3.6328µm;
the wavefront of the input
collimated
1. He-Ne
laser λ = 0,6328µmstate 2.
Telescope
opticallaser
system
Laser rod
4. Flashlamp
be the benchmark
to evaluate
variation
may 6.occur.
We
set the
wavefront
optical
3. Laser
rod; of8.the
beam before
it travels into
laserwhich
5. the
Reflective
chamber 2. Telescope
Microlens
arraysystem;
7. CCD
Camera
Computer
4. Flashlamp; 5. Reflective chamber; 6. Mirod
to
be
the
benchmark.
The
speinput collimated beam before itBecause
travelstheinto
thecrolens
laser
rod7. rod
to be4.0mm,
the benchmark.
Theadditional
diameter
of
thearray;
laser
doesn’t require
CCDisCamera;
8.itComputer.
cialized software allows to save the
optical
system
rebuild images
on thein
sensors.
He-Ne Figure
laser source
with suitable
specialized measured
softwareresults
allows
to files.
save
the2tomeasured
results
excelA files.
2 and
in excel
Fig.
optical
and Fig. 3 demonstrate
onesystem
of ourprovides collimated beam. Thorlab’s sensors can set any wavefront to
be the benchmark to evaluate variation which may occur. We set the wavefront of the
input collimated beam before it travels into the laser rod to be the benchmark. The
-1
specialized software allows to save the measured results in excel files. Figure 2 and
-0.8
figure 3 demonstrate one of our measured results.
-0.6
figure 3 demonstrate
one of our measured results.
measured results.
-0.4
-1
-0.2
-0.8
0
-0.6
0.2
-0.4
-0.2
0.4
0
0.6
0.2
0.8
0.4
0.6
1
1
0.5
0
(a) Wavefront
-0.5
0.8-1
1
1
0.5
0
-0.5
-1
(a) Wavefront
FigFig.
3 : 3.
The
measured
resultofof
wavefront
(a)Zernike
and
Zernike
coefficient
(b)
obtained
by
The
measured result
wavefront
and
coefficient
by coefficient
using
Fig
3 : The (a)
measured
result of
wavefront(b)
(a) obtained
and Zernike
(b) obtained by
wavefront sensor.
using wavefront sensor
using wavefront sensor
The measured results of Zernike coefficients (Fig. 3b) proved that thermal effects in laser
rod not only acts as a convex lens but also cause many other optical aberrations and its contributions to wavefront distortion are significant.
3
III. ROLE OF MATHEMATICAL MODEL IN PROCESSING EXPERIMENTAL
RESULTS
Our objective is to determine the heat generated in a flashlamp-pumped laser rod. In order to
use the measured results, we need to evaluate the relation between variations of wavefronts and the
generated heat. In some special cases, when wavefront is expressed in series of Zernike functions,
we need to determine the correlation between values of Zernike coefficients and generated heat.
3
242
PHAM VU THINH AND LE NGOC ANH
Most of the experimental research on measuring thermal effects in solid laser rod are based
on explicit expressions which obtained by W. Koechner [8]. To yield these explicit expressions,
Koechner has assumed that the generated heat is uniformly distributed within laser rod. Based
on theory of heat transfer in solid state material, Koechner has proved that heat distribution in
thermal equilibrium state has the shape of parabolic curve which centered at symmetric axis and
approaches zero on either sides, which means that the highest temperature points locate on the
symmetric axis of the laser rod. Temperature gradient causes intense local heating that induces a
refractive index gradient. As a result, the beam travelling through the laser rod undergoes the same
effects as of a convex lens. The explicit expressions obtained by Koechner allow us to determine
the dependency of focal length of thermal lens on generated heat density in laser rod. The Koechner’s assumptions may lead to rather simple explicit expressions, however, these assumptions are
rarely applied to optical pumped solid laser in effect because to achieve uniform heat distribution
in laser rod, we need to use some special techniques. In practical application, the heat source
distribution is so complicated to be described by explicit expressions. On the other hand, even in
the case of high repetition rate solid laser, the pumping pulse width is significantly shorter than the
time interval between two pulses. According to solid laser calculation models, pumping process
is often considered to be adiabatic [1], which means that the role of heat transfer and dissipation is
almost insignificant. Heat transfer and dissipation occur within time interval between two pulses.
This is a practical and reasonable assumption. Because pumping process is adiabatic, temperature
gradient is generated in laser rod due to pumping process rather than to heat transfer and dissipation process. Temperature gradient in laser rod is caused by heterogeneous heat distribution
and heat accumulation during pumping process. For this reason, the variations caused by thermal
effects do not depend on heat power, they depend on thermal energy and its distribution in laser
rod, which means that it depends on the distribution of absorbed energy. The laser rod may focus
(when heat is distributed mainly on the symmetric axis) or diverge (when heat is distributed on the
edge) the beam.
Because explicit expressions cannot be derived for absorbed energy distribution, the relation
between measured results and generated thermal energy of each quantron configuration can only
be obtained from calculation models and numerical methods.
The process of building calculation model for quantron in flashlamp-pumped Nd:YAG
solid-state laser using Monte – Carlo ray tracing method with Zemax software is presented in
details in [9, 10]. Using this calculation model, we can rebuild wavefront of the output-collimated
beam after it travels through a pumped laser rod using finite element method for each quantron
configuration. As for illustration, we show the calculations of 3 quantron configurations whose
reflector chamber has cylinder, ellipsoid and oval shape. The parameters of laser rod, flash lamp,
cooling liquid system, distance between flash lamp and laser rod, reflective index of reflector
chamber are the same for all experiments. The length of major axis and minor axis of the ellipsoid
correspondingly are 18.356 mm and 21.0 mm. The center of flash lamps and center of laser rod
locate at the foci of the ellipsoid. The radius of two halves of the oval is 6.0 mm. The distance
between centers of two halves of the oval is 9.0 mm. The center of laser rod and center of the
flash lamp locate on the center of two halves of the circle. Zernike coefficient is calculated based
on the obtained wavefront. In order to calculate Zernike coefficient, we use some Matlab functions coded by S. Carey [11]. The resolution of these calculations corresponds to the resolution of
wavefront sensor (the cross section is 4.0 mm×4.0 mm, the number of microlenses is 29×29). The
lation between generated thermal energy within laser rod and Zernike coefficient
C
an be expressed as:
E h = α .C20
(1
here α is rate coefficient which depends on specific quantron configurations. Th
ROLE OF CALCULATION MODEL IN EXPERIMENTALLY DETERMINING THERMAL ENERGY ...
lation between Zernike coefficient
C
0
2
and focal length
f
243
is determined b
results proved that values of Zernike coefficient depend linearly on generated thermal energy in
0
laser rod.coefficient
Fig. 4 demonstrates
dependency of Zernike coefficient C20 (according to OSA/ANSI
Cthe
alculating Zernike
2 from the expression [7]:
standard [12], which corresponds to the index of 5) on generated thermal energy. The coefficients
C20 are calculated in µm and energy2 is calculated in J. The coefficients C20 are considered critical
r0
because it allows to evaluate the impacts
of laser rod as a lens. This coefficient is positive when
f
=
laser rod acts as a concave lens and is 0negative when laser rod acts as a convex lens. In general
4 3Cthermal
2
cases, the relation between generated
energy within laser rod and Zernike coefficient C20
can be expressed as:
Eh = α.C20
(1)
where α is rate coefficient which depends on specific quantron configurations. The relation between Zernike coefficient C20 and focal length f is determined by calculating Zernike coefficient
C20 from the expression [7]:
f
r02
(2)
f= √ 0
4 3C2
C20
in which, r0 is radius of the pupil of wavefront sensor.
Some other experimental methods can only determine f rather than rebuild wavefront. For
this reason, the Zernike coefficient C20 is specially concerned in thermal effects investigations.
(2
which, r0 is radius of the pupil of wavefront sensor.
Some other experimental methods can only determine
avefront. For this reason, the Zernike coefficient
rather than rebuil
is specially concerned in therma
fects investigations.
Thermal energy generated in laser rod (J)
0
Fig 4 : Dependence
Zernike
coefficient
ongenerated
generated
energy
Fig. 4. of
Dependence
of Zernike
coefficient C
C202 on
thermal thermal
energy and its
distri- and its distribution
bution within laser rod.
within laser rod
Fig. 4 shows that in the case of using ellipsoid reflector chamber, the laser rod acts as a
convex lens due to thermal effects while in the case of using spherical and oval reflector chamber,
Figure 4 shows that in the case of using ellipsoid reflector chamber, the laser ro
cts as a convex lens due to thermal effects while in the case of using spherical an
val reflector chamber, the laser rod acts as a concave lens. The focusing power o
Table 1 : Dependence of Zernike coefficient C20 on generated thermal energy Eh
Eh(J)
0
2
1
2
3
244
(µm)
-0.0287
Based
4
5
6
7
8
9
10
-0.2006
-0.2292
-0.2579
-0.2866
PHAM VU THINH AND LE NGOC ANH
-0.0573
-0.0860
-0.1146
-0.1433
-0.1719
the laser rod acts as a concave lens. The focusing power of laser rod in the case of using oval
reflector is greater than that of the case using spherical reflector. The rate of generated thermal
energy Eh to Zernike coefficient C20 must be calculated for each specific quantron configuration. In
the case of using circle quantron (as illustrated in Fig. 1), from the calculation model, we achieve
on
the obtained
the results
in the Table 1:values in the table 1, rate coefficient α in expression
Table 1. For
Dependence
of Zernikein
coefficient
C20 on
generated
thermal energy
Eh . of
calculated to be -34.965.
example,
case of
the
measured
value
Eh (J)
1
2
3
4
5
7
8
be6 6.3636
the generatedCthermal
energy
0
h is calculated
-0.0287
-0.0573 E-0.0860
-0.1146 -0.1433to -0.1719
-0.2006 J.
-0.2292
2 (µm)
9
-0.2579
C20
(1) is
is 0.182µm,
10
-0.2866
Based (the
on the number
obtained values
the Table 1, the rate
α in expression
(1) is
The resolution
of inmicrolenses
in coefficient
a unit area)
of wavefront
sensor
calculated to be -34.965. For example, in case of the measured value of C20 is 0.182µm, the
generated thermal
energy
is calculated
to be
6.3636 J.
may have profound
impact
onEh the
rebuilt
wavefront
and hence it affects remarkably on
the
The resolution (the number of microlenses in a unit area) of wavefront sensor may have
profoundly
impact
on the rebuilt wavefront and hence it affects remarkably on the calculated
calculated Zernike
coefficients.
Zernike coefficients.
0
Fig. 5. Dependence of Zernike coefficients C2 on
the resolution and generated thermal energy.
Fig 5.Dependence
of Zernike coefficients C20 on
the resolution and generated thermal energy
Fig. 5 shows the results of C20 coefficients against changes in the number of microlenses
of each isthe
4.0 mm×4.0
with different generated
thermal
energy (from
1 to number of
Figure (area
5 shows
resultsmm)
of inCa20rowcoefficients
against
changes
in the
10 J) in the laser rod. These results proved
that Zernike coefficients become stable when we
several hundreds of microlenses in a row. Clearly the more heat energy is generated, the
microlenses use
(area
of each
is 4.0mm×4.0mm)
in a commercial
row withShack-Hartmann
different generated
thermal
more
distinctive
the Zernike
coefficients are. For typical
sensors at
number of microlenses per mm rarely exceeds 10-20, calculation for each specific
energy (frompresent,
1 tothe10J)
in the laser rod. These results proved that Zernike coefficients
sensor are critical to analyze the measured results.
become stable when we use several hundreds of microlenses in a row. Clearly the
more heat energy is generated, the more distinctive the Zernike coefficients are. For
typical commercial Shack-Hartmann sensors at present, the number of microlenses per
mm rarely exceeds 10-20, calculation for each specific sensor are critical to analyze
ROLE OF CALCULATION MODEL IN EXPERIMENTALLY DETERMINING THERMAL ENERGY ...
245
IV. CONCLUSIONS
Calculation models and numerical methods play an important role in determining the relation between experimental results and generated thermal energy when the distribution of absorbed
energy in laser rod is complicated and cannot be described by explicit expressions. The calculation
must be processed for each specific quantron configuration and wavefront sensor.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
A. V. Mezenov, L. N. Soms, A. I. Stepanov, Mechanical engineering, Leningrad (1986) 199 (in Russian)
P. V. Thinh, L. N. Anh, L. V. Dai, Journal of Military Science and Technology 8 (2013) 34 - 40.
A. A. Solviev, Dissertation, Nizhny Novgorod (2002) (in Russian)
S. L. Mikerin, I.G Pantrikova, V.D. Ugozaev, Computer Optics 26 (2004) 37-47 (in Russian)
S. Chenais, F. Druon, F. Balembois, G. Lucas-Leclin, Y. Fichot, P. Georges, R. Gaume, B. Viana, G. P. Aka, D.
Vivien, Opt. Mat. 22 (2003) 129–137.
D. Stuéinskas, R. Antipenkov, A. Varanavicius, Lithuanian Journal of Physics 50 (2010) 191–199.
. J. H. Sung, T.M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, J. Lee, Journal of the Korean Physical Society 55 (2009)
495-500.
W. Koechner, Appl. Opt. 9 (1970) 2548 - 2553.
P. V. Thinh, L. N. Anh, L. V. Dai, Journal of Military Science and Technology 8 (2013) 24 - 27.
P. V. Thinh, L. N. Anh, L. V. Dai, Journal of Military Science and Technology 8 (2013) 28 - 33.
http://www. Mathworks.com/matlabcentral/fileexchange/28665 - zernike3.
American National Standards Institute: Methods for Reporting Optical Aberrations of Eyes (ANSI Z80.28-2004,
2004).