...... =

retractor Report 4300

g of ThermaiLensing

Side and End-Pumped

Solid-State Laser Rods

nt G. Brackett

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LASFR P,ODS M.q. [_c, si_ (Hdmpton Univ.)

102 p CSCL 20F

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NASA Contractor Report 4300

Modeling of Thermal Lensing

in Side and End-Pumped

Finite Solid-State Laser Rods

Vincent G. Brackett

Hampton University

Hampton, Virginia

Prepared for

Langley Research Center

under Grant NAG1-666

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Division

) 1990

Table of Contents

Page

List

List

Chapter

I.

2.

of Tables ........................................ v

of Figures ....................................... vi

Introduction ................................. 1

Temperature Distribution Model ............... 4

A. Single Pulse Operations ................. 6

B. Multiple Pulse Operations ............... 13

C. Summary of the Temperature

Distribution Model ...................... 19

3. Thermal Lensing .............................. 20

A. Introduction to Thermal Lensing ......... 20

B. One Dimensional Theory of Thermal

Lensing ................................. 22

C. Two Dimensional Model Formulation ....... 33

4. Analysis and Discussion of Thermal Lensing... 46

A. Model Validation ........................ 47

B. Analysis of Thermal Lensing in EndPumped Nd:YAG and Ti:AL203 Laser Rods... 52

5. Summary and Conclusions ...................... 62

Appendixes

A. Computer Program Listing ..................... 65

i. Functional Description .................. 65

PRECEDING PAGE BLANK NOT FILMED

iii

fAGk.L/._.JNRNnON J 
Table of Contents (cont.)

S ,

Page

2, Environmental Characteristics ........... 65

3. Thermal Lensing Model Program Listing... 66

Program Input Files .......................... 78

i. Thermal Lensing Model Input File #i ..... 78

2. Thermal Lensing Model input File #2 ..... 78

C. Thermal Lensing Model Output File ............ 83

References ............................................ 92

iv
Table

3-1.

3-2.

4--i.

Tables

Block of Radial and Axial Temperature

Profiles for a Nd:YAG Crystal ..............

Block of Radial and Axial Index of

Refractions for the Temperatures in

Table 3-1 ..................................

End-Pumped Laser Rod Characteristics .......

Page

34

35

52

V
Figures

Figure

2--I. Thermal Relaxation of a Single Input

Pulse in a Cylindrical Laser Rod ...........

2-2. Heat Propagation in Nd:YAG Laser Rod

(End-Pumped, Single Pulse with a

Uniform Cross Section Pumping Beam) .......

2-3. Heat Propagation in Ti:AL203 Laser Rod

(End-Pumped, Single Pulse with a

Gaussian Cross Section Pumping Beam) ......

2-4 Temperature Profiles for Various Durations

of a Single Pump Pulse for Nd:YAG

(At Center of Pump-End Face) ...............

2--5. Temperature Profiles for Various Durations

of a Single Pump Pulse for Ti:AL203

(At Center of Pump-End Face) ...............

2--6. Thermal Relaxation in a Solid Cylinder

with Pulse Input Intervals near the

Thermal Relaxation Times ...................

2--7 . Thermal Buildup when Pulse Interval is Small

Compared to Thermal Relaxation Time ........

2-8. Variation of Temperature for Nd:YAG

(At Center of Pump-End Face) ...............

2-9. Extreme Temperature for Nd:YAG

(At Center of Pump-End Face) ...............

2-10. Variation of Temperature for Ti:AL203

(At Center of Pump-End Face) ...............

2-11. Extreme Temperature for Ti:AL20 3

(At Center of Pump-End Face) ...............

3--1. Radial Temperature Distribution of

a Nd:YAG Crystal ...........................

Page

7

9

10

12

12

14

15

16

16

17

18

24

vi
Figures (cont.)

Figure

3-2. Crystal Orientation for a Nd:YAG Rod

(left) and Orientation of Indicatrix

of a Thermally Stressed Rod in a Plane

Perpendicular to the Rod Axis (right) ......

3--3. Predicted Thermal Focusing Effects in

Nd:YAG .....................................

3-4. Types of Pump Beam Profiles ................

3--5. Partial Illustration Of the Thermal

Lensing Matrix with a Series Of Thin

Lens Overlaid ..............................

4-1. Thermal Focusing in Nd:YAG Rod .............

4-2. Transient Thermal Focal Length for

a Side-Pumped Nd:YAG Rod ...................

4--3. Focal Length vs Cooling in SidePumped Nd:YAG Rod ..........................

4--4. Single Pulse Transient of Thermal Focal

Length in Nd:YAG Crystal ...................

4--5.

4-6.

Single Pulse Transient of Thermal Focal

Length in Ti:AL203 Crystal .................

Transient Thermal Focal Length for a

Multiple Pulsed, End-Pump Nd:YAG Rod .......

4--7 .

4--8.

Transient Thermal Focal Length for a

Multiple Pulsed, End-Pumped Ti:AL203 Rod...

Thermal Focusing in End-Pumped Nd:YAG Rod..

4-9. Thermal Focusing in End-Pumped

Ti:AL203 Rod ...............................

4-10. Focal Length vs Side Cooling in

End-Pumped Nd:YAG Rod ......................

4-11. Focal Length vs Side Cooling in

End-Pumped Ti:AL203 Rod ....................

Page

26

32

37

40

48

50

51

53

54

55

56

57

58

60

6O

vii

Abstract

Modeling of Thermal Lensing in Side and End-Pumped

Finite Solid-State Laser Rods

Student: Vincent G. Brackett

Date of Birth: December 17, 1942

Department: Mathematics

Advisor: Dr. U.O.Farrukh

An analytical expression for approximating the timedependent thermal focal length in finite solid-state laser

rods has been derived. The analysis is based on the temperature variation of the material refractive index caused

by optical pumping of these rods. Several quantities were

found to be relevant to this analysis. These quantities

were the specific thermal profiles of the rods, type of optical pumping employed, type of cooling scheme employed

(side and end-cooling parameters), and the specific material characteristics of the rods. The Thermal Lensing Model

was formulated using the geometric ray tracing approach.

The focal lengths are then approximated, by calculating the

phase shift in the index of refraction, as the different

rays of an incident plane wave are tracked through a lenslike crystal medium. The approach also applies in the case

of Gaussian or parabolic pump beams. It is shown that the

prediction of thermal focal length is in good quantitative

agreement with experimentally obtained data.

PRECEDING PAGE BLANK NOT FILMED

ix

PA_J.__._./i I INTENTIONALLY 8LAN_

CHAPTER 1

Introduction

The next generation of solid-state lasers (to be used

for remote sensing of the atmosphere from space platforms),

are under development. Various new lasing materials and

laser configuations are being investigated. Currently, the

solid-state, end-pumped diode lasers are the most promising

candidates due to their small size, reliability, performance, low power and cooling requirements, etc. Analyzing

thermal lensing effects for these end-pumped configuations,

using various cooling schemes are of interest to the designers of laser systems. Theoretical formulation, numerical calculations, and experimental data on thermal lensing

does not exist for end-pumped configuations.

The management of thermal energy deposited in a solidstate material is critical in the design of laser systems. 1

Any solid laser material operating in either pulsed or continuous wave (CW) mode of operation dissipates an appreciable amount of heat. The heat arises from non-radiative

energy transitions as the pump beam traverses the material.

Stated very briefly, when a material is excited to provide

more atoms (or molecules) in a higher energy state than in

some lower state, the material will be capable of amplifying radiation at the frequency corresponding to the ener-
gy level difference. 2 The acronym LASER derives its name

from this process: "Light Amplification by Stimulated Emission of Radiation."

It's during this process that some fraction of the total energy of the pump beam is absorbed by the lasing material. This absorbed energy is then dissipated as heat. In

the cylindrical geometries normally used, the heat is extracted from the surface of the cylinder, thereby generating a radial thermal gradient. Both radial and axial

gradients are present in end-pumped lasers. Employment of

active cooling to dissipate heat from laser crystals will

also affect the thermal gradients. Therefore, the thermal

gradients present in laser rods are governed by many factors. These include: geometries of the rod in use, type of

laser material, thermal diffusivity of the material, and

type of pumping and active cooling schemes employed. A

Temperature Distribution Model has been developed to model

the time dependent temperature distributions of finite

solid-state laser rods that include the above factors.l, 3

The change in temperature, within a laser rod, causes

thermal defocusing and distortion of the laser beam. This

is due to the temperature variation of the material refractive index. These thermal gradients cause stresses within

the laser material. The generated stresses induce birefringence and non-uniform expansion (bulge) in the laser
rod. Collectively, the above factors of temperature variation of the refractive index, induced birefringence, and

bulge are called thermal lensing. Thermal lensing is the

defocusing and distortion of a laser beam transmitted

through a material medium, due to the non-uniform heating

of the material. 4

In this thesis, using the Temperature Distribution

Model, thermal lensing effects will be analyzed to develop

a Thermal Lensing Model. The Thermal Lensing Model will be

employed to predict changes in the focal length and resultant distortions due to the time dependent temperature variations of the refractive index for two common laser materials that are under evaluation. These materials are

Neodymium doped Yttrium Aluminum Garnet (YAG) (Y3AL5OI2)

and Titanium doped Aluminum Oxide (Sapphire) (AL203). From

this point onward, these two materials will be called

Nd:YAG and Ti:AL203 respectively. They will be evaluated

for various pumping (side and end) and cooling schemes.

The induced birefringence and bulge caused by stress in the

laser material will be discussed. However, these factors

will not be implemented in the first version of the Thermal

Lensing Model as they are minor in nature.

In the next chapter, the Temperature Distribution Model will be discussed. It's from this model's output that

the time dependent temperature profiles are derived. These

profiles serve as input to the Thermal Lensing Model.

3
CHAPTER2

Temperature Distribution Model

The problem of managing the thermal energy deposited

within a solid-state laser material under active pumping

has long been of critical importance in the design of laser

systems. 1 Koechner's paper on the transient temperature

profile in solid-state lasers was the original work in this

area. 5 In it, the time dependent temperature distribution

of an infinitely long laser rod with a uniform heat source

was determined. This configuation was adequate to treat

the flashlamp pumping of relatively long laser rods, and

rods with little or no end cooling employed. However, this

treatment has become inapplicable to other pump configuations. Recently, several new solid-state laser materials

have been introduced, each with their own thermal deposition mechanisms. 6 Additionally, several other novel laser

techniques such as end-pumping with laser diodes have also

been developed. For these systems, the assumptions of a

uniform distribution of pump energy and an infinite rod

length are no longer valid. The one dimensional approach

of Koechner 5 does not apply when the length of the laser

rod is comparable to its diameter. Furthermore, the longitudinal symmetry of an infinite rod is broken when it is

end-pumped.

4
It was for these reasons that the Temperature Distribution Model was developed. The time dependent tempe£ature

distribution of a cylindrical rod of finite length has been

derived by Farrukh et al. in two separate papers, i'3 Additionally, they studied the evolution of thermal profiles

for various pumping and cooling schemes. The general expression of the temperature distribution is given as: 7

T(r,z] = _ C_Onff] _(z) * fit} [2-1)

n,m= !

where Cn is the radial eigenfunction,_/m is the axial eigenfunction, Cnm is the pump pulse expansion coefficient,

and f(t) is the time dependence function. The time dependence function f(t) for the on and off periods o£ the

pumping pulse is given, respectively as: 7

Tnm

f(t] - K * Lamda *

ncyclel rnm ( e- t-ncycler nm+ offtime 1

"It

cycle e Tnm

{ i-.ncycle ) + 1 - e rnm ._

[2-2]

f[t) -.

Tnn_.

K • Lamda

rnm t-ncycle * offtlme 1 - e r

( cycl_____e_e ) , e nm 1 - e rnm

[2-3]
where K is the inverse of diffusivity, Lamda is the thermal

conductivity, and _nm is the time constant for specific

eigenfunctions.

The expression for temperature distribution is general

and applies to various situations such as pulsed or continuous operations of the laser. A variety of different pumping and cooling schemes including end-pumping by a Gaussian beam were also discussed. To gain an appreciation for

the capability of this model, we will look at two of the

more interesting cases that this model can be used to evaluate. These are end-pumped single and multiple pulse operations for two common solid-state laser materials (Nd:YAG

and Ti:AL203).

A. Single Pulse Operations

If a laser rod is pumped by a single pump pulse, a

transient thermal profile will be established. This is the

result of a fast pump-introduced heating process followed

by a slow recovery of the rod to its ambient temperature.

Although this thesis will primarily address long pump pulses, it is appropriate to distinguish the difference between short and long pump pulses. If the duration of the

pumping pulse is very short in comparison to the laser rod

relaxation time, then the heat energy deposited in the rod

is considered to be deposited instantaneously. (The laser

rod relaxation time is defined as the time required for the

6
rod to relax to its ambient temperature.) Rod relaxation

times are approximately 3.5 seconds for Nd:YAG and 300ms

for Ti:AL203 for typical rod dimensions. When a pulse is

characterized as short or long, this characterization depends on the laser rod material. If the pump pulse duration is less than one percent of the relaxation time of the

rod, then the pump pulse is considered to be short. In

this case, as previously stated, the heat energy is deposited instantaneously. An example of this will be shown in

a later graph. For long pulse operations, the heat energy

being deposited builds up and relaxes simultaneously during

the pumping pulse.

Figure 2-1 is an illustration of the transient nature

of temperature distribution in an optically pumped cylindrical laser rod for a long pumping pulse. During the

Temperatur_

/" // / / ' '' ," ,"

I / / / II " ," ""

' I I 1 I

Figure 2-1. Thermal Relaxation of a Sinq!e Input

Pulse in a Cylindrical Laser Rod z'''

7
pulse, heat energy is absorbed by the laser crystal. At

the end of the pump pulse, the temperature of the rod has

increased above the ambient in a spatially non-uniform manner. Since only the surface of the rod is in contact with

the coolant, a radial and axial heat flow develops during

and at the end of the pump pulse. The initial non-uniform

spatial temperature distribution changes to a nearly parabolic temperature profile which then decays at a rate determined by the rod specific relaxation time constant.

Since lasing action occurs during or at the end of the pump

pulse, the optical distortion in single pulsed lasers

arises from the thermal gradients established during the

pumped cycle.

Figure 2-2 and 2-3 are examples of the Temperature

Distribution Model output for the time dependent heat propagation in end-pumped Nd:YAG and Ti:AL20 3 laser rods for a

pump beam with spatially circular and Gaussian profiles respectively, with moderate side cooling. In both of these

figures, the energy absorbed as heat, is iJ/cm 3 at the

front end of the rod for the circular pump and at the center of the front end for the Gaussian pump. Figure 2-2 was

chosen as an example because it shows the impact of side

cooling in generating radial gradients in a Nd:YAG rod after only 20ms into its thermal relaxation cycle. Also in

Figure 2-2, the temperature at the center of rod has not

8
1.0 I

0.5-

0.0- 0.8 0.6

-0.5 -

-1.0 , i

0.0 0.2

0.4 0.2

0.4 O.6

Relative Axial Distance

Time=0.0 seconds

Temp rise of:

, I , I

0.8 1.0

,-I

0.5

0.0

-0.5

-1.0 ,. [ , I ,

0.0 0.2 0.4

Time=0.02 seconds

Temp rise of:

0.2

0.6 0.8 1.0

Reative Axial Distance

Figure 2-2. Heat Propagation in Nd:YAG Laser Rod

(End-Pumped, Sin_l]].e Pulse with a

Uniform Cross Section Pumpinq Beam)

9
.4

O_

1.0

Time=O.O seconds

0.5 _-"____emp rise of:

0.0 _ 0.2 _' 0.1

#,

-0.5

-t.0

0.0 0.2 0.4 0.6 0.8

Relative Axial Distance

.0

°m

1.0

0.5

0.0

-0.5

-1.0

0.0

Time=O.02 seconds

Temp rise of:

0.1

[ , 1 __L I _ 1 _ .

0.2 0.4 0.6 0.8 1.0

Relative Axial Distance

Figure 2-3. Heat PropAgation in Ti:AL_Q 3 Laser Rod

LEnd-Pumped, Sinqle Pulse with a

Gaussian Cross Section Pumping Beam)

1o
appreciably relaxed toward the ambient temperature at the

20ms point in time. Figure 2-3 was chosen as an example to

illustrate the difference between a circular and Gaussian

pumping beam. This figure also shows that after only 20ms

into the thermal relaxation cycle, the Ti:AL203 laser rod

has significantly relaxed toward its ambient temperature.

However, the thermal energy content in the Gaussian pump

case is only about 1/8 of that of the circular pump case.

Figure 2-4 is an example of the model output of the

temperature rise above ambient for five different pulse

durations for a Nd:YAG laser rod with one pulse of 50mJ of

absorbed energy. The Nd:YAG laser was end-pumped with a

Gaussian beam and subjected to moderate side cooling. The

rod ends were not cooled. The Ims pulse in this figure is

an example of a short pulse where the heat energy is deposited instantaneously. The remaining pulses in this figure

are examples of relatively long pulses. In Figure 2-4, the

rod has essentially relaxed to the ambient temperature approximately 3.5 seconds after completion of the pumping

pulse.

Figure 2-5 is an example of the model output of the

temperature rise above ambient for five different pulse

durations for a Ti:AL203 laser rod using 50mJ of absorbed

energy per pulse. The Ti:AL203 laser rod was end-pumped

with a Gaussian beam and subjected to moderate sideII
O

cm

_D

td

(9

7

<

T

O

[i.

LO

4.00

5.00

2.00

1.00

0.00

I StngJe Putoe .

- 50n_ (absorbed)

Ims IOres

' Z 4 I II • 4 • " • • 4 • •

0.001 0.01 0.1

TIME (SEC)

Figure 2-4. Temperature Profiles for Various

Durations of a Single Pump Pulse for

Nd:YAG (At Center of Pump-End Face)

20.00

0

_D

Ld

O

Z

<

I

(D

O_

W

IB.00

12.00

8.00

4.00

Slngle Pulse

Energy - 50mJ (ab_rb_x:l)

0.01 0.1 1

TIME (SEC)

Figure 2-5. Temperature Profiles for Various

Durations of a Sinqle Pump Pulse for

Ti:AL20 3 IAt Center of Pump-End Face)

12
cooling. The rod ends were not cooled. The ims pulse in

this figure is an example of a short pulse where the heat

energy is deposited instantaneously. The remaining pulses

in this figure are examples of relatively long pulses. In

Figure 2-5, the rod has essentially relaxed to thermal

equilibrium approximately 300ms after completion of the

pumping pulse.

Comparison of Figures 2-4 and 2-5 also shows that the

Ti:AL203 rod has a greater temperature rise than the Nd:YAG

rod under the same conditions of absorbed energy, pulse

duration, and cooling scheme. These results are not unexpected as the thermal conductivity of Ti:AL203 is higher

than Nd:YAG.

B. Multiple Pulse Operations

In the previous section, we discussed pump induced

thermal profiles that occur for single-pulse operations.

The pulse input intervals are long with respect to the

thermal relaxation times of the laser rod. Additionally,

the rod has returned to the ambient temperature before the

onset of the next pumping pulse. If the laser is operated

at a repetition rate at which there exists a residual temperature distribution from the previous pulse, a temperature buildup in the rod will occur. The temperature at

specific points will increase until a steady-state condition is reached. Figure 2-6 is an illustration of the tem13
Time /

/

l

Tempera ture

-_ Pulse in

laser (h Pulse in

Figure 2-6. Thermal Relaxation in a SolidCzlinder

with Pulse Input Intervals near the

Thermal Relaxation Times 2

perature profile of a laser operated at a low repetition

rate.

Figure 2-7 is an illustration of thermal buildup at a

pulse interval smaller than the thermal relaxation time.

Under these conditions, the laser will soon establish a

steady-state operating condition where the radial and axial

rate of flow of heat out of the rod will equal the rate of

heat deposition. During the transition period, the rod

radial and axial thermal profiles will continue to'change

until the steady-state equilibrium temperature is reached.

In the extreme case, when the interval between pulses be--

14

ORIGINAL PAGE IS

OF POOR QUALITY
Time

Temperature

Figure 2-7. Thermal Buildup when Pulse Interval is

Small Compared to Thermal Relaxation Time 2

comes negligibly small compared to the thermal relaxation

time, the thermal profile of the laser rod will approach

the profiles 6f continuously pumped rods with the same total power deposition. The transient temperature distributions, as the rod approaches thermal equilibrium, has a

large effect on the laser energy output and beam divergence

of the laser 2. These transient periods will be discussed

in the next chapter on thermal lensing.

Figures 2-8 and 2-9 are examples of a Gaussian endpumped Nd:YAG laser being operated at a high pulse repetition rate. Figure 2-8 depicts the detailed buildup of tem15 ORIGINAL PAGE IS

OF POOR QUALITY
10.00

O

CD

v

LU

(9

Z

<

T

O

0_

bJ

8.00

6.00

4.00

_00

o.o_.d

S_ab|llzes at approx .3.5 sec

M|n = 70.3 deg C

Max == 72.5 deg C

_cle I

l ] i

© 0.005

Energy -==50mJ/pu[se (obsorbed) PW 2ms

PRF' = lO0/pps

I I l i

0.010 0.015

TIME (SEC)

Figure 2-8, Variation of Temperature for Nd:YAG

_At Cen{er of Pump-End Face[

0.020

C)

ET_

q)

"0

v

(9

Z

<

I

0

D_

W

F8O

60

40

20

t

Energy = 50mJ/putse (ob,_)

PW =- 2ms

PRF =- 100/ppe

_ob,,z= Mtn ot==y_.9x deg,:,.5C,_

Max = 72.5 deg C

0.05 ' to% ' 0._5 ' 0.20....' 0]5 ' 0.!

TIME (SEC)

Figure 2-9. Extreme Temperature for Nd:YAG

(At Center of Pump-End Face)

16
perature for two pump cycles. Figure 2-9 shows the maximum

and minimum temperatures within each pulse cycle. These

graphs show that the Nd:YAG laser will reach a steady-state

temperature in approximately 3.0 seconds. The estimated

maximum temperature of the Nd:YAG rod is approximately

72.5°C above the ambient at the center of the rod pump-end

face.

This knowledge of the maximum tem.nerature is critical

to laser design. For example, if the temperature exceeds

the stress fracture limit of the rod, the laser crystal

will fracture. The stress fracture temperature limit for

Nd:YAG is approximately I15°C. 2

Figure 2-10 and 2-11 are examples of the output from

30.00

24.00

o

cD

IB.0o

V

h]

<9

Z

<" 12.00

I

(.9

Q_

"_ 6.00

hJ

I---

o.°_.o

Energy - 50rr_//pulse(obsorb_d)

== 2rag

PRF = lO0/pp#

z I

)0 0.005 l J .L 1 i

0.010 0.015

TIME (SEC)

0.020

Figure 2-10. Variation of Temperature for Ti:AL20 3

_(At Center of Pump-End Face)_

17
8O

O

8O

"o

ILl

Z 4O

<

I

0

£L

20

L_

Energy- 50mJ/pulae (abeorbed)

PW -, 2m8

PRF - lO0/pps

t I 1 I z I , ] t I t ]

0.05 O.10 O.15 0.20 0.25 0._

TIME (SEC)

Figure 2-11. Extreme Temperature for Ti:AL203

iAt Center of Pump-End Face)

the Temperature Distribution Model of the temperature rise

due to accumulation effects of multipuise operations for

Gaussian end-pumped Ti:AL203 rods. As can be seen, the rod

reaches a steady-state temperature in approximately 300ms

and has a maximum temperature of approximately 42°C at the

center of the pump-end face.

The above presented example profiles from the Temperature Distribution Model are just a few of the many profiles

that can be generated. Other profiles, such as radial and

axial temperature profiles versus time, can also be generated for a variety of pumping and cooling schemes.

18
C. Summary of the Temperature Distribution Model

The general theory describing the transient thermal

profiles in CW and pulse-pumped lasers for a large variety

of operating parameters are presented in Koechner 2,5 and

Farrukh et al. 1,3,7 These general theories were implemented by Farrukh into a general analytical model. The

model then generates thermal profiles for a large variety

of finite solid-state lasers, with user specified operating

parameters. Examples of typical profiles were presented

throughout this section to give the reader a feel for the

capabilities of this model.

These time-dependent derived thermal profiles are

critical in determining thermal lensing effects which depend upon the spatial distribution of temperature and its

gradients. 8

In the next chapter, the theory of thermal lensing and

formulation of the Thermal Lensing Model will be discussed.

The model takes as its input, the time-dependent temperature profiles from the Temperature Distribution Model outlined above. The Thermal Lensing Model is then used to

estimate the focal lengths and distortions present in finite solid-state Nd:YAG and Ti:AL20 3 laser rods.

19
CHAPTER 3

Thermal Lensing

A. Introduction to Thermal Lensing.

Thermal lensing is the distortion and defocusing of a

laser beam transmitted through material medium, due to the

non-uniform heating of the material. 4 Any solid-state laser material operating in either the steady-state, pulsed,

or CW mode of operation must dissipate an appreciable

amount of heat. This is the result of a fast pump-introduced heating process (where some fraction of the total

pump energy is deposited in the laser rod as heat), followed by a slow recovery of the rod to its ambient temperature. In the cylindrical rod geometries normally used, the

heat is removed on the surface of the rod. This process

generates radial and axial thermal gradients in the rod.

The change in temperature, within a laser rod, causes

thermal defocusing and distortion of the laser beam. This

is due to the temperature variation of the material refractive index. Additionally, these thermal gradients cause

stresses within the laser material. The generated stresses

induce birefringence and non-uniform elongation of the rod

and bulging of the ends. Collectively, these above factors

of £emperature variation of the refractive index, birefringence, and bulge are referred to as thermal lensing. From

2O
Koechner 9, the temperature dependent variation of the refractive index constitutes the major contribution to thermal lensing. The stress dependent variation of the refractive index (birefringence) modifies the focal length

about 20 percent. 9 The effects of end-face curvature,

caused by an elongation of the rod on thermal lensing is

less than six percent. 9

The resulting distortion and defocusing of laser beams

have been observed in a variety of experiments. Several

theoretical treatments of thermal lensing have been given.

These include geometrical-optics I0, wave-optics II, and the

vector Kirchhoff diffraction formulation of thermal fensing 4,11 to account for thermally-induced birefringence.

In the general theory for the formulation of the Thermal Lensing Model, various effects of thermal lensing will

be discussed. However, in the model, only thermal lensing

due to the temperature variation of the material refractive

index using ray-trace methodology will be implemented. The

ray-trace method was chosen as the simplest approach to obtain a working model. This method makes maximum use of the

temperature profiles generated by the Temperature Distribution Model previously discussed. This approach also accounts for 75-80 percent of the thermal lensing effects

present in side and end-pumped laser rods. In general,

the model will give the user a first approximation of the

21
thermal focal length andthe degree of distortion expected

for rods of specific dimensions and specified operating parameters. The input to this approach will be the temperature profiles generated by the Temperature Distribution

Model.

In the next section, a detailed one dimensional theory

of thermal lensing from Koechner 2,9 will be presented. The

presented theory will outline the effects of temperature,

birefringence, and bulge that contribute to thermal lensing

for a side-pumped Nd:YAG laser rod.

B. One Dimensional Theory of Thermal Lensing.

The one dimensional theory of thermal lensing is

looked at because the basic underlying theory and mathematical formulation can be modified and applied to the problem

of developing the Thermal Lensing Model.

Consider the case of the CW side-pumped Nd:YAG rod

where the heat generated within the laser rod, due to the

optical pumping, are removed by a coolant flowing along the

cylindrical rod surface. As a result of the assumption of

uniform internal heat generation and uniform cooling along

the cylindrical surface of an infinitely long rod, the temperature profiles will be strictly radial. The end effects

and small variation of coolant temperature in the axial direction are neglected here. The radial temperature distribution in a very long cylindrical rod is obtained from the

22
one dimensional heat conduction equation: 13

d2T AT 0

+ _ + - 0 [3-I} dr 2 rut K

where K is the material thermal conductivity and Q is the

rate at which heat is generated per unit volume. The solution to equation (3-1) gives the temperature at any point

along a radius of length r. With the boundary condition

T(r o) for r=r o, where T(r o) is the temperature at the rod

surface and ro is the radius of the rod, then:

T(r] : T[ro] + _Kfro 2 - r 2} [3-2]

This temperature profile is parabolic, with the

highest temperature being at the center of the rod. The

heat generated per unit volume, for uniform heat absorbtion, can be expressed as:

Pa

0 = 7nr_L {3-3]

where Pa is the total power absorbed by the rod; ro is the

rod radius; and L is the length of the rod. The temperature difference between the rod surface and the center is

given by :

Pa

T[O) - T[r o] = _tuKL [3-4]

The transfer of heat between the rod and the coolant

23
creates a temperature difference between the rod surface

and the coolant. A steady-state equilibrium temperature

will be reached when the rate of the total internal generation (Pa) of heat is equal to the rate at which heat is removed from the rod by the coolant. For a typical Nd:YAG

rod, the equilibrium temperature is reached about 3 1/2

seconds after laser turn-on. Figure 3-1 depicts an example

of the radial temperature profile of a 20 pulse per second,

side-pumped, Nd:YAG laser at thermal equilibrium.

4.00 r

0

ED

-O

"-_ 3.00

Ld

O

Z

<

I

O 2.00

Ld

<

rr 1.00

hJ

EL

W

0.000.!-------Li Radius of Rod: 25rnm

I Length of Rod: 7.6cm

; .

I = I _j J • 1 1

0.2 0.4 0.6 0.9 1.0

RELATIVE RADIAL

Radial Temperature Distribution

for a Nd:YAG Crystal

Figure 3-1.

F

The heat is generated at the rate of 0.3 Joules per pulse

and negligible cooling was assumed at the ends of the rod.

With these conditions, axial thermal gradients do not exist

24
in this rod.

These temperature gradients generate mechanical radial, tangential, and axial stresses in the laser rod, since

the hotter inside area of the rod is constrained from expansion by the cooler rod surface. Normally, these stresses are calculated when one needs to find the birefringence

and the stress fracture limit of the crystal. For Nd:YAG

rods, Koechner 2,9 has provided a sample of these calculations. For example, if the approximate temperature differential between the center and the surface of the rod is

i15°C, then the crystal may fracture. The equations for

these calculations are derived by Timoshenko and

Goodier. 15

Having discussed the basic theory of one dimensional

temperature distribution and the mechanical stresses caused

by the thermal gradients, our attention can now be turned

to the photoelastic effects (birefringence). The mechanical stresses generate thermal strains in the rod, which in

turn produce refractive index variations via the photoelastic effect. The refractive index of a medium is specified by the indicatrix. In its most general case, the

indicatrix is in the shape of an ellipsoid. A change of

refractive index due to strain is given by a small change

in the shape, size, and orientation of the indicatrix. 16

Since Nd:YAG is a cubic crystal, the indicatrix is a

" 25
x

X

,'_'¢

\ _"

Figure 3-2. Crystal Orientation for a Nd:YAG Rod (!eft__

and Orientation of Indicatrix of a Thermally

Stressed Rod in a Plane Perpendicular to the

Rod Axis _i_.ht_ 2

sphere. Under stress, the indicatrix becomes an ellipsoid.

Nd:YAG rods are grown with the cylindrical axes along the

[i!I] direction. An illustration of this is depicted in

Figure 3-2. The light propagates in this direction, and

thus the change of the refractive index along the [iii]

axis is of interest.

Since the transverse stresses are in the radial and

tangential direction (see Figure 3-2), the local indicatrix

also orients its axis in those directions. In a cylindrical coordinate system, the photoelastic changes in the refractive index for the radial and tangential polarizations

are only obtained after a considerable amount of tensor

26

ORIGINAL PAGE IS

OF POOR QUALITY
analysis. This is normally done using the vector Kirchhoff

diffraction formulation approach. Various treatments of

these involved birefringence calculations for Nd:YAG are

covered in detail in references 4, 8, 12, and 17. The expressions for the radial and tangential refractive index

changes for Nd:YAG are given by 8,17 as:

_Q_ 2

= -1⁄2n_ x-%r _0_ 2 {3-6]

where 4n r and _n_ are the radial and tangential stress dependent changes of the refractive index, respectively; n o

is the refractive index of the rod at its ambient temperature; e is the thermal coefficient of expansion for

Nd:YAG; Q is the heat generated per unit volume; and K is

the thermal conductivity of Nd:YAG. The coefficients C r

and C_ are functions of the electrooptical coefficients

that were determined from tensor analysis for Nd:YAG. Both

coefficients were calculated by Koechner 2 for Nd:YAG.

Their values are listed as C r = 0.017 and C_ = -0.0025 respectively.

Having explored the temperature and stresses in a

Nd:YAG rod, the optical distortions which are a result of

both the thermal gradients and the stresses can now be discussed. The change in the refractive index can be separated into a temperature and a stress-dependent contribu27
tion. 2 Therefore:

n[r]= no + An[r]T + Anfr]s [3-7]

where n(r) is the radial variation of the refractive index;

n o is the refractive index of the material at ambient temperature; 6n(r)T and An(r) S are the temperature and stressdependent changes of the refractive index, respectively.

The temperature dependent change of the refractive

index can be expressed as: 2

An(r]T = IT[r] - T[O]]_T [3-81

where T(r) and T(0) are the temperature at the radius r and

center of the rod, respectively. With the aid of equations (3-2 through 3-4) and equation (3-8), we obtain: 2

Q

An[r]T - 4K d_Tr2 [3-9]

Comparing equations (3-5, 3-6, and 3-9), we see that the

refractive index in a laser rod shows a quadratic variation

with radius. An optical beam propagating along the rod axis also develops a quadratic spatial phase variation. This

perturbation is equivalent to the effect of a spherical

lens 14 The focal length of a lens-like medium whose index

of refraction is assumed to vary according to: 2

28
where b measures the degree of variation of the index of

refraction is given by: 14

f - F2

4noL

[3-11]

The total variation of the refractive index is obtained by introducing equations (3-5, 3-6, and 3-9) into

(3-7). This gives us: 2

n[r] = no I - 2no dT +

(3-12]

As was previously discussed, the change in the refractive index, due to thermal strain, is dependent on the radial (r) and tangential (¢) stresses where Cr= 0.017 and

C¢= -0.0025. Therefore, the expression for n(r) in (3-12)

yeilds two values. Comparing equations (3-i0) and (3-12)

yields the focal length (f') equation: 2

-I

The first term in the parentheses is the temperature dependent variation of the refractive index. The second term

contains the stress dependent variations of the refractive

index. Because of the birefringence, Nd:YAG has two effective focal lengths. One is due to the radial electrooptic

29
(Cr) coefficient and the second is due to the tangential

(C¢) coefficient.

In our final expression for the focal length of a

Nd:YAG rod, we must also include the effect of rod end distortion (bulge). The stresses, that cause bulging, are

found to occur within a region of approximately one radius

from the ends for Nd:YAG. 18 This deviation from flatness

of the rod ends is obtained from: 2

L[r] = aLolT[r] - T[O)] [3-143

where e is the thermal coefficient of expansion for Nd:YAG

and L o is the length of the end section of the rod where

expansion occurs. With L 0 = r o ind equation (3-2), we

obtain: 2

Qr2

Err) = -taro-4-R- [3-IS)

The contribution to the focal length of the rod caused

by end-face curvature is given by the thick-lens formula of

geometric optics 16 as:

f' = 2[n_'l] [3-16]

where the radius of end-face curvature is defined as R = -

(d2L/dr2) -I. From the expressions (3-14, 3-15, and 3-16)

follows the focal (f") length of the rod caused by bulge: 2

3O
f": K[aOro[n ° - 1}]-1 113-17]

Inserting equation (3-17) into (3-13) gives us the final

expression for the focal length of a Nd:YAG rod: 2

[ KA + act,_ no

-I

ar°[n°L- l] ) (3-183

where A is the rod cross-sectional area and Pa is the power

absorbed.

This expression accounts for the temperature, stressdependent variation of the refractive index, and end-face

curvature. If one introduces the appropriate material parameters for Nd:YAG into equation (3-18), then one finds

that the temperature dependent variation of the refractive

index constitutes the major contribution to thermal fensing. 2 The stress dependent variation of the refractive

index and end-face curvature modifies the focal length by

approximately 20 percent and less than 6 percent, respectively. 2

Figure 3-3 depicts the graph of the approximation of

the focal lengths, using equation (3-18), for a side-pumped

Nd:YAG rod. The focal lengths were calculated using the

following parameters: length and radius of the rod was

7.6cm and 25mm, respectively; dn/dt, the material variation

of the refractive index was 8.9 x 10-6; index of refraction

31
0.18

U3

n," 0.16

I.LJ

I--

EL. 0.14

0

E3 0.12

T 0.10

t--

(.3

Z 0.08

ILl

_J

d 0.06

<

C3 0.04

0

LL

0.02

0.(30

+ Temp, Rodlol Birefringence,

- ond Bulge

" o Temp, TangenUol /_

_ BJrefringence, Bulge / _

2 4 6 8 10 12 14 16 18 20

PUMP FREQUENCY (HZ)

Figure 3-3. Predicted Thermal Focusinq Effects

in Nd:YAG

(n o ) was 1.825; thermal conductivity (K) was 0.i03; e, the

thermal coefficient for expansion was 7.9 x 10-6; the radial (Cr) and tangential (C_) variation Of the refractive index were 0.017 and -0.0025, respectively; and, the power

absorbed (Pa) by the rod (as heat), was 6.119w/cm 3.

This completes the one dimensional theory of thermal

lensing for a Nd:YAG rod. The same analysis holds for

Ti:AL203 and other laser materials. As was previously

stated, the temperature dependent variation of the refractive index will be the only aspect implemented in the Thermal Lensing Model. The implementation of birefringence and

bulge effects will be left as future enhancements to the

32
model. In the next section, the formulation of the Thermal

Lensing Model is developed.

C. Two Dimensional Model Formulation.

Formulation of the thermal lensing ray-trace model

requires basic understanding of various optical theories.

These include the theories of ray tracing 16, wave optics in

crystals II,19, Fresnel diffraction theory of waves in lenslike mediums 14, geometrical-optics I0, etc. It was not the

purpose of this thesis to become trapped into the finer details of the referenced optic theories. Nor do we intend

to repeat the derivations of the referenced theories. If

more detailed information is desired, then we refer the

reader to the attached list of references and other literature on the subject.

The basic task of this research is to formulate a computer ray-trace model that allows for a first approximation

of focal length. The model is to be based on the temperature variation of the refractive index. The approximations

of focal length are then obtained on specific crystals

operated with specified parameters. This will be done by

modifying the focal length expression (3-11) for a thin

lens. The focal lengths are then approximated, by calculating the phase shift in the index of refraction, as the

different rays of a plane wave front are tracked through a

lens-like crystal medium. The localized change in the ray

3S
direction as it transverses a lens-like medium is assumed

to obey Snell's law which is given by: 19

n_sinOo = n2sinO _

[3-19]

where n I and n 2 are the index of refractions of two adjacent mediums, respectively; and 8 i and 80 are the input and

output angles of the ray at the boundary between the adjacent mediums.

To assist in the formulation of the model, it is appropriate to introduce, and explain, two tables and a figure before proceeding. Table 3-1 depicts an illustration

of the output block of radial and axial temperature profiles, provided by the Temperature Distribution Model.

This output is based on a 20pps, side-pumped, Nd:YAG laser.

The laser was operated with moderate side cooling and rod

Table 3-1. Block of Radial and Axial Temperature

Profiles for a Nd:YAG Crystal

Temperature Distribution

Radial Diffusivity ............. 26.95

Radius of Rod (cm) ............. 0.25

Length of Rod (cm) ............. 7.6

Time (sec) ..................... 3.5

R\L 0.00 ... 0.50 ... 1.00

0.00

0.20

0.40

0.60

0.80

1.00

2.9139

2.8020

2.4660

1.9047

1.1276

0.2248

2.9139

2.8020

2.4660

1.9047

1.1276

0.2248

2.9139

2.8020

2.4660

1.9047

1.1276

0.2248

34
ends were not cooled. In the vertical direction, are the

radial temperatures starting from the center of the rod

(top) and increments outward to the surface of the rod

(bottom). In the horizontal direction, are the axial temperatures starting from the front face (left) of the rod

and increments to the back end (right) of the rod. In this

particular example, a set of I0 equall_ spaced increments

was chosen to represent the 25mm x 7.6cm rod in both the

axial and radial directions. Additionally, in Table 3-1,

the temperatures are observed only to vary in the radial

direction. The displayed block of temperatures in Table

3-1 is an example of a side-pumped Nd:YAG rod at thermal

e_uilibrium, when rod ends aren't cooled.

Table 3-2 displays a block of radial and axial index

of refractions for the temperatures presented in Table 3-1.

Table 3-2. Block of Radial and Axial Index of

Refraction for the Temperatures in

Table 3-i

R\L

0.00

0.20

0.40

0.60

0.80

1.00

Index of Refractions

0.00 . . . 0.50 1.00

1.825026 1.825026

1.825025 1.825025

1.825022 1.825022

1.825017 1.825017

1.825010 1.825010

1.825002 1.825002

1.825026

1.825025

1.825022

1.825017

1.825010

1.825002

35
The block of temperature variations of the Index of Refractions were calculated from the expression:

n(r, zl = no + aT[r,z] [3-20]

where n o is the index of refraction for the rod at its ambient temperature and e is the dn/dt specified for the rod.

Table 3-2, shows that the index of refraction is only

changing in the radial direction which was expected for a

side-pumped Nd:YAG rod with no end cooling employed. For

end-pumped rods, the index of refraction would change in

both the radial and axial direction. The incremental partition of the index of refractions in Table 3-2 are the

same as the partitions of temperatures displayed in Table

3-1.

The user specified partition, selected for use in the

Temperature Distribution Model, governs the partition of

the Thermal Lensing Model. The partition of i0 equally

spaced increments are normally chosen because it is more

than adequate for first approximations. Increasing the

number of partitions does increase the accuracy of the approximation. However, the time required to run the Temperature Distribution Model increases linearly as the number of partitions increases. Comparison of the temperature

outputs for a partition of i0 and 20 equally spaced increments showed a mere 0.4 percent difference in the approxi36
mated temperatures.

Figure 3-4 illustrates the three types of pumping beam

profiles used to pump a laser rod. All three types of pumping profiles can be used to end pump a rod. (Normally,

only the circular and parabolic beam profiles are used to

describe the distribution of heat deposition in side-pumped

operations.)

Parabolic

Crystal Rod

Circu ar Gaussian

Figure 3-4. Types of Pump Beam Profiles

The above mentioned types of pump beam profiles have

been implemented in the Temperature Distribution Model. In

the thermal lensing ray-trace model, the focal length of

the rod irrespective of the type of pumping beam used can

be approximated. However, only the circular pump beam has

been implemented for distortion calculations. Implementation of distortion calculations for parabolic and Gaussian

beam profiles have been left as a future model enhancement.

Table 3-1 represents a matrix of temperatures for a specif37
ic rod operating with user specified parameters. It serves

as one of the primary inputs to the Thermal Lensing Model.

A radially dependent phase delay is experienced by a

wave front propagating through a medium with a varying index of refraction n. Thus, a thin slab of the crystal medium can be approximated as a thin lens. The lensing model

traces wave fronts propagating through the rod starting

from several radial positions at the rod front face. As

the ray propagates through the rod, a series of thin lenses

are setup and the temperature dependence of the index of

refraction is calculated at specified locations. Using

these calculations, a focal length associated with each

tracing ray can be estimated. An average focal length for

the rod is obtained by taking the mean of the focal lengths

of the individual tracing rays. The specified axial increment of the temperature matrix determines the number of equivalent thin lens placed in series that the tracing ray

must pass through. For example, with an axial incremental

spacing of 0.I, as depicted in Table 3-1, 20 equivalent

thin lens are placed in series from the front to the back

of the rod.

A plane wave propagating through a lens-like medium

acts as a spherical thin lens and diverges. Using this

fact, the thin lens focal expression (3-11) was changed to:

[rro ]2

fr = - _zL_[g{r,z} - _[0,z)] {3-21]

38
where r o and r are the exiting relative radius of the ray

and the radius of the rod in cm, respectively; z is the

specified axial increment of the rod (dimensionless)_ L is

the length of the rod in cm; and nr(r,z ), no(0,z ) are the

temperature dependent index of refraction for the radial

point of interest and for the center of the rod, respectively. The focal length expression (3-21) can be applied

upon determination of the quantities r o, nr(r,z), and

no(0,z). Calculating these scalar quantities are the essence of the thermal lensing ray-trace model.

The radial variation in temperature, as confirmed by

use of curve fitting algorithms, is quadratic in nature.

This fact allowed the selection and development of a binomial interpolation algorithm to obtain values of temperature at points other than the selected matrix points.

The binomial interpolation algorithm is based on the approximation:

ax 2 + bx + c = T (3-22)

where a,b,and c are constant coefficients, x is either the

radial (r) or axial (z) coordinate, and T is the temperature. By invoking this algorithm using a set of three collinear temperature (input) values along with their coordinate specifications, we can obtain a value for the temperature at any other point not specified in the temperature

matrix.

39
ri+2

ri+l

T

,l

T3 T3c

oTIO

nl n2

nl

T2c

T6

n2

T6c

o

n I n2

TllI

)---

T5c

series of thin lens

Tlc iT4 T4c

o. o

nl

T9

n2

TI2

T8

T1 T 7

ri )

zj zj+ 1 zj+ 2

Figure 3-5. Partial Illustration of the Thermal

Lensing Matrix With a Series of Thin

Lens Overlaid

The temperature matrix (obtained from the input file)

is illustrated in Figure 3-5. A wave front ray will be

traced starting at the ri+ 1 radial input position through

a series of equally spaced (Az/2) thin lenses. The ri+ 1 ray

was selected to illustrate the definition of the input and

output angles that occur as the ray passes through a thin

lens. These angles are necessary to calculate the exiting

radius of the ray (ro) for use in the focal length expression (3-21) and distortion calculations.

The rI ray (center of the rod) is also traced to ob4O
tain a summation of the temperature variation of the index

of refraction at the center of the rod, no(O,z). As illustrated in Figure 3-5, temperatures T 1 through T 9 are obtained from the temperature matrix provided as input. Invoking the binomial interpolation algorithm allowed the

calculation of the intermediate temperatures Tlc through

T6c for use in the thermal lensing scheme. This completes

the basic setup of the thermal lensing scheme and provides

the framework to proceed with the ray-tracing of specific

input rays.

A collimated plane wave entering the rod will have an

input angle (8i) equal to zero. As it passes through the

first lens, the angle (8 m) between the normal to the surface of constant temperature and the rod axis will be calculated at Az/2. To calculate 8m, we determine first the

surface of constant temperature by obtaining the r and z

derivatives of expression (3-22). The derivatives are given by:

dT(r) 2ar i + b [3-23] dr =

dT[Zldz= 2a(_J + h [3-24]

where ri is the input radius, Az is the axial incremental

spacing, and a,b are the calculated coefficients using the

appropriate radial and axial temperature points in the in41
i

terpolation algorithm. The angle 8m at Az/2 is then calculated using the trigonometric expression:

[3-25_

On finding the angle 8m, the next step is to determine the

specific values of the index of refraction on each side of

the imagined lens. Due to the fact that the temperature

dependePt index of refraction changes very little between

adjacent points (<< i), it was decided to compute the index

of refraction on each side of 8m according to the following

expressions:

n_ = no ÷ _ 2 [3-261

n_ = no + _ (T m 2+ Ts_ [3-27]

where n o is the index of refraction of the rod at its ambient temperature; e is the specified dn/dT of the rod; and

T 2, T 5, and TM (same as T2c in the first stage of tracing)

are the specific temperature points before, at, and after

the position of the imagined lens.

By invoking Snell's law, the output angle (8o) of the

ray passing through the thin lens can be obtained by substitution in the expression:

m

42
[3-28]

After finding the output angle, the change in output radius

(to) can be calculated from the trigonometric expression:

ro = _tan 80 + rm

[3-29]

where rm is the midpoint radial position of the r_y (same

as ri+ 1 for initial entry of the ray into the rod). The

output angle (8 o) and output radius (r o) are then sent to a

recursive numerical algorithm to refine the output values

at _z to meet operator specified accuracy criteria for 8 o .

The phase shift in the temperature variation of the

index of refraction can be calculated, for the initial

entry of the ray into the rod, and is proportional to the

expression:

[r o - rm]r [3-30]

and for later increments of the rod to:

+ rilr + / [r°- rm]r

nr[r,z] = nj 1 [rm -

[__ZL) 2 n2_'+ (____L) 2

[3-31]

43
The individual calculations from each lens, as the ray propagates through the rod, are retained and summed to determine the value for nr(r,z) which is then used in the focal

length expression (3-21).

One-to one distortion mapping of a ray propagating

through a lens-like medium is a straight forward calculation when dealing with side and end-pumped circular beams.

The input irradiance (1) of the collimated plane wave is

constant as it enters the crystal medium. The incremental

change in irradiance is then calculated at the output end

of the rod from the expression:

r_[i+l] - r_[i] [3-32]

AI = r_[i+l] - r_[i}

where ro, r i are the output and input radius of the ray being traced, respectively. The percentage of total distortion (D), for each ray, is then calculated from the expression:

D = lO0- (r°2[i+llr_+]]- r_[i+l})

[3-33]

This completes the formulation of the thermal lensing

ray-trace model. Now, it's just a formality of computer

record keeping to invoke the binomial interpolation

algorithm, using r o, to calculate TI0, TII, and TI2, reset

44
the output radius (r o) of the previous rod segment equal to

the input radius (r i) of the next rod segment, reset the

output angle (8 0) equal to the input angle (ei), and increment forward to the next imagined thin lens. The calculations are repeated until the ray exits the rod. When the

ray exits the rod, the focal length and distortion associated with the ray are calculated and retained. Then, the

ray-trace program increments upward in the matrix of temperatures to trace the next input ray. Upon completion of

the ray tracing for a block of temperatures, the mean of

the individual focal lengths and distortion present for

that block of temperatures are then calculated. The model

then increments forward to the next block of input temperatures (at a later time) and repeats the process until the

input file of temperatures are exhausted and a time dependent average focal length are obtained.

The Thermal Lensing Model program code is contained in

Appendix A. An example of the lensing model inputs and

output are enclosed in Appendix B and C, respectively.

45
CHAPTER4

Analysis and Discussion of Thermal Lensing

The next generation of solid-state lasers (used for

remote sensing of the atmosphere from space platforms), are

under development. Various new lasing materials and laser

configuations are being investigated. Currently, the solid-state, end-pumped diode lasers are the most promising

candidates due to their small size, reliability, performance, low power and cooling requirements, etc. Analyzing

thermal lensing effects for these end-pumped configuations,

using various cooling schemes are of interest to the designers of laser systems. Theoretical formulations, numerical calculations and experimental data do not exist for

end-pumped configuations.

Before proceeding with the analysis of thermal lensing

in end-pumped Nd:YAG and Ti:AL203 laser rods, an assessment

of the validity of the Thermal Lensing Model was conducted.

The validation was based on the comparison of model results

with the experimental data for a repetitively pulsed sidepumped Nd:YAG laser. This case was interesting from two

points. It was the only laser found, with actual experimental data on thermal focal length. The second point of

interest is that the laser was side-pumped. The later

point allowed for the extrapolation of Koechner's predic46
tion which are based on a CW side-pumped Nd:YAG laser.

Thus, the model results were compared to both the experimental results and Koechner's theoretical formulation for a

long continuously side-pumped rod.

A. Model Validation.

Research of the literature shows that theoretical formulation and experimental data on thermal lensing effects

do not exist for repetitive pulsed end-pumped lasers due to

their recent development. In fact, very little data exist

on repetitively pulsed side-pumped lasers. The vast majority of data published such as theoretical formulations,

theoretical calculations, experimental results, etc. are

based on CW side-pumped lasers dating back to the early

1970's. Most of these early analyses were conducted by Koechner 5,9 and are still being referenced to as the authoritative source on the subject today.

A recent experiment, depicting the results of thermal

focusing for a repetitive pulsed side-pumped Nd:YAG laser

has been completed. 20 The laser rod used in the experiment

had a length of 7.6cm, a radius of 25mm, was side-pumped

uniformly, and cooled along its length (ends were not

cooled). Each pump pulse had a duration of 130_s and an

energy level of i0 Joules. Experimental calculations indicated that approximately 3.5 percent of the pump energy was

absorbed into the rod as heat.

47
Both, the Temperature Distribution Model and Thermal

Lensing Model were executed using the Nd:YAG material characteristics and the laser configuation of the experimental

setup. The modeling and experimental results, along with

Koechner's prediction of focal length from equation (3-18)

are shown in Figure 4-1. (It should be noted that Koechnet's prediction, used in Figure 4-1, did not include birefringence and bulge effects.)

0.14

O9

Or"

LJ 0.12

F--

EL

_o o,lo

C3

_-I: 0.08

(_9

7

Ld 0.06

_J

,-J

< 0.04

0

0

b_

0.02

0.00

-- Pulse width = 130us

- : MEx°d_Impr_ndt_t'°r;.u,ts -_'x-'..--- ._-'/_

-- o KoechnePs prediction /,e /Oj_per pulse

- // // NOTE: B[refrlngence end Bulge effe.c__ .._

/ // h,_,e 6or been implemented in this

- +"/ verB[on of the model.

-J_--_I i I l J__-L_ t _ J_ J___.L_ I_ I l l

2 4 6 8 10 12 14 16 18 20

PUMP FREQUENCY (HZ)

Figure 4-1. Thermal Focusinq in Nd:YAG Rod.

Figure 4-1 shows that the model's prediction of the

thermal focal length are in good agreement with the experimental results and Koechner's theoretical prediction of focal length for an equivalent power problem. Thus, the ray48
trace model, based on the thermal gradients provided by the

Temperature Distribution Model, will give a fairly accurate

first order approximation.

Transient thermal focal length calculations of a sidepumped Nd:YAG rod do indicate an appreciable deviation from

the standard steady-state power calculations using Koechher's prediction equation (3-18). Starting from an inactive condition, the Nd:YAG rod thermal focal length did

stabilize at about 3.5 seconds. However, the steady-state

calculations from the model at the end of each pump pulse

indicates a higher focusing power of about 15-25 percent at

frequencies of 4-20 hertz over the results as predicted by

Koechner. This was not unexpected as the model was executed in an repetitive pulsed mode. Where as, Koechner's

prediction are based on a thermally stabilized CW pulsed

(same power) rod.

The transient side-pumped thermal focal length varies

from 28 meters at lOOms to 6.97 meters at the steady-state

equilibrium condition. The time dependent focal length, as

shown in Figure 4-2, was computed at the end of each pump

pulse for the above described laser rod. The rod was pumped by a 20 Hz, I0 Joule per pulse flash lamp. The horizontal line was obtained by using Koechner's formula for a

thermally stabilized CW pulsed (same power) rod whose surface temperature was kept at a fixed value.

4g
..-,, 0.18

03

[.d 0.16

I--

(3_

0 0.14

_ OA2

I

r._ O.lO

z 0.08

0.06

g 0.04

LL

0,02

0.000,

Pulse width -,, 130us

Pulses per emc -, 20

Energy ,= lOj per pulse

s t_lodel pre_lCtlon

o Koechneg= llme

J = l_J 1 = I I I _[ J = [ i I

0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.8

TIME (SEC)

Figure 4-2. Transient Thermal Focal Lenqth for

a Side-Pumped Nd:YAG Rod

Thermally induced focusing varies significantly in the

time interval between pulses. Even at a 10Hz repetition

frequency, the variation in the Nd:YAG thermally induced

focal length was approximately I0 percent.

The model was also used to estimate the impact of surface cooling on the focal length. Focal length was found

to be very sensitive in the low end of the surface cooling

parameter. Figure 4-3 displays the results of focal length

versus cooling in side-pumped Nd:YAG rods. The cooling

parameter is a dimensionless quantity obtained by multiplying the length of the rod by the surface heat transfer

5O
coefficient and dividing by the bulk thermal conductivity

of the rod material. Koechner's formulation, by comparison, is for a fixed surface temperature.

0.18

W_" 0.16

0 0.14

0.12

I

F- 0.10

L9

Z

LJ 0.08

__1

0.06

<

<9

O 0.04

U-

"_ 0.02

i

Sido Cooling Only

Energy ,= lOJ Per Pulse

130 us Pulse

20 pps

0.00_10 i 0I ¢ 10I = 201 I 30L=_j_l 40 i 50[ =_ 60 I 70I I 80I J 90! I 1O0I__

SURFACE COOLING PARAMETER

Figure 4-3. Focal Lenqth vs Coolinq in SidePumped Nd:YAG Rod

Figure 4-3 is also important to system designs. It

shows that above a certain point, very little is gained

from additional cooling. Therefore, the most efficient

cooling for the laser is critical for spaced based applications. Power consumption in space platforms are one of the

most critical design parameters. Additional unneeded

cooling may strain the limited power resources available in

space.

51
One-to-one mapping of intensity in the repetitive pulsed side-pumped Nd:YAG laser showed distortion to be negligible during the rod transient period. This result was

expected due to the presence of only radial thermal gradients (no axial gradients) as a result of the uniform pumping of the rod and the absence of end-cooling.

B. Analysis of Thermal Lensing in End-Pumped Nd:YAG and

Ti:AL203 Laser Rods.

Thermal lensing calculations have been obtained for

the two end-pumped laser rod configuations depicted in Table 4-1 below. Both of these laser crystals are currently

undergoing feasibility experiments in the laboratory at

NASA Langley Research Center. It should be noted here that

the comparison is not between crystals of different dimensions, but between cyystals of practical dimensions. Both

Table 4-1. End-Pumped Laser Rod Characteristics

Parameters Nd:YAG Ti :AL203

Rod Length (cm)

Rod Radius (cm)

Thermal Diffusivity (cm2/s)

Thermal Conductivity (W/cm OK)

Material dn/dt

Index of Refraction

optical Attenuation

Pump Wavelength (microns) _

Pump Energy Absorbed (J/cm 3)

Pump Pulse Duration (seconds)

1.0

0.4

26. 947

0.103

8.9x10 -6

1.825

3.0

1.15

0.050

0.010

0.6

0.2

8.87

0.330

1.28xi0 -5

1.7654

3.0

0.6438

0.050

0.010

crystals were end-pumped by a Gaussian beam where the ratio

L

52
of the beamwidth to the rod radius is 0.5. With the exception of the rod material characteristics, all other parameters such as side and end cooling, energy absorbed, pump

pulse duration, etc. were the same for both rods. Both,

the Temperature Distribution Model and the Thermal Lensing

Model were executed using the appropriate material characteristics of the rod and the laser configuation depicted in

Table 4-1.

Figures 4-4 and 4-5 presents the single pulse transient behavior of thermal focal length for Nd:YAG and

Ti:AL20 3 crystals, respectively. Both rods were end-pumped

b0

nz

w

bQ_

O

C_

7-

O

Z

W

_J

_J

<

O

O

u_

w--

0.03

0.02

0.0I

0.00

0.001

Single lOma pulse

50mJ Absorbed

0.01 0.1

TIME(SEC)

Figure 4-4. Single Pulse Transient of Thermal

Focal Lenqth in Nd:YAG Crystal

with a single pulse of 10ms duration. The energy absorbed

53
"_" 0.40

U)

cY

taJ

b--

EL

--- o.._o

-rI.--

0

Z 0.20

LLI

._J

._J

<

C) 0.10

0

LL

0.00

0.001

t Single lOres pulse

50rnJ Abaorbed

1 I

1 4. I • I 4 $ • I 4.

0.01 0.1

TZME(SEC)

I I I I 1_

• t

Figure 4-5. Sinqle Pulse Transient of Thermal

Focal Lenqth in Ti:AL_Q$ C__r_sty__l

into the rod as heat was assumed to be 50mJ per pulse. Moderate side cooling parameter of 20 was employed. Ends

were not cooled. (The cooling parameter is a dimensionless

quantity obtained by multiplying the length of the rod by

the surface heat transfer coefficient and dividing by the

bulk thermal conductivity of the rod material.) Starting

from an inactive condition, the thermal focal length for

Nd:YAG ranged from 205 meters at 2ms to the minimum of 44.8

meters at 10ms when the pump pulse was turned off. For

Ti:AL203, the thermal focal length ranged from 18 meters at

Ims to 2.87 meters at 10ms. Although some residual heat

still exist in the rods, as the rods relaxes to its ambient
condition, the thermal focal lengths are no longer measurable 1.30 seconds and 250ms after pump shutdown for Nd:YAG

and Ti:AL203 rods, respectively. For all practical purposes, as far as focal length is concerned, the rods have

returned to their ambient state at this point.

The transient thermal focal length profiles for Nd:YAG

and Ti:AL203 during multiple pulse operations are shown in

Figures 4-6 and 4-7. As seen in Figure 4-6, the Nd:YAG rod

0.24

O3

PK

i--

O- 0.18

O

0.12

g 0.06

L

,/

o.ooo. - '

50mJ Absorbed

20pps

1Ores pulse

iii

/

I i I i I i I i I i I i I i [ i L

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

TIME (SEC)

Figure 4-6. Transient Thermal Focal Length for

a Multiple Pulsed_ End-PumR Nd:YAG Rod

J

thermal focal length stabilized at about 3.5 seconds. For

Ti:AL203, the rod thermal focal length stabilized at about

300ms. In both rods, it was found that the thermal focal

lengths stabilized at the same time the temperatures became

55
0.50

O3

CE

bJ 0.40

0_

O

E3

0.30

T

NO

X

I,I 0.20

-J

_J

<

O

0 O.lO

LL

50rnJ Absorbed

20pp8

1Orris pulse

i 1 = I ¢ I = I

0.10 0.20 0.30 0.40

TIME (SEC)

Figure 4-7. Transient Thermal Focal length for a

Multi21e Pulsed, End-pumped Ti:AL20 3 Rod

stabilized which was expected. The Nd:YAG transient endpumped thermal focal length varied from 44 meters at 10ms

to 6.11 meters at the steady-state equilibrium condition.

For Ti:AL203, the thermal focal length varied from 2.87 meters at 10ms to 2.66 meters at steady-state equilibrium.

These time dependent focal lengths, shown in the solid

lines, in Figures 4-6 and 4-7 were computed at the end of

each pump pulse for the described laser rods. Both rods

were pumped at a 20 Hz rate. Energy absorbed into the rods

as heat was assumed to be 50 mJ per pulse.

Thermally induced focusing varied significantly in the

56
time interval between pulses as shown by the dashed lines

in Figures 4-6 and 4-7. The variation was approximately 26

and 75 percent for Nd:YAG and Ti:AL203, respectively. The

time dependent focal length between pulses were computed at

the start of each pump pulse for the above rods. Not accounting for the temporal variation of the thermally induced focal length during the pulse relaxation period, may

lead to significant errors during design.

The next factor looked at was thermal focusing versus

pump frequency and the impact of side and end cooling on

thermal focal lengths. Figures 4-8 and 4-9 depict the results for Nd:YAG and Ti:AL203, respectively. Pump frequen0.18

Of

I.d

I---

1:1.

0

C3

0.12

I

I--

0

7

bJ

...J

-J 0.06

.<

0

LL

0.00

50mJ Absorbed

10me Pulse

J T._ __

/ _ _ ._ _ Cooling

,'t_ I i I I I _.L___ I , 1 = L I I I I , I ] 1

2 4 6 8 10 12 14 16 18 20

PUMP FREQUENCY (HZ)

Figure 4-8. Thermal Focusinq in EndPumped Nd:YAG Rod

57
0.50

O9

nD2

FO_ 0.40

0

a

T 0.30

F--

CD

Z

LJA

._j 0.20

_J

<

0

O 0,10

I,

0.00

SMe and End

Cooling

Side Cooling Only

50mJ Absorbed

lomb, Pulae

__t J l _1 t I t I J I , 1 = t.

2 4 6 B 10 12 14 16 18 20

PUMP FREQUENCY (HZ)

Figure 4-9. Thermal Focusinq in EndPumped Ti:AL203 Rod

cy was varied between 1 and 20 HZ. For each frequency, the

rod was allowed to stabilize (3.5s for Nd:YAG and 300ms for

Ti:ALgO3) before calculations were made. In both figures,

moderate side cooling parameter of 20 was employed. For

calculations involving both side and end cooling, end

cooling was set equal to the side cooling. As shown in

Figure 4-8, the impact of end cooling caused a significant

decrease in the Nd:YAG thermal focal length above 2 Hz.

This result was expected as the radial and axial temperature gradients near the rod ends become smoother due to end

cooling:

58 =
For Ti:AL203, very little impact from end cooling was

experienced until the 12 HZ pump frequency was reached.

However, in comparison to Nd:YAG where the thermal focal

length was significantly decreased, the opposite effect was

obtained for Ti:AL203. End cooling of Ti:AL203 crystals

causes small increases in the thermal focal length as the

pump frequency is increased. This result was not expected.

However, upon further analysis of the temperature profiles

for Ti:AL203, it was apparent that the radial and axial

temperature gradients near the ends were greater due to end

cooling.

Finally, the model was used to estimate the impact of

surface cooling on the induced thermal focal lengths for

end-pumped Nd:YAG and Ti:AL20 3 laser crystals. The rods

were pumped at a 20 HZ rate and energy absorbed into the

rod as heat was assumed at 50mJ per pulse. Figures 4-10

and 4-11 presents the results of focal length versus side

cooling for these configuations. The focal length was

found to be very sensitive only on the low end of the surface cooling parameter for both rods. Therefore, sizing

end-pumped lasers for correct cooling is critical for spaced based applications as previously explained.

Distortion calculations were not performed on these

end-pumped lasers as the function for mapping of Gaussian

beams has not been implemented in the model (left as a fu59
0.30

bO

n_

W 0.24

FO_

0

v 0.18

-r

I--

(_9

Z

W 0.12

._1

._J

<=:

0

O 0.06

LL

,¢-..

Side Cooling Only

50mJ Absorbed

lOms Pulse

20 pps

0.00 ._-J-_--J _L J I J I f I I 1 J I J__L_J___L_J_ I , 1

-]u u ]u 20 ,30 40 50 60 70 80 90 100

SURFACE COOLING PARAMETER

Figure 4-10. Focal Lenqth vs Side Cool inq

in End-Pumped Nd:YAG Rod

0.50

09

n_

w 0.40

Q_

O

c3

"-/ 0.30

T

F'-

(-9

Z

W 0.20

-3

_.J

.<

(..)

O 0.10

LL

,¢...

Side Cooling 0nly

50mJ Absorbed

10me Pulse

20 pps

0.00_10 J 1 L 1 , 1 , ! ...= 1 , I I I ,__I J 1 , I l_J__._ 0 10 20 30 40 50 60 70 80 90 100

SURFACE COOLING PARAMETER

Figure 4-11. Focal Lenqth vs Side Cooli_ in

End-Pumped Ti:AL2Q 3 Rod

6O
ture model enhancement). Unlike the side-pumped configuation, previously discussed, where distortion was nonexistent at thermal equilibrium, preliminary indications

are that distortion in the end-pumped configuations will be

on the order of 3-5 percent at thermal equilibrium.

61
CHAPTER 5

Summary and Conclusions

An analytic expression for approximating the time-dependent thermal focal length in finite solid-state laser

rods has been derived. The analysis is based on the temperature variation of the material refractive index caused

by optical pumping of these rods. This analysis is also

applicable to situations where the laser rod is side or

end-pumped by circular, parabolic, or Guassian beam profiles. Short or long pulse, continuous wave (CW), or repetitive pulsed operations are also applicable. The repetitive pulsed, end-pumped solid-state diode laser configuations are of current research interest as the next generation of lasers will come from this category.

Several quantities were found to be relevant to this

analysis. These quantities were the specific thermal profiles of the rods, type of optical pumping employed, type

of cooling scheme employed (side and end-cooling parameters), and the specific material characteristics of the

rods.

The Thermal Lensing Model was formulated using geometric ray tracing based on the focal length expression for a

thin lens. The focal lengths are then approximated, by

calculating the phase shift in the index of refraction, as

62
the plane wave is tracked through a lens-like crystal medium.

Thermally induced temperature profiles, for end-pumped

laser rods operating with specified parameters, are approximated by the Temperature Distribution Model. These thermal profiles are then inputted to the developed Thermal

Lensing Model. The Thermal Lensing Model is then used to

approximate the transient thermal focal length and estimate

the final rise in thermal focal length for the laser rod

under evaluation. Various pumping and cooling schemes can

be evaluated for their effect on the induced thermal focal

length. For circular pump beam profiles, distortion calculations are also approximated by the model.

The numerical examples for end-pumped, repetitively

pulsed Nd:YAG and Ti:AL203 laser rods showed the following

major results.

a. The minimum induced thermal focal length is

achieved at the time of pump pulse turn off

for both rods operating under single long

pulse conditions.

b. For multiple pulse operations, the minimum

induced thermal focal length was achieved when

the rod reached its thermal equilibrium condition (3.5 seconds for Nd:YAG and 300ms for

Ti:AL203).

63
c. Thermally induced focusing varied significantly in the time interval between pulses (26

percent for Nd:YAG and 75 percent for Ti:-

AL203).

d. Employment of end-cooling significantly decreased thermal focusing in Nd:YAG above the 2

Hz pump frequency. For Ti:AL203, the opposite

effect was found. Small increases were

noticed in the thermal focal length for pump

frequencies above 12 Hz.

e. Under conditions when only the side cooling

was varied, the induced thermal focal length

was found to be very sensitive at the low end

of the scale for both rods.

The model was validated against experimental data and

theoretical calculations of thermal focal length for a repetitive pulsed, side-pumped Nd:YAG laser. It is seen that

the prediction of thermal focal length is in good quantitative agreement with experimentally observed behavior. Thus,

the Thermal Lensing Model, based on the thermal profiles

provided by the Temperature Distribution Model, will give a

fairly accurate first order approximation of the thermal

focal length.

64
Appendix A

Computer Program Listing

I. Functional Description

The Thermal Lensing Model (THERLENS.FOR) is not a

standalone computer program. Its primary input is dependent on the output of a previously developed theoretical

model titled: Temperature Distribution Model for finite optically pumped laser rods. The output of the Temperature

Distribution Model (blocks of temperature data versus

time), along with other thermal lensing constants (see Appendix B), are inputted to the lensing model. The model

then calculates the finite changes in focal length and distortion effects due to the temperature gradients within the

laser rod. Appendix C depicts the Thermal Lensing Model

output.

Currently, the lensing model can calculate the thermal

lensing effects for side and end-pump configurations and

distortion approximations for circular pump beams only.

(Note: Distortion calculations for parabolic and Gaussian

pump beams will be implemented in the future.) The optical

axis of the laser rod is assumed to be along the axis of

the rod.

2. Environmental Chsracteristics

65
This program was written in the Fortran language. It

was compiled and ]inked using the Microsoft Fortran 77

Optimizing Compiler Version 4.10. The Thermal Lensing Model can be executed on any IBM/IBM compatible 286/386AT micro-computer with a 287/387 Math Coprocessor. This version

can also be executed on the VAX 750 mini-computer. The

source program size on the AT-machine is 46,527 bytes and

the executable program is 69,621 bytes. The enclosed program code contains all changes through March I, 1990.

. Thermal Lensing Program Listing

(Program listing starts on next page)

66
Source

1

2

3

4

5

6

7

8

9

i0

ii

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

Line

$DEBUG

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

Thermal Lensing Program Listing

03-01-90

18:42:07

Microsoft FORTRAN Optimizing Compiler Version 4.10

A COMPUTERPROGRAMSIMULATIONHODELTO COMPUTETHERMALLENSINGFROM

TEMPERATUREDISTRIBUTION PROFILESINOPTICALLYPUMPEDLASERCRYSTAL

RODS.(YY.FOR)

THERLENS.FOR: Version 1.0

Language: Microsoft Fortran77 Version 4.10

Written by: Vincent G. Brackett

Graduate Student Applied Mathematics

Hampton University

Hampton, Va. 23668

Thesis Advisor: Dr. Usamah O. Farrukh

Department of Engineering

Hampton University

For: NASA Langley Research Center

Hampton, Va. 23666

GRANT: NAG 1-666

Monitor: Philip Brockman

Date: Last Revised: March i, 1990

THIS PROGRAM USES THE TEMPERATURE OUTPUT PROFILES OF A PREVIOUSLY

DEVELOPED COMPUTER SIMULATION MODEL (THAT MODEL PREDICTS THE

NORMALIZED TEMPERATURE DISTRIBUTIONS VERSUS TIME IN FINITE SIDE AND

END-PUMPED LASER CRYSTAL RODS) TO PREDICT THERMAL LENSING (FINITE

CHANGE IN FOCAL LENGTHS) CAUSED BY THE RESULTANT TEMP GRADIENTS IN

THOSE RODS. THERMAL DISTORTION OF THE PUMP BEAM ARE CALC AND AVG

VALUES OF THE FOCAL LENGTHS AND DISTORTION ARE PRESENTED.

CAUTION: The matrix mesh of the Temperature Distribution Model and

this model must be the same size. To change matrix mesh

size, only the parameter statement needs to be changed.

67
44

45

46

47

48

49

5O

51

52

53

54

55

56

57

58

59

6O

61

62

63

64

65

66

67

68

69

7O

71

72

73

74

75

76

77

78

79

8O

81

82

83

84

85

86

87

88

89

9O

91

92

93

94

PARAMETER(N=II,M=II)

DIMENSION TEMP(N,M),INDEX(N,M),RAD(M),LON(N),DISTOR(M)

DIMENSIONANGI(M),ANGM(M),ANGO(M),REND(M),FOCAL(M)

REALAMBDA,LAMBDL,K,LRMIN,LRMAX,LR,LENG,LON,IOR,INDEX,NI,N2,LONI

DOUBLEPRECISIONP!,SUMIOR,SUM!NT,SIORO,SIORJ,SINTJ,ANGIJ,ANGOJ

DOUBLEPRECISIONRI,RM,RO,Z,A,B,C

CHARACTER*I XX(140)

COMMON /BLOCKI/IOR,DT,ACC,NI,N2,ZI,RI,RM,TM,LON!,I

COMMON /BLOCK2/RI,R2,R3,ZI,Z2,Z3

COMMON /BLOCK3/TI,T2,T3,T4,TL,T6,T7,T8,T9

COMMON /BLOCK4/TIC,T2C,T3C,T4C,T5C,T6C,TIO,TII,TI2

OPEN<4,FILE:'THERLENi. INP_,STATUS-" 'OLD')

oPEN(5, FILE:ITM LEN2 INPI,S ATUS:IOLD'I OPEN(6,FILE=THERLENS18U:,ST_TU__aNOWN )

...(PROCESSING OF INPUT FILE ONE)...

WRITE(6,*)' '

WRITE(6,*)' ......THERMAL LENSING OUTPUT FILE ......'

WRITE(6,*)' '

WRITE(6,*)' .........INPUT FILE _i ..........'

WRITE(6,*)' '

READ(4,'(80AI)') (XX(1),I=l,80)

WRITE(6,*) (XX(I),I=I,80)

READ(4,'(80AI)') (XX(1),I=I,80)

WRITE(6,*) (XX(!),I=l,80)

READ(4,*) WL

WRITE(6,*) '..WAVELENGTH OF INTEREST IN UM..(WL).: ',WE

READ(4,*)TOI

WRITE(6 *)'..TEMPERATURE OF INTEREST IN DEG C..(TOI).: ',TOI

READ(4,*) IOR

WRITE(6 *) '..INDEX OF REFRACTION..(IOR).: ',IOR

READ(4,*) DT

WRITE(6 *) '..ALPHA (dn/dt)..(DT).: ',DT

READ(4,*) TNF

WRITE(6 *) '..TEMP NORMALIZATION FACTOR..(TNF).: ',TNF

READ(4,*) TINF

WRITE(6 *) '..TIME NORMALIZATION FACTOR..(TINF).: ',TINF

READ(4,*) IDATA

WRITE(6 *) '..0=RELATIVE TEMP; 1=ACTUAL TEMP..(IDATA).: ',IDATA

READ(4,*) Zl

WRITE(6,*) '..AXIAL INCREMENT FACTOR..(ZI).: ',ZI

READ(4,*) POWER

WRITE(6,*) '..POWER MULTIPLICATION FACTOR..(POWER).: ',POWER

READ(4,*)Ace

68
%

95

96

97

98

99

i00

I01

102

I03

104

I(]5

106

107

I08

109

ii0

IIi

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

"141

142

143

144

C

C

C

5

i0

C

15

16

C

20

C

WRITE(6,*)'..OUTPUT ANGLE ACCURACY CRITERIA..(ACC).:',ACC

READ(4,*) TPUMP

WRITE(6,*) '..TYPE OF PUMPING (0=side, I=end)..:',TPUMP_

WRITE(6,*) '.................................................... '

WRITE(6,*) '.................................................... '

...(PROCESSING OF INPUT FILE TWO)...

WRITE(6,*)' '

WRITE(6,*)' .......INPUT FILE _2 ......'

_RITE(6,*)''

JJ=0

CONTINUE

IFf(JJ/3)*3.NE.JJ) GOTO 10

READ(5,100,ERR=86) (XX(I),I=l,132)

WRITE(6,100) (XX(I),I=l,132)

CONTINUE

IF(JJ.NE.0) GO TO 20

IF(TPUMP.EQ.0) THEN

DO 15 J=l,80

READ(5,100,ERR=87) (XX(I),I=l,132)

WRITE(6,100) (XX(I),I=l,132)

CONTINUE

ELSEIF(TPUMP,EQ,I) THEN

DO 16 J=l,81

READ(5,100,ERR=87) (XX(I),I:l,132)

_RITE(6,100) <XX(1),I:l,n2)

CONTINUE

ENDIF

JJ=JJ+l

P=P+I

READ(5,10(],ERR=88) (XX(1),I=I,132)

WRITE(6,*)'

READ(5,100)

WRITE(6,1(]0)

READ(5,100)

WRITE(6,100)

READ(5,105)

_ITE(6,1O5)

R_AD(5,105)

WRITE(6,105)

READ(5,105)

WRITE(6,105)

READ(5,105)

WR!TE(6,105)

BLOCK OF TEMPERATURE PROFILES FROM INPUT FILE

(XX(I),I=l,132)

(XX(I),I=I,132)

(XX(1),I=I,132)

(XX(1),I:I,132)

(xx(I),I:l,5o) DIFF

(XX(I),I=l,50) DIFF

(xx(i),_=l,5o) RADI

(xx(1),i:1,5o) RADI

(xx<i),i=l,so) LENG

(XX(1),I=l,50) LENG

(xx(i),i=t,5o) SEC

(xx(i),i=],5o) SEC

#2'

DO 25 J=l,2

READ(5,100) (XX(I),I=l,132)

69
145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

16i

162

163

164

165

166

167

168

169

170

171

172

175

174

175

176

177

17B

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

WRnE(6,100)(XX(1),1:1,1n)

25 CONTINUE

READ(5,iI5) (XX(1),I:I,7),(LON(I),I=I,N)

WRITE(6,115) (XX(1),I:I,7),(LON(1),I=I,N)

DO 30 J=I,M

READ(5,125) (XX(1),I:I,2),RAD{J),(TEMP(I,J),I=I,N)

WRITE(6,125)

3G CONTINUE

DO 40 J=I,M

DO 35 I=I,N

TEMP(I,J)=TEMP(I,J)*POWER

IF(IDATA.EQ.0) TEHP(I,J):TEMP(I, )*TNF

35 CONTINUE

40 CONTINUE

WRITE(6,*)' '

WRnE(S,*)'

_RITE(6,*)' '

WRITE(6,*)'

WRITE(6,*)' '

DO 50 J=I,M

DO 45 I:I,N

INDEX(I,J):IOR÷(DT*TEMP(I,J))

45 CONTINUE

WRITE(6,125) (XX(1),I=I,2),RAD(J),(INDEX(I,J),I=I,N)

50 CONTINUE

C

C

C

PI=3.141592653589793D00

AREA=PI*ZI**2

SIOR0=Q

DO 60 J=I,M-I

• A_J=O.0DO0

SIORJ=0.0D00

SINTJ=0.OD00

DO 55 I=I,N-I

IF(I.EQ.10) GOTO 65

IF(J.EQ.I) RI=-RAD(J+I)

IF(J.OT.I) RI:RAD(J-I)

R2:RAD(J)

R3=RAD(J+I)

IF(J.EQ.I) TI:TEMP(I,J+I)

IF(J.GT.I) TI:TF/4P(I,J-I)

IF(J.EQ.i) T4=TEMP(I+I,J+I)

IF(J.OT.I) 4:TEM (I+I,J-I)

IF(J.EQ.I) T7=TEMP(I+2,J÷I)

IF(J.GT.I) T7=TEHP(I+2,J-I)

T2=TEMP(I,J)

THERMAL LENSING PROGRAM OUTPUT DATA'

INDEX OF REFRACTION MATRIX '

...(CALC OF LENSING AND DISTORTION)...

7O
195

196

i97

198

199

2OO

201

202

2O3

204

2O5

206

207

2O8

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

65

TL=TEMP(I+I,J)

T8=TEMP(I+2,J)

T3=TEMP(!,J+I)

T6=TEMP(I+I,J+IJ

T9:TEMP(I+2,J+I)

ZI:LON(1)

Z2=LON(I+I)

Z3=LON(I+2)

Z=LON(1)+(Zl/2)

CALL INTERP(Z,ZI,Z2,Z3,T,TI,T4,T7,A,B,C)

TIC=T

CALL INTERP(Z,ZI,Z2,Z3,T,T2,TL,T8,A,B,C)

T2C=T

CALL INTERP(Z,ZI,Z2,Z3,T,T3,T6,TS,A,B,C)

T3C=T

z:noN(I+l)+(Zl/2)

CALL INTERP(Z,ZI,Z2,Z3,T,TI,T4,T7,A,B,C)

T4C=T

CALL INTERP(Z,ZI,Z2,Z3,T,T2,TL,T8,A,B,C)

TLC=T

CALL INTERP(Z,ZI,Z2,Z3,T,T3,T6,T9,A,B,C)

T6C=T

CONTINUE

LONI:LON(1)

IF(I.EQ,I)THEN

CALL INITIAL(RO,ANGOJ)

SUMIOR=NI+N2*SQRT(I+((((RO-RM)*RADI)/{ZI/2*LENG))**2))

SIORJ=SIORJ+SUMIOR

IF(J.EQ.I) SIOR0=SIORJ

CALL MIDPT(RO,ANGOJ)

ELSEIF(I.LT.N-I) THEN

CALL FINAL(RO,ANGOJ)

SUMIOR:NI*SQRT(I+((((RM-RI)*RADI)/(ZI/2*LENG))*'2)) +

N2'SQRT(I+((((RO-RM)*RADI)/(ZI/2*LENG})*'2))

SIORJ:SIORJ+SUMIOR

IF(J.EQ.I) SIOR0=SIORJ

CALLMIDPT(RO,ANQOJ)

ELSEIF(I.EQ.N-I) THEN

CALL FINAL(RO,ANGOJ)

REND(J):RO

SUMIOR=NI*SQRT(I+((((RM-RI)*RADI)/(ZI/2*LENG))**2)) +

N2*SQRT(I+((((RO-RM)*RADI)/(ZI/2*LENG))**2))

SIORJ=SIORJ+SUMIOR

Nl:N2

IF(J.EQ.I) THEN

DISTOR(J)=0

ELSEIF(J.GT.I) THEN

DISTOR(J)=IO0*((RO**2-R2**2)/R2**2)

ANOO(J):ASlN(NI*SlN(ANGOJ))

ENDIF

71

ORIGINAL PAGE IS

OF POOR QUALITY
245

246

247

248

249

25O

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

27i

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

IF(J.EQ.I) SIOR0=SIORJ

IF(SIORJ-SIOR0.EQ.0) THEN

FOCAL(J)=0

ELSEIF(SIORJ-SIOR0.NE.0) THEN

FOCAL(J)=-(RO*RADI)**2/(ZI*LENG*(SIORJ-SIOR0))

ENDIF

ENDIF

55 CONTINUE

60 CONTINUE

WRITE(6,*)' '

WRITE(6,*)' '

WRITE(6,135) WL,TOI,SEC

WRITE(6,*)' '

WRITE(6,*)' RAY ANGLE OUT RADIUS

1 ' DISTORTION '

WRITE(6,*)' (no.) (radians) (cm)

l '

WRITE(6,*)' ...................

, .......... f

DO 70 J=I,M

WRITE(6,130) RAD(J),ANGO(J),REND(J),FOCAL(J),DISTOR(J)

70 CONTINUE

WRITE(6,*)' '

AFOCAL=0

DO 75 J=I,M

AFOCAL=AFOCAL+FOCAL(J)

75 CONTINUE

IF(AFOCAL.EQ.0) GOTO 76

AFOCAL:AFOCAL/(M-2)

76 CONTINUE

ADISTOR=0

DO 80 J=I,M

IF(DISTOR(J).GT.ADISTOR) ADISTOR=DISTOR(J)

80 CONTINUE

WRITE(6,140) AFOCAL/100

WRITE(6,145) ADISTOR

WRITE(6,*)' '

FOCAL '

L_G(cm)'

WRITE(6,*)' .. END THERMAL LENSING OUTPUT..'

WRITE(6,*)' ......................................................

WRITE(6,*)' '

IF(AFOCAL.EQ.O) THEN

IF(P.GT.I) THEN

WRITE(6,*)'...PROGRAM TERMINATED. LASER ROD HAS ESSENTIALY',

' RELAXED TO STATIC CONDITIONS.'

GOTO 90

ENDIF

ENDIF

GOTO 5

86 CONTINUE

WRITE(6,*)

GOTO 90

' ERROR ON LINE 109'

72
=

296

297

298

299

3O0

301

302

303

304

3O5

306

307

3O8

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

C

C

C

C

C

C

87 CONTINUE

WRITE(6,*) '

GOTO 90

88 CONTINUE

WRITE(6,*) '

ERROR ON LINE i16/121'

ERROR ON LINE 127'

i00 FORMAT(132AI)

105 FORMAT(5OAI,EII.6)

115 FORMAT(7AI,14FI2.2)

125 FO}_4AT(2AI,F5.2,14EI2.5)

130 FORMAT(7X,F5.2,17EI5.7)

135 FORMAT(//,22X,'THERMAL LENSING CALCUATIONS',

1 /,15X,'WAVELENGTH OF INTEREST(um) ......

1 /,15X,'TEMPERATURE OF INTEREST(deg C)..

1 /,15X,'TIME (sec)......................

140

145

',E12.6,

',E12.6,

',E12.6)

FORMAT(TX,'AVERAGEOF THE FOCAL LENGTHS (meters) :',E15.6)

FORMAT(TX,'INCREMENTAL DISTORTION (%) RANGES FROM 0.0 TO ',El0.3)

90 CONTINUE

END

SUBROUTINE INITiAL(RO,ANGOJ)

This subroutine calculates the initial entry into the temperature

gradient matix and perdicts the initial angle and radius out for

the first axial increment of the laser rod.

REAL IOR,LONI,NI,N2

DOUBLE PRECISION ANGMJ,ANGIJ,ANGOJ,AI,A2,A3,RI,RM,RO,Z,A,B,C

COMMON /BLOCKI/IOR,DT,ACC.NI,N2,ZI,RI,RM,TM,LONI,I

COMMON /BLOCK2/RI,R2,R3,ZI,Z2,Z3

COMMON /BLOCK3/TI,T2,T3,T4,T5,T6,T7,T8,T9

COMMON /BLOCK4/TIC,T2C,T3C,T4C,TLC,T6C,TIO,TII,TI2

RI:R2

CALL INTERP(RI,RI,R2,R3,T,TIC,T2C,T3C,A,B,C)

DTR=(2*A*Ri)+B

Z=LONI+ZI/2

CALL INTERP(Z,ZI,Z2,Z3,T,T2,T5,T8,A,B,C)

DTZ=(2*A*Z)+B

IF(DTR.EQ.0) THEN

ANGMJ=0.0D00

ELSEIF(DTR.NE.O) TH_

;_GMJ:ASIN(I.O)

ENDIF

IF(DTZ.GE.I.0E-30) ANGMJ=ATAN(DTR/DTZ)

73

ORIGINAL PAGE IS

OF POOR QUALITY
346

347

348

349

35O

351

352

353

354

355

356

357

358

359

36O

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

38O

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

C

C

C

C

C

C

C

RM:R2

TM:T2C

NI=IOR+DT*((TM+T2)/2)

N2=IOR+DT*((TM+TL)/2)

AI=SIN(ANGMJ-ANGIJ)

A2:(N2/NI)*AI

A3=ASIN(A2)

ANGOJ:ANGM -A 

ANGOJ=ANGMJ-ASIN((N2/NI)*SIN(ANGMJ-ANGIJ))

RO=(ZI/2)*TAN(ANGOJ)+RM

CALL REFINE(RO,ANGIJ,ANGMJ,ANGOJ,T4,TL,T6)

RETURN

END

SUBROUTINE MIDPT(RO,ANGOJ)

This subroutine calculates the intermediate angles and predicts

the mid-point position of the next axial increment prior to

entry into the next increment of the temperature gradient matrix,

REAL IOR,LONI,NI,N2

DOUBLE PRECISION ANGMJ,ANGIJ,ANGOJ,AI,A2,A3,RI,RM,RO,Z,A,B,C

COMMON /BLOCKI/IOR,DT,ACC,NI,N2,ZI,RI,RM,TM,LONI,I

COMMON /BLOCK2/RI,R2,R3,ZI,Z2,Z3

COMMON /BLOCK3/TI,T2,T3,T4,TL,T6,T7,T8,T9

COMMON /BLOCK4/TIC,T2C,T3C,T4C,TLC,T6C,TIO,TII,TI2

RM:RO

ANGIJ:ANGOJ

CALL INTERP(RM,RI,R2,R3,T,TI,T2,T3,A,B,C)

TI0=T

CALL INTERP(RM,RI,R2,R3,T,T4,TL,T6,A,B,C)

TII:T

DTR=(2*A*RM)+B

CALL INTERP(RM,RI,R2,R3,T,T7,TS,T9,A,B,C)

TI2=T

Z=LONI+ZI

CALL INTERP(Z,ZI,Z2,Z3,T,TI0,TII,TI2,A,B,C)

DTZ=(2*A*Z)+B

TM=T

IF(DTR.EQ,O) THEN

ANGMJ=O.ODO0

ELSEIF(DTR.NE.O) THEN

ANGMJ=ASIN(I.O)

ENDIF

74
396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

C

C

C

C

C

C

IF(DTZ.GE.I.0E-30) ANGMJ=ATAN(DTR/DTZ)

CALL INTERP(RM,RI,R2,R3,T,T4C,TLC,T6C,A,B,C)

NI=N2

N2:IOR+DT*((TM+T)/2)

AI:SIN(ANGMJ-ANGIJ)

A2:(N2/NI)*AI

A3=ASIN(A2)

ANGOJ=ANGMJ-A3

ANGOJ=ANGMJ-ASIN((N2/NIJ*SIN(ANGMJ-ANGIJ))

RO=(ZI/2)*TAN(ANGOJ)+RM

CALL REFINE(RO,ANGIJ,ANGMJ,ANGOJ,T4C,TLC,T6C)

RETURN

END

SUBROUTINE FINAL(RO,ANGOJ)

This subroutine calculates the final angle and radius out for

all subsequent axial increments o£ the laser rod.

REAL IOR,LONI,NI,N2

DOUBLE PRECISION ANGMJ,ANGIJ,ANGOJ,AI,A2,A3,RI,RM,RO,Z,A,B,C

COMMON /BLOCKI/IOR,DT,ACC,NI,N2,ZI,RI,RM,TM,LONI,I

COMMON /BLOCK2/RI,R2,R3,ZI,Z2,Z3

COMMON /BLOCK3/TI,T2,T3,T4,TL,T6,T7,T8,T9

COMMON /BLOCK4/TIC,T2C,T3C,T4C,TLC,T6C,TI0,TII,TI2

RI:RM

RM=RO

ANGIJ:ANGOJ

IF(I.LT.10) CALL INTERP(RM,RI,R2,R3,T,TIC,T2C,T3C,A,B,C)

IF(I.EQ.10) CALL INTERP(RM RI,R2,R3,T,T4C,TLC,T6C,A,B,C)

DTR=(2*A*RM)+B

CALL INTERP(RM,RI,R2,R3,T,TI,T2,T3,A,B,C)

TIO=T

CALL INTERP(RM,RI,R2,R3,T,T4,TL,T6,A,B,C)

TII=T

CALL INTERP(RM,RI,R2,R3,T,T7,T8,T9,A,B,C)

TI2=T

IF(I.LT.IO) Z=LONI+Z!/2

IF(I.EQ.IO) Z=LONI+ZI/2

CALL INTERP(Z,ZI,Z2,Z3,T,TI0,TII,TI2,A,B,C)

DTZ=(2*A*Z)+B

IF(DTR.EQ.O) THEN

ANGMJ=O.ODO0

ELSEIF(DTR,NE,O) THEN

7_
446

447

448

449

450

45!

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

ANCHJ:ASI (I.0)

ENDIF

IF(DTZ.GE.I.0E-30) ANGMJ=ATAN(DTR/DTZ)

TM=T

NI:N2

IF(I.LT.10) N2=IOR+DT*((TM+TII)/2)

IF(I.EQ.i0) N2=IOR+DT*((TM+TI2)/2)

AI=SIN(ANGMJ-ANGIJ)

A2=(N2/NI)*AI

A3:ASIN(A2)

ANGOJ=ANGMJ-A3

ANGOJ=ANGMJ-ASIN((N2/NI)*SIN(ANGMJ-ANGIJ))

RO=(ZI/2)*TAN(ANGOJ)+RM

IF(I.LT.10) CALL REFINE(RO,AMGIJ,ANQMJ,AMGOJ,T4,TL,T6)

IF(I.EQ.10) CALL REFINE(RO,ANGIJ,ANGMJ,ANGOJ,T7,T8,T9)

RETURN

END

SUBROUTINE INTERP(R,RI,R2,R3,T,TI,T2,T3,A,B,C)

The subroutine INTERP is a binomial interpolation routine using

any three points in the R (radial) or Z (axial) direction to find

the coefficents of A,B, and C of the equation AX**2 + BX * C = T

so that a temperature T can be determined for any X (R or Z)

within the non-linear temperature matrix.

DOUBLE PRECISION R,A,B,C

A=((TI-T2)*(RI-R3)-(TI-T3)*(RI-R2))/((RI-R3)*(RI**2-R2**2)-

1 (RI-R2)*(RI**2-R3**2))

B=((TI-T3)*(RI**2-R2**2)-(TI-T2)*(RI**2-R3**2))/((RI-R3)*

1 (RI**2-R2**2)-(RI-R2)*(RI**2-R3**2))

C=TI-A*RI**2-B*RI

T:A*R**2+B*R+C

RETURN

END

SUBROUTINE REFINE(RO,ANGIJ,ANGMJ,ANGOJ,TI,T2,T3)

The subroutine REFINE is a continous numerical iteration routine

used to recalculate the final thermal lansing output angle for

each specified axial increment within each ray until the user

specified accuracy criteria for the output angle are met.

76
496 C

497

498 C

499

500

501 C

502

5O3

504 C

505

506

507

508

509

510

511

512

513

514 C

515

516

517

518

519 C

52O

521

REAL IOR,LONI,NI,N2

DOUBLE PRECISION ANGMJ,ANGIJ,ANGOJ,AI,A2,A3,RI,RM,RO

DOUBLE PRECISION ROO,D,A,B,C

COMMON /BLOCKI/IOR,DT,ACC,NI,N2,ZI,RI,RM,TM,LONI,I

COMMON /BLOCK2/RI,R2,R3,ZI,Z2,Z3

L:0

CONTINUE

L=L+I

CALL INTERP(RO,RI,R2,R3,T,TI,T2,T3,A,B,C)

N2=IOR+DT*((TM+T)/2)

AI=SIN(ANGMJ-ANGIJ)

A2=(N2/NI)*AI

A3:ASlN(A2)

ANGOJ=ANGMJ-A3

ANGOJ=ANGMJ-ASlN((N2/NII*SlN(ANGMJ-ANGIJ))

ROO:(ZI/2)*TAN(ANGOJ)÷RM

D=RO

RO=RO0

IF(ABS(D-ROO).GT.ACC) GOTO 5

RETURN

END

No errors detected

77
Appendix B

Program Input Files

1. Thermal Lensing Model Input File #1

This file (THERLENI.INP) contains thermal lensing

material characteristics and other constants required to

control the Thermal tensing Model. The example data

depicted below are for a pulsed side-pumped Nd:YAG rod.

Input File #1

THERMAL LENSTNC TNP[FF DATA FOR Nd'Y&G ROD

1.15

37.5

1.825

8.90E-06

0.33

0.05

I

0.]0

1.0

1.0E-06

0

; WAVELENGTH OF INTEREST (MICRONS)

, TEMPERATURE OF INTEREST (24-42 DEGREES C)

; INDEX OF REFRACTION

; ALPHA (DN/DT)

; TEMPERATURE NORMALIZATION FACTOR

; TIME NORMALIZATION FACTOR

, 0 - RELATIVE VALUES IN INPUT FLEE; 1 - AC'_L VALUES

• AXIAL INCREMENT

POWER MULTICATION FACTOR p

; OUTPUt ANGLE ACCURACY CRITERIA

; TYPE OF PUMPING (0=s_de, l=end)

2. Thermal Lensing Model Input File #2

This file (THERLEN2.1NP) is th&.output file from the

model titled: Temperature Distribution Model for finite

optically pumped laser rods. It contains that model's

control criteria and output temperature data blocks versus

time.

Those data blocks of temperatures contained within

Y8
this input file are used in the Thermal Lensing Model to

calculate the thermal lensing and pump beam distortion

effects. The example data depicted below are for a pulsed

side-pumped Nd:YAG laser at its steady-state equilibrium

temperature. Blocks of temperature data are shown from

3.4-3.6 seconds.

input File @2

Input Nd:YAG Side-Pump..MP(20)..GA=20..GL=None..Run=001 03/01/90

..... MIN., MAX. AND INCREMENT OF RADIUS OF ROD 2.500000E-01

2.500000E-01 2.500000E-01

..... MIN , MAX AND INCR. OF ASPECT RATIO (L/A) 30.400000

30.400000 50.000000

..... INVERSE OF RADIAL THERMAL DIFFUSIVITY 26.947000

..... THERMAL CONDUCTIVITY 1.030000E-01 1.030000E-01

..... MIN. RADIAL COOLING PARAMETER 20.000000

..... MAX. RADIAL COOLING PARAMETER 20.000000

TYPE OF SCALE I(LOG) OR O(LINEAR). 1

AND INCREMENT (SET TO i. IF LOG) 1.000000

........ MIN. AND MAX. FRONT COOLING RATIO OR VALUES

6.500000E-04

TYPE OF SCALE I(LOG) OR 0(LINEAR) i

AND INCREMENT (SET TO I IF LOG) 1.000000

........ MIN. AND MAX. BACK COOLING RATIO OR VALUES

6.500000E-04

TYPE OF SCALE 1(LOG) OR 0(LINEAR)

AND INCREMENT (SET TO I IF LOG)

........ TYPE OF DATA RATIO=O ACTUAL=]

PARAM=2

.............. PROPORTIONAL TO L/A=3

....... VARIATION OF END DATA SEPERATE=0

............................. TOGETHER=I

..... MAXIMUM NUMBER OF ROOTS ... RADIAL

..... MAXIMUM NUMBER OF ROOTS ... LONGIT

NUMBER OF ORTHOGONAL CHECKS RADIAL

NUMBER OF ORTHOGONAL CHECKS LONGIT

BESSEL ROOTS SCHEME ... 1 COMPI_ED

2 GIVEN

3

1

i. 000000

2

1

17

17

2

3

LL (GIVEN <21).

COMPUTED GE 21)

6.500000E-04

6.500000E-04

?g
ROOTSSCHEMEIS 2

RADIALANDLONG.INTEGRATIONDIVISIONS I000

TYPEOF PUMPBEAM...........

RADIAL EIGENFUNCTION -ORDER

SQUARE PULSE +0

GAUSSIAN +I

PARABOLIC +2

LONGIT EiGNFUNCTION -ORDER

CONSTANT +0

CONSTANT & ATTENUATED +i

FOCUSED(DEFOCUSED) +2

FOCUSED & ATTENUATED +3

NOTE NO DIPPRACTION EFFECTS ALLOWED

GAUSSIAN ENVELOPE ASSUMED

PUMP BEAM CODE 0 0

....... ATTENUATION DATA..O=ATTEN..I=ATTEN*L 1

...... OPTICAL DATA 3.000000

..... NUMBER OF RADIAL PROFILES ... ii

..... NUMBER OF LONGIT PROFILES ... Ii

........ CONTINUOUS PUMPPING ... O

........ SINGLE PULSE ... 1

OR MULTI-PULSED ... 2 2

..... MIN. REL. TIME OF INTEREST 3.400000

..... MAX. REL. TIME OF INTEREST 3.600000

TYPE 0F SCALE I($OG) OR 0(LINEAR) 1

AND INCREMENT (SET TO I IF LOG) 1.000000

........ ,MULTIPLE PULSES ... 2 2

........ ONTIME PER PULSE 1.300000E-04

........ OFFTIME PER PULSE 4.987000E-02

........ ONTIME SAMPLING RATE 1

........ OFFTIME SAMPLING RATE 1

..... ABSORBED ENERGY PER PULSE J/CM3 1574.280000

.......... DEBUG DESIRED ... 1

............. NOT DESIRED .,. 0, NDEBUG =

..MIN, MAX, RADIAL CALCULATION RANGE.. O.O00000E+O0

..MIN, MAX LONGITUDINAL CALCULATION RANGE.. 0.O00000E+O0

..RADIAL REFERENCE POINTS... 0 O.000000E+O0

..AXIAL REFERENCE POINTS... O O.O00000E+O0

ROD ASPECT RATIO (L/A) .......... = 30.400000

i00

1

1.000000

1.000000

80
f

TEHPERATURE DISTRIBUTION

RADIAL DIFFUS]VITY ................. 269470E+02

RADIUS OF REX) ................. 250000

LENGTH OF ROD ................ 7.600000

TIME I_ SECONDS ................. 345013E+01

L/R .00 .I0 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .72382E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E÷00 .72383E+00 .72383E+00 .72383E+00 .72382E+00

.10 .71330E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E÷00 .71330E+00

.20 .69403E+00 o69404E*00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69403E+00

.30 .66320E+00 .66320E+00 .66320E+00 .66320E÷00 .66320E+00 .66320E+00 .66320E+00 .66320E+00 .66320E+00 .66320E+00 .66320E+00

.40 .62046E+00 .6201,6E*00 .62046E+00 .62046E÷00 .62046E+00 .62047E+00 .62046E*00 .620_6E+00 .62046E+00 .62046E+00 .62046E+00

.50 .56521E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E÷00 .56522E+90 .56522E+00 .56521E+00

.60 .49688E÷00 .496_9E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .49688_+00

.70 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E÷00 .41552E÷00 .41552E+00 .41552E+00 .41552E+00

.80 .32288E+00 .32288E+00 .32288E+00 .32288E+00 .32288E+00 .32288E+00 .32288E+00 .3220_E+00 .32288E+00 .32288E+00 .32288E+00

.90 .22190E+00 .22190E÷00 .22190E+00 .22190E÷00 .22190E+00 .22190E+00 .22190E+00 .22190E+00 .22190E+00 .22190E÷00 .22190E+00

1.00 .I0556E+00 .I0556E+OO .I0556E÷00 .10556E+00 .I0556E+00 .I0556E+00 .10556E+00 .10556E+00 .I0556E-00 .I0556E+00 .I0556E+00

TEHPERATURE DISTRIBUTION

RADIAL DIFFUS[V[TY ................. 269470E+02

RADIUS OF RO0 ................. 250000

lENGTH OF ROO ................ 7.600000

TIHE IN SECONDS ................. 350000E÷01

L/R .00 ,tO .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .64627E+00 64627E*00 .64627E+00 .64627E+00 .64627E+00 .64627E*00 .64627E+00 .64627E+00 .6462TE+00 .6462TE+00 .64627E+00

.10 .6_006E_00 .64007E+00 .64007E*00 .64007E*00 .64007E+00 .64007E+00 .64007E÷00 .64007E÷00 oEAOO7E+O0 .64007_÷00 .64000E+00

.20 62!44E_00 .62145E*09 .62145E*C0 o62145E+00 .62145E+00 .62145E÷00 .62145E',00 .62145E+00 .62145E÷00 .62;45E+00 .62144E÷00

.30 .59041E_00 .59041E*00 .59041E+00 .59041E÷00 .59041E+00 .59041E+00 .59041E+00 .59041E+00 .59041E+00 .59041E+00 .59041E+00

.40 54693E*00 .54694E+00 .54694E+00 .5469_E+00 .54694E÷00 .54694E+00 .54694E+00 .54694E+00 .54694E*00 .54694E+00 .54693E+00

.50 .49096E+00 .49096E÷00 .49096E_00 .49096E+0C .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00

.50 .42_44E+00 .42244E÷00 .42_44E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .422_4E*00 .42244E+00 .42244E+00

.70 .34169E+00 .34169E+00 .34169E+00 .34169E_00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00

.80 .25008E+00 .25008E+00 .25008E+00 .250_SE÷Oe .25008E+00 .25008E+00 .25008E+00 .25008E+00 .25008E*00 .25008E+00 ._5008E+00

.90 .15093E+0_ .I_093E+00 .15093E÷00 .15093E÷00 .15093E+00 .15093E+00 .15093E*00 .15093E+00 .!50_3E+00 .15093E+00 .!5093E+00

1.00 .49856E-01 .49856E-01 .49856E-01 .49857E-01 ._9857E-01 .49857E-01 .49857E-01 .49857E-01 .49056E-01 .49856E-01 .49856E-01

TEMPERATURE D{STRIBUTION

RADIAL D[FFUSIVITY ................. 269470Ee02

RADIUS OF ROD ................. 250000

LENGTH OF ROD ................ 7.600000

TIME IN SECONOS ................. 350013E+01

I/R .00 .I0 .20 .30 .40 .50 .60 .70 .00 .90 1.00

O0 .72382E+00 .72383E÷00 .72383Et00 .72383E+00 .72383E+00 .72383E+00 .72383E_00 .72383E+00 .72383E+00 .T2383E÷O0 .72382E+00

.tO .71330E+00 ,7133_E+00 .71331E+03 .71331E*00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E÷00 .71331E÷00 .71330E+00

.20 ,69403E÷00 .69434E+00 .69434E+00 .69404E+00 o69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+05 .69404E÷00 .69403E+00

.30 ,66320E+00 .66321E+00 .66321E+00 .6632tEe00 .66321E+00 .6_32IE+00 .66321E÷00 .66321E÷00 .66321E÷00 ,66321E+00 .66320E+00

.40 .62046E÷00 .62047£*00 .62047E÷00 .62047E+00 .62047E÷00 o62047E÷00 .62047E+00 .62047E÷00 .62047E+00 ,62047E÷00 .62046E+00

.50 .56521E÷00 .56_22E+00 .56522E*00 .56522E+00 .56522E÷00 .56522E+00 .56522E*00 .56522E+00 .56522E+00 .56522E+00 .56521E+00

.60 .496_8E*00 .49689E+00 .49689E+00 .49689E+00 .49689E÷00 .49(_09E+00 .49689E+00 .496_9E+00 ._gE+O0 .496_9E÷00 .49688E+00

.70 .4!552E+00 .41552E+00 .41552E+00 .4_552E+00 .41552E÷00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E÷00 .4_552E+00

.BO .32288E+00 .3228_E÷00 .32288E÷00 .32288E+00 .322_8E÷00 .32288E+00 .32288E+00 .32288E+00 o32288E_00 .32288E+00 .32288E+00

.90 .22190E+00 .22190E÷00 .22190E÷00 .22190E÷00 .22190E*00 .2219_E+00 .22190E÷00 .22190E+00 .22100E+00 .22!90E÷00 .22_90E*00

l.O0 .I0556E+00 .10556E÷00 .I0556E+00 .I055_E+_0 .I0556E÷00 .I0556E÷00 .10556E*00 .I0556_,00 .I0556E_00 .I0556E÷00 .I0556E÷00

81

ORIGINAL PAGE IS

OF POOR QUALITY
TEMPERATURE DISTRIBUTION

RADIAL DIFFUSIVITY ................. 269470E+02

RADIUS OF ROD ................. 250000

LENGTH OF ROD ................ 7.bO00OO

TIME IN SECONDS ................. 355000E+01

L/R .OO .I0 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.DO .64627E+00 .64627E+00 .6,4627E+00 .64627E+00 .6.(,62BE+00 .64628E+00 .64628E+00 .64627E+00 .64627E+00 .6z,627E+O0 .64627"E+00

.ID .64006E+00 .64007E+00 .640071Z+00 .6,4007E+00 .6,4007E+00 .6-4007E*00 .64007E+00 .64007E+00 .64007E÷00 .64007E+00 .6,400,6E+00

.20 .62144E÷00 .62145E*00 .62145E+00 .62145E+00 .62145E+00 .62145E*00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62144E+00

.30 .59041E+00 .59041E+00 .59041E+00 .59041E÷00 .59042E+00 .59042E+00 .59042E+00 .59041E÷00 .59041E+00 .59041E+00 .59041E+00

.40 .54694E+00 .54694E+00 .54694E+00 .54694E÷00 .54694E÷00 .54694E+00 .54694E+00 .54694E+00 .54694E+00 .54694E+00 .54694E+00

.50 .49096E_00 .49097E÷00 .4909TE+00 .49097E+00 .49097E+00 .49097_+00 .49097E+00 .49097E*00 .49097E+00 .49097E+00 .49096E÷00

.60 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+O0 .42244E+00 .42244E+00 .42244E+00 .42244E÷00 .42244E+00

.70 .34169E+00 .34169E+O0 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E+00 .34169E÷00

.80 .25008E+00 .25008E+00 .25008E+00 .25008E+00 .25008E+00 .25008E÷00 .25008E+00 .25008E+00 .25008E+00 .25008E+00 .2500_E+00

.90 .15093E+00 .15093E+00 .15093E+00 .15093E*00 .15093E+00 .15093E+00 .15093E+00 .15093E+00 .15093E+00 .15093E+00 .15093E÷00

1.00 .49856E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49657E-01 .49857E-01 .49857E-01 .49856E-01

TEMPERATURE DISTRIBUTION

RADIAL DIFFUSIVITY ................. 269470E+02

RADIUS OF RO0 ................. 250000

LENGTH OF ROD ................ 7.600000

TIME IN SECONDS ................. 355013E*01

L/R .OO .10 .20 .30 .40 .50 .60 .70 .80 .90 4.00

.00 .72382E+00 ..72383E*00 .723a3E+O0 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .723B3E÷O0 .72383E+00 .72383E+00 .72382E+00

.I0 _71331E+00 .7!331E+00 .71331E+00 .71331E_00 .71331E÷00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E÷00 .71331E÷00

.20 .69403E+00 .69404E+00 .69404E+00 .69404E_00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E_00 .69404E+00 .69403E+00

.30 .66320E+00 .66321E÷00 .66321E+00 .66321E*00 ._321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66320E+00

.40 .62046E+00 .620/+7E÷00 .62047E+00 .62047E÷00 .6204TE÷00 .62047E+00 .62O47E+00 .620_TE÷DO .62O47E÷00 .62047E*00 .62046E+00

.50 .56521E+00 .56522E+O0 .56522E÷00 .56522E÷00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56521E+00

.60 .49688E+00 .496B9E+00 .49689E+00 .496.B9E+DO .496.BgE÷OO .49689E+00 .496,89E*00 .496,BgE+OO .49689E+00 .496.B9E+DO .49689E+00

.70 .41552E*00 .41552E+00 ._1552E400 .41552E+00 .41552E+00 .¢1552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00

.BO .32288E+00 .32288E+00 .32288E+00 .32288E+00 .3228,BE÷00 .32288E÷00 .32288E+00 .32288E+00 .32288E+O0 .32288E+00 .32288E*00

.90 .22190E_00 .22190E÷00 .22190E+00 .22190E+00 .22190E÷00 .22190E+00 .22190E+00 .22190E+00 .22190E+00 °22190E+00 .22190E+00

1.00 .I0556E+00 .I0556E+D0 .I0556E+00 .]0556E+00 .I0556E+00 .I0556E+00 .I0556E*00 .I0556E÷00 .I0556E+00 .1055_E÷00 .I0556E+00

TEMPERATURE D]STRIBUTION

RADIAL DIFFUSIVITY ................. 269470E÷02

RADIUS OF ROD ................. 250000

LENGTH OF ROD ................ 7.600000

TIME IN SECONDS ................. 360000E+01

L/R .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .64613E+00 .64613E+00 .64613E+00 .64613E+00 .64613E+00 .64613E+00 .64613E+00 .64613E+00 .64613E÷00 .64613E+00 .64613E+00

.10 .63993E+00 .63993E+00 .63994E+00 .63994E+00 .63994E+00 .63994E*00 .63994E+00 .63994E+00 .63994E+00 .63994E+00 .63_3E+00

.20 .62131E+00 .62132E+00 .62132E+00 .62132E+00 .62132E÷00 .62132E+00 .62132E+00 .62132E÷00 .62132E+00 .62132E+00 .62131E+00

.30 .59028E+00 .59028E+00 .59028E+00 .59028E+00 .59028E+00 .59028E+00 .59028E+00 .59028E÷00 .59028E÷00 .59028E+00 .59028E.+00

.40 .54680E+00 .54681E+00 .54681E+00 .54681E+00 .54681E+00 .54681E+00 .54681E+00 .54681E+00 .5468]E+00 .546,_IE÷00 .546_0E+00

.50 .49082E_00 .49083E+00 .49083E+00 .49083E+00 .49083E+00 .49083E+00 .49083E÷00 .49083E+00 .49083E÷00 .49083E+00 .49082E÷00

.60 .42230E+00 .42231E+00 .42231E+00 .42231E+00 .42231E+00 .42231E_00 .42231E÷00 .42231E+00 .42231E+00 .42231E+00 .42230E÷00

.70 .34155E+00 .34156E_0 .34156(!+00 .34156E+00 .34156E+00 .34156E÷00 .34155E+00 .34156E+00 .34156E+00 .34156E+00 .34155E+00

.80 .24995E+00 .24995E+00 .24995E+00 .2_995E+00 .24995E+00 .24995E+00 .24995E+00 .24995E+00 .24995E+00 .24995E÷00 .24995E÷00

.90 .150BOE+O0 .15080E+00 .15080E÷00 .15080E÷00 .15080E÷D0 .15080E+00 .150BOE+O0 .15080E+00 .15080E+00 .150BOE+O0 .15080E÷00

1.00 .49752E-01 .49753E-01 .49753E-01 .49T53E-01 .49753E-01 .49753E-01 .49753E-01 .497_3E-01 .49753E-01 .49753E-01 .49752E-01

82

ORIGINAL PAGE IS

OF POOR QUALITY
Appendix C

Program Output File

Thermal Lensing Model Output File

The Thermal Lensing Model output file (THERLENS.OUT)

rewrites the data from each of the input files. It then

calculates thermal lensing and pump beam distortion effects

for each block of temperature data obtained from the Temperature Distribution Model. Currently, the lensing model

is progran_ned to calculate i0 equally spaced rays transiting through the laser material. The output data from the

Thermal Lensing Model depicted below are for a pulsed sidepumped Nd:YAG laser using the input files in Appendix B.

Thermal Lensinq Model Output File

r

......... INPL_ FILE #i ..........

T[IERM_L LENSING INPLrI' DATA FOR Nd:YAG ROD

.WAVELENGTHOF _NT_EST IN ,_..(_L)" _.150000

.TEM_E[_TURE OF I_fEREST IN DEG C..{fOI).: 37.500000

.INDEX OP REFP_CTION..(IOR).: 1.825000

.ALPHA (dn/dt).. (DT) _ 900000E-06

TEMP NORmaLIZATION P%CTOR. •(TNF). : 3.300000E-01

[TIME NORMALIZATION FACTOR.. (TINF). : 5. 000000E-02

.0:RELATIVE TEMP; I:ACTUA 5 TEMP..(IDATA).: i

.AXIAL INCREMENT FACTOR..(ZI).: 1.000000E-01

.POWER MULTIPLICATION FACTOR..(POWER):: 1.000000

.OUTPUT _NGLE ACCURACY CRITERIA. iACC)." 1.000000E-06

.TYPE OF PUMPING (0=s_de, ]=end).°: 0 000000E+00

83 ORIGINAt PA_E IS

OF POOR QUALITY
....... INPUT FILE 4#2......

Input Nd:YAG Side-Pump..MP(20)..GA=20..GL=None..Run=O01 03/01/90

..... MIN., MAX. AND INCREMENT OF RADIUS OF ROD 2.500000E-01

2.500000E-01 2.500000E-01

..... MiN , MAX AND INCR. OF ASPECT RATIO (L/A) 30.400000

30.400000 50.000000

..... INVERSE OF RADIAL THERMAL DIFFUSIVITY 26.947000

..... THERMAL CONDUCTIVITY 1.030000E-OI 1.030000E-OI

..... MIN. RADIAL COOLING P_RAMETER 20.000000

..... MAX. RADIAL COOLING PARAMETER 20.000000

TYPE OF SCALE I($OG) OR O(LINEAR). i

AND INCREMENT (SET TO i. IF LOG) 1.000000

........ MIN. AND MAX. FRO_f COOLING _TIO OR VALUES

6.500000E-04

TYPE OF SCALE 1(LOG) OR 0(LINEAR) 1

AND INCREMENT (SET TO I IF LOG) 1.000000

........ MIN. AND _AX. BACK COOLING RATIO OR VALUES

6.500000E-04

TYPE OF SCALE I(LOG) OR O(LINEAR) 1

AND INCREMENT (SET TO 1 IF LOG) 1.000000

........ TYPE OF DATA RATIO=0 ACTUAL=I

PARAM=2

PROPORTIONAL TO L/A=3 2

....... VARIATION OF END DATA SEPERATE=O

............................. TOGETHER=I 1

..... MAXI_UMNUMBER OF ROOTS ... RADIAL 17

..... MAXIMUM _FUMBER OF ROOTS ... LONGIT 17

NUMBER OF ORTHOGONAL CHECKS RADIAL 2

NUMBER OF ORTHOGONAL CHECKS LONGIT 3

BESSEL ROOTS SCHEME ... 1 COMPUTED

2 GIVEN

3 ALL (GIVEN <21).

(COMPUTED GE 21)

ROOTS SCHEME IS _ 2

RADIAL AND LONG. INTEGRATION DIVISIONS i000

TYPE OF PUMP BEAM ...........

RADIAL EIGENFUNCTION -ORDER

SQUARE PULSE +0

GAUSSIAN +I

PARABOLIC +2

6.500000E-04

6.500000E-04

100

84
LONGIT EIGNFUNCTION -ORDER

CONSTANT +0

CONSTANT& ATTE_TATED +l

FOCUSED(DEFOCUSED) +2

FOCUSED & ATTENUATED +3

NOTE NO DIFFRACTION EFFECTS ALLOWED

G_]SSIAN ENVELOPE ASSUMED

PUMP BEAM CODE 0 0

....... ATTENUATION DATA..O=ATTEN.,I=ATTEN*L

...... OPTICAL DATA 3.000000

..... _TMBER OF RADIAL PROFILES .., 11

..... NUMBER OF LONGIT PROFILES ... il

........ COntINUOUS PUMPPING ... 0

........ SINGLE PULSE ... I

OR _]LTI-PULSED ... 2 2

..... MIN. REL. TIME OF INTERFST 3.400000

..... MAX. REL. TIME OF INTEREST 3,600000

TYPE OF SCALE I(LOG) OR O(LINEAR) i

AND INCREMENT (SET TO 1 IF LOG) 1.000000

........ MULTIPLE PULSES ... 2 2

........ ONTIME PER PULSE 1.300000E-04

........ OFFTIME PER PULSE 4,987000E-02

........ Ob_IME SAMPLING RATE I

........ OFFTIME S_MPLI_] RATE 1

..... ABSORBED [NERGY PER PULSE ]/CM3 1574.280000

.......... DEBUG DESIRED ... 1

............. NOT DESIRED ... 0, NDEBUG=

..MiN, MAX, RADIAL CALCULATION RANGE.. 0.000000E+00

..MIN, Y_X LONGITUDINAL CALCULATION RANGE. 0.000000E+00

..,RADIAL REFERENCE POINTS... 0 0,000000E+O0

..AXIAL REFERENCE POINTS... 0 0.000000E+00

ROD ASPECT RATIO (L/A) .......... : 30,400000

1.000000

io000000

85
BLOCK OF TEMPERATURE PROFILE5 FROH INPUT FILE #2

TEMPERATURE DISTRIBUTION

RADIAL OIFFUSIVITY ................. 269470E+02

RADIUS OF RCO ................. 250000E+00

LEHGTH OF ROD ................. 760000E_01

T[XE IN SECONDS ................. 34500qE*01

L/R .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .b4626E*00 .64627E+00 .6_627E*00 .64627E+00 .(_,627E+00 .64627E+00 .6.4627E+00 .64627E+00 .64627E+00 .6.4627E÷00 .64626E+00

.10 .64006E+00 .64006E+00 .6>4006E+00 .64005E+00 .64006E+00 ,64006E+00 .64006E+00 .64006E+00 .EAOO6E÷GO ,64006E+00 .(>4000E+O0

.20 .62I_4E+00 .62144E+00 .621_4E+00 .62144E*00 ._214_E_00 .62144E+00 .62144E+00 .6214_E+00 .62144E=00 .6214_E+00 .62144E*00

.30 .59040E+00 .59041E+00 .59041E+00 .59041E÷00 .59041E+00 .59041E*00 .59011E+00 .59041E+00 .59041E+00 .59041E+00 .59040E÷00

.40 .54693E÷00 .5469_E+00 .5_694E÷00 .54694E+00 .54694E+00 .54694E_00 .54694E+00 .54694E+00 .54694E_00 .54694E+00 .54693E*00

.50 ._9096E÷D0 .49090E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E+00 .49096E*00 .49096E_00 .49096E+00

.60 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244Ee00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E÷00

.70 .34169E+00 .34169E+00 .34169E+00 .34169E÷00 .34169E+00 .34169E÷00 .34169E_00 .34169E+00 .34169E+00 .34169E÷00 .3_.169E÷00

.80 .25008E÷00 .25008E+00 .25008E+00 .25008E+00 .25008E-00 .25008E+00 .25008E+00 .25008E+00 .25008E+00 .25008E+00 .25008E*00

.90 .15093E÷00 .15093E+00 .15093E+00 .15093E÷00 .15093E+00 .15093E+00 .15093E÷00 .15093E+00 .15093E_00 .15093E_00 .1509]E+00

1.00 ._9856E-01 .498S6E-01 .49856E-01 .49856E-01 .49856E-01 .49856E-0] .49856E-01 .49856E-01 .49856E-01 .49856E-01 .49856_-01

THERMAL IENS;NG PROGRAH OUTPUT DATA

[NOEX OF REFRACTION MATRIX

.00 ._8250E+0! .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E_01 .I0250E+01

.10 .18250E+01 ._8250E+01 .18250E+01 .18_SOE,01 .18250E+01 .18250E÷01 .18250E+01 .18250E+0| .18250E+01 .18250E+01 .18250E÷01

20 .18250E+01 .18250E+OI .182SOE+01 .18250E+01 .18250E+01 .18250E-01 .IS250E+01 .18250E+01 .18250E+01 .18250E*01 .18250E+01

.30 .IS_50E+01 .18250E.01 .18250E+01 .18250E÷01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E*01 .18250E+0! .18250E÷01

.40 .18250E+01 .15250E+01 .18250E+01 .18250E+01 .18250E*01 .18250E÷01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E÷01

.50 .182_0E_01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.60 .18250E*0_ .13250_+0_ .18_50E+01 .18250E,01 .!8250E+01 .18250E+01 .182SOE+01 .18250E_01 .18250E+01 .18250E+01 .18250E+01

.70 .I@250E+0! ._8250E+01 .18250E+01 .18250E,01 .182SOE+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E÷01 .18250E+01

.80 .18250E+01 .18250E+01 .18250E+01 .182_0E+01 18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.90 .18_50E-05 .18250E+01 .18250E*01 .;aZSOE+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01

100 .18250E+01 .18250E+01 .18250_+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E*01 .18250E+01 .182SOE÷01 .18250E÷01

THERMAL LENSING CALCUATIONS

_AVELENGTH OF INTEREST(L.m) ....... 115000E+01

TEMPERATURE OF IHTEREST(deg C).. ,375000E+02

TIME Csec) ....................... 345000E+01

RAY ANGLE OUT RAOIUS FOCAL DISTORTION

_no.) (radians) (cm) LENG{cm) (X)

.......................................

.00 .0000000E*GO .OO000OOE+O0 .0000000E+00 .0000000E+00

.I0 .0000000E+00 .1000000E+00 .0000000E+00 ,0000000E_OO

.20 .0000000E÷00 .2000000E÷00 .6898527E+03 .0000000E+00

.30 .0000000E+00 .3000000E*00 .7760843E+03 .1310063E-I0

,40 .0000000E+00 .4000000E÷00 .7884030E+03 .100T527E-lO

.50 .0000000E+00 .5000000E+00 .7839235E+03 .0000000E+00

.6G .O000000E+O0 .600_000E+00 .7760643E+03 .0000000E*00

.70 .0000000E+00 .7000000E+00 .7.582449t+03 .O000000E+O0

.80 .0000000E+00 .6000000E+00 .7612167E+03 .OGO0000E+O0

.93 .0000000E+00 .9000000E+00 .T55_OBOE+03 .0000000E+00

1.00 .0000000E+00 .O000000E+O0 .0000000E÷00 .0000000E+00

AVERAGE OF THE FOCAL LENG[HS (rr_ters) = .67765_E+01

INCREMENTAL DISTORT]OH (%) RAR_E$ ERO_t 0.0 TO .131E-I0

.. END THERMAL LENSING OUTPUT..

86

ORIGINAL PAGE L_

OF POOR QUALITY
BLOCK OF TEMPERATURE PRGFILES FROtt INPUT FILE #2

TEMPERATURE DISTRIBUTION

RADIAL D!FFUSIVIY_ ................. 269470E+02

RADIUS OF RO0 ................. 250000E+00

LENGTH OF ROB ................. 760000E+01

TIME IN SECONDS ................. 345013E+01

LIR .DO ,I0 .20 .30 ._0 .50 .60 .70 .80 .90 1.00

.00 .72382E+00 .72383E+00 .72383E_00 .72383E+DO .72383E+00 .72383E+00 .72383E+00 .72383E400 .T2383E*O0 .72383E+00 .72382E+00

.I0 .713]0E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71330E÷00

.20 .69403E*00 .69_04E+_0 .69404E+D0 .69404E+00 .69404_+00 .69404E+00 .69404E*00 .69404E_00 .69404E+00 .59404E+00 .69403E+00

.30 .66320E-00 .66320E+00 .66320E*00 .66320E+00 .66320E+00 .66320E÷00 .(xS3ZOE+O0 .66320E+00 ._5320E+00 .66320E*00 .66320E+00

.40 .62046E+00 .6204&E+C0 .62046E÷00 .62046E+00 .62046E+00 .62047E+00 .62046E+00 .62046E+00 .62046E+00 .62046E-00 .62046E+00

.50 .56521E*00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .55522E+00 .565_2E+00 .56522E+00 .56522E+00 .56522E÷00 .56521E+00

.60 .4Q588E+00 .49589E-00 .49689E+00 .49689E+00 .496_59E+00 .496<39Ee00 ._9689E+00 .49689E+00 ._9689E+00 .49689E+O0 .49688E+00

.70 .41552E+00 .41552E'00 .41552E+00 .4155_E+O0 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00

.BO .3228_E+00 .322_E_00 .32288E+00 .3228J_E+CO .3228_E÷00 .32288E+00 .32288E_00 .32288E+00 .322_E+O0 .32288E+00 .32288E+00

.ge .22_90E+00 .22|90E*00 .22190E+00 .22190E_00 .22190E+00 .22190E+00 .22190E÷00 .22190E÷00 .22190E+00 .22190E+00 .22190E+00

].00 .I0556E+00 .I0556E*00 .I0556E÷00 .10556E*Oa .I0555E+00 .I0556E_D0 .10556E+00 .!0556E+00 .I055&E+O0 ._0556E+00 .10556E÷00

THERMAL LEN_I_!G pROGRAM OUTPUT DATA

INOEX OF REFRACTION MATRIX

.00 .18250E+01 .18250E÷01 .18250E*01 .18250E*01 .t5250C.01 .18250E+01 .18250E+01 .18250E*01 .t8250E+01 .18250E+01 .18250E*01

.10 18250E+01 .182.50E÷01 .18250E+01 .18250E÷01 .18250E÷01 .18250E*B1 .18250E+0! .18250E+01 .18250E+0_ ._8250E+01 .18250E*0i

.20 .I8250E+OI ._8250E+01 .13250E+01 ._8250E+01 .I_250E+01 .16250E+0_ .18250E_01 .18250E+0! .18250E+01 .18250E÷01 .18250E+01

.30 .18250E+0I .!8250_01 .I8250E-01 .18250E+01 .t8250E*0t .18250E*01 .16250E-01 .18250E+01 .I_250E+01 .18250E*01 .18250E+01

.40 _1B?SOE÷Ol .18250E*01 .18250E+01 ._8250E+01 .!8250E+01 .18250E+01 .18250E+01 .1B250E+01 .18250E+01 .18250E+01 .18250E+01

.50 .IS250E÷01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E401 .18250E÷01 .18250E+01 .18250E+0i .18250E+01 .18250E+01

.60 .18250E÷01 .18250E+01 .18250E+0_ .18250E÷01 .18250E*01 .1825OE+01 .18250E+01 .18250E+01 .IB250E+01 .18250E÷_I .18250E÷01

.70 .18250E+0! .18_50E,0_ .18250E+0_ .i8250E+01 .182_0E+01 .18250E+01 .18250E+01 .18250E_01 .I_250E+01 .18250E_0! .18250E÷01

.80 .18250E*01 .i_250_+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .t8250E+01 .19250E+01 .18250E+01

.90 .18250E÷01 .18250E+01 .18250E+91 .18250E+01 .l_250E+0I .I8250E+01 I_250E<01 .18250E,01 .18250E*01 .18250E+01 .18250E+01

1.00 .182SeE+0_ .18250E-01 .18250E,01 .!8250E+01 .IBP5OE+01 .18250E÷01 .18250E+01 .18250E+01 .18250E_01 .18250E+01 .1E250_+01

F_

THERMAL I.ENSIN_ EALCUATIORS

WAVELENGTH OF INTEREST(Um) ....... I_5000E+01

TEMPERATURE OF _R_EREST(deg C).. .375000E_02

TIME (see) ....................... 345013E÷01

RAY ANGLE OJT RADIUS FOCAL DISTORTION

(no.) (radians) (c_) I.ENG(cm) C%)

.....................................

.00 .O00OO00E+O0 .O000300E_O0 .OOOO000E÷OO .CO0000OE+O0

.I0 .O000000E_O0 .IO000OOE÷O0 .3449263E÷03 .5773156E-11

.20 .O000000E_O0 .2GO00DOE_OO .6898527E+03 .139,=_881E'10

.30 .0000000E+00 .3000000E+0C .T760843E+03 .O0C0050E_O0

.40 ._OOGO00E÷O0 .4000000E+O0 .6898527E+03 .1353073E-08

.50 .O000000E+O0 .500000DE÷DO .7185965E+03 -.154765iE-10

.60 . O000000E+O0 .5000000E+00 .7304323E+03 .2216745E'10

.70 . OOO0000E+00 .7000000E*00 .T'348430_+03 .0000000E*00

.80 .O000000E*O0 .800OOOOE*03 .T358428E+03 .OOOO000E+O0

.90 .O000000E÷OO .90GOO0_E*OO .T55 I089E+03 GhOOOOOE+O3

1.00 .O000000E÷00 _O000000E_O3 .O000000E*O0 .OOCOGOOE+O0

AVERAGE OF THE _OCAL LENGTHS (meters) = .686171E_01

INCREMENTAL [.ISI'ORTIC_ (%7 RAUGES FROH 0.0 TO .I_3E-08

.. E_D THERMAL LENSING OUT_JT..

8?

ORIGINAL PAGE IS

OF POOR QUALITY
BLOCK OF TEMPERATURE PROFILES FROM INPUT FILE #2

tEMPERATURE DISTRIBUTION

RADIAL OIFFUSIVITY ................. 269470E+02

RADIUS OF RO_ ................. 250000E+00

LENGTH OF RO0 ................. 76000DE*Ol

TIME IN SECONDS ................. 35DOOOE+01

L/R .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 t.00

.00 .64627E+00 .64627E+00 .64627E+00 .64627E+00 .6Z,627E+00 .64627E+00 .(>4627E+00 .64627E+00 .6.4627E+00 .&4627E+00 .b4627E+00

.10 .6J, O06E*00 .64007E+00 .64007E*00 .64007E+00 .64007E+00 .64007E+D0 .64007E+0D .64007E+00 .64007E+00 .6.4007E+00 .6.'.,006_÷00

,20 .62144E*00 .62145E+OG .62145E+00 ,62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E*00 .62144E+00

.3O .59041E+00 .59041E*00 .59041E+00 .59041E+00 ,59041E+00 .59041E+00 ,59041E+DO .59041E+00 .59041E+00 .59041E+00 .59041E÷00

.40 .54693E÷00 .54694E÷00 .54694E+00 .54694E+00 ,54694E+00 .54694E+GO .54694E+00 .54694E*00 ,54694E+O0 ,54694E+00 .54693E+00

.50 .49096E+00 .49096E+00 .49096_+D0 .49096E+00 .49096E+00 .40096E+00 .49096E+00 .49096E+DO .49096E+00 .49096E+00 ,49096E+00

.60 .422_4E+00 .42244E+00 .42244E+00 .42244E+DO .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E+00 .42244E÷00 ,42244E+00

.70 .34169E+00 .34169E÷00 .34169E÷00 .34169E+00 .34169EeDO .34169E+O0 .34169E+DO .34169E÷00 .34169E+00 ,34169E+O0 .34169E+00

.80 .25DO8E+O0 .25008E+00 .25008E÷00 .250DOE+DO .25008E÷00 .25008E÷00 .2500BE+DO .25008E+O0 .25DOSE+DO .25008E+O0 .25008E+00

.90 .15003E÷00 .15093E+00 .15093E÷00 .15093E+00 ._5093E÷00 .15093E+00 .15093E*00 .15093E+00 .15093E+00 .15093E÷O0 ,15093E+00

1.00 .49856E-D1 +49856E-01 .49856E-01 .49857E-01 .49857E-01 .49857E-01 .49657E-01 .49857E-01 .49856E-0| .49856E-01 .49856E-01

THERMAL LENSI_G PROGRA_ OUTPUT DATA

INDEX OF REFRACTION MATRIX

,00 .18250E+01 ,18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 ,18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.I0 .18250E÷01 ,16250E÷01 .18250E+01 .18250E+01 ,18250E÷01 .18250E+01 .18250E+OI .1825DE+01 .18250E+01 ,18250E÷01 .1825DE+D|

.20 .18250E+01 .18250E+01 .IB250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+OI .18250E+01 .1825DE+01 .18250E+01 .18250E+01

.30 .18250E+01 .18250E+01 .18250E÷01 .18250E÷01 .1825DE+01 .18250E÷01 .18250E*01 .18250E+01 .18250E+0I .18250E+01 .t8250E_01

.40 .18250E+O1 18250E,01 .18250E÷01 .18250E+01 .18250E÷01 .18250E+01 .I8250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01

.50 .1625OE+01 1825DE÷D1 .18250E+01 .18250E_01 .18250E+01 .18250E*01 .18250E+01 .16250E+01 .18250E+01 .18250E+01 .18250E+01

.60 .18250E+01 16250E+01 .18250E+01 .18250E*01 .18250E+01 .18250E+01 .18250E+01 .18250E*01 .18250E+01 .18250E+01 .18250E+DI

,70 .18250E+01 18250E*01 ,I_250E+01 ,18250E+01 .1825DE+01 .IB25OE+01 ,16250E+01 .18250E+01 .18250E+01 ,1825DE+01 .IB250E÷01

.80 .18250E+0t 18250E+01 .18250E÷01 .18250E+01 .18250E÷01 .18250E÷01 .18250E+01 .18250E+01 .IBZ50E_01 .18250E+01 .18250E+01

.90 .18250E+01 1825CE+01 .1825GE+01 .18250E+01 .18250E+01 .18250E+01 .1825OE+01 .18250E+01 .18250E+D1 .18250E*01 .18250E+01

1,00 .18250E+01 18250E*01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .IB25OE+01 .18250E÷01

THERMAL LENSING CALCUATIORS

_AVELENGTH OF IHTEREST(um) ....... 115000E+01

TEMPERATURE OF INTEREST(deg C).. .37500DE+02

TIME (sec) ....................... 350000E÷01

RAY AN&LE OUT RADIUS FOCAL DI_TORTIOM

(no.) (radians) (cm) lE_(cm) (%)

.....................................

.00 .OOOO000E+O0 .O000000E+OO .O0000OOE+O0 .OOCO000E÷O0

.10 .00O0000E÷00 .100O000E+00 .00O0OO0E+00 .4551913E-tl

.2D .OOGOOOOE÷O0 .2000000E+OO .6898527E+03 -.1229572E-I0

.30 .0000000E+00 .30OO000E+00 .7'760843E+03 .O000000E+00

,40 .OO00000E+O0 .40DOOOOE÷O0 .7884030E+03 .979TTI5E-11

.5G .OOOOOOOE+O0 .5000000E+OO .7839235E+03 .OOOOOOOE+O0

.60 .OOOOOOOE+O0 ,60000DOE+DO .7760843E+03 .O00OO00E+O0

,70 .OO00000E+OO ,70OOOOOE+OO .7682449E+03 .DOOOOOOE+OO

.B0 .O000000EtOO .8000000E+0O .7612167E+03 .OOOO0OOE+OO

.90 .OOOO000E+O0 ,9000OOOE+O0 .7551089E+03 .O00000OE+OO

1.00 .0OOOOOOE+O0 .00O0O00E+O0 .0OGOOODE+0O .0000000E+00

AVERAGE OF THE FOCAL LENGTHS (meters} : .677658E+01

I_CREMEHTAL GI_TORTIO_ (%) RANGES FRC_ 0,0 TO .980E-11

.. END THERMAL LENSING OUTPUT..

88

ORIGINAL PAGE iS

OF POOR QUALITY
BLOCK OF TEMPERATURE PROFILES FROM INPUT FIL_ #2

TEMPERATURE DISTRIBUTION

RADIAL DIFFUSIVtTY ................. 269470E*02

RADIUS OF RO0 ................. 250000E+00

LENGTH OF RO0 ................. 760000E+01

TIME IN SECONDS ................. 350013E+01

L/R .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .72382E+OD .72383E+00 .72383E'00 .72383E+00 .72383E+00 .72_3.83E+00 .72383E+00 .7'2383E+00 .72383E+00 .72383E+00 .72382E+00

.10 .71330E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E÷00 .71331E+00 ,71331E+00 .71331E+00 ,71331E+00 .71330E+00

.20 .69403E+00 .69404E+00 .69404E+00 .694046÷00 .69404E+00 .69404E*00 .69404E+00 .69404E+00 ,69404E+00 ,694046+00 o69403E+00

.30 .66320E÷00 .6632iE+00 .66321E+00 ,66321E+00 o66321E+00 .66321E+00 .66321E+00 ._321E*00 .&6321E+O0 ,6&321E+O0 .66320E+00

.40 .62046E+00 .62047E+00 .62G47E+O0 .62047E+00 ,_,?.047E+00 .62047E+00 .620476+00 .62047E+00 .62047E+00 ,62047E+00 .62046E+00

.50 .56521E+00 .56522E_00 .56522E+00 ,56522E*00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 ,56522E÷00 .56521E*00

.60 ,49688E*00 .496896+00 .49689E÷00 .49689E+00 .49689E÷OO ._9689E*00 .49689E+00 .49689E+00 .49689E+00 ,49689E+OO .49688E+00

.70 ,41552E÷00 .41552E+00 .41552E÷00 .41S52E+00 .41552E+00 .41552E+00 .415526÷00 .41552E+00 .41552E÷00 ,415526÷00 .41552E÷00

.BO .32288E+00 .32288E+00 322886÷00 .32288E+00 .32288E+00 ,322&BE+O0 .32288E+00 .32288E+00 .32288E+00 ,3228_E+00 .32288E*00

.90 ,22190E+00 .22190E+00 .22190E+00 .22190E+00 .221906+00 .22190E+00 .22190E+00 .22190E+00 .221_0E+00 ,22190E+00 .221POE+O0

1.0G ,I05566+0O .I0556E-00 .I0556E+00 ,IOS56E+00 ,I0556E+00 .10556E*00 .I0556E*00 .I0556E+00 .10556E+00 ,105566÷00 .I0556E+00

THERMAL LENSING PROGRAM OUTPUT DATA

INDEX OF REFRACTION MATRIX

.00 .18250E+01 .18250E+01 .18250E*01 .18250E*01 ,18250E+01 .182506+01 .18250E+01 ,18250E*01 .18250E+01 ,1625OE+01 .182506+01

.10 .18250E401 .18250E+01 .18250E-01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E÷01

.20 .18250E+01 .18250E+01 .18250E÷01 .18ZSOE+01 .18250E*01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E*01

.30 ,18250E*01 .18250E+01 .18250E*01 .1825DE+01 ,18250E÷01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.40 .18250E+01 .18250E+01 ,18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E*01 .182506÷01

.50 .18250E+01 .18250E÷Ot .18250E*01 .18250E÷01 .18250E+01 .18250E+01 .18250E*01 .18250E+01 .18250E,01 .18250E÷01 .18250E+0_

.60 .18250E÷01 .1825CE+01 ,18250E÷01 .18250E*01 .18250E÷01 ,182506÷01 .18250E*01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01

.70 .18250E+01 .18250E÷01 .18250E÷01 .18250E÷01 .18250E+01 .182506÷01 .18250E*01 .182506*01 .18250E÷01 .18250E÷01 .18250E÷01

.80 .18250E+01 .18250E+01 .18250E+01 .18250E+01 ,182506+01 .18250E*01 .18250E+01 .18250E÷01 ._8250E+01 .tSZ5HE+01 +18250E+01

.90 .18250E+01 .18250E*01 .1825OE+01 .18250E*01 .18250E*01 .18250E+01 .1825HE+01 ,18250E+01 ,18250E+01 .18250E÷01 .18250E+01

1.00 .18250E÷01 .18250E_01 .18250E+01 .1825CE+01 .18250E÷0I .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E*ql .1825HE+0I

TNERMAL LEASING CALCUATIORS

_AVELENGTH OF INTEREST(Um) ....... 115HOOE÷H1

TEMPERATURE OF INTEREST(deg C).. .3TSODOE÷O2

TIME (see) ....................... 350013E÷01

RAY ANGLE OL;T RADIUS FOCAL DISTORTIO_

(no.) (radians) (cm) LEHG(Cm) (%)

.....................................

.DO .OOO0000E+OO .O00HOOO6÷O0 .O00OO006+O0 .O00GOOOE+O0

.10 .OOOOOOOE÷OO ,IOOOOOOE+OO .3449263E+03 .5T(3156E-11

.20 .O000000E÷O0 .2HOOOOOE+OJ ,_98527E+03 -.471_8E-I1

.30 .O0000HOE+OD ,3000000E+HO .7760843E+03 .3552712E-ii

.40 .OO0000GE÷OO ,4OO0000E÷O0 .6_98527E+03 .0936493E-11

.50 .OOOOGOOE+O0 .5OO000HE+O0 .7185965E+03 -.666133_E-12

.60 .OODO000E+O0 .600HHOOE+O0 .7304323E+03 ,22_6745E-10

.70 .HOOOOOOE+O0 .700000OE+O0 .7348430E+03 .O00OO00E+OO

.BO ,OOOOODOE*HO .8000000E+O0 .7"358_2BE*03 .O000000E+O0

.90 .O000000E+O0 .OO0000E_O0 .7551089E+03 .O00OO00_+O0

1.00 .OOHHOOHE+OO .OOOO000E+HO ,OO00000E+O0 .O000COOE+00

AVERAGE OF THE FOCAL LENGTHS (meters) = .6_,171E+01

INCREMENTAL DISTORTION (%) RANGES ER(:_ 0.0 TO .222E-IG

.. END THERMAL LEN3ING OUTFUT..

89 ORIGINAL PAGE IS

OF POOR QUALITY
BLOCK OF TEHPERATURE PROFILES FROM INPUT FILE #2

TEMPERATURE DISTRIBUTION

RADIAL DIFFUSIVITY ................. 269470E+02

RADIUS OF RCO ................. 250000E*00

LENGTH OF RO0 ................. 760000E+01

TIME IN SECONDS ................. 355000E+01

L/R .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.00

.00 .64627E+00 .64627E+00 .64627E+00 .64627E+00 .6A628E+O0 .64628E+00 .64628E+00 .64627E+00 .64627E+00 .64627E+00 .6_627E+00

.10 .64006E+00 .64007E*00 .64007E*00 .64007E+DO .64007E÷00 .64007E*00 .64007E+00 .64007E+00 .64007E+00 .6_,007E+00 .64006E+00

.20 .62144E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62145E+00 .62144E*00

.30 .59041E+00 .59041E÷00 .59041E+00 .59041E+00 .59042E+00 .59042E+00 .59042E+00 .59041E+00 .59041E÷00 .59041E÷00 .59041E+00

.40 .54694E+00 .54694E+00 .54694E+00 .54694E÷00 .54694E÷00 .54694E_00 .54694E+00 .54694E+00 .54694E÷00 .54694E+00 .54694E÷00

.50 .49096E+00 o49097E+00 .490_7E+00 .49097E+00 .49097E+00 .49097E÷00 .49097E+00 .49097E÷00 .49097E+00 .49097E+00 o49096E+00

.60 .42244E+00 .42244E÷00 .422_4E+00 .42244E÷OD .42244E+00 .42244E+00 .42244E+00 .42244E÷00 o42244E÷00 o42244E+00 .42244E÷00

.70 .3_169E+30 .34169E+00 .34169E*00 .34169E÷00 .34169E+00 .34169E÷00 .34169E+00 .34169E+00 .34169E÷00 .34|69E+00 .34169E+00

.80 .25008E÷00 .2500BE÷DO .25008E+00 .25008E+00 .25008E+00 .25008E÷00 .25008E+00 .25008E+00 .25008E÷00 .25008E+00 .25008E÷00

.90 .15093E+00 .15093E+00 .15093E+00 .15093E+00 .15093E+00 .15093E÷00 .15093E+00 .15093E*00 .15093E+00 .15093E+00 .1509]E+00

1.00 .49856E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49857E-01 .49856E-01

THERMAL LENS[NG FROGRAM OUTPUT DATA

INDEX OF REFRACTION MATRIX

.00 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .1825OE+01 .18250E*01 .18250E÷01

.10 .18250E+01 .16250E+01 .18250E+01 .16250E+01 .16250E+01 .18250E+01 .18250E*01 .18250E÷01 .18250E+01 .18250E÷01 .18250E+01

.20 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01

.30 .18250E+01 .18250E+01 .18250E÷01 .18250E÷01 .18250E÷01 .18250E*01 .18250E+01 .18250E÷01 .18250E+01 .1825DE+01 .18250E+01

.40 .18250E÷DI .18250E,01 .18250E+01 .18250E+0| .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01

.50 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.60 .18250E+01 .18250E*01 .IB250E+01 .18250E+01 .18250E+01 .18250E_01 .18250E+01 .18250E+01 .18250E+01 .18250E*01 .18250E+01

.70 .18250E+0t .18250E+01 .18250E+01 .18250E+01 .1B250E+01 .18250E÷01 .18250E+01 .16250E+01 .18250E÷0! .18250E+01 .18250E+01

.80 .t8250E+0t .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E_0t .18250E+01 .18250E+01

.90 .16250E+01 .18250E_01 .18250E*01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01

1.00 .i8250E+0i .18250E+0I _8250E+01 .18250E÷01 .18250E+0t _i8250E;01 .18250E+01 .18250E+01 .18250E+01 .18250E+0I .18250E+0I

THERMAL LENSING CALCUATIONS

WAVELENGTH OF INTEREST(um) ....... 115000E+01

TEMPERATURE OF IRTEREST(deg C).. .37_000E+02

TIME (see) ....................... 355DOOE+01

RAY ANGLE OUT RADIUS FOCAL DISTORTION

(no.) (radians) (cm) LENG(cm) (%)

.....................................

.DO .OOO0000E+O0 .O000000E+O0 ,O000000E+OO .DOCOOOOE÷O0

.10 .O00OOOOE÷O0 .IO00000E÷O0 .O000000E+O0 .4551913E-11

.20 .O000000E+OO .2000000E+O0 .6898527E+03 -.1229572E-I0

.30 .OO00000E+O0 .3000000E+O0 .7760843E+03 .9963511E-09

.40 .O000000E+DO .4000000E+O0 ,78.B/*O30E+03 .O000OOOE+O0

.50 .DOOOOOOE+O0 .5000000E+O0 .7839235E+03 ".1709743E-11

.60 .O000000E÷OO .600000DE+O0 .TT60843E+03 .DOOOOOOE+O0

.70 .O_O0000E+O0 .7000000E+OO .76824_9E÷03 .OO00000E÷OO

.80 .OOO0000E÷O0 .8000000E+OO .7612167E+03 .CO00OOOE+DO

°90 .O000000E+OO .9000000E÷OO .7551089E*03 .O000000E+O0

1.00 ,O000000E*00 .O000000E÷O0 .O00OO00E÷O0 .O00000OE+O0

AVERAGE OF T_E FOCAL LENGTHS (meters) = .677658E÷DI

INCREHENTAL DISTORTION (_) RANGES FROM 0.0 TO .996E-09

.. END THERMAL LENSING OUIPUT..

9O OR,_GINAL PAGE IS

OF POOR QUALITY
BLOCK OF TEHPERATURE PROFILES FROH INPUT FILE #2

TEMPERATURE DISTRIBUTION

RADIAL OIFFUSIVITY ................. 269470E+02

RADIUS OF RCC ................. 250000E+OO

LERGTH OF ROO ................. 760000E+01

TIME IM SECONDS ................. 355013E+01

L/R .00 .I0 .20 .30 .40 .50 .60 .70 .80 .9C 1.00

.OO .72382E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72383E+00 .72382E+00

.10 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00 .71331E+00

.20 .69403E_00 .69404E+00 .69404E+O0 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69404E+00 .69403E+00

.30 .66320E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66321E+00 .66320E+00

.40 .62046E_00 .620_7E+00 .62047E+00 .62047E+00 .62047E+00 .62047E+00 .62047E+00 .62047E+00 .62047E+00 .62047E+00 .62046E+00

.50 .56521E+00 .36522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E+00 .56522E÷00 .56522E÷00 .56521E+00

.60 .49688E+00 .49689E+00 .49689E+00 .49689E+00 .49689E+00 .496J59E+O0 .49689E+00 .49689E+00 .49b_9_+00 .49_9E+00 .496_9E+00

.70 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E+00 .41552E÷00 .41552E+00 .41552E+00

.80 .3228_E+00 .3228,8E+00 .32288E÷00 .32288E+00 .32288E+00 .32288E+00 .32288E+00 .32288E+O0 .32288E+00 .32288E+00 .32288E+00

.90 .22190E_C0 .22190E+00 .22190E+00 .22190E+00 .22190E+00 .22190E+00 .22!90E+00 .22190E+00 .22190E+00 .22190E+00 .22190E+00

1.00 .I0556E_00 .I0556E_00 .I0556E+00 .10556E_O0 .I0556E+00 .I0556E+00 .I0556E+00 ,I0556E÷00 .I0556E+00 .I055_+00 .I0556E+00

THERMAL LERSIRG PROGRAM OIJTPUT DATA

IRDEX OF REFRACTION MATRIX

.00 .18250E+01 .18250E+01 .1825CE+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.10 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+0t .18250E+01 .18250E+01 .18250E+01

.20 .18250E+01 .18250E+01 ,18250E+0I .18250E+01 .18250E+01 .18250E*01 .18250E+01 .18250E+01 .18250E+0I .I8250E+0I .18250E+01

.30 .18250E+01 .I_250E-01 .18250E,01 .18250E+01 .18250E+01 .IB250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .1825bE+01

.40 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .I_250E+01 .18250E+01 .18250E+01

.50 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .16250E+01 .18250E+01 .18250E+01

._0 .15250E+CI .18250E+01 .I_250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01

.70 .18250E+01 .18250E_01 .18250E_01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .15250E+01

.BO .18250E_01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250£+01 .18250E÷01 .IB250E+01 .18250E+01 .18250E+01

.90 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .18250E+01 .18250E+01 .18250E÷01 .18250E+01 .IB250E-01 .18Z50E+01 .18250E+01

1.00 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E+01 .18250E-01 .I_250E+01 .18250E+01

THERMAL LENSING CALCUATION_

VAVELENOTH OF INTEREST(um) ....... I_5000E+01

TEMPERATURE OF IHTEREST(deg C).. .3750OOE+O2

TIHE (sec) ....................... 355013E+01

RAY _NGLE CXJT RADIUS FOCAL DISTORTION

(no.) (radians) (cm) LENG(cm) (%)

.....................................

.00 .O000000E+O0 .O000000E+O0 .O000000E+O0 .O000000E+O0

.I0 .O00OO00E+O0 .IO00000E+O0 .3449263E+03 .O000000E+O0

.20 .O000000E+CC .20GO000E+O0 .6_98527E+03 .2553516E-11

.30 .O000000E+O0 .3000000E+O0 .7760843E+03 .3552712E-11

.40 .O000000E+O0 .4000000E+O0 .6898527E+03 .0936493E-11

.50 .OOO0000E+O0 .5000000E+O0 .71859_5E+03 -.6661338E-12

.60 .O000OOOE+O0 .6000000E_O0 .7304323E+03 .2216745E-I0

.70 .O000000E+O0 .7000000E÷O0 .73_8430E_03 .O000000E+O0

.80 .O000000E+00 .8000000E+0C .7"358428E+C3 ,0000000E÷00

.90 .O00OOOOE+00 .900O000E+00 .7551089E_-03 .0000000E+00

1.O0 .OO00000E+00 .0000000E+00 .0000000E+00 .0000000E-00

AVERAGE OF THE FOCAL LENGTHS (meters) = .686171E+01

INCREMENTAL DISTORTION (_) RANGES FROH 0,0 TO ,222E-I0

.. END TRERMAL LEN;ING COTPUT..

gl

OR_,G|NAL PAGE IS

OF POOR QUALITY
References

,

,

,

,

w

,

°

Farrukh, U. O., Buoncristiani, A. M., and Byvik, C. E.,

An Analysis of the Temperature Distribution in Finite

Solid-State Laser Rods, IEEE Journal of Quantum

Electronics, Vol. 24, No. ii, November 1988.

Koechner, Walter., Solid State Laser Engineering,

Springer-Verlag: Berlin, Heidalburg, New York, 1988.

Farrukh, U. O., Buoncristiani, A. M., and B_vik, C. E.,

Time Dependent Temperature Distribution in Pulsed Ti:

Sa_ap_P_h_e Lasers, Proceedings of SPIE-The International

Society for Optical Engineering, Vol. 892, 1988.

Bendow, B. and Gianino, P. D., Thermal Lensing of Laser

Beams in Optically Transmittinq Materials - I. General

Formulation, Journal of Applied Physics, Vol. 2, pp.

I-I0, 1973.

Koechner, W., Transient Thermal Profiles in Qptically

Pumped Laser Rods, Journal of Applied Physics, Vol. 44,

No. 7, pp. 3162-3170, 1973.

Antipenko, B. M., Cross-Relaxation Schemes for Pumping

Laser Rods_, Soviet Physics, Technical Physics, Vol. 29,

No. 2, pp. 228-230, 1984.

Farrukh, U., Brackett, V., and Brockman, P., SteadyState Temperature in End-Pumped Repetitively Pulsed

Laser Rods, Technical Digest, Paper presented at the Optical

Society of America, 1989 Annual Meeting, 15-20 October 1989.

S o Foster, J. D. and Osterink, L. M., Thermal Effects in

Nd:YAG Lasers, Journal of Applied Physics, Vol. 41, No.

9, pp. 3656-3663, 1970.

, Koechner, W., Thermal Lensing in Nd:YAG Laser Rod,

Journal of Applied Optics, Vol. 9, No. ii, pp. 2548-

2553, Nov. 1970.

i0. Sparks, M., Optical Distortion by Heated Windows in

H_ Power Laser Systems, Journal of Applied Physics,

Vol. 42, No. 12, pp. 5029-5046, 1971.

Ii. Jasperse, J. R. and Gianino, P. D., Thermal Lensing in

Infrared Laser Window Materials, Journal of Applied

Physics, Vol. 43, No. 4, pp. 1686-1693, 1972.

92
References (cont.)

12. Bendow, B. and Gianino, P. D., Thermal Lensinq of Laser

Beams in Qptically Transmitting Materials - II. Numerical Computations, Journal of Applied Physics, Vol. 2,

pp. 71-90, 1973.

13. Carslaro, H. S. and Jaeger, J. C., Conduction of Heat

in Solids, Oxford University Press: London, 1948.

14. Kogelnik, H., Ima___gjgf Optical Modes - Resonators

with Internal Lenses, Bell Systems Technical Journal,

Vol. 44, pp. 455-494, 1965.

]5. Timoshenko, S. and Goodier, J. N., Theory of

Elasticity, McGraw-Hill: New York, 1951.

16. Born, M. and Wolf, E., Principles of Optics, Pergamon

Press: London, 1965.

17. Koechner, W. and Rice, D. K., Effect of Birefrinqence

on the Performance of Linearly Polarized YAG: Nd

Lasers, IEEE Journal of Quantum Electronics, Vol. QE6,

No. 9, pp. 557-566, September 1970.

18. Koechner, W., Absorbed Pump Power, Thermal Profiles and

Stresses in a CW Pumped Nd:YAG Crystal, Journal of

Applied Optics, Vo!. 9, No. 6, pp. 1429-1434, June

1970.

19. Yariv, A. and Yeh, P., Optical Waves in Crystals, John

Wiley & Sons: New York, 1984.

20. Williams-Burg, J. A. and Barnes, N. P., Laser Performance, Thermal Focusing and Depolarization Effects in

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Los Angeles, Ca., January 1990.

93
f

N/LRA

1. Report No.

NASA CR-4300

4. Title and Subtitle

Report Documentation Page

2. Government Accession No.

Modeling of Thermal Lensing in

Finite Solid-State Laser Rods

7. Author('s)

Vincent G. Brackett

Side and End-Pumped

9. Performing Organization Name and Address

Hampton University

Physics Department

Hampton, VA 23668

12. Sponsoring Agency Name and Address

NASA Langley Research Center

Hampton, VA 23665-5225

3. Recipient's Catalog No.

5. Report Date

June 1990

6. Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

590-31-31

11. Contract or Grant No.

NAG1-666

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring _gency Code

15. Su_mentaw Notes

Langley Technical Monitor: Philip Brockman

Thesis presented to The Graduate College, Hampton University, in partial

fulfillment of the requirements for the Degree of Master of Science.

16. Abstract

An analytical expression for approximating the time-dependent thermal

focal length in finite solid-state laser rods has been derived. The

anaiysls Is based on the temperatUre variation of the materlal refractive

index caused by optical pumping of these rods. Several quantities were

found to be relevant to this analysis. These quantities were the specific

thermal profiles of the rods, type of optical pumping employed, type of

cooling scheme employed (side and end-cooling parameters), and the

specific material characteristics of the rods. The Themal Lenslng Model

was formulated using the geometric ray tracing approach. The focal

lengths are then approximated, by calculating the phase, shift in the lndex

of refraction, as the different rays of an Incident plane wave are tracked

through a lens-like crystal medium. The approach also applies in the case

of Gausslan or parabolic pump beams. It ls shown that the prediction of

thermal focal length is in good quantitative agreement with experimentally

obtained data.

17. Key Word= (Suggested by Author(s))

optical pumping, solid-state lasers,

optical properties, rods, temperature

effects, pulsed lasers

18. Distr_utmn Statement

Unclassified - Unlimited

Subject Category 36

19. Security Classif. (of this report) 20. Security Classif. (of this _zga) 21. No. of page=

Unclassi fled Uncl assi fied lO4

YASA FORM IlI_ OCT 86

For _Je by tile N&ti(mal Technical Jliforn|_ttion _ervi('.e, Si)riiltZfichl, Virgiuis 22161-21 ?l

I 22. Price

A06

NASA-L&ngley, 19g0