model of thermally induced wavefront distortion and birefringence in side-pumped nd-doped yag and...

Model of thermally induced wavefront distortion andbirefringence in side-pumped Nd-doped YAG and

phosphate glass heat capacity rod lasers

Liang Liu,* Xiaobo Wang, Shaofeng Guo, Xiaojun Xu, and Qisheng LuOpto-Electronic Science and Engineering College, National University of Defense Technology,

Changsha 410073, China

*Corresponding author: [email protected]

Received 9 June 2010; revised 20 August 2010; accepted 20 August 2010;posted 23 August 2010 (Doc. ID 129818); published 22 September 2010

We develop an analytic model to describe the dynamic average thermal distortion and phase differencebetween the two principal polarizations in side-pumped Nd:YAG andNd:glass heat capacity rod lasers. Itcan be predicted that the average thermal distortion is proportional to the temperature profile on thecross section from the analytic expression and, therefore, it is feasible to measure the temperature profileby wavefront sensing. In addition, temperature-dependent variation of the refractive index constitutesthe major contribution of the thermal lensing for Nd:YAG rod lasers. Temperature- and stress-dependentvariation of the refractive index constitute themajor contributions of the thermal lensing for Nd:glass rodlasers. In the case of the same pumping and cooling conditions, there are the same orders of depolariza-tion loss for Nd-doped YAG, LG-680, LG-750, LG-760, and LG-770 glass rod lasers. © 2010 OpticalSociety of AmericaOCIS codes: 140.3580, 140.6810, 140.3530.

1. Introduction

High-power solid-state lasers with excellent beamquality show a wide variety of applications in indus-trial, military, medical, and scientific research.However, thermal effects [1,2] seriously limit thesolid-state lasers from being scaled to high averagepower. Many techniques have been developed to mi-tigate the thermal distortion of the beam, such asdisk lasers [3], zigzag slab lasers [4,5], and so on.When solid-state lasers work in the steady-state re-gime, the waste heat deposits in the active medium,and it is removed from the surface. Since the tem-perature of the cooled surface is lower than the tem-perature inside the active medium, tensile stresspresents on the cooled surface. Average power lasersare designed not to exceed a critical tensile stress va-lue, or the medium will fracture. Thus, the averagepower heat removal effects constitute an intrinsic

limit to the steady-state average power that thesolid-state laser can put out.

In the 1990s, Lawrence Livermore National La-boratory developed solid-state heat capacity lasers[6]. Because the active medium is not cooled duringthe lasing cycle, the resulting surface stresses arecompressive, not tensile. In addition, the path tohigher power can be achieved by increasing the pulserepetition rate. Therefore, the operation of a solid-state laser in the heat capacity mode is a novel ap-proach to the development of systems that providehigh power, compactness, and, most importantly,scalability.

Because the power output declines obviously andthe pump threshold rises sharply with increasingtemperature in heat capacity lasers [7–9], accuratetemperature measurement in active media is essen-tial to heat capacity laser design. Interferometrictechniques [10] have been successfully used for manyyears to measure temperature changes in activemedia. However, high sensitivity to vibration andturbulence, high expense, and bulk data processing

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make it unfit for real-time measurement. In 2007, amethod was presented to measure the temperaturedistribution through a Shack–Hartmann (SH) wave-front sensor [11,12], which operated at a much fastermeasurement rate and was vibration insensitivecompared with interferometric techniques. Goodagreement between theory and experiment con-firmed that it was a good choice for measuring tem-perature change in real time.

To our knowledge, at present, there are no analyticmodels describing the relationship between thermaldistortion and thermal profile in heat capacity lasers.In this paper, an analytic model is developed to de-scribe the average thermal distortion and phase dif-ference between the two principal polarizations inside-pumped Nd:YAG and Nd:glass heat capacityrod lasers. The model indicates the feasibility of tem-perature measurement by wavefront sensing. An ex-planation of the agreement between the theory andexperiment in Refs. [11,12] is presented. The differ-ences between Nd:YAG and Nd:glass heat capacitylasers are also discussed.

2. Temperature and Stress Distribution in Side-Pumped Rod Lasers

Suppose internal heat generation and cooling is uni-form along the cylindrical axis; then the heat flow isstrictly radial. Thus, the three-dimensional problemis turned into a two-dimensional problem, and thetransient thermal profile in a solid rod under repeti-tive pumping can be written by [13]

Tðr; tÞ ¼ 2QVcγð1þ g=2Þ

X∞n¼1

expð−β2nt=τÞ

×ð1þ gÞβnJ1ðβnÞ − 2gJ2ðβnÞ

ðA2 þ β2nÞJ20ðβnÞ

×1 − expð−Mβ2ntp=τÞ1 − expð−β2ntp=τÞ

× J0ðβnr=r0Þ; ð1Þ

whereQ is the total heat absorbed by the active med-ium,V is the rod volume, c is the specific heat, γ is themass density, g is a measure of the pumping nonuni-formity, M is the number of shots, and tp is the pulseinterval. J0, J1, and J2 are Bessel functions of thefirst kind and the zero, first, and second orders, re-spectively. The dimensionless parameter A, whichspecifies the cooling condition of the rod, is givenas A ¼ r0h=k, where r0 is the rod radius, h is the sur-face heat transfer coefficient, and k is the thermalconductivity. The thermal time constant τ of therod is given as τ ¼ r20cγ=k. The parameter βn are theroots of the eigenfunction equation βnJ1ðβnÞ ¼AJ0ðβnÞ.

Because temperature is symmetrical about theaxis and independent of the axial coordinate z inthe rod, in the case of free ends, the expressionsfor radial stress σr, tangential stress σθ, and axialstress σz under the condition of plane strain are givenby Timoshenko and Goodier [14]:

σr ¼αE1 − v

�1

r20

Zr0

0Trdr −

1

r2

Zr

0Trdr

�; ð2Þ

σθ ¼αE1 − v

�1

r20

Zr0

0Trdrþ 1

r2

Zr

0Trdr − T

�; ð3Þ

σz ¼ σr þ σθ: ð4ÞUpon substituting Eq. (1) into Eqs. (2)–(4), we have

σrðr; tÞ ¼2αE1 − v

QVcγð1þ g=2Þ

X∞n¼1

�expð−β2nt=τÞ




×�J1ðβnÞβn

−

r0J1ðrβn=r0Þrβn

��; ð5Þ

σθðr; tÞ ¼2αE1 − v

QVcγð1þ g=2Þ

X∞n¼1





×�J1ðβnÞβn

þ r0J1ðrβn=r0Þrβn

− J0ðβnr=r0Þ��

;

ð6Þ

σzðr; tÞ ¼2αE1 − v

QVcγð1þ g=2Þ

X∞n¼1





×�2J1ðβnÞ

βn− J0ðβnr=r0Þ

��; ð7Þ

where α is thermal expansion coefficient, ν is thePoisson ratio, and E is Young’s modulus.

3. Thermal Distortion in Side-Pumped Rod Lasers

Because the refractive index of an optic mediumchanges with temperature and stress, thermal dis-tortion presents when the beam propagates throughthe active medium. In Koechner’s theory [1] of

5246 APPLIED OPTICS / Vol. 49, No. 28 / 1 October 2010

thermal effects, a thermal lens consists of threeparts: the temperature- and stress-dependent varia-tion of the refractive index, and the distortion of theend-face curvature of the rod. The contribution of theend effect is less than 6% in steady-state rod lasers.Based on the fact that there are far fewer thermalgradients between the center of the rod and the sur-face in heat capacity lasers than in steady-statelasers, the end effect constitutes an ignorable contri-bution to the optical path difference (OPD) due tothermal effects. Thus, thermal distortion can bewritten by

OPDðr; tÞ ¼ L½ΔnTðr; tÞ þΔnσðr; tÞ�; ð8Þwhere ΔnT and Δnσ are temperature- and stress-dependent variation of the refractive index respec-tively. The temperature-dependent change ofrefractive index can be expressed as

ΔnTðr; tÞ ¼dndT

ΔTðr; tÞ ð9Þ

where dn=dT is the thermo-optic coefficient andΔTðr; tÞ is the transient temperature profile, definedby ΔTðr; tÞ ¼ Tðr; tÞ − Tð0; tÞ.

The foregoing discussion shows the analysis ofΔnσðr; tÞ. Generally, the refractive index of a crystalis specified by an ellipsoid as follows [15]:

Bijxixj ¼ 1; ð10Þ

where the Einstein summation convention is usedand B is the relative dielectric impermeability tensordefined as the inverse matrix of the dielectric con-stant. The optically isotropic crystals become aniso-tropic when they are subject to stress, and Bij can beexpressed by

Bij ¼ B0;ij þ pijklεkl; ð11Þwhere p is the photoelastic tensor, ε is elastic straintensor induced by thermal stress, and B0;ij is given by

B0;ij ¼δijn

n0 þ dndT ½Tðr; θ; zÞ − T0�

o2 ; ð12Þ

where δij is the Kronecker delta function, n0 is therefractive index under temperature T0, andTðr; θ; zÞ is the temperature at point ðr; θ; zÞ.

The relation between elastic strains and stressesin a cylinder coordinate system is defined by Hooke’slaw [14] using Nye’s convention [15]:

εr ¼1Efσr − v½σθ þ σz�g; ð13Þ

εθ ¼1Efσθ − v½σr þ σz�g; ð14Þ

εz ¼1Efσz − v½σθ þ σr�g: ð15Þ

Nd:YAG is cubic, and the photoelastic tensor in thecrystal lattice system can be expressed through amatrix [15]:

0BBBBBB@

p011 p0

12 p012

p012 p0

11 p012

p012 p0

12 p011

p044

p044

p044

1CCCCCCA: ð16Þ

For Nd:YAG rods with their z axes along ½111�, thephotoelastic tensor can be expressed by [16]

0BBBBBB@

p11 p12 p13 p14 p15 0p12 p11 p13 −p14 −p15 0p13 p13 p33 0 0 0p14 −p14 0 p44 0 −p15

p15 −p15 0 0 p44 p14

0 0 0 −p15 p14 p66

1CCCCCCA: ð17Þ

The nonzero components in Eq. (17) are

p11 ¼ 12p011 þ

12p012 þ p0

44;

p12 ¼ 16p011 þ

56p012 −

13p044;

p13 ¼ 13p011 þ

23p012 −

23p044;

p33 ¼ 13p011 þ

23p012 þ

43p044;

p44 ¼ 13p011 −

13p012 þ

13p044;

p66 ¼ 16p011 −

16p012 þ

23p044;

p14 ¼ cosð3θÞ3

ffiffiffi2

p ð−p011 þ p0

12 þ 2p044Þ;

p15 ¼ sinð3θÞ3

ffiffiffi2

p ð−p011 þ p0

12 þ 2p044Þ;

ð18Þ

where θ is defined in Fig. 1.By substituting Eq. (17) into Eq. (11), we get the

suffix-reduced relative dielectric impermeabilitytensor:

0BBBBBBB@

Br

BθBz

BθzBzr

Brθ

1CCCCCCCA

¼

0BBBBBBB@

1=n20 þ p11εr þ p12εθ þ p13εz



000

1CCCCCCCA: ð19Þ

For the light propagating along the z axis, theelliptic equation at the plane perpendicular to thez axis is given by


½r; θ��Br BrθBrθ Bθ

��rθ

�¼ 1: ð20Þ

The eigenvectors of the above 2 × 2 matrix representthe directions of principal axes, in which there are nodepolarization effects. The corresponding eigenval-ues λ� represent the refractive indices n� at thesedirections:

n� ¼ 1=ffiffiffiffiffiλ�

p; ð21Þ

λ� ¼ 12

�ðBr þ BθÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBr − BθÞ2 þ 4B2

rθ

q �: ð22Þ

When a naturally polarized probe light and an SHwavefront sensor are employed, the thermal distor-tion due to the gross index change caused by thestress field can be approximated by averaging the op-tical path of the two electric field components. Thus,we define the average refractive index navg by

navg ¼ ðnþ þ n−Þ=2: ð23Þ

By substituting Eqs. (2)–(4), (13)–(15), and (19)–(22)into Eq. (23), we find that the average refractiveindex change associated with stress, defined byΔnσ ¼ navg − n0, is

Δnσ ¼n3σz6E

½−2p011 − 4p0

12 þ p044

þ ð4p011 þ 8p0

12 þ p044Þv�: ð24Þ

For convenience, the axial stress σz in Eq. (7) canbe expressed by

σzðr; tÞ ¼ f ðM; tÞ − αE1 − v

Tðr; tÞ; ð25Þ

where f ðM; tÞ is a function of time and pulse num-bers, independent of the radius. Thus, the dynamicwavefront profile can be written as

OPDðr; tÞ ¼ L

�dndT

þ αn3½2p011ð1 − 2vÞ þ 4p0

12ð1 − 2vÞ − p044ð1þ vÞ�

6ð1 − vÞ�ΔTðr; tÞ; ð26Þ

where L is rod length. For convenience, the secondterm of the parenthesized expression in Eq. (26) iswritten as Cs, which represents the contribution ofthermal stress.

In contrast to Nd:YAG, Nd:glass is isotropic andhas the same photoelastic tensor as that of Nd:YAG, with 2p0

44 ¼ p011 − p0

12. Therefore, the expressionof thermal distortion for Nd:YAG is also applicable to

Nd:glass. The dynamic wavefront profile for Nd:glasscan be written as

OPDðr; tÞ ¼ L

�dndT

þ αn3½p011ð1 − 3vÞ þ p0

12ð3 − 5vÞ�4ð1 − vÞ

�

×ΔTðr; tÞ: ð27Þ

Approximatively, the focal length caused by ther-mal effect can be predicted by

f ¼ r20=2 OPD ð28Þ

for side-pumped rod lasers operating in the pulsedregime, where the OPD is calculated from Eq. (26).

4. Thermally Induced Birefringence in Side-PumpedRod Lasers

From Eqs. (19) and (20), we find that plane waveslinearly polarized along the radial and tangential di-rections will not experience depolarization. When alinearly polarized beam that is not oriented in suchdirections propagates through the laser rods andthen a lossless analyzer, the fractional depolarizationloss is given as [1]

loss ¼ sin2ð2φÞsin2ðδ=2Þ; ð29Þ

where φ is the angle between the radial direction andthe polarized direction, and δ is the phase differencebetween the two principal polarizations, given as

δ ¼ 2πλ Lðnþ − n

−Þ: ð30Þ

Fig. 1. Cylindrical coordinate system for a YAG rod.


By substituting Eqs. (13)–(15) and (19)–(22) intoEq. (30), we get

δ ¼ −

2πλn30ðp0

11 − p012 þ 4p0

44Þð1þ vÞðσr − σθÞ6E

L:

ð31Þ

By substituting Eqs. (5) and (6) into Eq. (31), we have

δðr; tÞ ¼ 2πLλ

αn3ðp011 − p0

12 þ 4p044Þð1þ vÞ

3cγð1 − vÞ

×X∞n¼1

GðM; βnÞexp−β2nt=τJ2ðrβn=r0Þ; ð32Þ

where GðM; βnÞ is a function of shot numbers and βn,written as

GðM; βnÞ ¼Q

Vð1þ g=2Þð1þ gÞβnJ1ðβnÞ − 2gJ2ðβnÞ


×�1 − expð−Mβ2ntp=τÞ1 − expð−β2ntp=τÞ

�: ð33Þ

For Nd:glass rods, Eq. (32) can be written as

δðr; tÞ ¼ 2πLλ

αn3ðp011 − p0

12Þð1þ vÞcγð1 − vÞ

X∞n¼1

GðM; βnÞ

× exp−β2nt=τJ2ðrβn=r0Þ: ð34Þ

5. Discussion and Numerical Examples

A. Thermal Distortion

From Eq. (26), we find the wavefront profile is pro-portional to the temperature profile, and the thermaldistortion may be dramatically reduced if an activemedium with suitable properties is found. Table 1lists the physical properties of YAG and differentphosphate glasses. Using the physical constants inTable 1, we find that the contribution of the stress-dependent variation of the refractive index is greaterthan that of the temperature-dependent variation ofthe refractive index in Nd:glass rod lasers. Neverthe-less, the contribution of the temperature-dependentvariation of the refractive index is greater than thatof the stress-dependent variation of the refractiveindex in Nd:YAG rod lasers. This is because the val-ues of photoelastic coefficients for Nd:glass are muchlarger than the corresponding values for Nd:YAG. On

Table 1. Properties of Phosphate Glass [17] and YAG [1]

Material LG-680 LG-750 LG-760 LG-770 YAG

ρ (g=cm3) 2.54 2.83 2.6 2.585 4.56n 1.56 1.516 1.508 1.4996 1.818

dn=dT (×10−6) 2.9 −5:1 −6:8 −4:7 7.3α (×10−6) 10.18 13.01 15.04 13.36 7.8k (W=mK) 1.19 0.49 0.57 0.57 14C (J=g K) 0.92 0.72 0.77 0.75 0.59E (GPa) 90.1 50.1 53.7 47.29 280

v 0.242 0.256 0.267 0.253 0.24p0

11 0.11 0.20 0.19 0.16 −0:029p0

12 0.18 0.24 0.24 0.20 0.0091p0

44 −0:04 −0:02 −0:02 −0:02 −0:0615Cs (×10−6) 4.54 7.03 7.76 5.88 0.67

dn=dT þ Cs (×10−6) 7.44 1.93 0.96 1.18 7.97CB (×10−12) −2:09 −1:53 −2:26 −1:44 −2:55

Fig. 2. (Color online) Dynamic temperature distribution during the (a) pumping and (b) cooling process.


the other hand, the absolute thermo-optic coefficientfor Nd:glass is less than that of Nd:YAG. For an ac-tive medium with a positive proportion constant ofdn=dT þ Cs, thermal effects change from negativeto positive thermal lensing during the cycle fromthe pumping process to the cooling process.

To give a clear understanding of the above pro-blem, thermal distortion in Nd-doped LG-770 glassrod lasers is calculated. In our calculation, the laser

rod is 50 cm long with a radius of 2 cm. Suppose thepump field is uniform in the tangential direction. Therepetitive pulse rate is limited to 10 Hz with aFWHM of 460 μs. The total heat deposited Q inthe rod is 800 J and the surface heat transfer coef-ficient h is 1 W=m2K. The physical properties ofLG-770 are listed in Table 1.

Figure 2 illustrates the dynamic temperature dis-tribution during the pumping and cooling process. As

Fig. 3. (Color online) Dynamic OPD distribution during the pumping and cooling process: (a) OPD due to the temperature-dependentvariation of the refractive index, (b) average OPD due to the stress-dependent variation of the refractive index, and (c) OPD due to thetemperature- and stress-dependent variation of the refractive index (a-1), (b-1), (c-1) during the pumping process and (a-2), (b-2), (c-2)during the cooling process.


shown, a steady increase in temperature presentsduring the pumping process, and the rod surface ishotter than the center. During the cooling process,the temperature in the center first rises and thendecreases, while the temperature on the rod surfacedecreases. In the end, the rod surface is cooler thanthe center.

The corresponding thermal distortion is plotted inFig. 3. During the pumping process, the tempera-ture-dependent variation of the refractive indexleads to positive thermal lensing because of the ne-gative thermo-optic coefficient dn=dT. However, theaverage stress-dependent variation of the refractiveindex leads to negative thermal lensing because ofthe positive proportional constant Cs. Finally, theeffects of the temperature- and stress-dependentvariation of the refractive index lead to negativethermal lensing because of the positive proportionalconstant dn=dT þ Cs. During the cooling process, thenegative thermal lensing first degenerates and thenbecomes positive when the rod surface is cooler thanthe center.

In addition, it is valid to conclude from Eq. (26)that the temperature distribution on the cross sec-tion can be calculated by the measurement of thewavefront, if appropriate material parameters areintroduced. In 2007, a method for measuring thetemperature distribution through an SH wavefrontsensor [11,12] was presented. Good agreement be-tween theory and experiment confirmed that itwas a good choice for measuring the temperaturechange in real time. However, in Refs. [11,12], the ap-proximation of the thermal distortion was written by

OPDðr; tÞ ¼ L

�dndT

þ αðn − 1Þ�ΔTðr; tÞ; ð35Þ

and the proportional constant [dn=dT þ αðn − 1Þ] is1:4 × 10−6 for N31 phosphate glass [18]. Note thatEq. (35) is only valid in the absence of stress. Becausethe contribution of thermal stress is the major por-

tion of thermal distortion in glass rods, we considerEq. (35) not suitable for the calculation of thermaldistortion in Nd:glass rod lasers. Although we havededuced the analytic expression of thermal distor-tion in Nd:glass rod lasers, unfortunately, the photo-elastic tensor for N31 phosphate glass has not beentested and the values of parameters p0

11 and p012 are

unknown. Suppose N31 glass is consistent in naturewith the other phosphate glasses; then we get a pro-portional constant (dn=dT þ Cs) of the order of 10−6using the thermo-optic coefficient −4:3 × 10−6 of N31glass [18]. That is why good agreement between the-ory and experiment was presented in Refs. [11,12].Despite the fast measurement of temperature, thismethod is not suitable for steep thermal gradientsthat exist in stable surface cooling lasers due tothe limited spatial resolution of an SH sensor.

B. Thermally Induced Birefringence

As shown in Eq. (32), the phase difference betweenthe two principal polarizations is a function of phy-sical properties and pumping and cooling conditions,such as photoelastic coefficients, the Poisson ratio,the thermal expansion coefficient, and so on. Notethat the phase difference is proportional to the differ-ence of photoelastic coefficients for Nd:glass. It is pos-sible that birefringence caused by thermal stress canbe reduced as the difference of photoelastic coeffi-cients approaches zero.

Assume the second fraction inEq. (32) is denoted asCB. Using the physical constants in Table 1, we findthat YAG has the maximum absolute value of CB,while glasses have relative smaller absolute valuesof CB. However, it is not true to conclude thatNd:YAG rod lasers present a more serious thermalstress-induced birefringence effect than that ofNd:glass rod lasers under the samepumpingand cool-ing conditions, namely the same parameters of gand h.

By using Eqs. (32) and (34), calculations have beenperformed to determine the dynamic phase differ-ence pattern in Nd-doped LG-770 and YAG rod lasers

Fig. 4. (Color online) Dynamic OPD distribution between the two principal polarizations in (a) Nd:glass and (b) Nd:YAG rod lasers duringthe pumping process.


during the pumping process under the above-mentioned condition in Subsection 5.A. As shownin Fig. 4, the OPDs in Nd:glass and Nd:YAG rod la-sers are almost the same. From Eq. (31), we find thatthe values of n3

0ðp011 − p0

12 þ 4p044Þð1þ vÞ=6E are

−1:7 × 10−12 for LG-770 and −1:2641 × 10−12 forYAG. Since the thermal conduction of LG-770 glassis much less than that of Nd:YAG, a much greaterthermal gradient presents in Nd:glass rods than inNd:YAG. Generally, thermal stress is proportionalto thermal gradient in the solid material. However,it is apparent from Eqs. (2) and (3) that the effectcan be counteracted as the Young’s modulus of LG-770 glass is much less than that of Nd:YAG. Takingthe stress state in the end of the pumping process, forexample, the values of σr − σθ are 2:4 MPa in Nd:glass and 3:4 MPa in Nd:YAG. Thus, thermal bire-fringence effects in Nd-doped LG-770 glass andNd:YAG rod lasers present in the same degree.

Figure 5 shows the transmission profiles of a probelight (λ ¼ 1064 nm) passing through a Nd:YAG laserrod placed between two parallel polarizers. Thebright areas represent where low depolarizationlosses occur, and the dark areas represent wherehigh depolarization losses occur. As shown, depolar-ization loss occurs just after one pumping pulse, andit mainly presents near the surface of the rod. Sub-sequently, the areas where depolarization loss occursgradually converge to the center of the rod as thepulse numbers increase, and dark spots increase.During the following cooling process, the areas wheredepolarization loss occurs gradually diverge to thesurface of the rod, and dark spots decrease. In theend, the output beam is obtained in the form of across with a bright center.

Figure 6 illustrates the total depolarization lossesin different rod lasers. As discussed in this subsec-tion, there are the same orders of depolarization lossfor Nd-doped YAG, LG-680, LG-750, LG-760, andLG-770 glasses rod lasers.

6. Conclusion

Thermal effects are of great importance to high-power solid-state laser design. In this paper, themodel of OPD is studied for repetitively side-pumpedNd:YAG and Nd:glass rod lasers. The analytic ex-pression reveals a proportional relationship betweenaverage thermal distortion and the temperature pro-file, which indicates that wavefront sensors, such asSH sensors and wavefront curvature sensors, can beused in the measurement of temperature profiles. Itis notable that this method is not suitable for themeasurement of steep thermal gradients in CWrod lasers due to limited spatial resolution. In addi-tion, the contribution of stress to thermal effectscannot be ignored, especially in glass media. ForNd:YAG rod lasers, the contribution of temperature-dependent variation of the refractive index is greaterthan that of stress-dependent variation of therefractive index, but the reverse is true in Nd:glassrod lasers. In the case of the same pumping andcooling conditions, there are the same orders of depo-larization loss for Nd doped YAG, LG-680, LG-750,LG-760, and LG-770 glass rod lasers.

The validity of the theoretical model needs to beworked out by experimental evidence. Investigationsfor the analytic model are under progress in ourlaboratory and will be reported in future.

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17. Phosphate glass: LG-680, LG-750, LG-760, LG-770, http://www.schott.com/advanced_optics.

18. Nd-doped phosphate glass, http://www.laserglass.com.cn/english‑page/index/product/n31.htm.


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model of thermally induced wavefront distortion and birefringence in side-pumped nd-doped yag and...

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