Optics & Laser Technology 43 (2011) 622–629

Contents lists available at ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlastec

Stress-induced depolarization loss in a YAG zigzag slab

Liang Liu n, Shaofeng Guo, Qisheng Lu, Xiaojun Xu, Jinyong Leng, Jinbao Chen, Zejin Liu

Opto-electronic Science and Engineering College, National University of Defense Technology, Changsha 410073, China

a r t i c l e i n f o

Article history:

Received 2 December 2009

Received in revised form

22 August 2010

Accepted 2 September 2010Available online 22 September 2010

Keywords:

Solid-state laser

Thermal effect

Birefringence

92/$ - see front matter Crown Copyright & 2

016/j.optlastec.2010.09.002

esponding author. Tel.: +86 15973117064; fa

ail address: ll06b@126.com (L. Liu).

a b s t r a c t

The stress-induced depolarization loss in a [1 1 1] orientated YAG zigzag slab was studied. The process

to get correct piezo-optic tensor was given in detail. The results indicated that the relationships

between losses and cut angles varied with the change in the bounce numbers of light in zigzag

propagation through the slab. The loss mainly occurred in the area near the edges in width and the

mean depolarization loss was less than 3%. The coupling of a laser beam with adequate bounce number

and aspect ratio less than 1 on the entrance plane was found to reduce the depolarization loss.

Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Solid-state lasers with high average power have a wide varietyof applications due to the advantages of compactness, shortwavelength, and easy scaling to high power. However, thermaleffects [1] (including thermal lensing, thermal stress-inducedbirefringence) in the laser medium lead to strong opticalaberrations, and degrade the output power for the lasers withintracavity polarizers. Since a thin slab with a zigzag path caneliminates thermal effects [2–5], it is considered to be one of themost promising high-power solid-state laser proposals. In 2009, a105 kW solid-state laser system was demonstrated for the firsttime [6] by combining seven slab amplifier strains.

Ion-doped yttrium aluminum garnet (YAG) is widely used inhigh-power solid-state lasers due to excellent optical, mechanical,and thermal properties. Since 1970s, there have been manystudies on the stress-induced birefringence and depolarization inYAG slabs [2,7,8]. In 1995, Lu et al. [7] calculated the depolariza-tion loss in an ideal slab and the results indicated its dependenceon cut angle. In 2004, Chen et al. [8] pointed out the limitations ofRef. [2] and the errors in Ref. [7], and then studied [9] thedepolarization loss in slabs of finite dimensions. The resultsshowed the dependences of the loss on cut angle vary from placeto place within the gain medium. In 2007, Simmons et al. [10]corrected the errors in Ref. [8] and finally determined the correctvalue of the piezo-optic tensor.

To our knowledge, at present there is no research on theelimination of depolarization loss in zigzag slabs with differentbounce numbers. Furthermore, many papers [5,9,11] still quote

010 Published by Elsevier Ltd. All

x: +86 731 84514127.

the inaccurate piezo-optic tensor to study the stress-inducedbirefringence in YAG crystal. In this paper, we review the methodof calculating the depolarization loss and provide a process todetermine correct piezo-optic tensor. Two methods are used toconfirm the validity of the piezo-optic tensor from Ref. [10] andthat calculated by us. At first we compare the depolarization losscalculated using piezo-optic tensor with that calculated usingphotoelastic tensor. Then we compare the material parameterscalculated from compliance tensor with the values in [12]. In theend, the ability of eliminating thermal stress-induced depolariza-tion for different zigzag paths is analyzed.

2. Theory of depolarization loss

2.1. Depolarization loss

The refractive indexes of a crystal are specified by an ellipsoidas follows [13]

Bijxixj ¼ 1, i,j¼ 1,2,3 ð1Þ

where Einstein summation convention is used and B is therelative dielectric impermeability tensor defined as the inversematrix of the dielectric constant. Optically isotropic crystalsbecome anisotropic when they are subject to stress s, and Bij canbe expressed as

Bij ¼ B0,ijþpijklskl, i,j,k,l¼ 1,2,3 ð2Þ

where p is the piezo-optic tensor. If we work in terms of strains einstead of stresses s, Eq. (2) can be expressed as

Bij ¼ B0,ijþpijklekl ð3Þ

rights reserved.

zz″′

xx″′

γ

L

tβPin

Poutθ

φx

z″and zalong[111]

y″ along [121]y

x[100]

y [010]

z [001] x″ along[101]

Fig. 1. The coordinate systems used in the calculation. (a)The crystal lattice

coordinate system. (b) The Laboratory coordinate system. (c) The zigzag

coordinate system.

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629 623

B0,ij ¼dij

n0þdndT Tðx,y,zÞ�T0½ �

� �2ð4Þ

where p is photoelastic tensor, dij is the Kronecker delta function,n0 is the refractive index of YAG under temperature T0, dn/dT isthe thermo-optical coefficient, and T(x,y,z) is the temperature atpoint (x,y,z).

For convenience, the fourth-rank tensor can be described as asecond-rank tensor using Nye’s convention [13] (Appendix A).Thus, Eqs. (2) and (3) are changed to a matrix notation

Bi ¼ B0,iþpijsj, i,j¼ 1,2. . .,6 ð5Þ

and

Bi ¼ B0,iþpijej, i,j¼ 1,2. . .,6 ð6Þ

The abbreviation rules for the piezo-optic tensor p and thephotoelastic tensor p are given by

pmn ¼pijkl n¼ 1,2,3

2pijkl n¼ 4,5,6

(ð7Þ

Pmn ¼ pijkl, i,j,k,l¼ 1,2,3; m,n¼ 1,2,. . .,6 ð8Þ

Actually, the piezo-optic tensor can be calculated usingphotoelastic tensor p and compliance tensor S as follows

pmn ¼ pmlSln ð9Þ

where S is an inverse matrix of stiffness tensor C. The abbreviationrules for S and C are given by

Smn ¼

Sijkl m,n¼ 1,2,3

2Sijkl m,n¼ 4,5,6

4Sijkl m,n¼ 4,5,6

8><>: ð10Þ

Cmn ¼ Cijkl ð11Þ

It is notable that the stiffness and compliance tensors areinverse to each other, even in the reduced-suffix form. AlthoughRef. [10] pointed out that the stiffness tensor in Ref. [8] was incor-rectly inverted, no derivation and proof were provided in detail. Inorder to give a clear understanding of the errors in Ref. [8], detailedformula derivation and verification of calculating the compliancetensor are presented in Appendices B and C, respectively.

For the light propagating along the z-axis, the elliptic equationin the plane perpendicular to the z-axis becomes

½x,y�B11 B12

B21 B22

" #x

y

" #¼ 1 ð11Þ

The eigenvectors of the 2�2 matrix in Eq. (11) represent thedirections of principal axes, in which there are no depolarizationeffects. The refractive indexes n7 at these directions can becalculated by the corresponding eigenvalues l7

n7 ¼ 1=ffiffiffiffiffiffiffiffil7

qð12Þ

l7 ¼1

2B11�B227

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB11þB22Þ

2þ4B12B21

q� �ð13Þ

The depolarization loss is defined as the ratio of thedepolarized power to the initial linearly polarized power. For atop-hat-shaped beam with dimensions of w� t, we define themaximum and mean depolarization losses by

lossmax ¼maxflossðx,yÞg ð14Þ

lossmean ¼1

wt

Z w=2

�w=2

Z t=2

�t=2lossðx,yÞdxdy¼

1

NxNy

XNx ,Ny

i,j ¼ 1

lossðxi,yjÞ ð15Þ

where Nx, Ny are the numbers of sample points in x and y

directions, respectively, loss(x,y) is the depolarization loss for thelight entranced at point (x,y), given as [1]

lossðx,yÞ ¼ sin2ð2yÞsin2

ðj=2Þ ð16Þ

where y is the angle of the principal axes with respect to thelaboratory coordinate system, j is the phase difference betweenthe two principal polarizations. If light passes through the crystalwith a thickness Lg, the phase difference, expressed in radians, istherefore

j¼ 2pl

Lgðnþ�n�Þ ð17Þ

where l is the wavelength in vacuum.

2.2. Depolarization loss in zigzag slab

In practice, the calculation of depolarization loss is verycomplicated. In this section we demonstrate the basic process tocalculate the depolarization loss in zigzag slabs. Fig. 1(a) showsthe crystal lattice coordinate system where the stiffness andphotoelastic tensors are reported [12]. The piezo-optic tensor inthis orientation can be calculated with Eq. (9). Fig. 1(b) shows thelaboratory coordinate system where the slab is orientated, so thatthe thickness is in the x direction, the width is in the y direction,and the length is in the z direction. For YAG slabs, the z-axis isparallel to the [1 1 1] crystalline axis. The cut angle f defines theorientation of the material cut from the crystal boule. Tempera-ture, elastic thermal stress and strain are calculated in thisorientation by finite element analysis (FEA) with the soft programAnsys. Moreover, the piezo-optic tensor is expressed in thisorientation through coordinate transformation (Appendix C).Thus, we get the dielectric impermeability Bl in laboratorycoordinate system with Eq. (2).

Table 1 lists the nonzero components of piezo-optic tensor inlaboratory system, including the values in Refs. [7,8,10] and thatcalculated by us. Note the values we calculated are virtuallyidentical to that in Ref. [10], and the differences accrue from thedifferent values of nonzero stiffness components, which refer toRef. [14] in Ref. [10], but Ref. [12] in our calculation. Detailedcalculation process is presented in Appendix C. Table 2 lists thenonzero components of photoelastic tensors in laboratory system.

Table 1The piezo-optic coefficients of YAG in the laboratory system (Units: m2/N).

Lu et al.’s [7] Chen et al.’s [8] Simmon et al.’s [10] Ours

p11 �0.30�10�12�1.10�10�12

�3.02�10�13�3.04�10�13

p12 0.11�10�12 3.79�10�13 1.11�10�13 1.12�10�13

p13 0.17�10�12 7.07�10�13 1.72�10�13 1.74�10�13

p33 �0.36�10�12�1.43�10�12

�3.63�10�13�3.66�10�13

p44 �0.15�10�12�8.27�10�13

�2.92�10��13�2.91�10�13

p66 �0.21�10�12�1.48�10�12

�4.13�10�13�4.15�10�13

p14 �0.09�10�12cos(3f) �9.27�10�13cos(3f) �1.71�10�13cos(3f) �1.76�10�13cos(3f)

p15 �0.09�10�12sin(3f) �9.27�10�13sin(3f) �1.71�10�13sin(3f) �1.76�10�13sin(3f)

Table 2The photoelastic coefficients of YAG in the laboratory coordinate system.

p00

11 p00

12 p00

13 p00

33 p00

44 p00

66 p00

14 p00

15

�0.07 0.02 0.04 �0.09 �0.03 �0.05 �0.02cos(3f) �0.02sin(3f)

Fig. 2. Slab laser with angled end caps showing the parameters used in the

calculation (units: mm, 1).

Table 3Parameters used for modeling.

Thermal conductivity 14 W m K�1

Coefficient of thermal expansion 7.7�10�6 K�1

Young’s modulus 2.80�105 N mm�2

Poisson’s ratio 0.24

Absorption coefficient 0.69 cm�1

Refractive index 1.82

Thermo-optic coefficient 7.3�10�6 K�1

Fractional heat load 0.272

Thin film convection coefficient 10 W/cm2 K

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629624

Fig. 1(c) shows the laser beam propagating through the slab atan angle g with respect to the z-axis in the x–z plane. Thedielectric impermeability Bz in zigzag coordinate system can becalculated through coordinate transformation

Bz ¼ RðgÞBlRð�gÞ ð18Þ

RðgÞ ¼cosg 0 �sing0 1 0

sing 0 cosg

0B@

1CA ð19Þ

where R(g) is the transformation matrix from the laboratorycoordinate system to the zigzag coordinate system. Based on thesubmatrix of Bz, the refractive indexes n7 at the directions ofprincipal axes can be obtained through Eqs. (11)–(13). Along thezigzag optical path, the ray between incidence point Pin andtransmission point Pout is divided into several sections. At eachlocation, the Jones matrix is calculated to represent the thermalstress-induced polarization rotation and phase retardation. Thus,the phase difference between each polarization can be found bymultiplying the Jones matrices in order.

For slabs with y+b4p/2, shown in Fig. 1(c), the followingcondition should be satisfied [15]:

LTR¼L

t¼Nrtanb�

1

tanyð20Þ

where LTR is the slab’s length-to-thickness ratio, Nr is the bouncenumber of the light in zigzag propagation, b is the reflection angleon the total internal reflection surface, and y is the acute anglebetween the sloped edge of the angled end cap and the lengthdirection of the slab. Otherwise, the output beam will beseparated into two sections. It should be noted that Nr is an eveninteger, and the beam cannot cover the entire slab even though itcan cover the whole entrance plane for a conduction-cooled endpumped slab (CCEPS) with y¼p/4.

3. The depolarization loss in a slab with the z-axis in [1 1 1]

In the previous literature [2–3,7–9], the plane strain approx-imation is often used in the thermal stress analysis in rod and slablasers. Actually, it is no longer valid in a slab of finite dimensions.In this section, we calculate the depolarization loss in CCEPSwithout plane strain approximation.

As shown in Fig. 2, the slab consists of three parts: one dopedsection and two un-doped sections with 451 angled end caps. Thediode light propagates along the length of the slab by TIR from

both ends for efficient and uniform end-pumping. Heat depositedin the slab is given by

Q ðzÞ ¼ aZP0

wte�aðLd�zÞ þe�aðLdþ zÞ� �

ð21Þ

where P0 is the pump power at each end, a is the absorptioncoefficient at pump wavelength, Z is the ratio of pump power beingtransferred to heat, w is the width of the slab, t is the thickness ofthe slab, and Ld is the half length of the doped region. Heat isremoved from the two largest faces with an area of 50�5.6 mm2.The parameters used for modeling are listed in Table 3.

The temperature and von Mises stress distributions in the slabat P0¼1000 W are simulated and shown in Fig. 3, based on theassumption that all surfaces except the cooling areas are adiabaticand there is no constraint on the slab. The maximum temperatureand von Mises thermal stress are 80.9 1C and 74 MPa, respec-tively.

The contour plots of temperature and stress on the bondingplane are shown in Fig. 4. Fig. 4(a) shows that thermal gradientonly exists in the x direction due to the assumption of uniformheat deposition on the xy plane and cooling in x direction. Greatthermal gradient leads to great stress gradient near the edges in x

direction, which is shown in Fig. 4(b). In addition, free expansionalso leads to great stress gradient near the edges in y direction.Fig. 4(d)–(h) show the components of stress on the bonding plane.The x, y, and z components of stress are symmetric about thecenter lines x¼0.085 cm and y¼0.28 cm. The yz shear stress issymmetric about the center line x¼0.085 cm. The xz shear stressis symmetric about the center line y¼0.28 cm. The xy shear stressare symmetric about the center of the cross-section.

Fig. 3. The temperature and von Mises stress distribution in the slab. (a) Temperature

distribution (1C). (b) Von Mises stress distribution [units in CGS system].

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629 625

In order to assess how much error will be introduced using thefalse piezo-optics tensors, we calculate the depolarization lossesusing the values in Refs. [7,8] and that calculated by us, which arelisted in Table 1. Since there is no dispute about the values ofphotoelastic tensor, we can consider the results calculated usingphotoelastic tensor as the criteria. The corresponding depolariza-tion loss distributions on the output plane are shown in Fig. 5.Although all results show that the major depolarization lossesoccur near the edges in y direction, the magnitudes are quitedifferent. The predicted maximum and mean depolarizationlosses are 1.49%, 79.78%, 5.52%, 5.79%, and 0.21%, 12.34%, 0.77%,0.81%, calculated using the piezo-optics tensor of Lu et al. [7],Chen et al. [8], ours, and photoelastic tensor, respectively. It isapparent that our results are closest to the results calculatedusing photoelastic tensor. Therefore, the piezo-optics tensorcalculated by us is correct. So does the piezo-optics tensor inRef. [10]. However, the loss will be greatly overestimated usingthe parameters in Ref. [8] and underestimated using theparameters in Ref. [7].

Although a large stress gradient exists near the edges in both x

and y direction, only the effect of stress gradient in x direction can

be well averaged out by zigzag propagation. Thus, the loss mainlyoccurs near the edges along the y axis.

4. The depolarization loss as a function of bounce number andcut angle in zigzag slab

For a slab of finite dimensions, there is a wide choice of cutangles and bounce numbers [15]. Since p14 and p15 (Eq. (C.7))depend on the cut angle, so does the dielectric impermeability.Furthermore, the dielectric impermeability depends on the stressalong the optic path. Therefore, the depolarization depends onboth the cut angle and the bounce number. In this section, westudy the ability of eliminating thermal depolarization loss atdifferent cut angles and bounce numbers.

Fig. 6 shows the depolarization loss distribution at f¼601 andNr¼22. The maximum and mean depolarization losses are 10.44%and 1.05%, respectively. Compared with the case of f¼301 andNr¼22 in Fig. 5, variation in cut angle not only increases thedepolarization loss, but also changes the distribution pattern.Although the loss still mainly occurs near the edges along the y

axis, it becomes asymmetric about the center line x¼0.085 cm. Asindicated in Eqs. (C.7) and (5), variation in the cut angle changesnot only the piezo-optic tensor, but also the dielectric imperme-ability tensor due to the diverse intensity distribution of the sixstress components. Consequently, change in the depolarizationloss inevitably occurs.

Fig. 7 shows the mean depolarization loss as a function of cutangle at different bounce numbers. Since the piezo-optic tensor isa periodic function of cut angle, the figure only shows the meandepolarization loss in one period. As shown, the mean depolar-ization losses are less than 3% in Nd3 +:YAG zigzag slab, and theydepend on both the cut angle and the bounce number. Fordifferent bounce numbers, the relations between the meandepolarization losses and cut angles are quite different. The meandepolarization losses reach the minimum values at f¼301 in thecases of Nr¼12, 14, 18, and 20, whereas at f¼901 in the cases ofNr¼16, 22, and 24. Thus, the suitable cut angle is 301 or 901.

Fig. 8 shows the depolarization loss distribution at f¼301 andNr¼12. The maximum and mean depolarization losses are 7.29%and 0.71%, respectively. A little increase in loss accrues from thedecrease in bounce number, and the loss becomes more concen-trated on the four corners of the entrance plane, in comparison withthat at f¼301 and Nr¼22. This is because a change in bouncenumber alters the stress state along the optic path.

Fig. 9 shows the mean depolarization loss as a function ofbounce number at f¼301, 601, and 901. As shown, thedepolarization losses decrease overall, but fluctuate irregularlyas bounce number increases.

5. Conclusions

The cut angle and the number of bounces of light in zigzagpropagation through the slab are two important parameters for thedesign of slab lasers. Using finite element analysis, we calculated thetemperature and thermal stress in a YAG slab with the z-axisoriented along [1 1 1]. Based on these results, we numericallycalculated the stress-induced depolarization loss under different cutangles and bounce numbers using the piezo-optic tensor. Fordifferent bounce numbers, although the relations between the meandepolarization losses and cut angles are different, the meandepolarization losses are less than 3% in Nd3+:YAG slab, and thelosses occur mainly near the edges in width. For a certain cut angle,although the depolarization loss shows an overall downward trendwith irregular fluctuation as bounce number increases, a larger

Fig. 4. Contour plots of the temperature and thermal stresses on the bonding plane. (a) Temperature. (b) Von Mises stress. (c) x-component of stress. (d) y-component of

stress. (e) z-component of stress. (f) yz shear stress. (g) xz shear stress. (h) xy shear stress.

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629626

bounce number leads to relative lower loss. Therefore, using a laserbeam with adequate bounce number and aspect ratio less than 1 onthe entrance aperture would be preferable in order to reduce thedepolarization loss.

Appendix A

Nye’s reduction rule is [12]

e11 e12 e13

e21 e22 e23

e31 e32 e33

0B@

1CA¼

e11

2e6

1

2e5

1

2e6 e2

1

2e4

1

2e5

1

2e4 e3

0BBBBBB@

1CCCCCCA

ðA:1Þ

s11 s12 s13

s21 s22 s23

s31 s32 s33

0B@

1CA¼

s1 s6 s5

s6 s2 s4

s5 s4 s3

0B@

1CA ðA:2Þ

Pmn ¼ Pijkl ðm,n¼ 1,2,3,4,5, or 6, i,j,k,l¼ 1,2 or 3Þ ðA:3Þ

Appendix B. The reduction rule of compliance, stiffness, andpiezo-optic tensor

The stress tensor s and strain tensor e satisfy the relation

sij ¼ Cijklekl, eij ¼ Sijkl skl ði,j,k,l¼ 1,2,or 3Þ ðB1Þ

where C is the stiffness tensor and S is the compliance tensor.Because of symmetry, the number of independent components ofe and s is reduced to 6, and it is possible to use a 6�1 matrix todescribe e and s. Thus, we have

Cijkl ¼ Cjikl ¼ Cjilk ¼ Cijlk, Sijkl ¼ Sjikl ¼ Sjilk ¼ Sijlk ðB:2Þ

Consequently the number of independent component of C andS is reduced to 36 and it is possible to use a 6�6 matrix todescribe C and S.

According to Nye’s reduction rule, Eq. (B.1) becomes

sm ¼ Cmnen, em ¼ Smnsn ðm,n¼ 1,2,3,4,5, or 6Þ ðB:3Þ

Fig. 5. The depolarization loss distributions at f¼301 and Nr¼22. (a) the result using the piezo-optics tensor of Lu et al. [7]. (b) The result using the piezo-optics tensor of

Chen et al. [8]. (c) The result using the piezo-optics tensor of ours. (d) The result using the photoelastic tensor.

0 30 60 90 1200

0.5

1

1.5

2

2.5

3

φ [degree]

mea

n de

pola

rizat

ion

Loss

[%]

Nr=12Nr=14Nr=16Nr=18Nr=20Nr=22Nr=24

Fig. 7. Mean depolarization loss as a function of the cut angle.

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629 627

Comparing (B.1) with (B.3), we get the reduction rules of C and S

Smn ¼

Sijkl m,n¼ 1,2,3

2Sijkl m,n¼ 4,5,6

4Sijkl m,n¼ 4,5,6Cmn ¼ Cijkl

8><>: ðB:4Þ

Fig. 6. The depolarization loss distribution at f¼601 and Nr¼22.

In the crystal coordinate system, there are only 3 independentcomponents in the stiffness and compliance tensor for YAGcrystal. For simplicity, we use a, b, c, d, e, and f instead of themshown in Eq. (B.6).

Fig. 8. The depolarization loss distribution at f¼301 and Nr¼12.

8 10 12 14 16 18 20 22 240

2

4

6

8

10

12

Nr

mea

n de

pola

rizat

ion

Loss

[%]

φ=30φ=60φ=90

Fig. 9. Mean depolarization loss as a function of the bounce number.

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629628

According to Eq. (B.1), we find

sij ¼ Cijkl ekl ¼ Cijkl Sklmn smn ðB:5Þ

Substituting Eq. (B.4) into (B.5), we obtain

s11

s22

s33

s23

s31

s12

s32

s13

s21

266666666666666664

377777777777777775

¼

a b b 0 0 0 0 0 0

b a b 0 0 0 0 0 0

b b a 0 0 0 0 0 0

0 0 0 c 0 0 c 0 0

0 0 0 0 c 0 0 c 0

0 0 0 0 0 c 0 0 c

0 0 0 c 0 0 c 0 0

0 0 0 0 c 0 0 c 0

0 0 0 0 0 c 0 0 c

266666666666666664

377777777777777775

�

d e e 0 0 0 0 0 0

e d e 0 0 0 0 0 0

e e d 0 0 0 0 0 0

0 0 0 f 0 0 f 0 0

0 0 0 0 f 0 0 f 0

0 0 0 0 0 f 0 0 f

0 0 0 f 0 0 f 0 0

0 0 0 0 f 0 0 f 0

0 0 0 0 0 f 0 0 f

266666666666666664

377777777777777775

s11

s22

s33

s23

s31

s12

s32

s13

s21

266666666666666664

377777777777777775

ðB:6Þ

which implies

adþ2be¼ 1

ðaþbÞeþbd¼ 0

4cf ¼ 1

8><>: ðB:7Þ

Similarly, according to Eq. (B.3), we find

sm ¼ Cmn en ¼ Cmn Snp sp ðB:8Þ

Substituting Eq. (B.7) into (B.8), we obtain

a b b 0 0 0

b a b 0 0 0

b b a 0 0 0

0 0 0 c 0 0

0 0 0 0 c 0

0 0 0 0 0 c

2666666664

3777777775�

d e e 0 0 0

e d e 0 0 0

e e d 0 0 0

0 0 0 4f 0 0

0 0 0 0 4f 0

0 0 0 0 0 4f

26666666664

37777777775¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

2666666664

3777777775

ðB:9Þ

Note that the reduced stiffness and compliance tensors are stillinverse to each other. Therefore, the compliance tensor calculatedin Ref. [8], in which the last three diagonal elements in reducednotation are 4, is incorrect.

The unreduced-suffix matrix of the piezo-optic tensorcan be calculated from photoelastic p and compliance tensorS using

pijkl ¼ pijmnSmnkl ðB:10Þ

where p satisfies pijkl¼pjikl¼pijlk¼pjilk. According to Nye’s reduc-tion rule, p can be written as

pijkl ¼ pmn ði,j,k,l¼ 1,2,3; m,n¼ 1,2,. . .,6Þ ðB:11Þ

Thus, the piezo-optic tensor can be expressed in brief

pmn ¼ pmlSln ðB:12Þ

Substituting Eqs. (B.4) and (B.11) into (B.10) and (B.12), wefind that the pmn are related to the pijkl by

pmn ¼pijkl n¼ 1,2,3

2pijkl n¼ 4,5,6

(ðB:13Þ

or

p11 p12 p13 p14 p15 p16

p21 p22 p23 p24 p25 p26

p31 p32 p33 p34 p35 p36

1

2p14

1

2p24

1

2p34 p44 p45 p46

1

2p15

1

2p25

1

2p35 p45 p55 p56

1

2p16

1

2p26

1

2p36 p46 p56 p66

2666666666666664

3777777777777775

ðB:14Þ

Note pijapji.

Appendix C. The calculation of piezo-optic tensor inlaboratory coordinates system

In crystal coordinate system, the nonzero elements of photo-elastic tensor of YAG are [12]

pu11 pu12 pu12

pu12 p0

11 p0

12

pu12 pu12 pu11

pu44

pu44

pu44

0BBBBBBBBB@

1CCCCCCCCCA

ðC:1Þ

where

Pu11 ¼�0:029, Pu12 ¼ 0:0091, Pu44 ¼�0:0615 ðC:2Þ

Similarly, the nonzero values of stiffness of YAG in the crystalcoordinate system are [12]

L. Liu et al. / Optics & Laser Technology 43 (2011) 622–629 629

Cu11 ¼ 3:49� 1011N=m2

Cu12 ¼ 1:21� 1011N=m2

Cu44 ¼ 1:14� 1011N=m2 ðC:3Þ

which are quoted from Ref. [16], whereas in Refs. [8,10] they are

Cu11 ¼ 3:33� 1011N=m2

Cu12 ¼ 1:11� 1011N=m2

Cu44 ¼ 1:15� 1011N=m2 ðC:4Þ

which are quoted from Ref. [14]. Thus, the compliance coefficientsin crystal coordinate system may be calculated with Eq. (C.3),giving

Su11 ¼ 3:49�10�12m2=N

Su12 ¼�8:98�10�13m2=N

Su44 ¼ 8:77�10�12m2=N ðC:5Þ

Substituting Eqs. (C.2), (C.5) into (B.12), we get the piezo-opticcoefficients in crystal coordinate system

pu11 ¼�1:18�10�13m2=N

pu12 ¼ 4:96�10�14m2=N

pu44 ¼�5:40�10�13m2=N ðC:6Þ

The crystal coordinate system is transformed to laboratorysystem through a matrix

A¼

1ffiffiffi2p cosf�

1ffiffiffi6p sinf

2ffiffiffi6p sinf �

1ffiffiffi2p cosf�

1ffiffiffi6p sinf

�1ffiffiffi2p sinf�

1ffiffiffi6p cosf

2ffiffiffi6p cosf

1ffiffiffi2p sinf�

1ffiffiffi6p cosf

1ffiffiffi3p

1ffiffiffi3p

1ffiffiffi3p

0BBBBBBB@

1CCCCCCCAðC:6Þ

where f is the cut angle defined in Fig. 1(b).Through the coordinate transformation, we get the piezo-optic

tensor in laboratory system

p11 ¼1

2pu11þ

1

2pu12þ

1

2pu44

p12 ¼1

6pu11þ

5

6pu12�

1

6pu44

p13 ¼1

3pu11þ

2

3pu12�

1

3pu44

p33 ¼1

3pu11þ

2

3pu12þ

2

3pu44

p44 ¼2

3pu11�

2

3pu12þ

1

3pu44

p66 ¼1

3pu11�

1

3pu12þ

2

3pu44

p14 ¼ 2�cosð3fÞ

3ffiffiffi2p ð�pu11þpu12þppu44Þ

p15 ¼ 2�sinð3fÞ

3ffiffiffi2p ð�pu11þpu12þpu44Þ ðC:7Þ

Table 1 lists the values we calculated and that in Refs. [7,8,10].Note the values we calculated are virtually identical to that in

Ref. [10].

According to the orthotropic material’s stress–strain relation,it is easy to get the relation among Young’s modulus, shearmodulus, Poisson’s ratio, and compliance tensor

E1 ¼1

S11, E2 ¼

1

S22, E3 ¼

1

S33

G23 ¼1

S44, G13 ¼

1

S55, G12 ¼

1

S66u21

E2¼

u12

E1¼�S12,

u23

E2¼

u32

E3¼�S23,

u13

E1¼

u31

E3¼�S13 ðC:8Þ

Substituting Eq. (C.4) into (C.8), we have

E¼ 286:7GPa, G¼ 114GPa, u¼ 0:257 ðC:9Þ

This is consistent with the data given in the authoritativebook [12]

E¼ 280GPa, G¼ 113GPa, u¼ 0:24 ðC:10Þ

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