Abstract

Non-radially symmetric residual birefringence in laser rods due to pump edge effects is analyzed both theoretically and experimentally. For cubic crystals such as yttrium aluminum garnet (YAG), this depolarization has a unique sixfold symmetry that is closely connected to the crystal’s cubic symmetry. While this depolarization is small compared to that observed with linear or circular polarizations, it is measurable and important when using radial or azimuthal polarizations in rods generating heat powers in excess of 70 W/cm. A simple criterion was defined in order to help estimate the amount of depolarization in a high-power laser rod system.

© 2009 Optical Society of America

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References

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  1. J. D. Foster and L. M. Osterink, “Thermal effects in Nd:YAG laser,” J. Appl. Phys. 41, 3656-3663 (1970).
    [CrossRef]
  2. W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557-566 (1970).
    [CrossRef]
  3. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807-809 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
    [CrossRef]
  6. J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
    [CrossRef]
  7. M. S. Roth, V. Romano, T. Feurer, and T. Graf, “Self-compensating amplifier design for cw and Q-switched high-power Nd:YAG lasers,” Opt. Express 14, 2191-2196 (2006).
    [CrossRef] [PubMed]
  8. M. Sovizi and R. Massudi, “Study of thermal effects, considering birefringence, on phase distortion of beam in a side pumped Nd:YAG rod using BEM,” Opt. Commun. 275, 206-212 (2007).
    [CrossRef]
  9. Z. Li, X. Huai, L. Wang, and Y. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34, 187-189 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. Y. Lumer, I. Moshe, Z. Horovitz, S. Jackel, G. Machavariani, and A. Meir, “Thermally induced birefringence in nonsymmetrically pumped laser rods and its implications for attainment of good beam quality in high-power, radially polarized lasers,” Appl. Opt. 47, 3886-3891 (2008).
    [CrossRef] [PubMed]
  14. R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
    [CrossRef]
  15. W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. 61, 758-766 (1971).
    [CrossRef]
  16. I. Mukhin, O. Palashov, and E. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express 17, 5496-5501 (2009).
    [CrossRef] [PubMed]
  17. Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
    [CrossRef]
  18. D. R. Lovett, Tensor Properties of Crystals (IOP, 1989).
  19. M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. II, Chap. 33.
  20. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (Dover, 1988).
  21. V. Parfenov, V. Shashkin, and E. Stepanov, “Numerical investigation of thermally induced birefringence in optical elements of solid-state lasers,” Appl. Opt. 32, 5243-5255 (1993).
    [CrossRef] [PubMed]
  22. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468-1470 (2007).
    [CrossRef] [PubMed]
  23. R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
    [CrossRef]

2009 (2)

2008 (3)

Y. Lumer, I. Moshe, Z. Horovitz, S. Jackel, G. Machavariani, and A. Meir, “Thermally induced birefringence in nonsymmetrically pumped laser rods and its implications for attainment of good beam quality in high-power, radially polarized lasers,” Appl. Opt. 47, 3886-3891 (2008).
[CrossRef] [PubMed]

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

2007 (5)

2006 (2)

M. S. Roth, V. Romano, T. Feurer, and T. Graf, “Self-compensating amplifier design for cw and Q-switched high-power Nd:YAG lasers,” Opt. Express 14, 2191-2196 (2006).
[CrossRef] [PubMed]

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

2005 (1)

2003 (1)

2002 (1)

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

1995 (1)

Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

1993 (1)

1971 (1)

1970 (2)

J. D. Foster and L. M. Osterink, “Thermal effects in Nd:YAG laser,” J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557-566 (1970).
[CrossRef]

Bass, M.

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. II, Chap. 33.

Boley, B. A.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (Dover, 1988).

Bonnefois, A. Montmerie

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Bourderionnet, J.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Brignon, A.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Chen, W.

Dong, S.

Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Feldman, R.

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

Feurer, T.

Foster, J. D.

J. D. Foster and L. M. Osterink, “Thermal effects in Nd:YAG laser,” J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

Frey, R.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Gan, A.

Gilbert, M.

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Golan, Y.

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

Graf, T.

Horovitz, Z.

Huai, X.

Huignard, J. -P.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Jackel, S.

Khazanov, E.

Koechner, W.

W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. 61, 758-766 (1971).
[CrossRef]

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557-566 (1970).
[CrossRef]

Lebiush, E.

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

Leibush, E.

Li, L.

Li, Z.

Lovett, D. R.

D. R. Lovett, Tensor Properties of Crystals (IOP, 1989).

Lü, Q.

Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Lumer, Y.

Machavariani, G.

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

Massudi, R.

M. Sovizi and R. Massudi, “Study of thermal effects, considering birefringence, on phase distortion of beam in a side pumped Nd:YAG rod using BEM,” Opt. Commun. 275, 206-212 (2007).
[CrossRef]

Meir, A.

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

Moshe, I.

Mukhin, I.

Osterink, L. M.

J. D. Foster and L. M. Osterink, “Thermal effects in Nd:YAG laser,” J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

Paiken, Y.

Palashov, O.

Parfenov, V.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

Ramírez-Sánchez, V.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

Rice, D. K.

W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. 61, 758-766 (1971).
[CrossRef]

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557-566 (1970).
[CrossRef]

Romano, V.

Roth, M. S.

Shashkin, V.

Shi, P.

Shimony, Y.

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

Sovizi, M.

M. Sovizi and R. Massudi, “Study of thermal effects, considering birefringence, on phase distortion of beam in a side pumped Nd:YAG rod using BEM,” Opt. Commun. 275, 206-212 (2007).
[CrossRef]

Stepanov, E.

Tao, Y.

Thro, P. -Y.

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Wang, L.

Weiner, J. H.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (Dover, 1988).

Weulersse, J. -M.

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Wittrock, U.

Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557-566 (1970).
[CrossRef]

J. Appl. Phys. (1)

J. D. Foster and L. M. Osterink, “Thermal effects in Nd:YAG laser,” J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

J. Phys. Chem. Solids (1)

R. Feldman, Y. Shimony, E. Lebiush, and Y. Golan, “Effect of hot acid etching on the mechanical strength of ground YAG laser elements,” J. Phys. Chem. Solids 69, 839-846 (2008).
[CrossRef]

Opt. Commun. (4)

A. Montmerie Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

M. Sovizi and R. Massudi, “Study of thermal effects, considering birefringence, on phase distortion of beam in a side pumped Nd:YAG rod using BEM,” Opt. Commun. 275, 206-212 (2007).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976-1980 (2008).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effect in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95-101 (1995).
[CrossRef]

Opt. Lett. (4)

Other (3)

D. R. Lovett, Tensor Properties of Crystals (IOP, 1989).

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. II, Chap. 33.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (Dover, 1988).

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Figures (11)

Fig. 1
Fig. 1

(a) Symmetry of [111] oriented cubic crystal; (b) projection to the X Y plane.

Fig. 2
Fig. 2

Simulated rod piece. Pumped section (dark) is 3 cm long. Unpumped edges (light) are 1 cm long each.

Fig. 3
Fig. 3

Thermal and mechanical simulation results for pumping with unpumped edges. (a) Temperature; (b) normal elastic strain; (c) shearing strain. (d) Shearing strain for end pump (right side) and with unpumped edges (left side) for comparison. Maximum values of normal and shearing strains are of the same order-of-magnitude.

Fig. 4
Fig. 4

Shearing strain term ε r z at cross section of pump edge.

Fig. 5
Fig. 5

Radial depolarization term Δ B r φ at cross section of pump edge.

Fig. 6
Fig. 6

(a) Map of deviation angle from azimuthal polarization, (b) orthogonal (radial) portion of the beam, analogous to output intensity using crossed polarizers.

Fig. 7
Fig. 7

Depolarization from radial/azimuthal polarization versus effective depolarization heat load.

Fig. 8
Fig. 8

Depolarization from radial/azimuthal polarization versus length of smooth pump edge. Rod parameters are the following: 1 cm diameter, 100 W/cm heat load at rod center.

Fig. 9
Fig. 9

Experimental results versus heat power (a) map of deviation angle from azimuthal polarization, (b) orthogonal (radial) portion of the beam, analogous to output intensity using crossed polarizers.

Fig. 10
Fig. 10

Experimental and simulation results of output polarization after high-power laser rod versus heat power.

Fig. 11
Fig. 11

Depolarization versus rod length; heat power density maintained constant.

Equations (30)

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ε = ( ε r r ε r φ ε r z ε r φ ε φ φ ε φ z ε r z ε φ z ε z z ) .
B i j = B 0 , i j + Δ B i j = B 0 , i j + P i j k l ε k l ,
Δ B r r = 1 6 ( ε r ( 3 P 11 + 3 P 12 + 6 P 44 ) + ε φ ( P 11 + 5 P 12 2 P 44 ) + ε z ( 2 P 11 + 4 P 12 4 P 44 ) ) 2 3 ( P 11 P 12 2 P 44 ) ( cos ( 3 φ ) ε φ z + sin ( 3 φ ) ε r z ) ,
Δ B φ φ = 1 6 ( ε r ( P 11 + 5 P 12 2 P 44 ) + ε φ ( 3 P 11 + 3 P 12 + 6 P 44 ) + ε z ( 2 P 11 + 4 P 12 4 P 44 ) ) + 2 3 ( P 11 P 12 2 P 44 ) ( cos ( 3 φ ) ε φ z + sin ( 3 φ ) ε r z ) ,
Δ B r φ = 2 3 ( P 11 P 12 2 P 44 ) ( sin ( 3 φ ) ε φ z cos ( 3 φ ) ε r z ) + 1 3 ( P 11 P 12 + 4 P 44 ) ε r φ .
σ x x σ y y = f ( r ) ( x 2 y 2 ) ,     σ x y = f ( r ) x y ,
ε y z = 2 6 ( s 11 s 12 + 1 2 s 44 ) ( σ x x σ y y ) ,
ε y z = 2 3 ( s 11 s 12 + 1 2 s 44 ) σ x y ,
ε r z = cos ( φ ) ε x z + sin ( φ ) ε y z sin ( 3 φ ) ,
ε φ z = sin ( φ ) ε x z + cos ( φ ) ε y z cos ( 3 φ ) .
depol = f ( P h D rod / L rod ) ,
P ̃ D = P h D rod / L rod ,
( ε x x α x x Δ T ε y y α y y Δ T ε z z α z z Δ T 2 ε y z α y z Δ T 2 ε x z α x z Δ T 2 ε x y α x y Δ T ) = ( s 11 s 12 s 13 s 14 s 15 s 16 s 21 s 22 s 23 s 24 s 25 s 26 s 31 s 32 s 33 s 34 s 35 s 36 s 41 s 42 s 43 s 44 s 45 s 46 s 51 s 52 s 53 s 54 s 55 s 56 s 61 s 62 s 63 s 64 s 65 s 66 ) ( σ x x σ y y σ z z σ y z σ x z σ x y ) .
ε ̃ = D σ ̃ + α ̃ Δ T ,
D = ( s 11 s 12 s 16 s 21 s 22 s 26 s 61 s 62 s 66 ) ( s 13 s 14 s 15 s 23 s 24 s 25 s 63 s 64 s 65 ) ( s 33 s 34 s 35 s 43 s 44 s 45 s 53 s 54 s 55 ) 1 ( s 31 s 32 s 36 s 41 s 42 s 46 s 51 s 52 s 56 ) ,
α ̃ = ( α x x α y y α x y ) ( s 13 s 14 s 15 s 23 s 24 s 25 s 63 s 64 s 65 ) ( s 33 s 34 s 35 s 43 s 44 s 45 s 53 s 54 s 55 ) 1 ( α z z α y z α x z ) .
( ε x x α x x Δ T ε y y α y y Δ T ε z z α z z Δ T 2 ε y z α y z Δ T 2 ε x z α x z Δ T 2 ε x y α x y Δ T ) = ( s 11 s 12 s 12 0 0 0 s 12 s 11 s 12 0 0 0 s 12 s 12 s 11 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 ) ( σ x x σ y y σ z z σ y z σ x z σ x y ) .
ε i j = s i j k l σ k l .
s i j k l = R i m R j n R k p R l q s m n p q ,
R = ( 1 2 0 1 2 1 6 2 6 1 6 1 3 1 3 1 3 ) .
s 11 = 1 2 s 11 + 1 2 s 12 + 1 4 s 44 ,     s 12 = 1 6 s 11 + 5 6 s 12 1 12 s 44 ,
s 13 = 1 3 s 11 + 2 3 s 12 1 6 s 44 ,     s 14 = 2 3 ( s 11 s 12 + 1 2 s 44 ) ,
s 33 = 1 3 s 11 + 2 3 s 12 + s 44 ,     s 44 = 4 3 ( s 11 s 12 + 1 4 s 44 ) ,
s 66 = 2 3 ( s 11 s 12 + s 44 ) ,
s 11 = s 22 ,     s 13 = s 23 ,     s 44 = s 55 ,     s 14 = s 24 = 1 2 s 56 ,
s 15 = s 16 = s 25 = s 26 = s 34 = s 35 = s 36 = s 45 = s 46 = 0.
D = ( s 11 s 13 2 s 33 s 14 2 s 44 s 12 s 13 2 s 33 + s 14 2 s 44 0 s 12 s 13 2 s 33 + s 14 2 s 44 s 11 s 13 2 s 33 s 14 2 s 44 0 0 0 s 66 4 s 14 2 s 44 ) .
D 33 = 2 ( D 11 D 12 ) ,
2 ε x x y 2 + 2 ε y y x 2 2 2 ε x y x y = 0 ,
( s 11 s 13 2 s 33 s 14 2 s 44 ) 2 ( σ x x + σ y y ) + α ( 1 s 13 s 33 ) 2 T = 0.

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