Abstract

Calculations and experimental measurements of the thermally induced strain and birefringence are presented for a diode-pumped Nd:YAG rod that is encapsulated in a prismatic pump light collector. A numerical model is developed to determine the spatiotemporal stress-induced strain distribution across the prism, index-matching fixant, and laser rod, and the birefringence that arises from the stress-induced strain within the laser rod. Calculations of the birefringence are compared with polarscopic measurements and display good agreement. Support for the rod on all sides is provided by the prism and fixant, and the distribution and degree of the stress-induced strain (and birefringence) within the laser rod are therefore influenced by the geometry and composition of the prism and fixant. These strains are thermomechanical in origin and are primarily a function of the elastic modulus of the fixant and the temperature of the system. Such stress-induced strains are additional to those strains that are produced from temperature gradients across the laser rod and result from the laser rod being constrained from expanding. Collectors utilizing index-matching fluid as the encapsulant display the smallest measure of birefringence relating to the temperature gradients in the rod. However, for collectors utilizing solid fixants (with significant elastic modulus), an increase in the birefringence results. In this case collector designs that have the laser rod located in a symmetrically shaped prism are effective in reducing the nonuniform pressures on the sides of the rod and therefore the birefringence. 1996 Optical Society of America r

© 1996 Optical Society of America

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References

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  1. L. R. Marshall, A. Kaz, R. L. Burnham, “Highly efficient TEM00 operation of a transversely diode-pumped Nd:YAG laser,” Opt. Lett. 17, 186–188 (1992).
  2. R. J. Koshel, I. A. Walmsley, “Modeling of the gain distribution for diode pumping of a solid-state laser rod with nonimaging optics,” Appl. Opt. 32, 1517–1527 (1993).
  3. J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.
  4. S. D. Jackson, J. A. Piper, “Theoretical modelling of a diode-pumped Nd:YAG laser with a solid nonfocusing pump light collector,” Appl. Opt. 33, 2273–2283 (1994).
  5. S. D. Jackson, J. A. Piper, “Thermal modelling of solid nonfocussing pump light collectors used for diode-pumped Nd:YAG lasers,” Appl. Opt. 34, 2012–2023 (1995).
  6. J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).
  7. J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.
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  9. F. W. Quelle, “Thermal distortion of diffraction-limited optical elements,” Appl. Opt. 5, 633–637 (1966).
  10. W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt. 9, 1429–1434 (1970).
  11. J. D. Forster, L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
  12. W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarised YAG:Nd lasers,” IEEE J. Quantum Electron. QE-6, 557–566 (1970).
  13. S. Timoshenko, “Plane stress and plane strain,” in Theory of Elasticity (McGraw-Hill, New York, 1934), pp. 12–26.
  14. V. Parfenov, V. Shashkin, E. Stepanov, “Numerical investigation of thermally induced birefringence in optical elements of solid-state lasers,” Appl. Opt. 32, 5243–5255 (1993).
  15. Specification sheet data, Epoxy Technology Inc., 14 Fortune Drive, Billerica, Mass. 01821.
  16. C. A. Harper, ed., Handbook of Plastics and Elastomers (McGraw-Hill, New York, 1975), Chap. 3, p. 56.
  17. Specification sheet data, Dow Corning Corp., Midland, Mich. 48640.
  18. Specification sheet data, Scott Garsco Pty. Ltd., P.O. Box 174, Terrey Hills, N.S.W. 2084.
  19. S. S. Ballard, J. S. Browder, “Thermal properties,” in Volume IVOptical Materials Part 2: Properties, M. J. Weber, ed., CRC Handbook of Laser Science and Technology (CRC Press, Boca RatonFla., 1986), pp. 51–54.
  20. G. Sewell, “PDE2D: easy-to-use software for general two-dimensional partial differential equations,” Adv. Eng. Software 17, 105–112 (1993).
  21. J. F. Nye, “Natural and artificial double refraction. Second-order effects,” in Physical Properties of Crystals (Clarendon, Oxford, 1985), pp. 235–259.
  22. W. Koechner, “Heat removal,” in Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), pp. 350–401.
  23. W. G. Bickley, “The distribution of stress round a circular hole in a plate,” Philos. Trans. R. Soc. London Ser. A 227, 383–415 (1928).
  24. G. B. Jeffery, “Plane stress and plain strain in bipolar co-ordinates,” Philos. Trans. R. Soc. London Ser. A 221, 265–293 (1920).
  25. S. Timoshenko, Theory of Elasticity (McGraw-Hill, New York, 1934), Chap. 3, p. 55.
  26. C. A. Harper, ed., Handbook of Plastics and Elastomers (McGraw-Hill, New York, 1975), Chap. 1, p. 63.

1995 (2)

J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).

S. D. Jackson, J. A. Piper, “Thermal modelling of solid nonfocussing pump light collectors used for diode-pumped Nd:YAG lasers,” Appl. Opt. 34, 2012–2023 (1995).

1994 (1)

1993 (3)

1992 (1)

1971 (1)

1970 (3)

W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt. 9, 1429–1434 (1970).

J. D. Forster, L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarised YAG:Nd lasers,” IEEE J. Quantum Electron. QE-6, 557–566 (1970).

1966 (1)

1928 (1)

W. G. Bickley, “The distribution of stress round a circular hole in a plate,” Philos. Trans. R. Soc. London Ser. A 227, 383–415 (1928).

1920 (1)

G. B. Jeffery, “Plane stress and plain strain in bipolar co-ordinates,” Philos. Trans. R. Soc. London Ser. A 221, 265–293 (1920).

Ballard, S. S.

S. S. Ballard, J. S. Browder, “Thermal properties,” in Volume IVOptical Materials Part 2: Properties, M. J. Weber, ed., CRC Handbook of Laser Science and Technology (CRC Press, Boca RatonFla., 1986), pp. 51–54.

Bickley, W. G.

W. G. Bickley, “The distribution of stress round a circular hole in a plate,” Philos. Trans. R. Soc. London Ser. A 227, 383–415 (1928).

Browder, J. S.

S. S. Ballard, J. S. Browder, “Thermal properties,” in Volume IVOptical Materials Part 2: Properties, M. J. Weber, ed., CRC Handbook of Laser Science and Technology (CRC Press, Boca RatonFla., 1986), pp. 51–54.

Burnham, R. L.

Cai, Y.

J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

Dawes, J. M.

J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

Dekker, P.

J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

Forster, J. D.

J. D. Forster, L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).

Jackson, S. D.

S. D. Jackson, J. A. Piper, “Thermal modelling of solid nonfocussing pump light collectors used for diode-pumped Nd:YAG lasers,” Appl. Opt. 34, 2012–2023 (1995).

S. D. Jackson, J. A. Piper, “Theoretical modelling of a diode-pumped Nd:YAG laser with a solid nonfocusing pump light collector,” Appl. Opt. 33, 2273–2283 (1994).

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

Jeffery, G. B.

G. B. Jeffery, “Plane stress and plain strain in bipolar co-ordinates,” Philos. Trans. R. Soc. London Ser. A 221, 265–293 (1920).

Kaz, A.

Knowles, D. S.

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

Koechner, W.

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

W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt. 9, 1429–1434 (1970).

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarised YAG:Nd lasers,” IEEE J. Quantum Electron. QE-6, 557–566 (1970).

W. Koechner, “Heat removal,” in Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), pp. 350–401.

Koshel, R. J.

Marshall, L. R.

Nye, J. F.

J. F. Nye, “Natural and artificial double refraction. Second-order effects,” in Physical Properties of Crystals (Clarendon, Oxford, 1985), pp. 235–259.

Osterink, L. M.

J. D. Forster, L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).

Parfenov, V.

Piper, J. A.

Piper, J.A.

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

Quelle, F. W.

Rice, D. K.

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

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarised YAG:Nd lasers,” IEEE J. Quantum Electron. QE-6, 557–566 (1970).

Sewell, G.

G. Sewell, “PDE2D: easy-to-use software for general two-dimensional partial differential equations,” Adv. Eng. Software 17, 105–112 (1993).

Shashkin, V.

Stepanov, E.

Timoshenko, S.

S. Timoshenko, “Plane stress and plane strain,” in Theory of Elasticity (McGraw-Hill, New York, 1934), pp. 12–26.

S. Timoshenko, Theory of Elasticity (McGraw-Hill, New York, 1934), Chap. 3, p. 55.

Walmsley, I. A.

Adv. Eng. Software (1)

G. Sewell, “PDE2D: easy-to-use software for general two-dimensional partial differential equations,” Adv. Eng. Software 17, 105–112 (1993).

Appl. Opt. (6)

IEEE J. Quantum Electron. (1)

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarised YAG:Nd lasers,” IEEE J. Quantum Electron. QE-6, 557–566 (1970).

J. Appl. Phys. (1)

J. D. Forster, L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

J. M. Dawes, P. Dekker, Y. Cai, “Q-switching of a diode-pumped Nd:YAG laser with low uniform gain characteristic,” Opt. Commun. 115, 617–625 (1995).

Opt. Lett. (1)

Philos. Trans. R. Soc. London Ser. A (2)

W. G. Bickley, “The distribution of stress round a circular hole in a plate,” Philos. Trans. R. Soc. London Ser. A 227, 383–415 (1928).

G. B. Jeffery, “Plane stress and plain strain in bipolar co-ordinates,” Philos. Trans. R. Soc. London Ser. A 221, 265–293 (1920).

Other (12)

S. Timoshenko, Theory of Elasticity (McGraw-Hill, New York, 1934), Chap. 3, p. 55.

C. A. Harper, ed., Handbook of Plastics and Elastomers (McGraw-Hill, New York, 1975), Chap. 1, p. 63.

J. M. Dawes, P. Dekker, Y. Cai, D. S. Knowles, S. D. Jackson, J.A. Piper, “Q-switched diode-pumped Nd:YAG lasers,” in Advanced Solid-State Lasers, A. A. Pinto, T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), p.50.

Specification sheet data, Epoxy Technology Inc., 14 Fortune Drive, Billerica, Mass. 01821.

C. A. Harper, ed., Handbook of Plastics and Elastomers (McGraw-Hill, New York, 1975), Chap. 3, p. 56.

Specification sheet data, Dow Corning Corp., Midland, Mich. 48640.

Specification sheet data, Scott Garsco Pty. Ltd., P.O. Box 174, Terrey Hills, N.S.W. 2084.

S. S. Ballard, J. S. Browder, “Thermal properties,” in Volume IVOptical Materials Part 2: Properties, M. J. Weber, ed., CRC Handbook of Laser Science and Technology (CRC Press, Boca RatonFla., 1986), pp. 51–54.

J. M. Dawes, S. D. Jackson, Y. Cai, P. Dekker, J.A. Piper, “Diode-pumped Nd:YAG lasers using solid nonfocussing collector geometry,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1992), p. 219.

J. F. Nye, “Natural and artificial double refraction. Second-order effects,” in Physical Properties of Crystals (Clarendon, Oxford, 1985), pp. 235–259.

W. Koechner, “Heat removal,” in Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), pp. 350–401.

S. Timoshenko, “Plane stress and plane strain,” in Theory of Elasticity (McGraw-Hill, New York, 1934), pp. 12–26.

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Figures (12)

Fig. 1
Fig. 1

Schematic of the single-sided pumping arrangement with the Cartesian coordinate system used for the numerical modeling.

Fig. 2
Fig. 2

Calculated three-dimensional plots of (a) the radial component of the stress-induced (S-I) strain within the prism, excluding the fixant and rod; (b) the tangential component of the stress-induced strain within the prism, excluding the fixant and rod and viewed from the opposite side of (a); (c) the strain gradient within the prism, excluding the fixant and rod.

Fig. 3
Fig. 3

Calculated three-dimensional plots of (a) the radial component of the stress-induced (S-I) strain within the fixant, (b) the tangential component of the stress-induced strain within the fixant, (c) the strain gradient within the fixant.

Fig. 4
Fig. 4

Calculated topographical plots of (a) the radial component of the stress-induced strain within the rod, (b) the tangential component of the stress-induced strain within the rod, (c) the strain gradient within the rod. The values on the contours represent the values for stress-induced strain multiplied by 10−6.

Fig. 5
Fig. 5

Calculations of A d for a number of collector combinations as a function of time from diode switch on. Calculations (solid curves) and experimental measurements (circles and triangles are for BK-7 and sapphire prisms, respectively) of A d as a function of time from diode switch on are for the fluid–BK-7 and for the fluid–sapphire collector combinations. Included in the inset are the calculations of A d for silicone–BK-7 and silicone–sapphire collector combinations.

Fig. 6
Fig. 6

Calculated contour plots of (a) the transmitted intensity for the fluid–BK-7 collector combination, (b) the steady-state ΔOPL distribution across the laser rod for the fluid–BK-7 collector combination, (c) the transmitted intensity for the epoxy–BK-7 collector combination. Note that the plane of the polarizer is oriented with the x axis.

Fig. 7
Fig. 7

Schematic of the polarscopic experiment: TM, A, PM, LPF, and CR represent the turning mirror, aperture, power meter, low-pass filter, and chart recorder, respectively.

Fig. 8
Fig. 8

A d d (a) calculations (solid curve) and experimental measurements (circles) from diode switch on for the silicone–BK-7 collector combination and (b) experimental measurements for an epoxy–BK-7 collector combination from diode switch on.

Fig. 9
Fig. 9

Calculations of the radial component of the stressinduced (S-I) strain (solid curves), the tangential component of the stress-induced strain (long-dashed curves), and strain gradient (short-dashed curves) for variation in the elastic modulus of (a) the fixant material, (b) the prism material.

Fig. 10
Fig. 10

Calculations of the steady-state value for A d as a function of the elastic modulus of the fixant material for collectors made up of both BK-7 and sapphire prisms. Note that the other material parameters for the fixant are consistent with the epoxy material.

Fig. 11
Fig. 11

Calculated topographical plots (for the right-rectangular prism with a square cross section) of (a) the radial component and (b) the tangential component of the stress-induced strain within the prism, (c) the strain gradient within the prism. Note that the values on the contours represent the values for stress-induced strain multiplied by 10−6.

Fig. 12
Fig. 12

Calculated topographical plots (for the right-rectangular prism with a square cross section) of (a) the radial component and (b) the tangential component of the stress-induced strain within the rod, (c) the strain gradient within the rod. Note that the values on the contours represent the values for stress-induced strain multiplied by 10−6.

Tables (1)

Tables Icon

Table 1 Mechanical Constants

Equations (30)

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σ x x x + σ x y y = 0 ,
σ x y x + σ y y y = 0 ,
ε x x = U x α ( x , y ) [ T ( x , y ) T 0 ] ,
ε y y = V y α ( x , y ) [ T ( x , y ) T 0 ] ,
ε x y = 1 2 ( V x + U y ) ,
σ x x = E ( x , y ) [ 1 + υ ( x , y ) ] [ 1 2 υ ( x , y ) ]          × { [ 1 υ ( x , y ) ] ε x x + υ ( x , y ) ε y y } ,
σ y y = E ( x , y ) [ 1 + υ ( x , y ) ] [ 1 2 υ ( x , y ) ]           × { υ ( x , y ) ε x x + [ 1 υ ( x , y ) ] ε y y } ,
σ x y = E ( x , y ) 2 [ 1 + υ ( x , y ) ] ε x y ,
ε x x = 1 + υ ( x , y ) E ( x , y )          × { [ 1 υ ( x , y ) ] σ x x υ ( x , y ) σ y y } ,
ε y y = 1 + υ ( x , y ) E ( x , y )          × { [ 1 υ ( x , y ) ] σ y y υ ( x , y ) σ x x } ,
ε x y = 2 [ 1 + υ ( x , y ) ] E ( x , y ) σ x y .
σ x x n x + σ x y n y = 0 ,
σ x y n x + σ y y n y = 0 ,
ε r = ε x x   cos 2   θ+ε y y   sin 2   θ+ε x y   cos   θ sin θ,
ε θ =   cos  θ  U +   sin  θ V x 2 + y 2        + ε x x   sin 2   θ +  ε y y   cos 2   θ -  ε x y   sin  θ cos θ,
B i j x i x j = 1 ,
B i i = 1 n 0 2 + Δ B i i ,
B i j = Δ B i j ,
B x , y = B 0 + 1 2 ( Δ B x x + Δ B y y )           ± 1 2 [ ( Δ B x x Δ B y y ) 2 + 4 Δ B x y 2 ] 1 2 .
ΔOPL = ( B x B y ) ( L λ ) ,
A d = S I ( x , y )   sin 2    2 [ θ  ( x , y ) γ ]   sin 2 [ δ ( x , y ) / 2 ] d S S I ( x , y ) d S ,
ε ( x , y ) = [ ε x x ε x y 0 ε x y ε y y 0 0 0 0 ] ,
ε ( x , y ) = U 1 ε ( x , y ) U ,
U = [   cos  α  cos  β   sin  α  cos  β sin  β sin  α   cos  α 0   sin  β  cos  α   sin  α  sin  β   cos  β ] .
ε n ( x , y ) = [ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] ,
Δ B m ( x , y ) = p m n ( x , y ) ε n ( x , y ) ,
p m n ( x , y ) = [ p 11 p 12 p 12 0 0 0 p 12 p 11 p 12 0 0 0 p 12 p 12 p 11 0 0 0 0 0 0 p 44 0 0 0 0 0 0 p 44 0 0 0 0 0 0 p 44 ] ,
p 11 = 0.0290 , p 12 = + 0.0091 , p 44 = 0.0615.
Δ B ( x , y ) = U Δ B ( x , y ) U 1 ,
Δ B ( x , y , z ) = [ Δ B x x Δ B x y Δ B x z Δ B x y Δ B y y Δ B y z Δ B x z Δ B y z Δ B z z ] .

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