Abstract

We develop an analytic model to describe the dynamic average thermal distortion and phase difference between the two principal polarizations in side-pumped Nd:YAG and Nd:glass heat capacity rod lasers. It can be predicted that the average thermal distortion is proportional to the temperature profile on the cross section from the analytic expression and, therefore, it is feasible to measure the temperature profile by wavefront sensing. In addition, temperature-dependent variation of the refractive index constitutes the major contribution of the thermal lensing for Nd:YAG rod lasers. Temperature- and stress-dependent variation of the refractive index constitute the major contributions of the thermal lensing for Nd:glass rod lasers. In the case of the same pumping and cooling conditions, there are the same orders of depolarization loss for Nd-doped YAG, LG-680, LG-750, LG-760, and LG-770 glass rod lasers.

© 2010 Optical Society of America

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References

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  1. W. Koechner, Solid-State Laser Engineering (Springer, 1999).
  2. W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9, 2548–2553 (1970).
    [CrossRef] [PubMed]
  3. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
    [CrossRef]
  4. H. Injeyan and C. S. Hoefer, “End pumped zig-zag slab laser gain medium,” U.S. patent 6,094,297 (25 July 2000).
  5. S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.
  6. C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
    [CrossRef]
  7. A. C. Erlandson and G. F. Albrecht, “Model predicting the temperature dependence of the gain coefficient and the extractable stored energy density in Nd Phosphate glass lasers,” J. Opt. Soc. Am. B 9, 214–222 (1992).
    [CrossRef]
  8. A. Rapaport, S. Zhao, G. Xiao, A. Howard, and M. Bass, “Temperature dependence of the 1.06μm stimulated emission cross section of neodymium in YAG and in GSGG,” Appl. Opt. 41, 7052–7057 (2002).
    [CrossRef] [PubMed]
  9. W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
    [CrossRef]
  10. G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2738 (1967).
    [CrossRef]
  11. X. Wang, X. Xu, Q. Lu, and E. Xi, “Shack–Hartmann wavefront sensor measurement for dynamic temperature profiles in heat capacity laser rods,” Appl. Opt. 46, 2963–2968(2007).
    [CrossRef] [PubMed]
  12. W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).
  13. W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
    [CrossRef]
  14. S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).
  15. J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1992).
  16. Q. LÜ, U. Wittrock, and S. Dong, “Photoelastic effects in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95–101 (1995).
    [CrossRef]
  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.

2009 (1)

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
[CrossRef]

2007 (2)

X. Wang, X. Xu, Q. Lu, and E. Xi, “Shack–Hartmann wavefront sensor measurement for dynamic temperature profiles in heat capacity laser rods,” Appl. Opt. 46, 2963–2968(2007).
[CrossRef] [PubMed]

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).

2002 (1)

2000 (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

1995 (2)

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

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

1992 (1)

1973 (1)

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

1970 (1)

1967 (1)

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2738 (1967).
[CrossRef]

Albrecht, G. F.

Baldwin, G. D.

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2738 (1967).
[CrossRef]

Bass, M.

Burt, W.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Campbell, B. E.

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

Chui, F.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Contag, K.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Conway, G.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Dong, S.

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

Dulaney, J. L.

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

Epstein, H. M.

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

Erlandson, A. C.

Giesen, A.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Goodier, J. N.

S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).

Harkenrider, J.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Hoefer, C. S.

H. Injeyan and C. S. Hoefer, “End pumped zig-zag slab laser gain medium,” U.S. patent 6,094,297 (25 July 2000).

Hoffmaster, D.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Howard, A.

Hügel, H.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Injeyan, H.

H. Injeyan and C. S. Hoefer, “End pumped zig-zag slab laser gain medium,” U.S. patent 6,094,297 (25 July 2000).

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Koechner, W.

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9, 2548–2553 (1970).
[CrossRef] [PubMed]

W. Koechner, Solid-State Laser Engineering (Springer, 1999).

Larionov, M.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Long, W.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Lu, Q.

LÜ, Q.

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

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1992).

Palese, S.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Qisheng, L.

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
[CrossRef]

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).

Rapaport, A.

Riedel, E. P.

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2738 (1967).
[CrossRef]

Stewen, C.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Tapos, F.

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

Timoshenko, S.

S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).

Walters, C. T.

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

Wang, X.

Wittrock, U.

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

Xi, E.

Xiao, G.

Xiaobo, W.

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
[CrossRef]

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).

Xiaojun, X.

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
[CrossRef]

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).

Xu, X.

Zhao, S.

Appl. Opt. (3)

Chin. J. Lasers (1)

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Effect of thermally induced change of stimulated emission cross section in heat capacity lasers,” Chin. J. Lasers 36, 43–46 (2009).
[CrossRef]

High Power Laser Part. Beams (1)

W. Xiaobo, X. Xiaojun, and L. Qisheng, “Dynamic temperature profiles in heat capacity laser rods,” High Power Laser Part. Beams 19, 589–592 (2007).

IEEE J. Quantum Electron. (1)

C. T. Walters, J. L. Dulaney, B. E. Campbell, and H. M. Epstein, “Nd-glass burst laser with kW average power output,” IEEE J. Quantum Electron. 31, 293–300 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1KW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

J. Appl. Phys. (2)

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2738 (1967).
[CrossRef]

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Laser Technol. (1)

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

Other (7)

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

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

W. Koechner, Solid-State Laser Engineering (Springer, 1999).

S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951).

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1992).

H. Injeyan and C. S. Hoefer, “End pumped zig-zag slab laser gain medium,” U.S. patent 6,094,297 (25 July 2000).

S. Palese, J. Harkenrider, W. Long, F. Chui, D. Hoffmaster, W. Burt, H. Injeyan, G. Conway, and F. Tapos, “High brightness, end-pumped, conduction cooled Nd:YAG zig-zag slab laser architecture,” in Advanced Solid State Lasers, C.Marshall, ed., Vol. 50of OSA Trends Optics Photonics (Optical Society of America, 2001), pp. 41–46.

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Figures (6)

Fig. 1
Fig. 1

Cylindrical coordinate system for a YAG rod.

Fig. 2
Fig. 2

Dynamic temperature distribution during the (a) pumping and (b) cooling process.

Fig. 3
Fig. 3

Dynamic OPD distribution during the pumping and cooling process: (a) OPD due to the temperature-dependent variation of the refractive index, (b) average OPD due to the stress-dependent variation of the refractive index, and (c) OPD due to the temperature- 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.

Fig. 4
Fig. 4

Dynamic OPD distribution between the two principal polarizations in (a) Nd:glass and (b) Nd:YAG rod lasers during the pumping process.

Fig. 5
Fig. 5

Transmission profiles through parallel polarizers for Nd:YAG rod from a top-hat pump profile.

Fig. 6
Fig. 6

Dynamic depolarization loss in Nd:glass and Nd:YAG rod lasers during the pumping process.

Tables (1)

Tables Icon

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

Equations (35)

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T ( r , t ) = 2 Q V c γ ( 1 + g / 2 ) n = 1 exp ( β n 2 t / τ ) ( 1 + g ) β n J 1 ( β n ) 2 g J 2 ( β n ) ( A 2 + β n 2 ) J 0 2 ( β n ) × 1 exp ( M β n 2 t p / τ ) 1 exp ( β n 2 t p / τ ) × J 0 ( β n r / r 0 ) ,
σ r = α E 1 v ( 1 r 0 2 0 r 0 T r d r 1 r 2 0 r T r d r ) ,
σ θ = α E 1 v ( 1 r 0 2 0 r 0 T r d r + 1 r 2 0 r T r d r T ) ,
σ z = σ r + σ θ .
σ r ( r , t ) = 2 α E 1 v Q V c γ ( 1 + g / 2 ) n = 1 { exp ( β n 2 t / τ ) × ( 1 + g ) β n J 1 ( β n ) 2 g J 2 ( β n ) ( A 2 + β n 2 ) J 0 2 ( β n ) × 1 exp ( M β n 2 t p / τ ) 1 exp ( β n 2 t p / τ ) × ( J 1 ( β n ) β n r 0 J 1 ( r β n / r 0 ) r β n ) } ,
σ θ ( r , t ) = 2 α E 1 v Q V c γ ( 1 + g / 2 ) n = 1 { exp ( β n 2 t / τ ) × ( 1 + g ) β n J 1 ( β n ) 2 g J 2 ( β n ) ( A 2 + β n 2 ) J 0 2 ( β n ) × 1 exp ( M β n 2 t p / τ ) 1 exp ( β n 2 t p / τ ) × ( J 1 ( β n ) β n + r 0 J 1 ( r β n / r 0 ) r β n J 0 ( β n r / r 0 ) ) } ,
σ z ( r , t ) = 2 α E 1 v Q V c γ ( 1 + g / 2 ) n = 1 { exp ( β n 2 t / τ ) × ( 1 + g ) β n J 1 ( β n ) 2 g J 2 ( β n ) ( A 2 + β n 2 ) J 0 2 ( β n ) × 1 exp ( M β n 2 t p / τ ) 1 exp ( β n 2 t p / τ ) × ( 2 J 1 ( β n ) β n J 0 ( β n r / r 0 ) ) } ,
OPD ( r , t ) = L [ Δ n T ( r , t ) + Δ n σ ( r , t ) ] ,
Δ n T ( r , t ) = d n d T Δ T ( r , t )
B i j x i x j = 1 ,
B i j = B 0 , i j + p i j k l ε k l ,
B 0 , i j = δ i j { n 0 + d n d T [ T ( r , θ , z ) T 0 ] } 2 ,
ε r = 1 E { σ r v [ σ θ + σ z ] } ,
ε θ = 1 E { σ θ v [ σ r + σ z ] } ,
ε z = 1 E { σ z v [ σ θ + σ r ] } .
( p 11 p 12 p 12 p 12 p 11 p 12 p 12 p 12 p 11 p 44 p 44 p 44 ) .
( p 11 p 12 p 13 p 14 p 15 0 p 12 p 11 p 13 p 14 p 15 0 p 13 p 13 p 33 0 0 0 p 14 p 14 0 p 44 0 p 15 p 15 p 15 0 0 p 44 p 14 0 0 0 p 15 p 14 p 66 ) .
p 11 = 1 2 p 11 + 1 2 p 12 + p 44 , p 12 = 1 6 p 11 + 5 6 p 12 1 3 p 44 , p 13 = 1 3 p 11 + 2 3 p 12 2 3 p 44 , p 33 = 1 3 p 11 + 2 3 p 12 + 4 3 p 44 , p 44 = 1 3 p 11 1 3 p 12 + 1 3 p 44 , p 66 = 1 6 p 11 1 6 p 12 + 2 3 p 44 , p 14 = cos ( 3 θ ) 3 2 ( p 11 + p 12 + 2 p 44 ) , p 15 = sin ( 3 θ ) 3 2 ( p 11 + p 12 + 2 p 44 ) ,
( B r B θ B z B θ z B z r B r θ ) = ( 1 / n 0 2 + p 11 ε r + p 12 ε θ + p 13 ε z 1 / n 0 2 + p 12 ε r + p 11 ε θ + p 13 ε z 1 / n 0 2 + p 13 ε r + p 13 ε θ + p 33 ε z 0 0 0 ) .
[ r , θ ] [ B r B r θ B r θ B θ ] [ r θ ] = 1.
n ± = 1 / λ ± ,
λ ± = 1 2 [ ( B r + B θ ) ± ( B r B θ ) 2 + 4 B r θ 2 ] .
n avg = ( n + + n ) / 2.
Δ n σ = n 3 σ z 6 E [ 2 p 11 4 p 12 + p 44 + ( 4 p 11 + 8 p 12 + p 44 ) v ] .
σ z ( r , t ) = f ( M , t ) α E 1 v T ( r , t ) ,
OPD ( r , t ) = L [ d n d T + α n 3 [ 2 p 11 ( 1 2 v ) + 4 p 12 ( 1 2 v ) p 44 ( 1 + v ) ] 6 ( 1 v ) ] Δ T ( r , t ) ,
OPD ( r , t ) = L [ d n d T + α n 3 [ p 11 ( 1 3 v ) + p 12 ( 3 5 v ) ] 4 ( 1 v ) ] Δ T ( r , t ) .
f = r 0 2 / 2 OPD
loss = sin 2 ( 2 φ ) sin 2 ( δ / 2 ) ,
δ = 2 π λ L ( n + n ) .
δ = 2 π λ n 0 3 ( p 11 p 12 + 4 p 44 ) ( 1 + v ) ( σ r σ θ ) 6 E L .
δ ( r , t ) = 2 π L λ α n 3 ( p 11 p 12 + 4 p 44 ) ( 1 + v ) 3 c γ ( 1 v ) n = 1 G ( M , β n ) exp β n 2 t / τ J 2 ( r β n / r 0 ) ,
G ( M , β n ) = Q V ( 1 + g / 2 ) ( 1 + g ) β n J 1 ( β n ) 2 g J 2 ( β n ) ( A 2 + β n 2 ) J 0 2 ( β n ) × [ 1 exp ( M β n 2 t p / τ ) 1 exp ( β n 2 t p / τ ) ] .
δ ( r , t ) = 2 π L λ α n 3 ( p 11 p 12 ) ( 1 + v ) c γ ( 1 v ) n = 1 G ( M , β n ) exp β n 2 t / τ J 2 ( r β n / r 0 ) .
OPD ( r , t ) = L [ d n d T + α ( n 1 ) ] Δ T ( r , t ) ,

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