Abstract

A hologram generally is formed in a photorefractive crystal by the interaction of the grating formulation that is due to the space-charge field that grows from the beam interference pattern and of the beam energy exchange that results from two-beam coupling in the grating. Analytic expressions for the grating recording formed by this interaction can be obtained only by simultaneous solution of the material equations and the coupled-wave equations. Using Moharam’s space-charge field function, we deduce both the exact steady-state analytic solution for two-beam coupling under any boundary-light modulation and constant-light excitation and the approximate analytic solutions with distance-dependent light excitation. Based on Kukhtarev’s idea of three-step calculation, the formulation of a hologram under any light modulation and excitation is further solved analytically. These steady-state analytic solutions understanding are useful in arriving at a clearer and more exact understanding of photorefractive two-beam coupling and holographic recording.

© 2000 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  2. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
    [CrossRef]
  3. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications. I (Springer-Verlag, Berlin, 1988), Chaps. 2, 4, and 6.
  4. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications. II (Springer-Verlag, Berlin, 1988), Chaps. 2–4.
  5. M. Jeganathan and L. Hesselink, “Diffraction from thermally fixed gratings in a photorefractive medium: steady state and transient analysis,” J. Opt. Soc. Am. B 11, 1791–1799 (1994).
    [CrossRef]
  6. B. Liu, L. Liu, and L. Xu, “Characteristics of recording and thermal fixing in lithium niobate,” Appl. Opt. 37, 2170–2176 (1998).
    [CrossRef]
  7. J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formulation in photorefractive crystals,” Opt. Acta 332, 397–408 (1985).
    [CrossRef]
  8. L. B. Au and L. Solymar, “Transients in photorefractive two-wave mixing: a numerical study,” Appl. Phys. B 49, 339–342 (1989).
    [CrossRef]
  9. J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
    [CrossRef]
  10. M. Jeganathan, M. C. Bashaw, and L. Hesselink, “Evolution and propagation of grating envelopes during erasure in bulk photorefractive media,” J. Opt. Soc. Am. B 12, 1370–1383 (1995).
    [CrossRef]
  11. P. Buchhave, “Computer simulation of multiple dynamic photorefractive gratings,” J. Opt. Soc. Am. B 15, 1865–1870 (1998).
    [CrossRef]
  12. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [CrossRef]
  13. H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), Chap. 3.
  14. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  15. M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
    [CrossRef]
  16. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
    [CrossRef]
  17. Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
    [CrossRef]
  18. L. Liu and X. Liu, “Matrixing coupled wave theory of photorefractive hologram recorded by two-beam coupling,” J. Mod. Opt. 40, 2257–2265 (1993).
    [CrossRef]
  19. M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
    [CrossRef]

1998 (2)

1997 (1)

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

1996 (1)

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

1995 (1)

1994 (2)

M. Jeganathan and L. Hesselink, “Diffraction from thermally fixed gratings in a photorefractive medium: steady state and transient analysis,” J. Opt. Soc. Am. B 11, 1791–1799 (1994).
[CrossRef]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

1993 (1)

L. Liu and X. Liu, “Matrixing coupled wave theory of photorefractive hologram recorded by two-beam coupling,” J. Mod. Opt. 40, 2257–2265 (1993).
[CrossRef]

1992 (1)

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

1989 (2)

L. B. Au and L. Solymar, “Transients in photorefractive two-wave mixing: a numerical study,” Appl. Phys. B 49, 339–342 (1989).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

1985 (1)

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formulation in photorefractive crystals,” Opt. Acta 332, 397–408 (1985).
[CrossRef]

1979 (4)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Au, L. B.

L. B. Au and L. Solymar, “Transients in photorefractive two-wave mixing: a numerical study,” Appl. Phys. B 49, 339–342 (1989).
[CrossRef]

Bashaw, M. C.

Belic, M. R.

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

Bledowski, A.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

Buchhave, P.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Heaton, J. M.

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formulation in photorefractive crystals,” Opt. Acta 332, 397–408 (1985).
[CrossRef]

Hesselink, L.

Jaric, M. V.

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

Jeganathan, M.

Jeong, J. S.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Kim, H. S.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

Kim, J. C.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

Kim, Y. H.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Kukhtarev, V.

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Kwak, C. H.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Lee, E. H.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Lee, Y. H.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

Liu, B.

Liu, L.

B. Liu, L. Liu, and L. Xu, “Characteristics of recording and thermal fixing in lithium niobate,” Appl. Opt. 37, 2170–2176 (1998).
[CrossRef]

L. Liu and X. Liu, “Matrixing coupled wave theory of photorefractive hologram recorded by two-beam coupling,” J. Mod. Opt. 40, 2257–2265 (1993).
[CrossRef]

Liu, X.

L. Liu and X. Liu, “Matrixing coupled wave theory of photorefractive hologram recorded by two-beam coupling,” J. Mod. Opt. 40, 2257–2265 (1993).
[CrossRef]

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Markov, V. B.

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Meyer, M.

M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Munser, R.

M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Otten, J.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

Park, S. Y.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Petrovic, M.

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

Ringhofer, K. H.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

Rupp, R. A.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

Solymar, L.

L. B. Au and L. Solymar, “Transients in photorefractive two-wave mixing: a numerical study,” Appl. Phys. B 49, 339–342 (1989).
[CrossRef]

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formulation in photorefractive crystals,” Opt. Acta 332, 397–408 (1985).
[CrossRef]

Soskin, M. S.

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Suh, H. H.

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Timotijevic, D.

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Wurfel, P.

M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Xu, L.

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

L. B. Au and L. Solymar, “Transients in photorefractive two-wave mixing: a numerical study,” Appl. Phys. B 49, 339–342 (1989).
[CrossRef]

Comput. Phys. Commun. (1)

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, “Dynamic holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187–191 (1992).
[CrossRef]

Ferroelectrics (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystal. II. Beam coupling-light amplification,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Appl. Phys. (1)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electronic transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

J. Mod. Opt. (1)

L. Liu and X. Liu, “Matrixing coupled wave theory of photorefractive hologram recorded by two-beam coupling,” J. Mod. Opt. 40, 2257–2265 (1993).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Acta (1)

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formulation in photorefractive crystals,” Opt. Acta 332, 397–408 (1985).
[CrossRef]

Opt. Commun. (3)

M. R. Belic, D. Timotijevic, M. Petrovic, and M. V. Jaric, “Exact solution to photorefractive two-wave mixing with arbitrary modulation depth,” Opt. Commun. 123, 201–206 (1996).
[CrossRef]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Y. H. Lee, Y. H. Kim, J. C. Kim, H. S. Kim, H. H. Suh, and E. H. Lee, “Photorefractive grating formation in the region of large intensity modulation,” Opt. Commun. 144, 70–74 (1997).
[CrossRef]

Phys. Status Solidi A (1)

M. Meyer, P. Wurfel, and R. Munser, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Other (3)

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), Chap. 3.

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications. I (Springer-Verlag, Berlin, 1988), Chaps. 2, 4, and 6.

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications. II (Springer-Verlag, Berlin, 1988), Chaps. 2–4.

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Figures (6)

Fig. 1
Fig. 1

Arrangement for observation of photorefractive holographic recording and two-beam coupling.

Fig. 2
Fig. 2

Intensity modulation m(z) as a function of the equivalent propagation distance Γz with various light-excitation efficiencies p: (a) for I20I10 and m0=0.8, (b) for I20>I10 and m0=0.5. Heavy solid curves represent z-dependent light-excitation efficiency.

Fig. 3
Fig. 3

(a) Definition of Γz(0.5)a for I20I10; (b) definitions of Γz(0.5)b and Γz(1) for I20>I10.

Fig. 4
Fig. 4

Shifting ratio as a function of boundary intensity modulation m0 and light-excitation efficiency p for (a) I20I10 and (b) I20>I10.

Fig. 5
Fig. 5

Constants in reciprocal approximation: (a) ca for I20I10 and (b) cb for I20>I10.

Fig. 6
Fig. 6

Geometry of the kth thin layer divided from a crystal slab.

Tables (1)

Tables Icon

Table 1 Analytic Intensity Modulations Owing to Two-Beam Coupling

Equations (69)

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Ej=Ij(z) exp[ωt-ik·r-iφj (z)],
I(z)=I0(z){1+m(z)cos[Kx-φ(z)]},
Esc=ED sin[Kx-φ(z)-ϕ0]1+p(z)m(z)cos[Kx-φ(z)-ϕ0]-VL-EV×1-[1-p(z)2m(z)2]1/21+p(z)m(z)cos[Kx-φ(z)-ϕ0],
p(z)=τg0τg0+nD,
g0=NDsI0(z)τ,nD=βτND,τ=1γRNA.
Esc=Ee f(pm)cos[Kx-φ(z)-ϕ0],
f(pm)=1-(1-p2m2)1/2pm,
tan ϕ0=EDV/L-EV,
Ee=2[(V/L-EV)2+ED2]1/2.
dI1dz=12Γf(pm)I1I2-αcos θI1,
dI2dz=-12Γf(pm)I1I2-αcos θI2,
dφ1dz=-12Γf(pm)I2I1,
dφ2dz=-12Γf(pm)I1I2,
Γ=2πλn1 sin ϕ0cos θ,Γ=πλn1 cos ϕ0cos θ,
n1=n03γeff,
d(I1+I2)dz=-αcos θ,
I(z)=I0 exp(-αz/cos θ).
I1(z)=½I0 exp(-αz/cos θ){1±[1-m(z)2]1/2},
I2(z)=½I0 exp(-αz/cos θ){1-(±)[1-m(z)2]1/2},
dmdz=±12Γ f (pm)1-m2,
dφ(z)dm=2ΓΓ1m.
φ(z)=-2ΓΓlnmm0.
p(z)=sI0 exp(-αz/cos θ)sI0 exp(-αz/cos θ)+β.
ma(z)=1-1-(1-1-m02)×exp-12Γz21/2.
mb(z)=1-1-(1-1-m02)exp12Γz21/2.
Γz(1)(P=1)=2 ln 11-1-m02.
mc(z)=1-1-exp-12Γ[z-z(1)]21/2.
m=[1-tan2(u)cot2(ξ)]1/2,
Γz=-2 lntan(π/4+u/2)tan(π/4+u0/2)×sin[(ξ-u)/2]sin[(ξ-u0)/2]sin[(ξ+u0)/2]sin[(ξ+u)/2]1/p,
u0=arctanp1-p21-m02.
Γz=2 lntan(π/4+u/2)tan(π/4+u0/2)×sin[(ξ-u)/2]sin[(ξ-u0)/2]sin[(ξ+u0)/2]sin[(ξ+u)/2]1/p.
Γz(1)=2 ln1tan(π/4+u0/2)×sin[(ξ+u0)/2]sin[(ξ-u0)/2]1/p.
Γz=-2 lntan(π/4+u/2)×sin[(ξ-u)/2]sin[(ξ+u)/2]1/p+Γz(1).
m(z)=2m0[exp(¼Γz)(1+1-m02)+exp(-¼Γz)×(1-1-m02)]-1,I20I10,
m(z)=2m0[exp(¼Γz)(1-1-m02)+exp(-¼Γz)×(1+1-m02)]-1,I20>I10.
Γz(0.5)a=2 ln 1-1-m021-1-(m0/2)2,
Γz(0.5)b=-2 ln[(1-1-m02)(1-1-0.52)].
u(0.5)a=arctan p1-p21-(m0/2)2,
u(0.5)b=arctan p1-p21-0.52.
log[ΔΓz j(p)]=log[Γz(0.5)j(p)]-log[Γz(0.5)j(p=1)],
ma(z)=(1-{1-(1-1-m02)×exp[-Γz/2ΔΓza(p)]}2)1/2.
mb(z)=(1-{1-(1-1-m02)×exp[Γz/2ΔΓzb(p)]}2)1/2.
Γz(1)=2ΔΓzb ln 11-1-m02,
mc(z)=(1-{1-exp[Γ(z-z(1))/2ΔΓzb]}2)1/2.
ΔΓz j(p)c j/p.
ma(z)=1-1-(1-1-m02)×sI0+βsI0 exp(-αz/cos θ)+β-Γ cos θ2aca21/2,
mb(z)=1-1-(1-1-m02)×sI0+βsI0 exp(-αz/cos θ)+βΓ cos θ2acb21/2,
z(1)=-cos θαln1+βsI02I0I02cbαΓ cos θ-βsI0.
mc(z)
=1-1-sI0 exp(-αz(1)/cos θ)+βsI0 exp(-αz/cos θ)+β-Γ cos θ2αcb21/2.
η=sin πn1 f (L)2λ cos θ2,
f(L)=0L M(z)dz,
M(z)=1-[1-p(z)2m(z)2]1/2p(z)m(z).
pmdm1-m2[1-1-(pm)2]=±Γ2dz.
ln(1-1-m2)=±Γ2z+c,
m(z)=1-1-exp±Γ2z-c21/2,
tan(u)=tan(ξ)1-m2,
tan(ξ)=p1-p2.
pmdm1-m2[1-1-(pm)2]
=1cos(u)-1cos(u)-1-p2du,
lntan(π/4+u/2)sin[(ξ-u)/2]sin[(ξ+u)/2]1/p=±Γ2z+c.
m(z)=2exp±Γz4+c+exp±Γz4-c-1.
I1(k+1)I2(k+1)=exp(-αΔz/cos θ)A11(k)A12(k)A21(k)A22(k)I1(k)I2(k),
11-exp[-p(zk)ΓΔz/2c]0exp[-p(zk)ΓΔz/2c],I2(k)<I1(k),
exp[p(zk)ΓΔz/2c]01-exp[p(zk)ΓΔz/2c]1,I2(k)>I1(k),
l=k0 exp(-αΔz/cos θ)11-exp[-p(zl)ΓΔz/2ca]0exp[-p(zl)ΓΔz/2ca]
=exp-αl=0K Δz/cos θ×11-exp-l=0K p(zl)ΔzΓ/2ca0exp-l=0K p(zl)ΔzΓ/2ca.
i=1Kp(zi)Δz=0z p(z)dz=cos θαln sI0+βsI0 exp(-αz/cos θ)+β.
I1(z)I2(z)=exp(-αz/cos θ)×11-sI0+βsI0 exp(-αz/cos θ)+β-Γ cos θ2aca0sI0+βsI0 exp(-αz/cos θ)+β-Γ cos θ2aca×I10I20.

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