Abstract

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack–Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.

© 2005 Optical Society of America

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2003

2002

2000

1999

1998

1997

1995

1994

N. C. Mehta, C. W. Allen, “Dynamic compensation of atmospheric turbulence with far-field optimization,” J. Opt. Soc. Am. A 11, 434–443 (1994).
[CrossRef]

Feature issue, R. Benedict, J. Breckinridge, D. Fried, eds., “Atmospheric-Compensation Technology,” J. Opt. Soc. Am. A 11 (1994).

1993

1991

V. V. Vorob’ev, “Thermal blooming of laser beams in atmosphere,” Prog. Quantum Electron 15(1–2), 1–152 (1991).
[CrossRef]

J. F. Holms, “Enhancement of backscattered intensity for a bistatic lidar operating in atmospheric turbulence,” Appl. Opt. 30, 2643–2646 (1991).
[CrossRef]

1987

A. A. Vasil’ev, M. A. Vorontsov, I. A. Kudryashov, V. I. Shmalhauzen, “Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions,” Sov. J. Quantum Electron. 17, 1106–1107 (1987).
[CrossRef]

1985

P. E. Wolf, G. Maret, “Weak localization and coherence backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

1984

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz’minskii, V. I. Shmalhauzen, “Speckle-effects in adaptive optical systems,” Sov. J. Quantum Electron. 14, 761–766 (1984).
[CrossRef]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831 (1984).
[CrossRef]

1982

Yu. A. Kravtsov, A. I. Saichev, “Effect of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp., 494–508 (1982).
[CrossRef]

1981

V. V. Kolosov, A. V. Kuzikovskii, “On phase compensation for refractive distortions of partially coherent beams,” Sov. J. Quantum Electron. 8, 490–494 (1981).

1978

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977

1970

V. V. Vorob’ev, “Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index,” Radiophys. Quantum Electron. 13, 1053–1060 (1970).

1966

V. V. Tamoikin, A. A. Fraiman, “Statistical properties of field scattered by rough surface,” Radiophys. Quantum Electron. 11, 56–74 (1966).

Aksenov, V.

Allen, C. W.

Banakh, V.

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Dedham, Mass., 1987).

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, Vol. XXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1991), pp. 65–197.

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1980).

Billman, K. W.

K. W. Billman, “Multi-beam illuminator laser,” U.S. patent5,734,504 (1998).

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Carhart, G. W.

Cauwenberghs, G.

Cohen, M.

Davidson, F. M.

DuChateau, P.

P. DuChateau, D. Zachmann, Applied Partial Differential Equations (Dover, Mineola, N.Y., 2002).

Ellerbroeck, B.

Ellerbroek, B. L.

Fraiman, A. A.

V. V. Tamoikin, A. A. Fraiman, “Statistical properties of field scattered by rough surface,” Radiophys. Quantum Electron. 11, 56–74 (1966).

Fried, D. L.

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1980).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Gracheva, M. E.

M. E. Gracheva, A. Gurvich, C. C. Kashkarov, V. V. Pokasov, “Scaling relationships and their experimental verification under strong intensity scintillations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1979).

Gurvich, A.

M. E. Gracheva, A. Gurvich, C. C. Kashkarov, V. V. Pokasov, “Scaling relationships and their experimental verification under strong intensity scintillations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1979).

Hammond, J.

J. Hammond, “Scintillation smoothing of pulsed laser beams,” (1984).

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).

Holmes, R. B.

Holms, J. F.

Ishimaru, A.

Karnaukhov, V. N.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz’minskii, V. I. Shmalhauzen, “Speckle-effects in adaptive optical systems,” Sov. J. Quantum Electron. 14, 761–766 (1984).
[CrossRef]

Kashkarov, C. C.

M. E. Gracheva, A. Gurvich, C. C. Kashkarov, V. V. Pokasov, “Scaling relationships and their experimental verification under strong intensity scintillations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1979).

Koivunen, A. C.

Kokorowski, S. A.

Kolosov, V. V.

V. V. Kolosov, A. V. Kuzikovskii, “On phase compensation for refractive distortions of partially coherent beams,” Sov. J. Quantum Electron. 8, 490–494 (1981).

V. V. Kolosov, R. P. Ratowsky, A. A. Zemlyanov, R. A. London, “X-ray laser coherence in the presence of density fluctuations,” in Hard X-Ray/Gamma-Ray and Neutron Optics, Sensors, and Applications, R. B. Hoover, F. P. Doty, eds., Proc. SPIE2859, 269–280 (1996).
[CrossRef]

Koriabin, A. V.

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controllable Optical Systems (Nauka, Moscow, 1988).

Kravtsov, Yu. A.

Yu. A. Kravtsov, “New effects in wave propagation and scattering in random media (a mini review),” Appl. Opt. 32, 2681–2691 (1993).
[CrossRef] [PubMed]

Yu. A. Kravtsov, A. I. Saichev, “Effect of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp., 494–508 (1982).
[CrossRef]

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, Vol. XXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1991), pp. 65–197.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Kudryashov, I. A.

A. A. Vasil’ev, M. A. Vorontsov, I. A. Kudryashov, V. I. Shmalhauzen, “Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions,” Sov. J. Quantum Electron. 17, 1106–1107 (1987).
[CrossRef]

Kuga, Y.

Kuz’minskii, A. L.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz’minskii, V. I. Shmalhauzen, “Speckle-effects in adaptive optical systems,” Sov. J. Quantum Electron. 14, 761–766 (1984).
[CrossRef]

Kuzikovskii, A. V.

V. V. Kolosov, A. V. Kuzikovskii, “On phase compensation for refractive distortions of partially coherent beams,” Sov. J. Quantum Electron. 8, 490–494 (1981).

LeBigot, E. O.

London, R. A.

V. V. Kolosov, R. P. Ratowsky, A. A. Zemlyanov, R. A. London, “X-ray laser coherence in the presence of density fluctuations,” in Hard X-Ray/Gamma-Ray and Neutron Optics, Sensors, and Applications, R. B. Hoover, F. P. Doty, eds., Proc. SPIE2859, 269–280 (1996).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Maradudin, A. A.

A. V. Shchegrov, A. A. Maradudin, E. R. Méndez, “Multiple scattering of light from randomly rough surfaces,” in Progress in Optics, Vol. XLVI (2000), pp. 117–241.

Maret, G.

P. E. Wolf, G. Maret, “Weak localization and coherence backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

McKechnie, T. S.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 4.

Mehta, N. C.

Méndez, E. R.

A. V. Shchegrov, A. A. Maradudin, E. R. Méndez, “Multiple scattering of light from randomly rough surfaces,” in Progress in Optics, Vol. XLVI (2000), pp. 117–241.

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Dedham, Mass., 1987).

Northcott, M.

O’Meara, T. R.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Ozrin, V. D.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, Vol. XXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1991), pp. 65–197.

Pearson, J. E.

Pedinoff, M. E.

Pennington, T. L.

Pokasov, V. V.

M. E. Gracheva, A. Gurvich, C. C. Kashkarov, V. V. Pokasov, “Scaling relationships and their experimental verification under strong intensity scintillations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1979).

Ratowsky, R. P.

V. V. Kolosov, R. P. Ratowsky, A. A. Zemlyanov, R. A. London, “X-ray laser coherence in the presence of density fluctuations,” in Hard X-Ray/Gamma-Ray and Neutron Optics, Sensors, and Applications, R. B. Hoover, F. P. Doty, eds., Proc. SPIE2859, 269–280 (1996).
[CrossRef]

Ricklin, J. C.

Rigaut, F.

Roggemann, M. C.

Rousset, G.

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 91–130.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Saichev, A. I.

Yu. A. Kravtsov, A. I. Saichev, “Effect of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp., 494–508 (1982).
[CrossRef]

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, Vol. XXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1991), pp. 65–197.

Schell, A. C.

A. C. Schell, The Multiple Plate Antenna (Ph.D dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1961).

Shchegrov, A. V.

A. V. Shchegrov, A. A. Maradudin, E. R. Méndez, “Multiple scattering of light from randomly rough surfaces,” in Progress in Optics, Vol. XLVI (2000), pp. 117–241.

Shmalhauzen, V. I.

A. A. Vasil’ev, M. A. Vorontsov, I. A. Kudryashov, V. I. Shmalhauzen, “Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions,” Sov. J. Quantum Electron. 17, 1106–1107 (1987).
[CrossRef]

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz’minskii, V. I. Shmalhauzen, “Speckle-effects in adaptive optical systems,” Sov. J. Quantum Electron. 14, 761–766 (1984).
[CrossRef]

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controllable Optical Systems (Nauka, Moscow, 1988).

Tamoikin, V. V.

V. V. Tamoikin, A. A. Fraiman, “Statistical properties of field scattered by rough surface,” Radiophys. Quantum Electron. 11, 56–74 (1966).

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Tikhomirova, O.

Vasil’ev, A. A.

A. A. Vasil’ev, M. A. Vorontsov, I. A. Kudryashov, V. I. Shmalhauzen, “Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions,” Sov. J. Quantum Electron. 17, 1106–1107 (1987).
[CrossRef]

Vorob’ev, V. V.

V. V. Vorob’ev, “Thermal blooming of laser beams in atmosphere,” Prog. Quantum Electron 15(1–2), 1–152 (1991).
[CrossRef]

V. V. Vorob’ev, “Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index,” Radiophys. Quantum Electron. 13, 1053–1060 (1970).

Vorontsov, M. A.

M. A. Vorontsov, G. W. Carhart, “Adaptive phase distortion correction in strong speckle-modulation conditions,” Opt. Lett. 27, 2155–2157 (2002).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

A. A. Vasil’ev, M. A. Vorontsov, I. A. Kudryashov, V. I. Shmalhauzen, “Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions,” Sov. J. Quantum Electron. 17, 1106–1107 (1987).
[CrossRef]

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz’minskii, V. I. Shmalhauzen, “Speckle-effects in adaptive optical systems,” Sov. J. Quantum Electron. 14, 761–766 (1984).
[CrossRef]

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controllable Optical Systems (Nauka, Moscow, 1988).

Welsh, B. M.

Wheelon, A.

A. Wheelon, Electromagnetic Scintillation. I. Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001).

Wild, W. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Wolf, P. E.

P. E. Wolf, G. Maret, “Weak localization and coherence backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Zachmann, D.

P. DuChateau, D. Zachmann, Applied Partial Differential Equations (Dover, Mineola, N.Y., 2002).

Zemlyanov, A. A.

V. V. Kolosov, R. P. Ratowsky, A. A. Zemlyanov, R. A. London, “X-ray laser coherence in the presence of density fluctuations,” in Hard X-Ray/Gamma-Ray and Neutron Optics, Sensors, and Applications, R. B. Hoover, F. P. Doty, eds., Proc. SPIE2859, 269–280 (1996).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

M. C. Roggemann, A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of the target-in-the-loop wave propagation arrangement. The propagation directions of the outgoing A and returned ψ waves are shown by dashed lines with arrows.

Fig. 2
Fig. 2

Schematic representation of the brightness-function trajectories departing (a) from the target plane point RL and (b) from the receiver aperture point R0. The trajectory L is an example of a trajectory that links a target hot-spot-area point with a receiver aperture point (an essential trajectory for analysis). In contrast, the trajectory L1 misses either the receiver aperture in (a) or the target hot-spot area in (b) and represents an example of a trajectory that is nonessential for analysis.

Fig. 3
Fig. 3

Target-induced aberration for a Gaussian spherical surface of curvature radius Rs versus the parameter μ=1+L/Rs for different ratios Dψ/bs—the ratio of the scattered-beam footprint at the receiver plane to the radius bs for the Gaussian target-plane intensity distribution.

Equations (76)

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2ik A(r, z, t)z+2A(r, z, t)+2k2n(r, z, t)A(r, z, t)=0,
A(r, z=0, t)=A0(r)exp[iu(r, t)],
ψ(r, z=L, t)=A(r, z=L, t)T(r, t),
T(r, t)=γ(r, t)exp[ikξ(r)+ikS(r, t)].
ξ(r1)ξ(r2)=Ks(r),
Ks(r)=exp(-r2/2ls2),
-2ik ψ(r, z, t)z+2ψ(r, z, t)+2k2n(r, z, t)ψ(r, z, t)=0,
P(t)=SRψ0(r, t)ψ0*(r, t)d2r
rjc(t)=sIj(rF, t)rFd2rFsIj(rF, t)d2rF,  j=1, , N,
{αj(t)}={rjc(t)}/F=k-1Ij(t)φj(t)Ij(t).
S(r, t)=(2ik)-1ψ0*(r, t)ψ0(r, t)-ψ0(r, t)ψ0*(r, t)=k-1I(r, t)φ(r, t).
α(r, t)S(r, t)/I(r, t)=k-1I(r, t)φ(r, t)/I(r, t).
Iψ(r, t)=ε|ψ0(r1, t)+ψ0(r2, t)exp(iφ0)|2,
Iψ(r, t)=2ε{ψ0(r1, t)ψ0*(r1, t)+2 Re[ψ0(r1, t)ψ0*(r2, t)exp(-iφ0)]}=2ε{I(r1, t)+I1/2(r1, t)I1/2(r2, t)×cos[φ(r1, t)-φ(r2, t)-φ0]}.
Γ(r1, r2, z, t)ψ(r1, z, t)ψ*(r2, z, t).
R=(r1+r2)/2,  ρ=(r1-r2).
Γ(ρ, R, z, t)ψ(R+ρ/2, z, t)ψ*(R-ρ/2, z, t).
P(t)=SRΓ(ρ=0, R, z=0, t)d2R.
α(R, t)S(R, t)/I(R, t)=(ik)-1ρ ln Γ(ρ0, R, z=0),
Iψ(R, t)=2ε{Γ(0, R, z=0)+Re[Γ(Δ, R, z=0)×exp(-iφ0)]}.
-2ik Γ(r1, r2, z, t)z+(12-22)Γ(r1, r2, z, t)+2k2[n(r1, z, t)-n(r2, z, t)]Γ(r1, r2, z, t)=0,
-ik Γ(ρ, R, z, t)z+RρΓ(ρ, R, z, t)+k2[n(R+ρ/2, z, t)-n(R-ρ/2, z, t)]Γ(ρ, R, z, t)=0.
Γ(r1, r2, L, t)=A(r1, L, t)A*(r2, L, t)×Ts(r1, r2, t)Γξ(r1, r2),
Γξ(r1, r2)=exp[ikξ(r1)-ikξ(r2)]
Ts(r1, r2, t)=γ(r1, t)γ(r2, t)×exp[ikS(r1)-ikS(r2, t)].
Γξ(r1, r2)=Γξ(ρ, R)exp[-ρ2/2lΓ2],
lΓ=λls/(2πσs).
A(r1, L, t)A*(r2, L, t)|A(R, L, t)|2=I(R, L, t)
Ts(r1, r2, t)γ2(R, t)exp[ikρRS(R, t)].
Γ(ρ, R, L, t)=γ2(R, t)exp[ikρRS(R, t)]×exp(-ρ2/2lΓ2)I(R, L, t).
n(z, R+ρ/2)-n(z, R-ρ/2)ρRn(z, R).
-Γ(ρ, R, z, t)z+1ik RρΓ(ρ, R, z, t)+ki ρRn(R, z, t)Γ(ρ, R, z, t)=0.
-B(κ, R, z, t)z+k-1κRB(κ, R, z, t)+kRn(R, z, t)κB(κ, R, z, t)=0
B(κ, R, z, t)=(2π)-2-+Γ(ρ, R, z, t)exp(-iκρ)d2ρ,
Γ(ρ, R, z, t)=-+B(κ, R, z, t)exp(iκρ)d2κ.
B(κ, R, L, t)=γ2(R, t)exp{-lΓ2[κ-kRS(R, t)]2/2}×I(R, L, t)sΓ,
I(R, z, t)=ψ(R, z, t)ψ*(R, z, t)=Γ(0, R, z, t).
I(R, z, t)=-+B(κ, R, z, t)d2κ.
B(κ0, R0, z=0, t)=B(κL, RL, z=L, t).
ddz B(κ(z, t), R(z, t), z, t)=0.
ddz B(κ(z), R(z), z, t)=Bz+BXXz+BYYz+Bkxkxz+Bkykyz=0.
ddz B(κ(z, t), R(z, t), z, t)=B(κ, R, z, t)z+dR(z, t)dz RB(κ, R, z, t)+dκ(z, t)dz κB(κ, R, z, t)=0.
dR(z, t)dz=-κ(z, t)k, dκ(z, t)dz=-kRn(R, z, t).
dR(z, t)dz=-θ(z, t),  dθ(z, t)dz=-Rn(R, z, t).
dR(z, t)dz=θ(z, t),  dθ(z, t)dz=Rn(R, z, t).
P(t)=SRΓ(ρ=0, R, 0)d2R=kSRd2RΘ0B(θ, R, 0, t)d2θ.
Iψ(R, t)=2εΘ0{B(θ, R, 0, t)×[1+cos(kθΔ-φ0)]}d2θ.
α(R, t)S(R, t)/I(R, t)=Θ0θB(θ, R, 0, t)d2θΘ0B(θ, R, 0, t)d2θ.
B(κ, R, L, t)=γ2(R, t)I(R, L, t)sΓ.
R(z)=R0+θ0z, or R(z)=R0+κ0(z/k),
B(θ0, R0, 0, t)=B(θ0, R0+Lθ0, L, t)=γ2(R0+Lθ0, t)I(R0+Lθ0, L, t)sΓ.
Γ(ρ, R0, 0, t)=sΓk2-+γ2(R0+Lθ0, t)I(R0+Lθ0, L, t)exp(ikρθ0)d2θ0.
Γ(ρ, R0, 0, t)=k2L2 sΓ exp-i kL ρR0×-+γ2(RL, t)I(RL, L, t)×expi kL ρRLd2RL.
I(R0, t)=Γ(ρ=0, R0, 0, t)=(k/L)2sΓ-+γ2(RL, t)I(RL, L, t)d2RL.
α(R0, t)=Θ0θ0γ2(RL, t)I(RL, L, t)d2θ0Θ0γ2(RL, t)I(RL, L, t)d2θ0,
α(R0, t)=-R0L+Rc(t)L,
Rc(t)=STRLγ2(RL, t)I(RL, L, t)d2RLSTγ2(RL, t)I(RL, L, t)d2RL.
φT(R0, t)φq(R0, t)+φt(R0, t)=-k R02L+kαc(t)R0,
Γφ(ρ, R, t)ψ0(R+ρ/2, t)ψ0*(R-ρ/2, 0, t)×exp{-i[φ(R+ρ/2, t)-φ(R-ρ/2, t)]}.
Γφ(ρ, R, t)Γ0(ρ, R, t)exp[-iρRφ(R, t)],
Γ0(ρ, R, t)ψ0(R+ρ/2, t)ψ0*(R-ρ/2, 0, t)
Bφ(θ, R, t)=B(θ+θφ, R, 0, t),
Bφ(θ0, R0, t)=B(θL, R0, 0, t)=B(θL, RL, L, t)=γ2(RL, t)I(RL, L, t)sΓ,
θL(R0, t)=θ0-k-1Rφ(R0, t).
α(R0, t)=1k Rφ(R0, t)-R0L+Rc(t)L,
φSH(R0, t)=φ(R0, t)+φT(R0, t)=φ(R0, t)-k R02L+kαc(t)R0.
B(θ, R, L, t)=γ2(R, t)×exp-(θ+R/Rs)22θs2I(R, L, t)sΓ,
α(R0, t)=-R0L+Q(R0, t)L,
Q(R0, t)=STRLγ2(RL, t)I(RL, L, t)exp[-(μRL-R0)2/2Dψ2]d2RLSTγ2(RL, t)I(RL, L, t)exp[-(μRL-R0)2/2Dψ2]d2RL,  μ=1+L/Rs.
α(R0, t)=1k Rφ(R0, t)-R0L+Q(R0, t)L.
φSH(R0, t)=φ(R0, t)+φq(R0)+φt(R0, t)+φab(R0, t).
Q(R0, t)Rc(t)-(2Dψ2)-1[R02Rc(t)-2μR0M2(t)+μ2M3(t)],
Mn(t)=STRLnγ2(RL, t)I(RL, L, t)d2RLSTγ2(RL, t)I(RL, L, t)d2RL, n=1, 2, 3.
α(R0, t)=k-1Rφ(R0, t)+1L Rc(t)-μ22Dψ2 M3(t)-1L 1-μDψ2 M2(t)R0-Rc(t)2Dψ2L R02.
φab(R0, t)=-kμM2(t)2LDψ2 R02-k6LDψ2 Rc(t)R03.
Q(R0, t)=R0Q=R01-μ(μ-1)+4(Dψ/bs)2μ2+4(Dψ/bs)2.

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