Abstract

Beam fanning in low-symmetry photorefractive crystals features nonuniform angular distribution of linear polarization. We study this phenomenon experimentally in monoclinic Sn2P2S6 crystals that belong to the m class of symmetry and compare the results with the predictions of a simple model that describes amplification of a weak scattered wave via nonlinear coupling to the intense incident wave.

© 2013 Optical Society of America

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References

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  1. V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
    [CrossRef]
  2. J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
    [CrossRef]
  3. S. Stepanov and M. Petrov, “Nonstationary holographic recording for efficient amplification and phase conjugation,” in Photorefractive Materials and Their Applications I: Fundamental Phenomena, P. Günter and J.-P. Huignard, eds. Vol. 61 of Topics in Applied Physics (Springer-Verlag, 1988), pp. 263–289.
  4. G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
    [CrossRef]
  5. R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
    [CrossRef]
  6. V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
    [CrossRef]
  7. A. Shumelyuk, A. Volkov, A. Selinger, M. Imlau, and S. Odoulov, “Frequency-degenerate nonlinear light scattering in low-symmetry crystals,” Opt. Lett. 33, 150–152 (2008).
    [CrossRef]
  8. B. Boulanger, Y. Petit, P. Segonds, C. Félix, B. Ménaert, J. Zaccaro, and G. Aka, “Absorption and fluorescence anisotropies of monoclinic crystals: the case of Nd:YCOB,” Opt. Express 16, 7997–8002 (2008).
    [CrossRef]
  9. A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.
  10. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Vol. 11 of Oxford Series in Optical and Imaging Sciences (Oxford University, 1996).
  11. A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
    [CrossRef]
  12. B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
    [CrossRef]
  13. A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
    [CrossRef]
  14. D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn2P2S6 at visible and infrared wavelengths,” Opt. Express 13, 2047–2057 (2005).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

2010 (1)

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

2008 (2)

2005 (1)

1996 (1)

B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
[CrossRef]

1995 (1)

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

1993 (1)

V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
[CrossRef]

1987 (1)

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

1984 (1)

R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
[CrossRef]

1982 (1)

1980 (1)

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Aka, G.

Anderson, D. Z.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Boulanger, B.

Cook, G.

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

Czaia, L.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Dorosh, I. R.

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Evans, D.

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

Feinberg, J.

Félix, C.

Goulkov, M.

B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
[CrossRef]

Grabar, A.

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Grousson, R.

R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
[CrossRef]

Grunnet-Jepsen, A.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Vol. 11 of Oxford Series in Optical and Imaging Sciences (Oxford University, 1996).

Guarino, A.

Günter, P.

D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn2P2S6 at visible and infrared wavelengths,” Opt. Express 13, 2047–2057 (2005).
[CrossRef]

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Haertle, D.

Hajfler, J.

Imlau, M.

Jazbinšek, M.

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Karabekian, S.

V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
[CrossRef]

Kuz’minov, Y. S.

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Mallick, S.

R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
[CrossRef]

Ménaert, B.

Montemezzani, G.

D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn2P2S6 at visible and infrared wavelengths,” Opt. Express 13, 2047–2057 (2005).
[CrossRef]

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Novikov, A.

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

Obukhovsky, V.

V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
[CrossRef]

Odoulov, S.

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

A. Shumelyuk, A. Volkov, A. Selinger, M. Imlau, and S. Odoulov, “Frequency-degenerate nonlinear light scattering in low-symmetry crystals,” Opt. Lett. 33, 150–152 (2008).
[CrossRef]

B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
[CrossRef]

V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
[CrossRef]

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
[CrossRef]

Oleinik, O.

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

Petit, Y.

Petrov, M.

S. Stepanov and M. Petrov, “Nonstationary holographic recording for efficient amplification and phase conjugation,” in Photorefractive Materials and Their Applications I: Fundamental Phenomena, P. Günter and J.-P. Huignard, eds. Vol. 61 of Topics in Applied Physics (Springer-Verlag, 1988), pp. 263–289.

Segonds, P.

Selinger, A.

Shumelyuk, A.

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

A. Shumelyuk, A. Volkov, A. Selinger, M. Imlau, and S. Odoulov, “Frequency-degenerate nonlinear light scattering in low-symmetry crystals,” Opt. Lett. 33, 150–152 (2008).
[CrossRef]

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Solymar, L.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Vol. 11 of Oxford Series in Optical and Imaging Sciences (Oxford University, 1996).

Stepanov, S.

S. Stepanov and M. Petrov, “Nonstationary holographic recording for efficient amplification and phase conjugation,” in Photorefractive Materials and Their Applications I: Fundamental Phenomena, P. Günter and J.-P. Huignard, eds. Vol. 61 of Topics in Applied Physics (Springer-Verlag, 1988), pp. 263–289.

Sturman, B.

B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
[CrossRef]

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

Tkachenko, N. V.

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Volkov, A.

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

A. Shumelyuk, A. Volkov, A. Selinger, M. Imlau, and S. Odoulov, “Frequency-degenerate nonlinear light scattering in low-symmetry crystals,” Opt. Lett. 33, 150–152 (2008).
[CrossRef]

Voronov, V. V.

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Vysochanskii, Y.

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

Webb, D. J.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Vol. 11 of Oxford Series in Optical and Imaging Sciences (Oxford University, 1996).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Zaccaro, J.

Zgonik, M.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Zozulya, A. A.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Appl. Phys. B (1)

A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010).
[CrossRef]

Ferroelectrics (1)

A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, “Beam-coupling, four-wave mixing and optical oscillation due to spatially-oscillating photovoltaic currents in lithium niobate crystals,” Ferroelectrics 75, 295–315 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

R. Grousson, S. Mallick, and S. Odoulov, “Amplified backward scattering in LiNbO3:Fe,” Opt. Commun. 51, 342–346 (1984).
[CrossRef]

V. Obukhovsky, S. Odoulov, and S. Karabekian, “Backward conical photorefractive scattering in LiNbO3,” Opt. Commun. 104, 123–128 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rep. (1)

B. Sturman, S. Odoulov, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1996).
[CrossRef]

Phys. Rev. A (1)

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. V. Voronov, I. R. Dorosh, Y. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
[CrossRef]

Other (4)

S. Stepanov and M. Petrov, “Nonstationary holographic recording for efficient amplification and phase conjugation,” in Photorefractive Materials and Their Applications I: Fundamental Phenomena, P. Günter and J.-P. Huignard, eds. Vol. 61 of Topics in Applied Physics (Springer-Verlag, 1988), pp. 263–289.

A. Grabar, M. Jazbinšek, A. Shumelyuk, Y. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications 2: Materials, P. Günter and J.-P. Huignard, eds. Vol. 114of Springer Series in Optical Sciences (Springer, 2007), pp. 327–362.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Vol. 11 of Oxford Series in Optical and Imaging Sciences (Oxford University, 1996).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of the scattering geometry. (a) The light beam with polarization adjusted by half-wave plate λ / 2 induces photorefractive scattering in SPS crystal. The intensity distribution of the beam fanning on the translucent screen Sc with the polarization selected by polarizer P is stored by camera DC. (b) The direction s of any scattering ray in air is characterized by angles ξ and ζ in the crystallographic frame ( x , y , z ) , while θ , φ are given in the optical frame ( x 1 , x 2 , x 3 ) related to the optical indicatrix.

Fig. 2.
Fig. 2.

Scattered light patterns on the screen recorded with the DC: (a) with no polarizer between the sample and the screen, and (b) with polarizer adjusted to cut the light with the polarization azimuth ψ = 42 ° . The horizontal width of each frame makes 70°.

Fig. 3.
Fig. 3.

Polarization distributions of the fanning light on the screen induced by a laser beam with polarization angle ψ p = 47 ° . Circles, triangles, squares, pentagrams, and diamonds denote measured spatial distributions of scattered light polarization angles ψ = 46 ° , 44 ° , 42 ° , 40 ° , and 38 ° . The thin gray lines show calculated data for the same set of polarization angles.

Fig. 4.
Fig. 4.

Calculated distribution of the polarization angle ψ of scattering rays produced by the incident wave with polarization angle ψ p = 47 ° .

Fig. 5.
Fig. 5.

Calculated distribution of the polarization angle ψ of scattering rays produced by the incident wave with polarization angle ψ p = 43 ° .

Fig. 6.
Fig. 6.

Scattered intensity versus scattering angle measured along two straight lines with constant polarization φ = 0 (curve A) and φ = 90 ° (curve B) shown in Fig. 4.

Fig. 7.
Fig. 7.

Angular dependence of the space charge field calculated for λ = 647 nm and 2 π s = 2 μm (doted line) and its linear approximation in an angular range from 15° to 60° (red solid line).

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

I s ( ) = I s 0 exp ( Γ )
Γ = 2 π λ cos ( θ i / 2 ) n s n p 2 r eff E sc ,
E sc = ( e s · e p ) k B T q K g 1 + ( K g s ) 2 .
s = ϵ ϵ 0 k B T q 2 N eff
K g = 4 π n s λ sin θ i 2
r eff = d p ( r ^ k g ) d s
[ s , [ s , B ^ d ] ] + d n 2 = 0 ,
s = ( θ i cos φ 1 1 2 θ i 2 θ i sin φ )
d 1 = ( 1 1 2 θ i 2 cos 2 φ θ i cos φ ( κ 1 2 ) θ i 2 sin 2 φ ) , d 3 = ( κ θ i 2 sin 2 φ θ i sin φ 1 1 2 θ i 2 sin 2 φ ) ,
κ = n 1 2 ( n 3 2 n 2 2 ) 2 n 2 2 ( n 3 2 n 1 2 ) = 1 1 cos 2 V ,
tan α = cos ( θ θ i ) tan α ,
p = x cos ψ + z sin ψ .
d = { ( y · [ s , d ] ) [ s , y ] + ( y · d ) [ [ s , y ] , y ] } [ s , y ] 2 .
Δ α = α α sin 2 θ θ i 2 sin 2 α .
ln I s ( θ ) Γ ( θ ) E s ( θ ) sin θ 1 + ( 2 π s / λ ) 2 sin 2 θ
ln I s ( θ ) = a b θ

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