Abstract

The problem of diffraction of an electromagnetic wave by a thick hologram grating can be solved by the famous Kogelnik’s coupled-wave theory (CWT) to a very high degree of accuracy. We confirm this finding by comparing the CWT and the exact result for a typical example and propose an explanation in terms of perturbation theory. To this end we formulate the problem of diffraction as a matrix problem following similar well-known approaches, especially rigorous coupled-wave theory (RCWT). We allow for a complex permittivity modulation and a possible phase shift between refractive index and absorption grating and explicitly incorporate appropriate boundary conditions. The problem is solved numerically exact for the specific case of a planar unslanted grating and a set of realistic values of the material’s parameters and experimental conditions. Analogously, the same problem is solved for a two-dimensional truncation of the underlying matrix that would correspond to a CWT approximation but without the usual further approximations. We verify a close coincidence of both results even in the off-Bragg region and explain this result by means of a perturbation analysis of the underlying matrix problem. Moreover, the CWT is found not only to coincide with the perturbational approximation in the in-Bragg and the extreme off-Bragg cases, but also to interpolate between these extremal regimes.

© 2014 Optical Society of America

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    [CrossRef]
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2013 (2)

K.-M. Voit and M. Imlau, “Holographic spectroscopy: wavelength-dependent analysis of photosensitive materials by means of holographic techniques,” MDPI Materials 6, 334–358 (2013).

M. Prijatelj, J. Klepp, Y. Tomita, and M. Fally, “Far-off-Bragg reconstruction of volume holographic gratings: a comparison of experiment and theories,” Phys. Rev. A 87, 063810 (2013).
[CrossRef]

2011 (1)

2008 (1)

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

2006 (1)

1999 (2)

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

1997 (1)

G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 55, 1035–1047 (1997).
[CrossRef]

1994 (2)

1990 (1)

1984 (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
[CrossRef]

1981 (2)

1978 (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. II. Beam coupling–light amplification,” Ferroelectrics 22, 961–964 (1978).
[CrossRef]

1973 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef]

Bieringer, S.

M. Imlau, S. Odoulov, T. Woike, and S. Bieringer, “Holographic data storage,” in Nanoelectronics and Information Technology, R. Waser, ed. (Wiley-VCH, 1999), pp. 727–750.

Blanche, P. A.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Blow, K. J.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Burr, G.

M. Imlau, M. Fally, H. Coufal, G. Burr, and G. Sincerbox, “Holography and optical storage,” in Springer Handbook of Lasers and Optics, F. Träger, ed. (Springer, 2007), pp. 1205–1249.

Cotter, D.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Coufal, H.

M. Imlau, M. Fally, H. Coufal, G. Burr, and G. Sincerbox, “Holography and optical storage,” in Springer Handbook of Lasers and Optics, F. Träger, ed. (Springer, 2007), pp. 1205–1249.

Ellis, A. D.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Fally, M.

M. Prijatelj, J. Klepp, Y. Tomita, and M. Fally, “Far-off-Bragg reconstruction of volume holographic gratings: a comparison of experiment and theories,” Phys. Rev. A 87, 063810 (2013).
[CrossRef]

M. Imlau, M. Fally, H. Coufal, G. Burr, and G. Sincerbox, “Holography and optical storage,” in Springer Handbook of Lasers and Optics, F. Träger, ed. (Springer, 2007), pp. 1205–1249.

Flores, D.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef]

Gaylord, T. K.

Gleeson, M. R.

Gu, T.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Guibelalde, E.

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
[CrossRef]

Guo, J.

Hilaire, P. St.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Imlau, M.

K.-M. Voit and M. Imlau, “Holographic spectroscopy: wavelength-dependent analysis of photosensitive materials by means of holographic techniques,” MDPI Materials 6, 334–358 (2013).

M. Imlau, M. Fally, H. Coufal, G. Burr, and G. Sincerbox, “Holography and optical storage,” in Springer Handbook of Lasers and Optics, F. Träger, ed. (Springer, 2007), pp. 1205–1249.

M. Imlau, S. Odoulov, T. Woike, and S. Bieringer, “Holographic data storage,” in Nanoelectronics and Information Technology, R. Waser, ed. (Wiley-VCH, 1999), pp. 727–750.

Kamenov, V. P.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

Kamshilin, A. A.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

Kelly, A. E.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Klepp, J.

M. Prijatelj, J. Klepp, Y. Tomita, and M. Fally, “Far-off-Bragg reconstruction of volume holographic gratings: a comparison of experiment and theories,” Phys. Rev. A 87, 063810 (2013).
[CrossRef]

Klonidis, D.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kowarschik, R.

Kukhtarev, N. V.

J. Strait, J. D. Reed, and N. V. Kukhtarev, “Orientational dependence of photorefractive two-beam coupling in InP:Fe,” Opt. Lett. 15, 209–211 (1990).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. II. Beam coupling–light amplification,” Ferroelectrics 22, 961–964 (1978).
[CrossRef]

Li, G.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Lin, W.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Manning, R. J.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. II. Beam coupling–light amplification,” Ferroelectrics 22, 961–964 (1978).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Moharam, M. G.

Montemezzani, G.

G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 55, 1035–1047 (1997).
[CrossRef]

Nejabati, R.

Nesset, D.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Nippolainen, E.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

Norwood, R. A.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

O’Mahony, M. J.

Odoulov, S.

M. Imlau, S. Odoulov, T. Woike, and S. Bieringer, “Holographic data storage,” in Nanoelectronics and Information Technology, R. Waser, ed. (Wiley-VCH, 1999), pp. 727–750.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. II. Beam coupling–light amplification,” Ferroelectrics 22, 961–964 (1978).
[CrossRef]

Peyghambarian, N.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Phillips, D. I. D.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Podivilov, E. V.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

Politi, C.

Poustie, A. J.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Prijatelj, M.

M. Prijatelj, J. Klepp, Y. Tomita, and M. Fally, “Far-off-Bragg reconstruction of volume holographic gratings: a comparison of experiment and theories,” Phys. Rev. A 87, 063810 (2013).
[CrossRef]

Prokofiev, V. V.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

Reed, J. D.

Ringhofer, K. H.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, “Exact solution of the Bragg-diffraction problem in sillenites,” J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

Rogers, D. C.

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, D. I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523–1528 (1999).
[CrossRef]

Rokutanda, S.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Russell, P.St.J.

P.St.J. Russell, “Optical volume holography,” Phys. Rep. 71, 209–312 (1981).
[CrossRef]

Shamonina, E.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, “Exact solution of the Bragg-diffraction problem in sillenites,” J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

Sheridan, J. T.

Simeonidou, D.

Sincerbox, G.

M. Imlau, M. Fally, H. Coufal, G. Burr, and G. Sincerbox, “Holography and optical storage,” in Springer Handbook of Lasers and Optics, F. Träger, ed. (Springer, 2007), pp. 1205–1249.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. II. Beam coupling–light amplification,” Ferroelectrics 22, 961–964 (1978).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Strait, J.

Sturman, B. I.

B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, E. Shamonina, V. P. Kamenov, E. Nippolainen, V. V. Prokofiev, and A. A. Kamshilin, “Theory of photorefractive vectorial wave coupling in cubic crystals,” Phys. Rev. E 60, 3332–3352 (1999).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, “Exact solution of the Bragg-diffraction problem in sillenites,” J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

Tay, S.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Thomas, J.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Tomita, Y.

M. Prijatelj, J. Klepp, Y. Tomita, and M. Fally, “Far-off-Bragg reconstruction of volume holographic gratings: a comparison of experiment and theories,” Phys. Rev. A 87, 063810 (2013).
[CrossRef]

Tunc, A. V.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

Uchida, N.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

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K.-M. Voit and M. Imlau, “Holographic spectroscopy: wavelength-dependent analysis of photosensitive materials by means of holographic techniques,” MDPI Materials 6, 334–358 (2013).

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S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
[CrossRef]

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S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
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Yamamoto, M.

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
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[CrossRef]

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[CrossRef]

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

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K.-M. Voit and M. Imlau, “Holographic spectroscopy: wavelength-dependent analysis of photosensitive materials by means of holographic techniques,” MDPI Materials 6, 334–358 (2013).

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[CrossRef]

S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St. Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451, 694–698 (2008).
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G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 55, 1035–1047 (1997).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Schematic sketch of the exact solution of the diffraction problem by the superposition of various generations. The thick red arrow at the left-hand side symbolizes an incoming plane wave. E+(n) and E(n) denote the nth generations of the special solutions of the Helmholtz equation inside the grating. The sign ± refers to the two possible directions of the wave vectors. E+(n) and E(n) are related by boundary conditions at z= and, similarly, E(n) and E+(n+1) by boundary conditions at z=0; see Eqs. (51) and (52). The superposition of the transmitted solutions at z>, symbolized by small pencils of rays at the right-hand side, can be measured and compared with the theoretical predictions.

Fig. 2.
Fig. 2.

Square amplitude |A1|2 of the first diffraction order of the transmitted wave according to the exact solution of the diffraction problem and the values shown in Table 1. The two independent variables are the thickness of the crystal in units of the wavelength λ0 and the angle θ0 of the incident plane wave in units of the Bragg angle θB.

Fig. 3.
Fig. 3.

Difference δ|A1|2 between the square amplitudes of the exact solution according to Fig. 2 and those of the corresponding CWT divided by the maximum of |A1|2 w. r. t. and fixed θ0. The independent variables are the same as in Fig. 2. Note that the relative difference increases with the deviation of θ0 from the Bragg angle but never exceeds 103.

Fig. 4.
Fig. 4.

Absolute square |A1(z)|2 of the m=1 component of the exact solution (61) and the approximate coordinates (72), (73) of its first maximum. The parameters are chosen according to Table 1 and θ0=(1/2)θB.

Fig. 5.
Fig. 5.

Absolute square |A1(z)|2 of the m=1 component of the exact solution (61) and the approximate coordinates (72), (73) of its first maximum. The parameters are chosen according to Table 1 and θ0=(1/2)θB.

Tables (1)

Tables Icon

Table 1. Parameters Used for the Exact Solution and the CWT Solution

Equations (75)

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ϵ=(n0(1+iκ0))2.
EI=(0EI0)eiωtexp(ik0(xsinθ0+zcosθ0)),
EIII=(0EIII0)eiωtexp(ik0(xsinθ0zcosθ0)).
EII=(0EII0)eiωtexp(ik(xsinθ+cosθ)qzcosθ).
ΔEII+k02ϵEII=0
k=12k01+2n02(1κ02)cos(2θ0)+ξ,
ξ=12(3+8n02(1+κ02+n02(1+κ02)24(1+2n02(1+κ02))cos2θ0)+cos4θ0)1/2,
q=kκ0(1+(kk0n0)2+κ02),
ksinθ=k0sinθ0(Snell's law),
EIIIEI=(k+iq)cosθ+k0cosθ0(k+iq)cosθ+k0cosθ0,
EIIEI=2k0cosθ0(k+iq)cosθ+k0cosθ0,
ϵ(x)=νZϵ˜νexp(iKνx).
E=(0EII0)eiωtexp(ik(xsinθ+cosθ)qzcosθ)mZAm(z)exp(imKx).
0=exp(ik(xsinθ+cosθ)qzcosθ)mZexp(imKx){[(iksinθimK)2+(ikq)2cos2θ]Am(z)+2(ikq)cosθAm(z)+Am(z)+k02νZϵ˜νAm+ν(z)}.
0=k2+(q22ikq)cos2θ+k02ϵ˜0.
ϵν={ϵ˜νifν0,0ifν=0,
0=mK(2ksinθmK)Am(z)+2(ikq)cosθAm(z)+Am(z)+k02νZϵνAm+ν(z),
0=2(ikq)cosθA(z)+A(z)+BA(z),
A(z)=(A1(z)A0(z)A1(z))
Bm,m+ν={k02ϵνifν0,mK(2ksinθmK)ifν=0,
Bϕ(μ)=b(μ)ϕ(μ),μZ.
A(z)=μZaμ(z)ϕ(μ).
0=2(ikq)cosθaμ(z)+aμ(z)+b(μ)aμ(z),μZ,
aμ(z)=cμexp(βμz),cμC,
βμ=(qik)cosθ(1±1b(μ)(qik)2cos2θ).
αμ±(qik)2cos2θbμ,such thatR(αμ)<0,
E+=(0E0)eiωtexp(ikxsinθ)mZexp(imKx)μZcμ+exp(αμz)ϕmμ,
E=(0E0)eiωtexp(ikxsinθ)mZexp(imKx)μZcμexp(αμz)ϕmμ.
aμ±(z)cμ±exp(±αμz)=aμ±(0)exp(±αμz),
Am±(z)μZcμ±exp(±αμz)ϕm(μ)=μZaμ±(z)ϕm(μ).
A±(z)=Φa±(z).
E±=(0E0)eiωtexp(ikxsinθ)mZexp(imKx)Am±(z).
Δ(z)μνδμνexp(αμz),
TzΦΔ(z)Φ1,
A+(z)=Φa+(z)=ΦΔ(z)a+(0)
=(ΦΔ(z)Φ1)A+(0)
=TzA+(0)=TzA+().
TzTu=Tz+u,andTz=Tz1.
A(z)=Φa(z)=ΦΔ(z)a(0)
=(ΦΔ(z)Φ1)A(0)
=TzA(0)=TzA().
Am+(0)(0)=δm0E,for allmZ.
Em+(n)(z)=(0E0)eiωtexp(ikxsinθ)exp(imKx)Am+(n)(z),
Em(n)(z)=(0ρE0)eiωtexp(ikxsinθ)exp(imKx)Am+(n)(2z),
Emτ(n)(z)=(0τE0)eiωtexp(ik0x(sinθmx+cosθm(z)).
ksinθmK=k0sinθm,
τ=(1+ρ)Am+(n)().
τ=(1ρ)1ik0cosθmμZcμ+αμexp(αμ)ϕm(μ)(1ρ)A˜m+(n)().
ρ=A˜m+(n)()Am+(n)()A˜m+(n)()+Am+(n)(),
τ=2A˜m+(n)()Am+(n)()A˜m+(n)()+Am+(n)().
Am(n)()=ρAm+(n)()=A˜m+(n)()Am+(n)()A˜m+(n)()+Am+(n)()Am+(n)().
Am+(n+1)(0)=ρAm(n)(0)=A˜m(n)(0)Am(n)(0)A˜m(n)(0)+Am(n)(0)Am(n)(0).
E=(0E0)eiωtexp(ikxsinθ)mZexp(imKx)nN(Am+(n)(z)+Am(n)(z)).
ϵ(x)=[(n0+n1cos(Kx+Φn))(1+i(κ0+κ1cos(Kx+Φκ)))]2,
K=2k0sinθB,
K=2ksinθb,
(bγB2b+γ00000bγB1b+γ00000bγ0b+γ00000bγB1b+γ00000bγB2b+γ).
U(0b+γbγ0),
p=(pm+1γb+bpm)(pm+1+γb+bpm).
ϕν(m)=δmν+γ(δν,m+1Ψm,ν+δν,m1Ψm,ν)+O(γ2).
A(z)=ΦΔ(z)Φ1A(0)=Δ0(z)A(0)+γ(Δ0(z)ΨA(0)+ΨΔ0(z)A(0))+O(γ2),
(Δ0(z))mm=exp((qik)2cos2θBmz).
Ψm,m±1=b±BmBm±1.
A±1(z)=b±Bm±1(eαzeμ±z)+O(γ2),
α(qik)cosθ,
μ±(qik)2cos2θB±1.
|eαzeμ±z|2=|e(α1+iα2)ze(μ1±+iμ2±)z|2=e2α1z+e2μ1±z2e(α1+μ1±)z×cos((α2μ2±)z).
0=α1e(α1μ1±)z±+μ1±e(α1μ1±)z±+A±sin((α2μ2±)z±+ϕ±),
A±(α1+μ1±)2+(α2μ2±)2,
ϕ±arctanα1+μ1±α2μ2±.
z±α1+μ1±+A±(π+ϕ±)(α1μ1±)2+A±(α2μ2±).
z±π+ϕ±α2μ2±;
|A±1(z±)|2|γb±B±1|2|eα1z±+eμ1±z±|2.
|b+B1bB1|282.1,
|A1(z+)A1(z)|251.5.

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