Abstract

An analytical model is developed to study layered structures of photorefractive thin films, a promising application of stratified volume holographic optical elements. The theory gives a closed-form solution for the first-order diffraction efficiency of such elements. A new holographic multiplexing scheme that uses the photorefractive phase shift induced by an applied electric field is also analyzed with this theory.

© 1994 Optical Society of America

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References

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  1. A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  3. B. Ya. Zel’dovich and T. V. Yakovleva, “Theory of a two-layer hologram,” Sov. J. Quantum Electron. 14, 323–328 (1984).
    [Crossref]
  4. B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
    [Crossref]
  5. A. R. Tanguay and R. V. Johnson, “Stratified volume holographic optical elements,” J. Opt. Soc. Am. A 3, P53 (1986).
  6. R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189–191 (1988).
    [Crossref] [PubMed]
  7. G. P. Nordin, R. V. Johnson, and A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A 9, 2206–2217 (1992).
    [Crossref]
  8. R. V. Johnson and A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
    [Crossref]
  9. A. Granger, L. Song, and R. A. Lessard, “Multiple beam generation using a stratified volume holographic grating,” Appl. Opt. 32, 2534–2537 (1993).
    [Crossref] [PubMed]
  10. Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
    [Crossref]
  11. K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
    [Crossref]
  12. S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
    [Crossref] [PubMed]
  13. Q. Wang, R. M. Brubaker, and D. D. Nolte, “Photorefractive quantum wells: transverse Franz–Keldysh geometry,” J. Opt. Soc. Am. B 9, 1626–1641 (1992).
    [Crossref]
  14. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. 1, Chap. 2, p. 46.
  15. Ref. 14, Vol. 2, Chap. 6, p. 205.
  16. N. V. Kukhtaharev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).
  17. A. G. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
    [Crossref]
  18. H. J. Eichler, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), Chap. 4, p. 100.
  19. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985), Chap. 7, p. 329.
  20. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [Crossref]
  21. F. H. Mok, “Angle-multiplexed storage of 5000 holograms in lithium niobate,” Opt. Lett. 18, 915–917 (1993).
    [Crossref] [PubMed]
  22. K. Curtis, C. Gu, and D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
    [Crossref] [PubMed]
  23. C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase-encoding technique,” Appl. Opt. 31, 5700–5705 (1992).
    [Crossref] [PubMed]
  24. L. Hesselink and M. C. Bashaw, “Optical memories implemented with photorefractive media,” Opt. Quantum Electron. 25, 5611–5661 (1993).
    [Crossref]
  25. J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
    [Crossref]

1993 (4)

1992 (4)

1991 (2)

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

1989 (1)

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

1988 (1)

1986 (2)

A. R. Tanguay and R. V. Johnson, “Stratified volume holographic optical elements,” J. Opt. Soc. Am. A 3, P53 (1986).

R. V. Johnson and A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[Crossref]

1984 (2)

B. Ya. Zel’dovich and T. V. Yakovleva, “Theory of a two-layer hologram,” Sov. J. Quantum Electron. 14, 323–328 (1984).
[Crossref]

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

1980 (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).

1978 (1)

A. G. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[Crossref]

1976 (1)

N. V. Kukhtaharev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

1969 (1)

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

Bashaw, M. C.

L. Hesselink and M. C. Bashaw, “Optical memories implemented with photorefractive media,” Opt. Quantum Electron. 25, 5611–5661 (1993).
[Crossref]

Bize, D.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985), Chap. 7, p. 329.

Brubaker, R. M.

Curtis, K.

Denz, C.

DiCarolis, S. A.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Ducharme, S.

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

Eason, R. W.

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

Eichler, H. J.

H. J. Eichler, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), Chap. 4, p. 100.

Fainman, S.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Feigelson, R. S.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Ford, J. E.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Glass, A. G.

A. G. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[Crossref]

Gower, M. C.

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

Granger, A.

Gu, C.

Hesselink, L.

L. Hesselink and M. C. Bashaw, “Optical memories implemented with photorefractive media,” Opt. Quantum Electron. 25, 5611–5661 (1993).
[Crossref]

Hiskes, R.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Jacowitz, R. D.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Johnson, R. V.

G. P. Nordin, R. V. Johnson, and A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A 9, 2206–2217 (1992).
[Crossref]

R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189–191 (1988).
[Crossref] [PubMed]

R. V. Johnson and A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[Crossref]

A. R. Tanguay and R. V. Johnson, “Stratified volume holographic optical elements,” J. Opt. Soc. Am. A 3, P53 (1986).

Kogelnik, H.

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

Kukhtaharev, N. V.

N. V. Kukhtaharev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Lee, S. H.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Lessard, R. A.

Lu, Z.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Mirovitskii, D. I.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

Moerner, W. E.

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

Mok, F. H.

Neurgaonkar, R. R.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Nolte, D. D.

Nordin, G. P.

Pauliat, G.

Psaltis, D.

Roosen, G.

Rostovtseva, N. V.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

Route, R. K.

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

Scott, J. C.

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

Serov, O. B.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

Song, L.

Taketomi, Y.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

Tanguay, A. R.

G. P. Nordin, R. V. Johnson, and A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A 9, 2206–2217 (1992).
[Crossref]

R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189–191 (1988).
[Crossref] [PubMed]

R. V. Johnson and A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[Crossref]

A. R. Tanguay and R. V. Johnson, “Stratified volume holographic optical elements,” J. Opt. Soc. Am. A 3, P53 (1986).

Tschudi, T.

Twieg, R. J.

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

Vainos, N. A.

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

Wang, Q.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985), Chap. 7, p. 329.

Yakimovich, A. P.

A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).

Yakovleva, T. V.

B. Ya. Zel’dovich and T. V. Yakovleva, “Theory of a two-layer hologram,” Sov. J. Quantum Electron. 14, 323–328 (1984).
[Crossref]

Yeh, P.

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

Youden, K. E.

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

B. Ya. Zel’dovich and T. V. Yakovleva, “Theory of a two-layer hologram,” Sov. J. Quantum Electron. 14, 323–328 (1984).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. E. Youden, R. W. Eason, M. C. Gower, and N. A. Vainos, “Expitaxial growth of Bi12GeO20thin-film optical waveguides using excimer laser ablation,” Appl. Phys. Lett. 59, 1929–1932 (1991).
[Crossref]

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (1)

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

J. Cryst. Growth (1)

Z. Lu, R. S. Feigelson, R. K. Route, R. Hiskes, S. A. DiCarolis, and R. D. Jacowitz, “Solid source MOCVD for the epitaxial growth of thin oxide films,” J. Cryst. Growth 128, 788–792 (1992).
[Crossref]

J. Opt. Soc. Am. A (2)

G. P. Nordin, R. V. Johnson, and A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A 9, 2206–2217 (1992).
[Crossref]

A. R. Tanguay and R. V. Johnson, “Stratified volume holographic optical elements,” J. Opt. Soc. Am. A 3, P53 (1986).

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

Opt. Eng. (2)

A. G. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[Crossref]

R. V. Johnson and A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

L. Hesselink and M. C. Bashaw, “Optical memories implemented with photorefractive media,” Opt. Quantum Electron. 25, 5611–5661 (1993).
[Crossref]

Opt. Spectrosc. (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).

Phys. Rev. Lett. (1)

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846–1849 (1991).
[Crossref] [PubMed]

Sov. J. Quantum Electron. (2)

B. Ya. Zel’dovich and T. V. Yakovleva, “Theory of a two-layer hologram,” Sov. J. Quantum Electron. 14, 323–328 (1984).
[Crossref]

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovtseva, and O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[Crossref]

Sov. Tech. Phys. Lett. (1)

N. V. Kukhtaharev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Other (5)

H. J. Eichler, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), Chap. 4, p. 100.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985), Chap. 7, p. 329.

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. 1, Chap. 2, p. 46.

Ref. 14, Vol. 2, Chap. 6, p. 205.

J. E. Ford, Y. Taketomi, S. H. Lee, D. Bize, R. R. Neurgaonkar, and S. Fainman, “Effects of applied voltage on holographic storage in SBN:60,” in Nonlinear Optical Properties of Materials, H. R. Schlossberg and R. V. Wick, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1148, 12–24 (1989).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

General configuration of a SVHOE: (a) writing geometry, (b) reading geometry.

Fig. 2
Fig. 2

Two-dimensional k-space diagram of the writing, the reading, and the diffracted wave vectors and the Bragg mismatch parameter.

Fig. 3
Fig. 3

Comparison between a SVHOE and a Fabry–Perot étalon. δ and δ ^ represent in both cases the phase retardation between adjacent beams. Ei is the incident field, Ed is the field diffracted by the SVHOE, Et is the field transmitted by the étalon, r is the reflectivity of the mirror.

Fig. 4
Fig. 4

Diffraction efficiency as a function of the Bragg mismatch parameter ξ for SVHOE’s of different number N of thin films and for the equivalent thick grating.

Fig. 5
Fig. 5

General concept of phase multiplexing in a SVHOE: the three types of curve (solid, short-dashed, and long-dashed) represent three holograms stored in the structure. Phase multiplexing is based on the fact that each hologram undergoes a different phase shift in each film.

Fig. 6
Fig. 6

Normalized diffraction efficiencies of seven stored holograms in SVHOE of seven films: optimum case. the numbers correspond to the hologram numbers given in Table 2.

Fig. 7
Fig. 7

Normalized diffraction efficiencies of four stored holograms in a SVHOE of seven films: photorefractive case. The numbers correspond to the hologram numbers given in Table 3.

Tables (3)

Tables Icon

Table 1 Maximum First-Order Diffraction Efficiency as a Function of N and Corresponding Grating Strength

Tables Icon

Table 2 Phase Pattern ϕj,k across the Structure for the Optimum Case and the Chosen Intensities Ak2 of Each Hologram

Tables Icon

Table 3 Field Pattern E0j,k (kV/cm) across the Structure for the Photorefractive Case and the Chosen Intensities Ak2 of Each Hologram

Equations (41)

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

E ( r ) = E A ( r ) + E B ( r ) = A exp [ i ( Θ A + k A · r ) ] + B exp ( i k B · r ) ,
k = k + k 2 - k 2 e z ,
r = x + z e z .
t j ( x ) = exp [ i k 0 d film j Δ n j ( x , z ) d z ] ,             j [ 0 , N - 1 ] ,
Δ n j ( x , z ) = ½ n e 3 r eff E SC j cos ( Θ A + z K z + q · x + ϕ j ) ,
q = k A - k B , K z = k A z - k B z
E SC j exp ( i ϕ j ) = - m [ E Q ( E 0 + i E D ) E Q + E D - i E 0 ] j ,
κ j = 2 π λ 0 Δ n j d film j = ½ k 0 d film j n e 3 r eff E SC j ,
t j ( x ) = exp [ i κ j cos ( Θ A + l j K z + q · x + ϕ j ) ] .
t j ( x ) = a j = - C a j j exp ( i a j q · x ) .
C a j j = i a j J a j ( κ j ) exp [ i a j ( Θ A + ϕ j + l j K z ) ] ,
E R ( r ) = R exp ( i k R · r ) ,
k R = k B + k 0 Ψ + [ ( ω 0 + Δ ω ) 2 c 2 - ( k B + k 0 Ψ ) 2 ] 1 / 2 e z
k S , n = k R + n q + [ ω 2 c 2 - ( k R + n q ) 2 ] 1 / 2 e z ,
Q f = 2 π d film λ 0 n e Λ g 2 ,
E S ( 1 ) ( x ) = t 0 ( x ) E R ( x , z = 0 ) = a 0 = - R c a 0 0 exp [ i ( a 0 q + k B + k 0 Ψ ) · x ] ,
U ( k ; x , z ) = U ( k ; x , 0 ) exp [ i ( k 2 - k 2 ) 1 / 2 z ] ,
E S ( N ) ( x ) = R exp [ i ( k B + k 0 Ψ ) · x ] × j = 0 N - 2 ( a j = - C a j j exp ( i a j q · x ) exp { [ i ( l j + 1 - l j ) ] × [ ω 2 c 2 - ( q k = 0 j a k + k B + k 0 Ψ ) 2 ] 1 / 2 } ) × a N - 1 = - C a N - 1 N - 1 exp ( i a N - 1 q · x ) .
η N , Q = | E S ( N ) E R ( z = 0 ) | j = 0 N - 1 a j = Q 2 .
η N , Q = | a 0 a 1 a N - 1 j = 0 N - 1 J a j ( κ j ) exp [ i a j ( ϕ j + l j K z ) ] × j = 0 N - 2 exp { [ i ( l j + 1 - l j ) ] × [ ω 2 c 2 - ( q k = 0 j a k + k B + k 0 Ψ ) 2 ] 1 / 2 } | 2 ,
j = 0 N - 1 a j = Q .
l j = j d N - 1 ,             j [ 0 , N - 1 ] .
j = 0 N - 1 a j = 1 ,
j = 0 N - 1 J a j ( κ j ) j = 0 N - 1 κ j a j 2 a j a j ! .
j = 0 N - 1 a j = 1.
η N , 1 = 1 4 | j = 0 N - 1 κ j exp ( i ϕ j ) exp [ i 2 j N - 1 f ( Ψ , Δ ω ) ] | 2 ,
f ( Ψ , Δ ω ) = d 2 { K z + [ ( ω 0 + Δ ω ) 2 c 2 - ( k B + k 0 Ψ ) 2 ] 1 / 2 - [ ( ω 0 + Δ ω ) 2 c 2 - ( k B + q + k 0 Ψ ) 2 ] 1 / 2 } = d 2 ( K z + K R z - K S z ) = d 2 Δ k ( Ψ , Δ ω )
η N , 1 = κ 2 4 | j = 0 N - 1 exp [ i 2 j N - 1 f ( Ψ , Δ ω ) ] | 2 = κ 2 4 sin 2 [ f ( Ψ , Δ ω ) N N - 1 ] sin 2 [ f ( Ψ , Δ ω ) 1 N - 1 ] .
η thick = lim N η N , 1 .
η thick = π 2 λ 0 2 | 0 d Δ n ( z ) exp [ i ϕ ( z ) ] exp [ i 2 f ( Ψ , Δ ω ) d z ] d z | 2 ,
η thick = ν 2 sin 2 f ( Ψ , Δ ω ) f 2 ( Ψ , Δ ω ) ,
ν = lim N κ N 2 = π Δ n d λ 0
f ( Ψ , Δ ω ) ξ = d 2 ( k A k A z - k B k B z ) · Ψ k 0 + d 2 k 0 Δ ω c ( 1 k B z - 1 k A z ) .
lim N η N , 1 = sin 2 ( π Δ n d λ 0 ) , lim N η N , Q = 0 ,             Q 1 ,
SNR ( k ) = η N , 1 ( k ) ( ξ = ξ k ) j = 1 j k M η N , 1 ( j ) ( ξ = ξ k ) ,
η N , 1 ( k ) = 1 4 | j = 0 N - 1 κ j , k exp ( i ϕ j , k ) exp ( i 2 j ξ N - 1 ) | 2 ,
ϕ j , k = j k 2 π N ,             κ j , k = κ k ,             j , k [ 0 , N - 1 ] .
η N , 1 ( k ) A k 2 sin 2 ( ξ N N - 1 + π k ) sin 2 ( ξ 1 N - 1 + π k N ) .
E SC j , k = m [ E Q 2 ( E 0 j , k 2 + E D 2 ) ( E Q + E D ) 2 + E 0 j , k 2 ] 1 / 2 ,
ϕ j , k = ϕ PR ( E 0 j , k ) = arctan E D E 0 j , k ( 1 + E D E Q + E 0 j , k 2 E D E Q ) ,
η N , 1 ( k ) A k 2 | j = 0 N - 1 [ E Q 2 ( E 0 j , k 2 + E D 2 ) ( E Q + E D ) 2 + E 0 j , k 2 ] 1 / 2 × exp [ i ϕ PR ( E 0 j , k ) ] exp ( i 2 j ξ N - 1 ) | 2 .

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