Abstract

Multiplexed holographic structures have been suggested to provide large capacity and parallel access as three-dimensional storage media. One of the most widely used techniques in the literature for analyzing such structures has been the coupled-wave analysis and its variations. Another approach that is becoming increasingly popular because of the ease with which it can be implemented is the beam propagation method (BPM). The BPM is quantitatively compared with the rigorous coupled-wave analysis for the cases of single and multiplexed gratings. Normal and off-normal incidence as well as TE and TM polarizations are considered for single (slanted and unslanted) and multiplexed gratings. It is shown that the BPM, even in its most rudimentary form, is a powerful and accurate calculational method that is especially suited for analyzing the many multiplexed grating diffraction problem.

© 1996 Optical Society of America

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  1. H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
    [CrossRef]
  2. E. Maniloff, K. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
    [CrossRef]
  3. P. Asthana, G. P. Nordin, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic optical interconnections,” Appl. Opt. 32, 1441–1469 (1993).
    [CrossRef] [PubMed]
  4. H. Tu, T. Tamir, H. Lee, “Multiple scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta. 25, 67–81 (1978).
    [CrossRef]
  9. T. Huang, K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31, 372–390 (1995).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planargrating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  11. T. K. Gaylord, M. G. Moharm, “Analysis and applications of optical difraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  12. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  13. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
    [CrossRef]
  14. E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled-wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
    [CrossRef] [PubMed]
  15. E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anistropic grating diffraction,” J. Opt. Soc. Am A 7, 1399–1420 (1990).
    [CrossRef]
  16. K. Tu, T. Tamir, “Wave diffraction by many superposed volume gratings,” Appl. Opt. 32, 3654–3660 (1993).
    [CrossRef] [PubMed]
  17. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  18. D. Yevick, L. Thylen, “Analysis of gratings by the beam propagation method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
    [CrossRef]
  19. P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
    [CrossRef]
  20. Y. Chung, N. Dagli, “An assesment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
    [CrossRef]
  21. A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
    [CrossRef]
  22. W. P. Huang, C. L. Xu, “A wide angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
    [CrossRef]
  23. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anistropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  24. R. V. Johnson, A. R. Tanguay, “Optical beam propagation method birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
  25. H. E. Hernandez-Figueroa, “Simple nonparaxial beam propagation method for integrated optics,” J. Lightwave Technol. 12, 644–649 (1994).
    [CrossRef]
  26. J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
    [CrossRef]
  27. G. P. Nordin, R. V. Johnson, A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A 9, 2206–2217 (1992).
    [CrossRef]

1995 (1)

T. Huang, K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31, 372–390 (1995).
[CrossRef]

1994 (2)

H. E. Hernandez-Figueroa, “Simple nonparaxial beam propagation method for integrated optics,” J. Lightwave Technol. 12, 644–649 (1994).
[CrossRef]

J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
[CrossRef]

1993 (2)

1992 (2)

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

W. P. Huang, C. L. Xu, “A wide angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
[CrossRef]

1991 (2)

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
[CrossRef]

E. Maniloff, K. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

1990 (3)

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anistropic grating diffraction,” J. Opt. Soc. Am A 7, 1399–1420 (1990).
[CrossRef]

Y. Chung, N. Dagli, “An assesment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

H. Tu, T. Tamir, H. Lee, “Multiple scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
[CrossRef]

1989 (2)

H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled-wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[CrossRef] [PubMed]

1988 (1)

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

1987 (1)

1986 (1)

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

1985 (1)

T. K. Gaylord, M. G. Moharm, “Analysis and applications of optical difraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (1)

1982 (2)

1981 (1)

1978 (2)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta. 25, 67–81 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

1977 (1)

1971 (1)

1969 (1)

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

Asthana, P.

Chung, Y.

Y. Chung, N. Dagli, “An assesment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Dagli, N.

Y. Chung, N. Dagli, “An assesment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Gaylord, T. K.

Glytsis, E. N.

Gu, X.

H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Hernandez-Figueroa, H. E.

H. E. Hernandez-Figueroa, “Simple nonparaxial beam propagation method for integrated optics,” J. Lightwave Technol. 12, 644–649 (1994).
[CrossRef]

Huang, T.

T. Huang, K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31, 372–390 (1995).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, “A wide angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
[CrossRef]

Jenkins, B. K.

Johnson, K.

E. Maniloff, K. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

Johnson, R. V.

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

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

Kaczmarski, P.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

Kogelnik, H.

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

Kong, J. A.

Kowarschik, R.

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta. 25, 67–81 (1978).
[CrossRef]

Lagasse, P. E.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

Lee, H.

H. Tu, T. Tamir, H. Lee, “Multiple scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
[CrossRef]

H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Magnusson, R.

Majd, M.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
[CrossRef]

Maniloff, E.

E. Maniloff, K. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

Moharam, M. G.

Moharm, M. G.

T. K. Gaylord, M. G. Moharm, “Analysis and applications of optical difraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Nordin, G. P.

Petermann, K.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
[CrossRef]

Psaltis, D.

H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Splett, A.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
[CrossRef]

Tamir, T.

Tanguay, A. R.

Thylen, L.

Tu, H.

Tu, K.

Wagner, K.

T. Huang, K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31, 372–390 (1995).
[CrossRef]

Weidman, D.

J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, “A wide angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
[CrossRef]

Yevick, D.

J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
[CrossRef]

D. Yevick, L. Thylen, “Analysis of gratings by the beam propagation method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
[CrossRef]

Yu, J.

J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

Electron. Lett. (1)

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Chung, N. Dagli, “An assesment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

T. Huang, K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31, 372–390 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large referactive index steps and large propagation distances,” IEEE Photon. Technol. Lett. 3, 1066–1068 (1991).
[CrossRef]

W. P. Huang, C. L. Xu, “A wide angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
[CrossRef]

J. Appl. Phys. (2)

H. Lee, X. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

E. Maniloff, K. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

J. Lightwave Technol. (2)

H. E. Hernandez-Figueroa, “Simple nonparaxial beam propagation method for integrated optics,” J. Lightwave Technol. 12, 644–649 (1994).
[CrossRef]

J. Yu, D. Yevick, D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” J. Lightwave Technol. 12, 797–802 (1994).
[CrossRef]

J. Opt. Soc. Am A (1)

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anistropic grating diffraction,” J. Opt. Soc. Am A 7, 1399–1420 (1990).
[CrossRef]

J. Opt. Soc. Am. (6)

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

Opt. Acta. (1)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta. 25, 67–81 (1978).
[CrossRef]

Opt. Eng. (1)

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

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharm, “Analysis and applications of optical difraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of wave diffraction by a dielectric volume containing two superposed gratings.

Fig. 2
Fig. 2

Diffraction efficiency (DE) versus grating thickness (d) plots of 0 order and +1 order for single unslanted grating where the incident field is (a) TE polarized and M 1 = 0.01 and (b) TE polarized and M 1 = 0.1. The grating is Bragg-matched for +1 order with period Λ = 5 μm. TM polarization results are similar.

Fig. 3
Fig. 3

Diffraction efficiency (DE) versus grating thickness (d) plots of 0 order and +1 order for single unslanted grating (M 1 = 0.01) where the incident field is (a) TE polarized and θ = 5°, (b) TE polarized and θ = 10°, and (c) TM polarized and θ = 10°. The gratings are Bragg matched for +1 order so that when θ = 5°, Λ = 2.868 μm and when θ = 10°, Λ = 1.440 μm.

Fig. 4
Fig. 4

Diffraction efficiency (DE) versus grating thickness (d) plots of 0 order and +1 order for single-slanted grating (M 1 = 0.01) where the incident field is (a) TE polarized and ϕ = 85°, (b) TE polarized and ϕ = 80°, and (c) TM polarized and ϕ = 80°. The gratings are Bragg matched for +1 order so that when ϕ = 85°, Λ = 2.868 μm and when ϕ = 80°, Λ = 1.440 μm.

Fig. 5
Fig. 5

Diffraction efficiency (DE) versus grating thickness (d) plots of (0, 0) order and (0, 1) order for two-multiplexed gratings where Δϕ is large and the incident field is (a) TE polarized, ϕ1 = 85° and ϕ2 = 90° and (b) TM polarized, ϕ1 = 85° and ϕ2 = 90°.

Fig. 6
Fig. 6

Diffraction efficiency (DE) versus grating thickness (d) plots of (0, 0)- and (0, 1)-orders for two-multiplexed gratings where Δϕ is small and the incident field is (a) TE polarized (0, 0)- and (1, 0)-orders with ϕ1 = 85° and ϕ2 = 85.5° and (b) TE polarized (0, 1)-and (1, −1)-orders with ϕ1 = 85° and ϕ2 = 85.5°. TM results look similar.

Fig. 7
Fig. 7

(a) Fourier coefficients of the various propagating orders that exist in the modulated region that contains 11 angularly multiplexed gratings. (b) Diffraction efficiency (DE) versus grating thickness (d) plot of the (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) and (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1) orders for the same case. The +1 orders for the other 10 gratings were similar to the eleventh grating.

Tables (2)

Tables Icon

Table 1 Periodicities (Λ) and Slant Angles (ϕ) of the 11-Multiplexed Gratings a

Tables Icon

Table 2 Error Bounds of the BPM for Single and Multiplexed Gratings: BPM versus RCWA Maximum Error

Equations (15)

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2 E ( x , z ) + k 0 2 ε ( x , z ) E ( x , z ) = 0 ,
ε ( x , z ) = ε 0 [ 1 + ε 1 ( x , z ) ] , ε 1 ( x , z ) = l = 1 2 M l cos ( K l x x + K l z z ) ,
E ( x , z + Δ z ) = P Q P E ( x , z ) ,
P = P TE = exp ( j Δ z 4 k 0 2 x 2 ) , Q = exp { j k 0 Δ z 2 [ ε ( x , z + Δ z / 2 ) / ε 0 1 ] } ,
2 H ( x , z ) + k 0 2 ε ( x , z ) H ( x , z ) ( Δ ε ε · ) H ( x , z ) = 0 ,
ε ε · H ( x , z ) ε 0 [ l = 1 2 K l x M l sin ( K l x x ) ] ε 0 [ 1 + l = 1 2 M l cos ( K l x x ) ] x H ( x , z ) C H x ,
P = P TM = exp ( j Δ z 4 k 0 2 x 2 + C x ) ;
E y ( x , z ) = i 1 = i 2 = S i 1 , i 2 ( z ) exp ( j σ i 1 , i 2 · r ) ,
D E = k x | E ( k x ) | 2 Re { cos θ ( k x ) } d k x / k x | E ( k x ) | 2 Re { cos θ ( k x ) } d k x .
r ( x , z ) = ε ( x , z ) ε ( x , z + Δ z ) ε ( x , z ) + ε ( x , z + Δ z ) ,
R ( k x , z ) = k z k z + Δ z k z + k z + Δ z ,
k z = k 0 2 ε r ( z ) k x 2 , k z + Δ z = k 0 2 ε r ( z + Δ z ) k x 2 , k x = k 0 sin ( θ ) 2 π m L .
η = γ 2 sin 2 γ 2 + ξ 2 γ 2 + ξ 2 ,
γ = π ε 0 M 1 d λ 0 cos θ , ξ = | sin θ sin θ B | π d Λ cos θ ,
Δ θ 2 ξ Λ π d .

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