Abstract

A multiple-scattering technique that was recently developed to evaluate wave diffraction by two superposed gratings is extended to situations in which there is an arbitrary number of gratings. In this approach the diffraction process can be represented in terms of a flow graph that serves as a template to construct algorithms for calculating the intensity of any diffracted order. We show that such calculations do not require a large computer memory if they are implemented by judiciously tracking the relevant diffracted order throughout the flow paths. Using two types of typical grating structures as examples, we also investigate the effect of the relative grating phase on the diffraction efficiency. We thus find that the multiple-scattering analysis can readily identify those grating structures that are sensitive to the relative phase relationship.

© 1993 Optical Society of America

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References

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  1. M. G. Moharam, “Cross talk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).
  2. 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]
  3. K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
    [Crossref]
  4. N. Tsuhada, R. Tsujinishi, K. Tomishima, “Effects of the relative phase relationships of gratings on diffraction from thick holograms,” J. Opt. Soc. Am. 69, 705–711 (1979).
    [Crossref]
  5. H. P. Herzig, P. Ehbets, D. Prpngue, R. Dandliker, “Fan-out elements recorded as volume holograms: optimized recording conditions,” Appl. Opt. 31, 5716–5723 (1992).
    [Crossref] [PubMed]
  6. A. Korpel, W. Bridge, “Monte Carlo simulation of the Feynman diagram approach to strong acousto-optic interaction,” J. Opt. Soc. Am. A 7, 1503–1508 (1990).
    [Crossref]
  7. matlab, The MathWorks, Inc., South Natick, Mass., 1992.
  8. K.-Y. Tu, T. Tamir, “Full-wave multiple scattering analysis of diffraction by superposed gratings,” submitted to J. Opt. Soc. Am. A.

1992 (1)

1990 (2)

A. Korpel, W. Bridge, “Monte Carlo simulation of the Feynman diagram approach to strong acousto-optic interaction,” J. Opt. Soc. Am. A 7, 1503–1508 (1990).
[Crossref]

K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
[Crossref]

1989 (1)

1979 (1)

Bridge, W.

Dandliker, R.

Ehbets, P.

Gaylord, T. K.

Glytsis, E. N.

Herzig, H. P.

Korpel, A.

Lee, H.

K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
[Crossref]

Moharam, M. G.

M. G. Moharam, “Cross talk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).

Prpngue, D.

Tamir, T.

K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
[Crossref]

K.-Y. Tu, T. Tamir, “Full-wave multiple scattering analysis of diffraction by superposed gratings,” submitted to J. Opt. Soc. Am. A.

Tomishima, K.

Tsuhada, N.

Tsujinishi, R.

Tu, K.-Y.

K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
[Crossref]

K.-Y. Tu, T. Tamir, “Full-wave multiple scattering analysis of diffraction by superposed gratings,” submitted to J. Opt. Soc. Am. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. 7, 1421–1436 (1990).
[Crossref]

N. Tsuhada, R. Tsujinishi, K. Tomishima, “Effects of the relative phase relationships of gratings on diffraction from thick holograms,” J. Opt. Soc. Am. 69, 705–711 (1979).
[Crossref]

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

Other (3)

matlab, The MathWorks, Inc., South Natick, Mass., 1992.

K.-Y. Tu, T. Tamir, “Full-wave multiple scattering analysis of diffraction by superposed gratings,” submitted to J. Opt. Soc. Am. A.

M. G. Moharam, “Cross talk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).

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Figures (10)

Fig. 1
Fig. 1

Geometry of wave diffraction by a thick dielectric layer containing more than 2 superposed gratings.

Fig. 2
Fig. 2

Flow chart for scattering (up to the m = 3 level) by 2 gratings.

Fig. 3
Fig. 3

Parallel search for scattered components at the m = 2 level for a 2-grating case.

Fig. 4
Fig. 4

Sequential search of scattered components belonging to the m = 2 level for a 2-grating case.

Fig. 5
Fig. 5

Wave-vector k diagrams for (a) recording and (b) reading out 12 superposed gratings Kμ, with μ = 1, 2,…, 12. In (b) the letters A through L (A, B, C, K, and L are shown) denote wave vectors for the diffraction orders (1 0 0 0 0 0 0 0 0 0 0 0), (0 1 0 0 0 0 0 0 0 0 0 0),…, (0 0 0 0 0 0 0 0 0 0 0 1), respectively. The angular separations between references waves are given by Δϕj for j =1, 2,…, 11.

Fig. 6
Fig. 6

Multiple-scattering results for 12 gratings arranged as in Fig. 5(b), for z0/d1 = 500, K1/k0 = 0.4, and Mμ = M = 1.65 × 10−4 for all μ, with the angle of K1 being ϕ = π/2. The variation of diffraction efficiencies is shown for various orders A, B,…, L versus angular separation Δϕj = Δϕ= const, for all j.

Fig. 7
Fig. 7

Wave-vector k diagrams for (a) recording and (b) reading out 9 superposed gratings. The angular separations between the 3 object and 3 reference waves have the same value of Δϕ. In (b) the wave vector A, B, C, D,…, I represent the diffracted orders (1 0 0 0 0 0 0 0 0), (0 1 0 0 0 0 0 0 0), (0 0 1 0 0 0 0 0 0),…, (0 0 0 0 0 0 0 0 1), respectively. Some of the major orders appear in pairs belonging to the same diffraction orders; these pairs are B with D, C with G, and F with H.

Fig. 8
Fig. 8

Multiple-scattering results for 9 gratings arranged as in Fig. 7(b). Here z0/d1 = 500, K1/k0 = 0.4, and Mμ =M = 1.65 × 10−4 for μ = 1, 2,…, 9, with the angle of K1 being ϕ = π/2. The variation of diffraction efficiencies is shown for the A, B,…, F orders versus angular separation Δϕ [given in Fig. 7(a)]. For this case the grating phases θμ, are all set equal to zero for all μ.

Fig. 9
Fig. 9

Multiple-scattering results for 9 gratings arranged as in Fig. 7(b). The variation of diffraction efficiency is shown versus angular separation Δϕ [given in Fig. 7(a)]. The parameters are the same as those in Fig. 8 except that the grating phases for the wave vectors A, B, C, D, E, F, G, H, and I are given by a set of random numbers (2.0485, 0.47229, 2.1405, 1.2121, 1.2181, 1.5700, 0.4635, 1.8447, and 2.6565, respectively).

Fig. 10
Fig. 10

Simplified k diagrams showing some of the diffraction orders involved in (a) the 12 gratings of Fig. 5 and (b) the 9-grating case of Fig. 7.

Equations (17)

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( r ) = 0 [ 1 + g ( r ) ] , g ( r ) = μ = 1 N M μ cos ( K μ · r + θ μ ) .
( 2 + k 0 2 ) E ( r ) = k 0 2 g ( r ) E ( r ) .
E ( r ) = E ( 0 ) ( r ) + k 0 2 G ( r , r i ) g ( r i ) E ( r i ) d r i ,
E ( r ) = m = 0 ν ( m ) E n ( m ) ,
n ( m + 1 ) = [ n 1 ( m + 1 ) , n 2 ( m + 1 ) , , n N ( m + 1 ) ] = { [ n 1 ( m + 1 ) + 1 , n 2 ( m + 1 ) , , n N ( m + 1 ) ] [ n 1 ( m + 1 ) 1 , n 2 ( m + 1 ) , , n N ( m + 1 ) ] for μ ( m + 1 ) = 1 [ n 1 ( m + 1 ) , n 2 ( m + 1 ) + 1 , , n N ( m + 1 ) ] [ n 1 ( m + 1 ) , n 2 ( m + 1 ) 1 , , n N ( m + 1 ) ] for μ ( m + 1 ) = 2 [ n 1 ( m + 1 ) , n 2 ( m + 1 ) , , n N ( m + 1 ) + 1 ] [ n 1 ( m + 1 ) , n 2 ( m + 1 ) , , n N ( m + 1 ) 1 ] for μ ( m + 1 ) = N ,
E n ( m ) = A n ( m ) Ψ n ( m ) exp ( i k ¯ n ( m ) · r ) ,
k ¯ n ( m ) = x ˆ u n ( m ) + z ˆ w n ( m ) ,
u n ( m ) = u 0 + n 1 ( m ) U 1 + n 2 ( m ) U 2 + + n N ( m ) U N ,
w n ( m ) = [ k 0 2 u n ( m ) 2 ] 1 / 2 .
A n ( m ) = q = 1 m α n ( q ) = q = 1 m k 0 2 M μ ( q ) z 0 4 ω n ( q ) exp [ i r ( q ) θ μ ( q ) ] ,
Ψ n ( m ) = L 1 { i m s ( s + i Δ n ( m ) ) q = 1 m 1 [ s + i ( Δ n ( m ) Δ n ( q ) ) ] } ζ = 1 ,
Δ n ( q ) = [ w n ( q ) υ n ( q ) ] z 0 ,
υ n ( q ) = υ 0 + n 1 ( q ) V 1 + n 2 ( q ) V 2 + + n N ( q ) V N ,
E n 1 , n 2 , , n N = m = 0 ν ( m ) E n 1 , n 2 , , n N ( m ) ,
E ( m ) = ν ( m ) E n ( m ) | E ( m ) | = μ ( m ) | E n ( m ) | .
| E ( m ) | max = ( 2 N α max ) m m ! .
| E ( m + 1 ) | max | E ( m ) | max = ( 2 N α max ) m + 1 ( m + 1 ) ! ( 2 N α max ) m m ! = 2 N α max m + 1 ,

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