Abstract

We utilize a novel multiple-scattering formalism to study cross-talk effects in volume holographic interconnections. Specifically, we explore the composition of that cross talk and evaluate its maximum expected values as functions of various design parameters such as dynamic range, minimum separation between interconnections, and hologram size. Examples are given for canonic interconnections that involve two superposed gratings, but the analysis and results are relevant to a broader class of high-capacity interconnects.

© 1992 Optical Society of America

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References

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  1. J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
    [CrossRef]
  2. D. Psaltis, N. Farhat, “Optical information processing based on an associative memory model of neural nets with thresholding and feedback,” Opt. Lett. 10, 98–100 (1985).
    [CrossRef] [PubMed]
  3. H. Lee, X.-g. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
    [CrossRef]
  4. 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).
  5. 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]
  6. K.-Y. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1436 (1990).
    [CrossRef]
  7. H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 53125316 (1989).
    [CrossRef]
  8. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]

1990

1989

H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 53125316 (1989).
[CrossRef]

H. Lee, X.-g. 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]

1985

1984

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

1975

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Athale, R. A.

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Farhat, N.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Gu, X.-g.

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

Kung, S. Y.

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Lee, H.

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

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

H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 53125316 (1989).
[CrossRef]

Leonberger, F. J.

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[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).

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Psaltis, D.

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

D. Psaltis, N. Farhat, “Optical information processing based on an associative memory model of neural nets with thresholding and feedback,” Opt. Lett. 10, 98–100 (1985).
[CrossRef] [PubMed]

Tamir, T.

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

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tu, K.-Y.

Appl. Opt.

H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 53125316 (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]

IEEE Trans. Microwave Theory Tech.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. A

Opt. Lett.

Proc. IEEE

J. W. Goodman, F. J. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VSLI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Other

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 (13)

Fig. 1
Fig. 1

(a) Holographic interconnection scheme between an input point i and an output point o. (b) Wave vectors for canonic interconnects with two channels.

Fig. 2
Fig. 2

Basic geometry of a volume hologram that consists of two superposed gratings that implement an interconnection involving two channels.

Fig. 3
Fig. 3

Flow chart showing multiple scattering of a single input wave by two gratings.

Fig. 4
Fig. 4

Vector diagrams illustrating higher-order diffracted waves, with wave vectors kn1,n2 shown on the left and actual propagation vectors k ¯ n 1 , n 2 shown on the right for (a) a 1 → 2 interconnect and (b) a 2 → 1 interconnect. As drawn, (a) and (b) describe complementary interconnections.

Fig. 5
Fig. 5

Reduced flow chart (up to level m = 5) for the channel established by K1 in a 1 → 2 interconnect. Bragg and off-Bragg transitions are indicated by solid and dashed lines, respectively.

Fig. 6
Fig. 6

Variation of diffraction efficiencies tn1,n2 and τn1,n2 versus angular separation Δϕ for a 1 → 2 interconnect where K1/k0 = 1.33, θ0 = 41.68°, z0/d1 = 3.25 × 104, M2 = 0 for t1,0, and M2 = M1 for τ1,0, with (a) M1 = 1.2 × 10−5 or (b) M1 = 8 × 10−6.

Fig. 7
Fig. 7

Variation of diffraction efficiencies tn1,n2 and τn1,n2 versus angular separation Δϕ for a 2 → 1 interconnect where K1/k0 = 1.33, θ0 = 41.68°, z0/d1 = 6.5 × 104, M2 = 0 for t1,0, and M2 = M1 for τ1,0, with (a) M1 = 6 × 10−5 or (b) M2 = 3 × 10−6.

Fig. 8
Fig. 8

Variation of angular separation Δϕ versus ideal efficiency t1,0 for fixed spectral cross talk γ1,0 = 0.5%, 1%, and 2% (solid lines), and spatial cross talk χ1,0 = 0.5%, 1%, and 2% (dashed lines). Here z0 = 6.5 × 104d1 is fixed and t1,0 is varied by changing M1 = M2 = M, K1/k0 = 1.33, and θ0 = 41.68°, and K2 is adjusted to maintain a Bragg condition at all Δϕ.

Fig. 9
Fig. 9

Variation of angular separation Δϕ versus ideal efficiency t1,0 for spectral cross talk δ1,0 = 1% (solid lines) in both 1 → 2 and 2 → 1 cases and spatial cross talk χ1,0 = 1% (dashed lines) in the 2 → 1 case. The parameters and their variations are the same as in Fig. 8 except that now three different values of z0 are considered, with ζ = z0/d1 = 6.5 × 104.

Fig. 10
Fig. 10

Variation of diffraction efficiency τn1,n2 versus angular separation Δϕ for complementary interconnects. Here M1 = M2 = M, K1/k0 = 1.33, and θ0 = 41.68° as in Fig. 8. Two values of z0 are plotted, with ζ = z0/d1 = 6.5 × 104, but Mz0/d1 = 0.39 is fixed in all the curves shown. The τ0,1 curves hold only for the 2 → 1 case and the t1,0 curve refers to M2 = 0.

Fig. 11
Fig. 11

Variation of diffraction efficiency τn1,n2 versus angular separation Δϕ for 1 → 2 interconnects, where K1/k0 = 1.33, θ0 = 41.68°, z0/d1 = 6.5 × 104, M1 = 4 × 10−6, and r = M2/M1 = 0.5, 1.0, and 2.5.

Fig. 12
Fig. 12

Variation of diffraction efficiency τn1,n2 versus angular separation Δϕ for 2 → 1 interconnects, where K1/k0 = 1.33, θ0 = 41.68°, z0/d1 = 6.5 104, M1 = 3 × 10−6, and r = M2/M1 = 0.5, 1.0, and 2.0.

Fig. 13
Fig. 13

Variation of angular separation Δϕ versus ideal efficiency t1,0 for spectral cross talk γ1,0 = 1% (solid lines) in both the 1 → 2 and 2 → 1 cases, and for spatial cross talk χ1,0 = 1% (dashed lines) in the 2 → 1 case. The grating parameters and their variation are the same as in Fig. 8 except that now r = M2/M1 = 0.1, 0.5, 1, 2, and 10.

Equations (34)

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k n 1 , n 2 = k i + n 1 K 1 + n 2 K 2 ,             with             n 1 , n 2 = 0 , ± 1 , ± 2 , .
= 0 [ 1 + g ( r ) ] ,             g ( r ) = M 1 cos ( K 1 · r ) + M 2 cos ( K 2 · r ) ,
E ( 0 ) = exp ( i k 0 , 0 · r ) ,
E ( r ) = m = 0 E ( m ) ( r ) ,
E ( m ) ( r ) = ν ( m ) E n ( m ) .
n ( q + 1 ) = [ n 1 ( q + 1 ) , n 2 ( q + 1 ) ] = { [ n 1 ( q ) + 1 , n 2 ( q ) ] [ n 1 ( q ) - 1 , n 2 ( q ) ] with μ ( q + 1 ) = 1 [ n 1 ( q ) , n 2 ( q ) + 1 ] [ n 1 ( q ) , n 2 ( q ) - 1 ] with μ ( q + 1 ) = 2 ,
E n ( m ) = A n ( m ) Ψ n ( m ) exp [ i k ¯ n ( m ) · r ] ,
k n 1 n 2 = x ^ u n 1 , n 2 + z ^ v n 1 , n 2 ,
k ¯ n 1 , n 2 = x ^ u n 1 , n 2 + z ^ w n 1 , n 2 ,
w n 1 , n 2 = ( k 0 2 - u n 1 , n 2 2 ) 1 / 2 .
p n 1 , n 2 = w n 1 , n 2 - v n 1 , n 2 ,
A n ( m ) = q = 1 m α n ( q ) ,
α n ( q ) = k 0 2 M μ ( q ) z 4 w n ( q ) ,
Ψ n ( m ) = L - 1 { i m s [ s + i p n ( m ) ] q = 1 m - 1 [ s + i ( p n ( m ) - p n ( q ) ) ] } 1 z m ,
E n 1 , n 2 = m = 1 ν ( m ) E n 1 , n 2 ( m ) ,
τ n 1 n 2 = w n 1 , n 2 w 0 , 0 E n 1 , n 2 2 .
I o = t o i I i ,
k i 1 2 = k 0 , 0 ,
k o 1 1 2 = k 1 , 0 = k 0 , 0 + K 1 ,
k o 2 1 2 = k 0 , 1 = k 0 , 0 + K 2 ,
t 1 , 0 = w 1 , 0 w 0 , 0 E B 1 + E B 2 + E B 4 2 ,
τ 1 , 0 = w 1 , 0 w 0 , 0 | j = 1 7 E B j + j = 1 7 E O j | 2
γ 1 , 0 = τ 1 , 0 - t 1 , 0 t 1 , 0 .
k o 2 1 = k i 1 2 ;             k i 1 2 1 = k o 1 1 2 ;             k i 2 2 1 = k o 2 1 2 .
k i 2 2 1 = k i 1 2 1 + K 1 - K 2 = k 0 , 0 + K 1 - K 2 = k 1 , - 1 .
( n 1 , n 2 ) 2 1 = ( 1 , 0 ) - ( n 1 , n 2 ) 1 2 .
E 1 , 0 2 1 = w 1 , 0 1 2 w 0 , 0 1 2 E 1 , 0 1 2 ,
χ 1 , 0 = 1 t 1 , 0 u ( n 1 , n 2 ) τ n 1 , n 2 ,
τ 0 , 1 w 0 , 1 w 0 , 0 [ α 0 , 1 2 sinc ( p 0 , 1 z 0 2 ) ] 2 ,
t 1 , 0 w 1 , 0 w 0 , 0 | α 1 , 0 | 2 .
χ 1 , 0 w 0 , 1 w 1 , 0 ( α 0 , 1 2 α 1 , 0 ) 2 sinc 2 ( p 0 , 1 z 0 2 ) ,
n ( q + 1 ) = [ n 1 ( q + 1 ) , n 2 ( q + 1 ) , n 3 ( q + 1 ) ] = { [ n 1 ( q ) + 1 , n 2 ( q ) , n 3 ( q ) ] [ n 1 ( q ) - 1 , n 2 ( q ) , n 3 ( q ) ] with μ ( q + 1 ) = 1 [ n 1 ( q ) , n 2 ( q ) + 1 , n 3 ( q ) ] [ n 1 ( q ) , n 2 ( q ) - 1 , n 3 ( q ) ] with μ ( q + 1 ) = 2 [ n 1 ( q ) , n 2 ( q ) , n 3 ( q ) + 1 ] [ n 1 ( q ) , n 2 ( q ) , n 3 ( q ) - 1 ] with μ ( q + 1 ) = 3 ,
g ( r ) = μ = 1 N g μ ( r ) ,
g μ ( r ) = M μ [ a μ 1 cos ( K μ · r ) + a μ 2 cos ( 2 K μ · r ) + a μ 3 cos ( 3 K μ · r ) + ] .

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