Abstract

We investigate the diffraction properties of vector synthetic gratings, which consist of two volume index gratings with different grating wave vectors, using coupled-wave analysis. These index gratings can be obtained in photorefractive materials or in Bragg cells with two driving acoustic waves. We consider the case of pseudo phase matching in which a Bragg mismatch exists in the scattering from the individual index gratings while a Bragg condition is satisfied for the scattering from the vector sum of the two index gratings. Analytic expressions for the diffraction efficiency as well as the amplitude of the three waves involved are obtained. The results are presented and discussed.

© 2000 Optical Society of America

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References

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  4. H. Lee, X. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
    [CrossRef]
  5. K. Tu, T. Tamir, and H. Lee, “Multiple scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
    [CrossRef]
  6. P. Asthana, G. P. Nordin, A. R. Tanguay, and B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic optical interconnections,” Appl. Opt. 32, 1441–1469 (1993).
    [CrossRef] [PubMed]
  7. H. Kogelnik, “Coupled wave theory for thick holographic gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
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    [CrossRef]
  9. T. C. Poon, M. R. Chatterjee, and P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
    [CrossRef]
  10. M. R. Chatterjee and S. T. Chen, “Multiple plane-wave scattering analysis of light diffraction by parallel Raman–Nath and Bragg ultrasonic cells with arbitrary frequency ratios,” J. Opt. Soc. Am. A 11, 637–648 (1994).
    [CrossRef]
  11. S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
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    [CrossRef]
  15. K. Tu and T. Tamir, “Wave diffraction by many superposed volume gratings,” Appl. Opt. 32, 3654–3660 (1993).
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  18. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
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  21. D. Yevick and L. Thylen, “Analysis of gratings by the beam propagation method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
    [CrossRef]
  22. J. Yu, D. Yevick, and D. Weidman, “A comparison of beam propagation and coupled-mode methods: application to optical fiber couplers,” Lightwave Technol. 12, 797–802 (1994).
    [CrossRef]
  23. S. Ahmed and E. N. Glytsis, “Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings,” Appl. Opt. 35, 4426–4435 (1996).
    [CrossRef] [PubMed]

1996

1995

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

1994

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

M. R. Chatterjee and S. T. Chen, “Multiple plane-wave scattering analysis of light diffraction by parallel Raman–Nath and Bragg ultrasonic cells with arbitrary frequency ratios,” J. Opt. Soc. Am. A 11, 637–648 (1994).
[CrossRef]

1993

1990

1989

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

1987

1986

1985

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

1982

1981

1978

1977

1975

1973

1969

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

Ahmed, S.

Alferness, R.

Asthana, P.

Banerjee, P. P.

Black, T. D.

Case, S. K.

Chatterjee, M. R.

Chen, S. T.

Feit, M. D.

Fleck Jr., J. A.

Gaylord, T. K.

Glytsis, E. N.

Green, M.

Gu, X.

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

Huang, T.

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

Jenkins, B. K.

Kaspar, F. G.

Kogelnik, H.

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

Larson, D. A.

Lee, H.

K. Tu, T. Tamir, and 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, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross-talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Magnusson, R.

Moharam, M. G.

Nordin, G. P.

Poon, T. C.

Psaltis, D.

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

Tamir, T.

Tanguay, A. R.

Thylen, L.

Torti, R. G.

Tu, K.

Wagner, K.

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

Wang, Y. J.

Weidman, D.

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

Yevick, D.

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

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

Appl. Opt.

Bell Syst. Tech. J.

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

IEEE J. Quantum Electron.

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

J. Appl. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Lightwave Technol.

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

Proc. IEEE

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

Other

See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (4)

Fig. 1
Fig. 1

Schematic drawing of the momentum diagram of the scattering by two index gratings.

Fig. 2
Fig. 2

Propagation of the three waves in the photorefractive medium with a perfect Bragg phase match (ΔαL=0). The parameters are the following: the angle of the two grating vectors is 12°, the grating periods of the two gratings are Λ1=Λ2=1 µm, and the wavelength of the incident beam is λ=522.64 nm. (a) κ12=κ23, (b) 2κ12=κ23, (c) κ12=2κ23. Note that the complete energy transfer is possible by a synthetic grating when the two gratings have the same coupling constants. Also note that the propagation of the diffraction beam (A3) is the same in cases (b) and (c), which means the diffraction properties of the synthetic grating are irrelevant to whether the incident beam is diffracted by the first grating first or by the second grating first, once the two gratings are given.

Fig. 3
Fig. 3

(a) Contour plot of the diffraction efficiency of a vector synthetic index grating in the ΔαLκL plane. Three-dimensional plots of the diffraction efficiency of (b) a vector synthetic index grating and (c) an individual index grating as functions of ΔαL and κL.

Fig. 4
Fig. 4

Diffraction spectra of a vector synthetic index grating and an individual grating near some 100% diffraction peaks. (a) The first 100% diffraction peak of the synthetic grating (m=0 and n=0 with κL=π) and the individual grating with κ12L=π/2. Note that an individual grating has a narrower bandwidth. (b) The third 100% diffraction peak of a synthetic grating (m=0 and n=2 with κL=5π) and an individual grating with κ12L=5π/2. Note that a vector synthetic index grating has a narrower bandwidth in this case.

Equations (41)

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n=n0+na cos Ka·r+nb cos Kb·r,
Ej=Aj(x) exp[i(ωt-kj·r)](j=1, 2, 3),
kj=αjβj0=k0cos θjsin θj0(j=1, 2, 3),
E=[A1(x) exp(-iα1x-iβ1y)+A2(x) exp(-iα2x-iβ2y)+A3(x) exp(-iα3x-iβ3y)]exp(iωx).
2E+n2k02E=0,
β2-β1=Kay,β3-β2=Kby,
β3-β1=Kay+Kby,
A1=-iκ12A2 exp(-iΔαax),
A2=-iκ21A1 exp(iΔαax)-iκ23A3 exp(-iΔαbx),
A3=-iκ32A2 exp(iΔαbx),
Δαa=α2-α1-Kax,
Δαb=α3-α2-Kbx,
κ12=πnaλ cos θ1,κ21=πnaλ cos θ2,
κ23=πnbλ cos θ2,κ32=πnbλ cos θ3.
cos θ1|A1(x)|2+cos θ2|A2(x)|2+cos θ3|A3(x)|2=const,
A1=-iκ12A2 exp(-iΔαx),
A2=-iκ21A1 exp(iΔαx)-iκ23A3 exp(iΔαx),
A3=-iκ32 A2 exp(-iΔαx),
Δα=Δαa=α2-α1-Kax.
A2-iΔαA2+κ2A2=0,
κ2=κ12κ21+κ23κ32.
A2(x)=expi Δα2x(C2 sin sx+C4 cos sx),
s2=κ2+Δα22.
A2(0)=-iκ21A1(0)-iκ23A3(0).
A1(x)=A1(0)-iκ120xA2(x) exp(-iΔαx)dx,
A3(x)=A3(0)-iκ320xA2(x) exp(-iΔαx)dx.
A1(0)=1,A2(0)=0,A3(0)=0,
A2(x)=-i κ21sexpi Δα2x sin sx.
A1(x)=κ21κ12sexp-i Δα2x s cos sx+(i/2)Δα sin sxκ2+κ23κ32κ2,
A3(x)=κ32κ12sexp-i Δα2x s cos sx+(i/2)Δα sin sxκ2-κ32κ21κ2.
ηs=cos θ3|A3(L)|2cos θ1|A1(0)|2,
ηs=cos θ3cos θ1κ32κ21κ22exp-i Δα2L(cos sL+i Δα2ssin sL)-12.
ηs=cos θ3cos θ1κ32κ21κ22(1-cos κL)2(Δα=0).
κL=(2n+1)π(n=0, 1, 2, 3, 4),
ηs max=1-na2 cos θ3-nb2 cos θ1na2 cos θ3+nb2 cos θ12.
ηs=cos θ3cos θ1κ32κ21κ222-2 cos sL cos Δα2L-1-Δα2s2 sin2 sL-2 Δα2ssin sL sin Δα2L.
ηs=cos θ3cos θ1κ32κ21κ224
ΔαL2=mπ(m=0, 1, 2, 3),
sL=(2n+1)π+mπ(n=0, 1, 2, 3).
κL=(2n+1)(2n+1+2m)π
(n=0, 1, 2, 3,,m=0, 1, 2, 3,).

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