Abstract

Bistability induced in a second-harmonic signal generated in the presence of the Kerr effect in a prism coupler is investigated. Using a rigorous solution provided by Maxwell's equations, we analyze this behavior, taking into account third-order nonlinearity (Kerr nonlinearity). Comparison with a Kerr-based bistable device shows the interest of quadratic nonlinearity for integrated optics. We show that the bistable behavior depends on the sign of third-order nonlinearity. The numerical simulations are presented for both an incident plane wave and a spatially limited incident beam.

© 1997 Optical Society of America

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References

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  1. S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).
  2. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  3. G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, and G. I. Stegeman, Opt. Lett. 14, 898 (1989).
    [CrossRef] [PubMed]
  4. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
    [CrossRef]
  5. J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
    [CrossRef]
  6. Y. Takeda, T. Motohiro, T. Hioki, and S. Noda, J. Opt. Soc. Am. B 12, 1905 (1995).
    [CrossRef]
  7. P. Vincent, H. Akhouayri, and M. Nevière, J. Opt. Soc. Am. B 8, 1149 (1991).
    [CrossRef]
  8. P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
    [CrossRef]
  9. H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
    [CrossRef]
  10. R. Reinisch, E. Popov, and M. Nevière, Opt. Lett. 20, 854 (1995).
    [CrossRef] [PubMed]
  11. M. Nevière, E. Popov, and R. Reinisch, Proc. SPIE 2532, 329 (1995).
    [CrossRef]
  12. C. Cabrillo and F. Bermejo, Phys. Rev. A 48, 2433 (1993).
    [CrossRef] [PubMed]
  13. P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
    [CrossRef]
  14. H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).
  15. E. Lantz, D. Métin, H. Cornet, and A. Lacourt, J. Opt. Soc. Am. B 11, 347 (1994).
    [CrossRef]
  16. G. M. Carter and Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
    [CrossRef]
  17. M. Haelterman, G. Vitrant, and R. Reinisch, J. Opt. Soc. Am. B 7, 1309 (1990).
    [CrossRef]
  18. G. Vitrant, M. Haelterman, and R. Reinisch, J. Opt. Soc. Am. B 7, 1319 (1990).
    [CrossRef]
  19. M. Nevière, E. Popov, and R. Reinisch, J. Opt. Soc. Am. B 12, 513 (1995).
    [CrossRef]
  20. M. Nevière, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), p. 123.

1995 (4)

R. Reinisch, E. Popov, and M. Nevière, Opt. Lett. 20, 854 (1995).
[CrossRef] [PubMed]

M. Nevière, E. Popov, and R. Reinisch, Proc. SPIE 2532, 329 (1995).
[CrossRef]

Y. Takeda, T. Motohiro, T. Hioki, and S. Noda, J. Opt. Soc. Am. B 12, 1905 (1995).
[CrossRef]

M. Nevière, E. Popov, and R. Reinisch, J. Opt. Soc. Am. B 12, 513 (1995).
[CrossRef]

1994 (1)

1993 (2)

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).

C. Cabrillo and F. Bermejo, Phys. Rev. A 48, 2433 (1993).
[CrossRef] [PubMed]

1991 (3)

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
[CrossRef]

P. Vincent, H. Akhouayri, and M. Nevière, J. Opt. Soc. Am. B 8, 1149 (1991).
[CrossRef]

1990 (2)

1989 (1)

1988 (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

1983 (1)

G. M. Carter and Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

1980 (1)

H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
[CrossRef]

1979 (1)

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
[CrossRef]

1975 (1)

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

Akhouayri, H.

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).

P. Vincent, H. Akhouayri, and M. Nevière, J. Opt. Soc. Am. B 8, 1149 (1991).
[CrossRef]

Assanto, G.

J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
[CrossRef]

G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, and G. I. Stegeman, Opt. Lett. 14, 898 (1989).
[CrossRef] [PubMed]

Bermejo, F.

C. Cabrillo and F. Bermejo, Phys. Rev. A 48, 2433 (1993).
[CrossRef] [PubMed]

Bermejo, F. J.

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Brand, H.

H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
[CrossRef]

Cabrillo, C.

C. Cabrillo and F. Bermejo, Phys. Rev. A 48, 2433 (1993).
[CrossRef] [PubMed]

Carter, G. M.

G. M. Carter and Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Chen, Y. J.

G. M. Carter and Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Churchill, G. G.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

Colet, P.

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Cornet, H.

Drummond, P. D.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
[CrossRef]

Ehrlich, J. E.

J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
[CrossRef]

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Garci´a Fernández, P.

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Gibbs, H. M.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Graham, R.

H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
[CrossRef]

Haelterman, M.

Hioki, T.

Lacourt, A.

Lantz, E.

Mc Neil, K. J.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
[CrossRef]

McCall, S. L.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

Métin, D.

Motohiro, T.

Nevière, M.

R. Reinisch, E. Popov, and M. Nevière, Opt. Lett. 20, 854 (1995).
[CrossRef] [PubMed]

M. Nevière, E. Popov, and R. Reinisch, Proc. SPIE 2532, 329 (1995).
[CrossRef]

M. Nevière, E. Popov, and R. Reinisch, J. Opt. Soc. Am. B 12, 513 (1995).
[CrossRef]

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).

P. Vincent, H. Akhouayri, and M. Nevière, J. Opt. Soc. Am. B 8, 1149 (1991).
[CrossRef]

M. Nevière, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), p. 123.

Noda, S.

Paumier, J. C.

Popov, E.

R. Reinisch, E. Popov, and M. Nevière, Opt. Lett. 20, 854 (1995).
[CrossRef] [PubMed]

M. Nevière, E. Popov, and R. Reinisch, J. Opt. Soc. Am. B 12, 513 (1995).
[CrossRef]

M. Nevière, E. Popov, and R. Reinisch, Proc. SPIE 2532, 329 (1995).
[CrossRef]

Reinisch, R.

San Miguel, M.

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Schenzle, A.

H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Stegeman, G. I.

J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
[CrossRef]

G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, and G. I. Stegeman, Opt. Lett. 14, 898 (1989).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Takeda, Y.

Toral, R.

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Venkatesan, T. N. C.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

Vincent, P.

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).

P. Vincent, H. Akhouayri, and M. Nevière, J. Opt. Soc. Am. B 8, 1149 (1991).
[CrossRef]

Vitrant, G.

Walls, D. F.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
[CrossRef]

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

G. M. Carter and Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Bull. Am. Phys. Soc. (1)

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, Bull. Am. Phys. Soc. 20, 636 (1975).

IEEE J. Quantum Electron. (1)

J. E. Ehrlich, G. Assanto, and G. I. Stegeman, IEEE J. Quantum Electron. 27, 809 (1991).
[CrossRef]

J. Lightwave Technol. (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

J. Opt. Soc. Am. B (6)

Nonlinear Opt. (1)

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, Nonlinear Opt. 5, 127 (1993).

Opt. Commun. (2)

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, Opt. Commun. 28, 255 (1979).
[CrossRef]

H. Brand, R. Graham, and A. Schenzle, Opt. Commun. 32, 359 (1980).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

C. Cabrillo and F. Bermejo, Phys. Rev. A 48, 2433 (1993).
[CrossRef] [PubMed]

P. Garcı́a Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, Phys. Rev. A 43, 4923 (1991).
[CrossRef]

Proc. SPIE (1)

M. Nevière, E. Popov, and R. Reinisch, Proc. SPIE 2532, 329 (1995).
[CrossRef]

Other (2)

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

M. Nevière, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), p. 123.

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the prism-coupler device: e2=0.9 µm, e3=0.7 µm, n1(ω)=n1(2ω)=1.0, n2(ω)=2.27+i2×10-4, n2(2ω)=2.38+i2×10-4, n3(ω)n3(2ω)=1.3+i3×10-4, n4(ω)=2.51, and n4(2ω)=2.52.

Fig. 2
Fig. 2

Reflectivity with a Kerr nonlinearity χ(3)=2×10-3 nm2/V2: δ=1×10-3 (solid curve) and δ=2×10-3 (dashed curve).

Fig. 3
Fig. 3

Reflectivity with a quadratic nonlinearity χ(2)=1×10-2 nm/V: δ=-1×10-3 (solid curve) and δ=-4×10-3 (dashed curve).

Fig. 4
Fig. 4

Nonlinear second-harmonic reflectivity (same parameters as Fig. 3).

Fig. 5
Fig. 5

Reflectivity with both of the nonlinearities: δ=-1×10-3, χ(2)=1×10-2 nm/V, χ(3)=2×10-4 nm2/V2 (solid curve), χ(3)=3×10-4 nm2/V2 (short-dashed curve), and χ(3)=5×10-4 nm2/V2 (long-dashed curve).

Fig. 6
Fig. 6

Reflectivity with both of the nonlinearities: δ=-1×10-3, χ(2)=1×10-2 nm/V, χ(3)=-2×10-4 nm2/V2 (solid curve), and χ(3)=-3×10-4 nm2/V2 (dashed curve).

Fig. 7
Fig. 7

Spatial variations of the reflected fundamental and harmonic fields. The incident field has a Gaussian shape with a width of 0.3 mm.

Fig. 8
Fig. 8

Reflected energy for the fundamental and harmonic frequencies: δ=-3×10-3, χ(2)=2×10-5 nm/V, and χ(3)=1×10-10 nm2/V2.

Equations (21)

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Pnl(2ω, r)=ε0[χ2ω(2):E(ω, r)E(ω, r)+χ2ω(3)E(2ω, r)E*(2ω, r)E(2ω, r)+2χ2ω(3)E(ω, r)E*(ω, r)E(2ω, r)],
Pnl(ω, r)=ε0[χω(2):E*(ω, r)E(2ω, r)+χω(3)E(ω, r)E*(ω, r)E(ω, r)+2χω(3)E(2ω, r)E*(2ω, r)E(ω, r)],
rotE(ω, r)=jωμ0H(ω, r),
rotH(ω, r)=-jω[ε0εωE(ω, r)+Pnl(ω, r)],
rotE(2ω, r)=2jωμ0H(2ω, r),
rotH(2ω, r)=-2jω[ε0εωE(2ω, r)+Pnl(2ω, r)].
Ex(ω)z=jωμ0Hy(ω),
Ex(ω)y=-jωμ0Hz(ω),
Hz(ω)y-Hy(ω)z=-jωε0ε(ω)Ex-jωPxNL(ω).
ΔEx(ω)+k2(ω)Ex(ω)=-μ0ω2PxNL(ω).
ΔE(ω)+kl2(ω)E(ω)
=-k02(ω){χ(2)E(2ω)E*(ω)+χ(3)×[|E(ω)|2+2|E(2ω)|2]E(ω)},
ΔE(2ω)+kl2(2ω)E(2ω)
=-k02(2ω){χ(2)E2(ω)+χ(3)×[2|E(ω)|2+|E(2ω)|2]E(2ω)},
Einc(y, z)=-+p(α-α0)exp(-iβ0y)exp(iαz)dα,
EΩ(y, z)=-+E˜Ω(α, y)exp(iαz)dα.
ΔE˜ω(α, z)+k2(ω)E˜ω(α, z)=-ωμ0P˜ωNL(α, z),
ΔE˜2ω(α, z)+k2(2ω)E˜2ω(α, z)=-2ωμ0P˜2ωNL(α, z).
P˜ωNL(α, z)=ωε(ω)[χ(2)E˜ω*(α, z)*E˜2ω(α, z)+2χ(3)E˜2ω(α, z)*E˜2ω*(α, z)*E˜ω(α, z)+χ(3)E˜ω(α, z)*E˜ω*(α, z)*E˜ω(α, z),
P˜ωNL(α, z)=ωε(2ω)[χ(2)E˜ω(α, z)*E˜ω(α, z)+χ(3)E˜2ω(α, z)*E˜2ω*(α, z)*E˜2ω(α, z)+2χ(3)E˜ω(α, z)*E˜ω*(α, z)*E˜2ω(α, z).
δ=k4(ω)[sin(θinc)-sin(θr)]/k0(ω).

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