Abstract

A new use of gratings in nonlinear optics is presented, i.e., the realization of bistable components. The device studied here consist of grating couplers ruled on a Kerr nonlinear medium, which use the guided-wave resonance to increase the local field and thus the nonlinearities. The result is that they are intrinsic bistable optical systems with high-speed, low pumping thresholds, and geometry well adapted to optical integration. First, a linear electromagnetic study of the devices is presented. It follows optimizing the grating parameters in order to get the best coupling between the incident beam and the guided mode inside the corrugated waveguide. Then a graphical construction is given that demonstrates the bistable character of the system in nonlinear optics. Next a nonlinear analysis of the devices is rigorously derived from Maxwell equations. It states precisely the predictions of the graphical construction and allows comparison of the effectiveness of the guided-wave resonance with the surface plasmon resonance in order to reduce the threshold of bistability.

© 1985 Optical Society of America

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  1. S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
    [Crossref]
  2. T. N. C. Venkatesan and S. L. McCall, Appl. Phys. Lett. 30, 282 (1977).
    [Crossref]
  3. A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).
  4. D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
    [Crossref]
  5. G. M. Wysin, H. J. Simon, and R. T. Deck, Opt. Lett. 6, 30 (1981).
    [Crossref] [PubMed]
  6. P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).
  7. P. Martinot, thèse 3e cycle (University of Paris XI, Orsay, France, June30, 1983).
  8. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
    [Crossref]
  9. R. Reinisch and M. Nevière, Phys. Rev. B 26, 5987 (1982).
    [Crossref]
  10. R. Reinisch and M. Nevière, Phys. Rev. B 28, 1870 (1983).
    [Crossref]
  11. M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
    [Crossref]
  12. M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
    [Crossref]
  13. M. Nevière, “The homogeneous problem,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [Crossref]
  14. R. Hulthen, Phys. Scr. 12, 342 (1975).
    [Crossref]
  15. M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
    [Crossref]
  16. R. Reinisch and M. Nevière, J. Opt. (Paris) 13, 81 (1982).
    [Crossref]
  17. M. Nevière and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
    [Crossref]
  18. R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454 (1979).
    [Crossref]
  19. P. Vincent, “Differential methods,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [Crossref]
  20. M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
    [Crossref]
  21. M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

1984 (1)

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

1983 (1)

R. Reinisch and M. Nevière, Phys. Rev. B 28, 1870 (1983).
[Crossref]

1982 (4)

R. Reinisch and M. Nevière, J. Opt. (Paris) 13, 81 (1982).
[Crossref]

M. Nevière and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
[Crossref]

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

R. Reinisch and M. Nevière, Phys. Rev. B 26, 5987 (1982).
[Crossref]

1981 (1)

1979 (3)

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[Crossref]

R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454 (1979).
[Crossref]

1977 (3)

M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
[Crossref]

T. N. C. Venkatesan and S. L. McCall, Appl. Phys. Lett. 30, 282 (1977).
[Crossref]

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

1976 (1)

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

1975 (1)

R. Hulthen, Phys. Scr. 12, 342 (1975).
[Crossref]

1974 (1)

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[Crossref]

1973 (2)

M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

Cadilhac, M.

M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

Carvalho, W.

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

Churchill, G. C.

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Deck, R. T.

Gibbs, H. M.

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Hermann, J. P.

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

Hulthen, R.

R. Hulthen, Phys. Scr. 12, 342 (1975).
[Crossref]

Jain, R. K.

R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454 (1979).
[Crossref]

Johnston, A.

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[Crossref]

Kaplan, A. E.

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

Klein, M. B.

R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454 (1979).
[Crossref]

Koster, A.

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

Laval, S.

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

Maloney, P. J.

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

Martinot, P.

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

P. Martinot, thèse 3e cycle (University of Paris XI, Orsay, France, June30, 1983).

Maystre, D.

M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
[Crossref]

McCall, S. L.

T. N. C. Venkatesan and S. L. McCall, Appl. Phys. Lett. 30, 282 (1977).
[Crossref]

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[Crossref]

Nevière, M.

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

R. Reinisch and M. Nevière, Phys. Rev. B 28, 1870 (1983).
[Crossref]

R. Reinisch and M. Nevière, Phys. Rev. B 26, 5987 (1982).
[Crossref]

R. Reinisch and M. Nevière, J. Opt. (Paris) 13, 81 (1982).
[Crossref]

M. Nevière and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
[Crossref]

M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
[Crossref]

M. Nevière, “The homogeneous problem,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

Paraire, N.

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

Petit, R.

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
[Crossref]

Reinisch, R.

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

R. Reinisch and M. Nevière, Phys. Rev. B 28, 1870 (1983).
[Crossref]

R. Reinisch and M. Nevière, Phys. Rev. B 26, 5987 (1982).
[Crossref]

R. Reinisch and M. Nevière, J. Opt. (Paris) 13, 81 (1982).
[Crossref]

M. Nevière and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
[Crossref]

Simon, H. J.

Smith, P. W.

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

Smith, S. D.

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[Crossref]

Tomlinson, W. J.

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

Venkatesan, T. N. C.

T. N. C. Venkatesan and S. L. McCall, Appl. Phys. Lett. 30, 282 (1977).
[Crossref]

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Vincent, P.

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

P. Vincent, “Differential methods,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

Wysin, G. M.

Appl. Phys. B (1)

P. Martinot, A. Koster, S. Laval, and W. Carvalho, Appl. Phys. B 29, 172 (1982).

Appl. Phys. Lett. (4)

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[Crossref]

T. N. C. Venkatesan and S. L. McCall, Appl. Phys. Lett. 30, 282 (1977).
[Crossref]

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[Crossref]

R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454 (1979).
[Crossref]

J. Opt. (Paris) (2)

M. Nevière, D. Maystre, and P. Vincent, J. Opt. (Paris) 8, 231 (1977).
[Crossref]

R. Reinisch and M. Nevière, J. Opt. (Paris) 13, 81 (1982).
[Crossref]

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[Crossref]

Opt. Commun. (2)

M. Nevière, R. Petit, and M. Cadilhac, Opt. Commun. 8, 113 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun. 9, 48 (1973).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (3)

R. Reinisch and M. Nevière, Phys. Rev. B 26, 5987 (1982).
[Crossref]

R. Reinisch and M. Nevière, Phys. Rev. B 28, 1870 (1983).
[Crossref]

M. Nevière and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
[Crossref]

Phys. Rev. Lett. (1)

S. L. McCall, H. M. Gibbs, G. C. Churchill, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Phys. Scr. (1)

R. Hulthen, Phys. Scr. 12, 342 (1975).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. Nevière, P. Vincent, N. Paraire, and R. Reinisch, “New use of gratings: bistable components,” Proc. Soc. Photo-Opt. Instrum. Eng. 503, 216 (1984).

Sov. Phys. JETP (1)

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

Other (3)

P. Martinot, thèse 3e cycle (University of Paris XI, Orsay, France, June30, 1983).

M. Nevière, “The homogeneous problem,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

P. Vincent, “Differential methods,” in The Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

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Figures (20)

Fig. 1
Fig. 1

Schematic representation of the two corrugated waveguides considered in the paper.

Fig. 2
Fig. 2

Zero-order transmitted efficiency curve for the grating with h = 0.08 μm, described in Fig. 1(a).

Fig. 3
Fig. 3

Absorbed efficiency inside the corrugated waveguide.

Fig. 4
Fig. 4

Same as Fig. 2, but for h = 0.07, 0.09, 0.12 μm.

Fig. 5
Fig. 5

Same as Fig. 3, but for h = 0.07, 0.09, 0.12 μm.

Fig. 6
Fig. 6

Zero-order transmitted efficiency curve as a function of ν2: θ = 25.07°; h = 0.12 μm.

Fig. 7
Fig. 7

Absorbed efficiency curve as a function of ν2: θ = 25.07°; h = 0.12 μm.

Fig. 8
Fig. 8

Field-intensity map inside the waveguide. (a) x = 0; (b) x = d/2.

Fig. 9
Fig. 9

(a) Absorbed, (b) reflected, and (c) transmitted efficiency curves as a function of ν2: θ = 24.7°; h = 0.12 μm. The straight lines representing Eq. (3) for different Pi are also plotted in (a).

Fig. 10
Fig. 10

Hysteresis loops in the (a) absorbed, (b) reflected, and (c) transmitted efficiencies for the device of Fig. 9.

Fig. 11
Fig. 11

Reflected zero-order efficiency as a function of incidence: h = 0.0410 μm; ν3 = 0.3 + i9.5 [curve (a)]; ν3 = i9.5 [curve (b)]; sapphire–silicon–silver device.

Fig. 12
Fig. 12

Field map inside silicon for x = 0. (a) ν3 = 0.3 + i9.5; (b) ν3 = i9.5.

Fig. 13
Fig. 13

Reflected zero-order efficiency as a function of incidence: h = 0.0226 μm; ν3 = 0.3 + i9.5.

Fig. 14
Fig. 14

Schematic representation of the planar waveguide excited by a prism coupler.

Fig. 15
Fig. 15

Reflectivity as a function of incidence for the device described in Fig. 14.

Fig. 16
Fig. 16

Field map for θ = θR and for x = 0 for the device described in Fig. 14.

Fig. 17
Fig. 17

Hysteresis loop for the device described in Fig. 14 and different angle offsets. (a) θθR = 0.022°; (b) θθR = 0.015°; (c) θθR = 0.011°.

Fig. 18
Fig. 18

Hysteresis loops for the air–silicon–sapphire device optimized in Fig. 4(c) (h = 0.12 μm). (a) Transmitted zero-order efficiency; (b) reflected zero-order efficiency; (c) sum of transmitted and reflected zero-order efficiencies.

Fig. 19
Fig. 19

Hysteresis loop for the zero-order reflected efficiency and the sapphire–silicon–silver device optimized in Fig. 11 (h = 0.0410 μm).

Fig. 20
Fig. 20

Same as Fig. 19 but for the grating in Fig. 13 (h = 0.0226μm).

Equations (28)

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A = 1 n ( E n r + E n t ) ,
ν 2 2 = ( ν 2 0 ) 2 ( 1 + α | E | 2 ) ,
ν 2 = ν 2 0 ( 1 + α 2 | E | 2 ) .
ν 2 = ν 2 = ν 2 0 + β A P i ,
ν 2 = ν 2 0 + β A P i ,
A = ν 2 ν 2 0 β P i .
A ( ν 2 ) = ν 2 ν 2 0 β P i .
× E ( r ) = j ω μ 0 H ( r ) ,
× H ( r ) = j ω D ( r ) ,
D ( r ) = 0 ( x , y ) E ( r ) + P NL ( r ) .
× ( × E ) = ω 2 μ 0 0 ( x , y ) E + ω 2 μ 0 P NL ( x , y ) .
P NL ( r ) = χ ( r ) ( 3 ) E ( r ) E ( r ) * E ( r ) ,
E ( r ) = E ( r ) e z .
Δ E + k 2 ( x , y ) E = ω 2 μ 0 P z NL ( x , y ) = ω 2 0 c 2 P z NL ( x , y ) ,
Δ E + k 0 2 ( n 0 2 + χ | E | 0 2 ) E = 0 ,
Δ E = k 0 2 n 0 2 ( 1 + α | E | 2 ) E = 0 ,
α ( x , y ) = χ ( x , y ) 0 n 0 2 ( x , y ) .
Δ E + k 0 2 n 2 ( x , y , E ) E = 0 .
Δ E 0 , 1 + k 0 2 n 0 2 E 0 , 1 = 0
Δ E p , 1 + k 0 2 n p , 1 2 E p , 1 = 0 ,
n , 1 = lim p n p , 1 .
n p , m + 1 2 = n 0 2 ( 1 + α | E p 1 , m + 1 | 2 ) p 1 ,
Δ E p , m + 1 + k 0 2 n p , m + 1 2 E p , m + 1 = 0 .
ñ p , m 2 = n p 1 , m 2 + ( n p , m 2 n p 1 , m 2 ) υ ,
P a = ω 0 2 | E | 2 ,
P a = P i A e cos θ inc .
ν 2 = ν 2 0 ( 1 + α ω 0 A cos θ inc e P i ) ,
ν 2 = ν 2 0 + n inc c cos θ inc 4 ω e ν 2 U i A ,

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