Abstract

A unified nonlinear modal theory that is valid not only for waveguide couplers but also for nonlinear Fabry–Perot resonators has recently been proposed. The validity of this new approach is numerically demonstrated by comparison with the well-known bidimensional theory. The unified modal theory appears to be a powerful tool for studying transverse effects in a broad class of nonlinear resonators. It has been used to derive the main characteristics of the stationary behavior of the optical resonators in the case of a local nonlinearity. In particular, the disappearance of optical bistability when the resonator is tilted is predicted. Optical bistability at nearly normal incidence is interpreted by the coupling between two resonantly excited counterpropagating modes that are at the origin of the required transverse feedback.

© 1990 Optical Society of America

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  1. J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
    [CrossRef]
  2. U. Olin, O. Sahlén, “Transverse effects in switching of bistable Fabry–Perot étalons filled with a saturable medium,” J. Opt. Soc. Am. B 4, 319 (1987).
    [CrossRef]
  3. D. Weaire, J. P. Kermode, “Dispersive optical bistability: numerical methods and definitive results,” J. Opt. Soc. Am. B 3, 1706 (1986).
    [CrossRef]
  4. J.V. Moloney, “Bistable behaviour of a detuned Fabry–Pérot étalon with a Gaussian input spatial profile under self-focusing and defocusing conditions,” Opt. Acta 29, 1503 (1982).
    [CrossRef]
  5. M. Haelterman, G. Vitrant, R. Reinisch, “Transverse effects in nonlinear planar resonators. I. Modal theory,” J. Opt. Soc. Am. B 7, 1309 (1990).
    [CrossRef]
  6. M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
    [CrossRef]
  7. G. M. Carter, Y. J. Chen, “Nonlinear optical coupling between radiation and confined modes,” Appl. Phys. Lett. 42, 643 (1983).
    [CrossRef]
  8. G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, G. I. Stegeman, “Nonlinear prism coupler with nonlocality,” Opt. Lett. 14, 898 (1989).
    [CrossRef] [PubMed]
  9. R. Reinisch, G. Vitrant, “Electromagnetic-resonance-induced optical response of a thin nonlinear dielectric film,” Phys. Rev. B 39, 5775 (1989).
    [CrossRef]
  10. W. Chen, D. L. Mills, “Optical response of a nonlinear dielectric film.” Phys. Rev. B 35, 534 (1987); J. Danckaert, H. Thienpont, I. Veretennicoff, M. Haelterman, P. Mandel, “Self-consistent stationary description of a nonlinear Fabry–Pérot,” Opt. Commun. 71, 317 (1989).
    [CrossRef]
  11. F. S. Felber, J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731 (1976).
    [CrossRef]
  12. W. J. Firth, I. Galbraith, E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005 (1985).
    [CrossRef]
  13. U. Olin, “Effects of diffraction and diffusion in dispersive optical bistability in Fabry–Perot étalons,” J. Opt. Soc. Am. B 5, 20 (1988).
    [CrossRef]
  14. H. F. Harmuth, “On the solution of the Schrödinger and the Klein Gordon equations by digital computers,” J. Math. Phys. 36, 269 (1957).
  15. M. Kubicek, “Dependence of solution of nonlinear systems on a parameter,” Assoc. Comput. Mach. Trans. Math Software, 2, 98 (1976).
    [CrossRef]
  16. H. B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977), 359–384
  17. L. A. Lugiato, C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896 (1988).
    [CrossRef] [PubMed]
  18. L. A. Lugiato, L. M. Narducci, “Nonlinear dynamics in a Fabry–Pérot resonator,” Z. Phys. B 71, 129 (1988).
    [CrossRef]

1990 (2)

M. Haelterman, G. Vitrant, R. Reinisch, “Transverse effects in nonlinear planar resonators. I. Modal theory,” J. Opt. Soc. Am. B 7, 1309 (1990).
[CrossRef]

M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
[CrossRef]

1989 (2)

G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, G. I. Stegeman, “Nonlinear prism coupler with nonlocality,” Opt. Lett. 14, 898 (1989).
[CrossRef] [PubMed]

R. Reinisch, G. Vitrant, “Electromagnetic-resonance-induced optical response of a thin nonlinear dielectric film,” Phys. Rev. B 39, 5775 (1989).
[CrossRef]

1988 (3)

U. Olin, “Effects of diffraction and diffusion in dispersive optical bistability in Fabry–Perot étalons,” J. Opt. Soc. Am. B 5, 20 (1988).
[CrossRef]

L. A. Lugiato, C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, L. M. Narducci, “Nonlinear dynamics in a Fabry–Pérot resonator,” Z. Phys. B 71, 129 (1988).
[CrossRef]

1987 (3)

W. Chen, D. L. Mills, “Optical response of a nonlinear dielectric film.” Phys. Rev. B 35, 534 (1987); J. Danckaert, H. Thienpont, I. Veretennicoff, M. Haelterman, P. Mandel, “Self-consistent stationary description of a nonlinear Fabry–Pérot,” Opt. Commun. 71, 317 (1989).
[CrossRef]

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

U. Olin, O. Sahlén, “Transverse effects in switching of bistable Fabry–Perot étalons filled with a saturable medium,” J. Opt. Soc. Am. B 4, 319 (1987).
[CrossRef]

1986 (1)

1985 (1)

1983 (1)

G. M. Carter, Y. J. Chen, “Nonlinear optical coupling between radiation and confined modes,” Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

1982 (1)

J.V. Moloney, “Bistable behaviour of a detuned Fabry–Pérot étalon with a Gaussian input spatial profile under self-focusing and defocusing conditions,” Opt. Acta 29, 1503 (1982).
[CrossRef]

1976 (2)

F. S. Felber, J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731 (1976).
[CrossRef]

M. Kubicek, “Dependence of solution of nonlinear systems on a parameter,” Assoc. Comput. Mach. Trans. Math Software, 2, 98 (1976).
[CrossRef]

1957 (1)

H. F. Harmuth, “On the solution of the Schrödinger and the Klein Gordon equations by digital computers,” J. Math. Phys. 36, 269 (1957).

Assanto, G.

Carter, G. M.

G. M. Carter, Y. J. Chen, “Nonlinear optical coupling between radiation and confined modes,” Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Chen, W.

W. Chen, D. L. Mills, “Optical response of a nonlinear dielectric film.” Phys. Rev. B 35, 534 (1987); J. Danckaert, H. Thienpont, I. Veretennicoff, M. Haelterman, P. Mandel, “Self-consistent stationary description of a nonlinear Fabry–Pérot,” Opt. Commun. 71, 317 (1989).
[CrossRef]

Chen, Y. J.

G. M. Carter, Y. J. Chen, “Nonlinear optical coupling between radiation and confined modes,” Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

English, J. H.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

Felber, F. S.

F. S. Felber, J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731 (1976).
[CrossRef]

Firth, W. J.

Galbraith, I.

Gossard, A. C.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

Haelterman, M.

M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
[CrossRef]

M. Haelterman, G. Vitrant, R. Reinisch, “Transverse effects in nonlinear planar resonators. I. Modal theory,” J. Opt. Soc. Am. B 7, 1309 (1990).
[CrossRef]

Harmuth, H. F.

H. F. Harmuth, “On the solution of the Schrödinger and the Klein Gordon equations by digital computers,” J. Math. Phys. 36, 269 (1957).

Jewell, J. L.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

Keller, H. B.

H. B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977), 359–384

Kermode, J. P.

Kubicek, M.

M. Kubicek, “Dependence of solution of nonlinear systems on a parameter,” Assoc. Comput. Mach. Trans. Math Software, 2, 98 (1976).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, L. M. Narducci, “Nonlinear dynamics in a Fabry–Pérot resonator,” Z. Phys. B 71, 129 (1988).
[CrossRef]

Marburger, J. H.

F. S. Felber, J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731 (1976).
[CrossRef]

McCall, S. L.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

Mills, D. L.

W. Chen, D. L. Mills, “Optical response of a nonlinear dielectric film.” Phys. Rev. B 35, 534 (1987); J. Danckaert, H. Thienpont, I. Veretennicoff, M. Haelterman, P. Mandel, “Self-consistent stationary description of a nonlinear Fabry–Pérot,” Opt. Commun. 71, 317 (1989).
[CrossRef]

Moloney, J.V.

J.V. Moloney, “Bistable behaviour of a detuned Fabry–Pérot étalon with a Gaussian input spatial profile under self-focusing and defocusing conditions,” Opt. Acta 29, 1503 (1982).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, “Nonlinear dynamics in a Fabry–Pérot resonator,” Z. Phys. B 71, 129 (1988).
[CrossRef]

Oldano, C.

L. A. Lugiato, C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

Olin, U.

Paumier, J. C.

Reinisch, R.

Sahlén, O.

Scherer, A.

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

Stegeman, G. I.

Tolley, M. D.

M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
[CrossRef]

Vitrant, G.

M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
[CrossRef]

M. Haelterman, G. Vitrant, R. Reinisch, “Transverse effects in nonlinear planar resonators. I. Modal theory,” J. Opt. Soc. Am. B 7, 1309 (1990).
[CrossRef]

G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, G. I. Stegeman, “Nonlinear prism coupler with nonlocality,” Opt. Lett. 14, 898 (1989).
[CrossRef] [PubMed]

R. Reinisch, G. Vitrant, “Electromagnetic-resonance-induced optical response of a thin nonlinear dielectric film,” Phys. Rev. B 39, 5775 (1989).
[CrossRef]

Weaire, D.

Wright, E. M.

Appl. Phys. Lett. (3)

J. L. Jewell, A. Scherer, S. L. McCall, A. C. Gossard, J. H. English, “GaAs-AlAs monolithic microresonator arrays,” Appl. Phys. Lett. 51, 94 (1987).
[CrossRef]

F. S. Felber, J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731 (1976).
[CrossRef]

G. M. Carter, Y. J. Chen, “Nonlinear optical coupling between radiation and confined modes,” Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Assoc. Comput. Mach. Trans. Math Software (1)

M. Kubicek, “Dependence of solution of nonlinear systems on a parameter,” Assoc. Comput. Mach. Trans. Math Software, 2, 98 (1976).
[CrossRef]

J. Appl. Phys. (1)

M. Haelterman, M. D. Tolley, G. Vitrant, “Transverse effects in optical bistability with the nonlinear Fabry–Pérot: a new theoretical approach,” J. Appl. Phys. 67, 2725 (1990).
[CrossRef]

J. Math. Phys. (1)

H. F. Harmuth, “On the solution of the Schrödinger and the Klein Gordon equations by digital computers,” J. Math. Phys. 36, 269 (1957).

J. Opt. Soc. Am. B (5)

Opt. Acta (1)

J.V. Moloney, “Bistable behaviour of a detuned Fabry–Pérot étalon with a Gaussian input spatial profile under self-focusing and defocusing conditions,” Opt. Acta 29, 1503 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

L. A. Lugiato, C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (2)

R. Reinisch, G. Vitrant, “Electromagnetic-resonance-induced optical response of a thin nonlinear dielectric film,” Phys. Rev. B 39, 5775 (1989).
[CrossRef]

W. Chen, D. L. Mills, “Optical response of a nonlinear dielectric film.” Phys. Rev. B 35, 534 (1987); J. Danckaert, H. Thienpont, I. Veretennicoff, M. Haelterman, P. Mandel, “Self-consistent stationary description of a nonlinear Fabry–Pérot,” Opt. Commun. 71, 317 (1989).
[CrossRef]

Z. Phys. B (1)

L. A. Lugiato, L. M. Narducci, “Nonlinear dynamics in a Fabry–Pérot resonator,” Z. Phys. B 71, 129 (1988).
[CrossRef]

Other (1)

H. B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977), 359–384

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Figures (7)

Fig. 1
Fig. 1

Physical structure under study. See text for definition of labels.

Fig. 2
Fig. 2

a, Comparison between the bidimensional theory [curves (a)–(c)] and the unified modal theory [curve (d)] for various resonator finesses. f(X = 0) is plotted versus g. Curves (a)–(c) correspond to increasing finesses and to the same modal curve (d). The numerical parameters are specified in Table 1. b, Comparison between the bidimensional theory and the unified modal theory for narrow incident beams. f(X = 0) is plotted versus g. The resonator finesse is Fc = 30.6, and its optical thickness is kyL = 9.9555π. The plane-wave response is plotted on curve (a), and curves (b)–(d) correspond to decreasing beam waists. The beam radius (at intensity 1/e) k y x 0 / 2 is, respectively, 12.58π, 6.29π, 3.15π for curves (b), (c), (d). For every curve the result of the bidimensional calculation (solid curves) is almost superimposed upon the result of the modal one (dashed curves).

Fig. 3
Fig. 3

Influence of the incident beam width X0, at normal incidence, on the response of the Fabry–Perot resonator, i.e., the transmitted amplitude at the center: a, positive nonlinearity; b, negative nonlinearity. The plane-wave response under the same conditions is reported for each. Δ = 3η.

Fig. 4
Fig. 4

Field maps of the transmitted beam: working points on the upper levels of the associated bistable curves of a, Fig. 3a for an incident amplitude g = 2.5; b, Fig. 3b for g = 3.

Fig. 5
Fig. 5

Influence of the angle of incidence on the response demonstrated for the case of a negative nonlinearity: Bistability disappears as α0 increases. The reduced half-width of the beam is, respectively, X0 = 4, 5.66, 10.45 for curves (a), (b), (c) of a and 20.4, 40.2 for curves (a), (b) of b. A comparison with the solution of Carter and Chen under the same conditions is shown as curve (c) of b. Δ = 3.4.

Fig. 6
Fig. 6

a, Output field maps associated with points A, B, C of curve (c) of Fig. 5a. b, Comparison between field maps associated with curves (b) and (c) of Fig. 5b for an incident beam amplitude, g = 5. The shape of the Gaussian incident beam is presented for comparison in both a and b.

Fig. 7
Fig. 7

Threshold for normalized angle (α0,t)2 plotted versus cavity detuning: a, positive nonlinearity; b, negative nonlinearity. Every curve corresponds to a specified value of the incident-beam width.

Tables (1)

Tables Icon

Table 1 Numerical Parameters for Curves (a)–(c) of Fig. 2a

Equations (31)

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2 j k y E f y + 2 j β 0 E f x + 2 E f x 2 = A NL ( | E f | 2 + 2 | E b | 2 ) E f ,
+ 2 j k y E b y + 2 j β 0 E b x + 2 E b x 2 = A NL ( | E b | 2 + 2 | E f | 2 ) E b ,
β 0 = ω c n i sin ( θ i ) = ω c n t sin ( θ t ) , k y = [ ( ω c n 0 ) 2 ( β 0 ) 2 ] 1 2 , A NL = 2 ω 2 c 2 n 0 n 2 .
E f ( x , 0 ) = R f E b ( x , 0 ) + T f A i ( x ) ,
E b ( x , y = L ) = R b E f ( x , y = L ) ,
A i ( x ) = A i exp { [ x cos ( θ i ) / x 0 ] 2 } ,
A t ( x ) = T b E f ( x , y = L ) .
E f , b ( x = x l , r , y ) = 0.
d 2 A t d x 2 + 2 j β 0 d A t d x + ( β 2 β 2 + η ζ 2 | A t ( x ) | 2 ) A t ( x ) + j σ A t ( x ) = j σ A i ( x ) .
β m 2 = ( ω c n 0 ) 2 ( m π L ) 2 , σ = | T f T b | l m 2 ,
l m = L m π , m is an integer .
ζ 2 = 3 1 + | R b | 2 | T b | 2 ( ω c ) 2 n 0 n 2 = 3 2 | A NL | 1 + | R b | 2 | T b | 2 .
r = | R f R b | , t = | T f T b | .
T = t e j ϕ / 2 1 r e j ϕ ,
ϕ ( β 2 ) = 2 L [ ( ω c n 0 ) 2 β 2 ] 1 / 2 .
γ 2 = ρ m 2 + j κ m 2 , ρ m 2 = ( ω c n 0 ) 2 ( 2 m π ) 2 [ ln ( r ) ] 2 4 L 2 , κ m 2 = m π ln ( r ) L 2 .
ϕ ( β 2 ) ϕ ( γ m 2 ) + d ϕ d β 2 | γ m 2 ( β 2 γ m 2 ) .
d 2 A t d x 2 + 2 j β 0 d A t d x + [ ρ m 2 β 0 2 + η ζ 2 | A t ( x ) | 2 ] A t + j κ m 2 A t = j ξ A i ( x ) ,
ξ = t r ( 1 ) m 2 m π + j ln ( r ) 2 L 2 .
T = j ξ ( ρ m 2 β 0 2 ) + j κ m 2 .
| ξ | t r m π L 2 m π log ( r ) L 2 .
Δ n = 3 2 n 2 1 + | R b | 2 | T b | 2 | A t | 2 .
X = κ m x ,
f = ζ κ m A t , g = ξ ζ κ m 3 A i ,
Δ = β 0 2 ρ m 2 κ m 2 ,
α 0 = β 0 κ m .
d 2 f d X 2 + 2 j α 0 d f d X + ( j Δ + η | f | 2 ) f = j g exp ( X 2 X 0 2 ) .
ρ m 2 β 0 2 2 β 0 ( γ m , r β 0 ) , κ m 2 2 β 0 γ m , i , | ξ | 2 β 0 γ m , i .
d A t d x = j [ ( γ m , r β 0 ) + j γ m , i + η ζ 2 2 β 0 | A t ( x ) | 2 ] A t + γ m , i A i ( x ) ,
d f d X = j [ Δ + j + η | f ( X ) | 2 ] f ( X ) + g ( X ) ,
Δ = β 0 γ m , r γ m , i , X = x γ m , i .

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