Abstract

The calculus of variations is applied to electromagnetic fields in a layered nonlinear structure supporting a guided wave. The system also includes a phase-conjugate mirror (PCM). By introducing a variational dimension and using a collection of plane waves as a trial function, we approximate the exact solution of the nonlinear Kerr–Maxwell equation. The formalism is new, and it involves the nonlinear interference of multiple plane waves. A simple analytical expression for the nonlinear field in the presence of the PCM is derived, and the fact that the scattered intensities may become bistable when the angle of incidence is varied is demonstrated. In particular, our theory predicts the angular bistability in the backscattering direction, where the effect of the guided waves is subtle. Our numerical results are also in good agreement with other theoretical approaches and with the experimental data.

© 2000 Optical Society of America

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References

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  1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  2. R. W. Boyd, Nonlinear Optics (Academic, London, 1992).
  3. S. Dutta Gupta, “Nonlinear optics of stratified media,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 1–84.
  4. G. I. Stegeman, “Comparison of guided wave approaches to optical bistability,” Appl. Phys. Lett. 41, 214–216 (1982).
    [CrossRef]
  5. M. B. Pande and S. Dutta Gupta, “Nonlinearity-induced resonances and optical multistability with coupled surface plasmons in a symmetric layered structure,” Opt. Lett. 15, 944–946 (1990).
    [CrossRef] [PubMed]
  6. W. Chen and D. L. Mills, “Optical response of a nonlinear dielectric film,” Phys. Rev. B 35, 524–532 (1987); “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
    [CrossRef]
  7. R. Weinstock, Calculus of Variations (McGraw-Hill, New York, 1952); K. Rektorys, Variational Methods in Mathematics, Science, and Engineering (Reidel, Boston, Mass., 1977).
  8. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  9. D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979); D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
    [CrossRef]
  10. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  11. See, for example, D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988); D. J. Kaup, T. I. Lakoba, and B. A. Malomed, “Asymmetric solitons in mismatched dual-core optical fibers,” J. Opt. Soc. Am. B 14, 1199–1206 (1997).
    [CrossRef]
  12. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  13. F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
    [CrossRef]
  14. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994); D. J. Kaup and B. A. Malomed, “Variational principle for the Zakharov–Shabat equations,” Physica D 84, 319–328 (1995).
    [CrossRef]
  15. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  16. P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
    [CrossRef]
  17. P. Dannberg and E. Broose, “Observation of optical bistability by prism excitation of a nonlinear film-guided wave,” Appl. Opt. 27, 1612–1614 (1988).
    [CrossRef] [PubMed]
  18. J. Jose and S. Dutta Gupta, “Phase conjugation induced distortion correction and optical multistability in enhanced back scattering in nonlinear layered media,” Opt. Commun. 145, 220–226 (1998).
    [CrossRef]
  19. R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
    [CrossRef]
  20. G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
    [CrossRef]
  21. T. A. Laine and A. T. Friberg, “Rigorous volume grating solution to distortion correction in nonlinear layered media near a phase-conjugate mirror,” Opt. Commun. 159, 93–98 (1999).
    [CrossRef]
  22. W. Dittrich and M. Reuter, Classical and Quantum Dynamics from Classical Paths to Path Integrals (Springer-Verlag, Berlin, 1996).

1999 (1)

T. A. Laine and A. T. Friberg, “Rigorous volume grating solution to distortion correction in nonlinear layered media near a phase-conjugate mirror,” Opt. Commun. 159, 93–98 (1999).
[CrossRef]

1998 (1)

J. Jose and S. Dutta Gupta, “Phase conjugation induced distortion correction and optical multistability in enhanced back scattering in nonlinear layered media,” Opt. Commun. 145, 220–226 (1998).
[CrossRef]

1997 (1)

F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
[CrossRef]

1991 (1)

1990 (1)

1988 (1)

1985 (2)

P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
[CrossRef]

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

1984 (1)

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1982 (1)

G. I. Stegeman, “Comparison of guided wave approaches to optical bistability,” Appl. Phys. Lett. 41, 214–216 (1982).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
[CrossRef]

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Arlot, P.

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

Broose, E.

Chilwell, J.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

Dannberg, P.

Desaix, M.

Dutta Gupta, S.

J. Jose and S. Dutta Gupta, “Phase conjugation induced distortion correction and optical multistability in enhanced back scattering in nonlinear layered media,” Opt. Commun. 145, 220–226 (1998).
[CrossRef]

M. B. Pande and S. Dutta Gupta, “Nonlinearity-induced resonances and optical multistability with coupled surface plasmons in a symmetric layered structure,” Opt. Lett. 15, 944–946 (1990).
[CrossRef] [PubMed]

Friberg, A. T.

T. A. Laine and A. T. Friberg, “Rigorous volume grating solution to distortion correction in nonlinear layered media near a phase-conjugate mirror,” Opt. Commun. 159, 93–98 (1999).
[CrossRef]

Jose, J.

J. Jose and S. Dutta Gupta, “Phase conjugation induced distortion correction and optical multistability in enhanced back scattering in nonlinear layered media,” Opt. Commun. 145, 220–226 (1998).
[CrossRef]

Koster, A.

P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
[CrossRef]

Laine, T. A.

T. A. Laine and A. T. Friberg, “Rigorous volume grating solution to distortion correction in nonlinear layered media near a phase-conjugate mirror,” Opt. Commun. 159, 93–98 (1999).
[CrossRef]

Laval, S.

P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
[CrossRef]

Lisak, M.

Martinot, P.

P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
[CrossRef]

Nurmanov, N. K.

F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
[CrossRef]

Pande, M. B.

Pic, E.

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

Reinisch, R.

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

Smith, S. D.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

G. I. Stegeman, “Comparison of guided wave approaches to optical bistability,” Appl. Phys. Lett. 41, 214–216 (1982).
[CrossRef]

Tsoy, E. N.

F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
[CrossRef]

Vitrant, G.

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

G. I. Stegeman, “Comparison of guided wave approaches to optical bistability,” Appl. Phys. Lett. 41, 214–216 (1982).
[CrossRef]

R. Reinisch, P. Arlot, G. Vitrant, and E. Pic, “Bistable optical behavior of a nonlinear attenuated total reflection device: resonant excitation of nonlinear guided modes,” Appl. Phys. Lett. 47, 1248–1250 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Martinot, A. Koster, and S. Laval, “Experimental observation of optical bistability by excitation of a surface plasmon wave,” IEEE J. Quantum Electron. QE-21, 1140–1143 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

T. A. Laine and A. T. Friberg, “Rigorous volume grating solution to distortion correction in nonlinear layered media near a phase-conjugate mirror,” Opt. Commun. 159, 93–98 (1999).
[CrossRef]

J. Jose and S. Dutta Gupta, “Phase conjugation induced distortion correction and optical multistability in enhanced back scattering in nonlinear layered media,” Opt. Commun. 145, 220–226 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Phys. Rev. E (1)

F. Kh. Abdullaev, N. K. Nurmanov, and E. N. Tsoy, “Variational approach to the problem of dark-soliton generation,” Phys. Rev. E 56, 3638–3644 (1997).
[CrossRef]

Other (11)

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994); D. J. Kaup and B. A. Malomed, “Variational principle for the Zakharov–Shabat equations,” Physica D 84, 319–328 (1995).
[CrossRef]

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

W. Dittrich and M. Reuter, Classical and Quantum Dynamics from Classical Paths to Path Integrals (Springer-Verlag, Berlin, 1996).

See, for example, D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988); D. J. Kaup, T. I. Lakoba, and B. A. Malomed, “Asymmetric solitons in mismatched dual-core optical fibers,” J. Opt. Soc. Am. B 14, 1199–1206 (1997).
[CrossRef]

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

R. W. Boyd, Nonlinear Optics (Academic, London, 1992).

S. Dutta Gupta, “Nonlinear optics of stratified media,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 1–84.

W. Chen and D. L. Mills, “Optical response of a nonlinear dielectric film,” Phys. Rev. B 35, 524–532 (1987); “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[CrossRef]

R. Weinstock, Calculus of Variations (McGraw-Hill, New York, 1952); K. Rektorys, Variational Methods in Mathematics, Science, and Engineering (Reidel, Boston, Mass., 1977).

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979); D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Geometry of the layered system. The refractive index of layer p (p=14) is denoted by np; the layer thickness, by dp; and the angle of the incident wave, by θ.

Fig. 2
Fig. 2

Scattered intensities in (a) the specular reflection direction and in (b) the backscattering direction, when the second layer is linear (γ=0).

Fig. 3
Fig. 3

Reflected intensities when (a) γ=3×10-15, (b) γ=4×10-15, and (c) γ=5×10-15. The solid curve corresponds to the linear result, γ=0.

Fig. 4
Fig. 4

Intensities in the phase-conjugate direction. The nonlinearity coefficient γ has the same values as in Fig. 3.

Equations (40)

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n2=n20+αI,
2Ez2+2Ey2+k02n22E=0,
2Ez2+2Ey2+k2E+γ|E|2E=0,
Ep(z, y)=[Ap exp(-ikzpz)+Bp exp(ikzpz)]exp(ikyy)+[A-p exp(ikzpz)+B-p exp(-ikzpz)]exp(-ikyy),
A-1=μA1*,
B1=μ(B-1)*,
2E(z)z2+kz2E(z)+γ|E(z)|2E(z)=0,
2E(z, ξ)z2+kz2E(z, ξ)+γ|E(z, ξ)|2E(z, ξ)=0,
E(z, ξ)=A(ξ)exp(-ikzz)+B(ξ)exp(ikzz),
S=Ldzdξ,
L=E(z, ξ)z2-kz2|E(z, ξ)|2-γ2|E(z, ξ)|4,
ddzLEz*+ddξLEξ*-LE*=0,
E(z, ξ)=A(ξ)exp(-ik-z)+B(ξ)exp(ik+z),
k-=m-q,k+=m+q,
SR=Ldξ,
L(ξ)=q2π02π/qL(z, ξ)dz.
I1=q2π02π/q E(z, ξ)zE*(z, ξ)zdz=q22(m-+m+)π{2(m-+m+)π[m-2|A(ξ)|2+m+2|B(ξ)|2]-2m-m+ Re[A*(ξ)B(ξ)]sin[2(m-+m+)π]+4m-m+ Im[A*(ξ)B(ξ)]sin2[2(m-+m+)π]}=q2[m-2|A(ξ)|2+m+2|B(ξ)|2]=k-2|A(ξ)|2+k+2|B(ξ)|2,
I2=-kz2 q2π02π/q|E(z, ξ)|2dz=-kz22(m-+m+)π{2(m-+m+)π(|A|2+|B|2)+2 Re(A*B)sin[2(m-+m+)π]-4 Im(A*B)sin2[2(m-+m+)π]}=-kz2(|A|2+|B|2).
I3=-γ2q2π02π/q|E(z, ξ)|4dz=-γ8(m-+m+)π×{4(m-+m+)π(|A|4+4|A|2|B|2+|B|4)+8 Re[AB*(|A|2+|B|2)]sin[2(m-+m+)π]+2 Re(A2B*2)sin[4(m-+m+)π]+16 Im[AB*(|A|2+|B|2)]sin2[(m-+m+)π]+4 Im(A2B*2)sin2[2(m-+m+)π]}=-γ2(|A|4+4|A|2|B|2+|B|4).
L(ξ)=I1+I2+I3=k-2|A|2+k+2|B|2-kz2(|A|2+|B|2)-γ2(|A|4+4|A|2|B|2+|B|4).
L|A(ξ)|2=k-2-kz2-γ[|A(ξ)|2+2|B(ξ)|2]=0,
L|B(ξ)|2=k+2-kz2-γ[2|A(ξ)|2+|B(ξ)|2]=0.
k-2=kz2+γ(|A|2+2|B|2),
k+2=kz2+γ(2|A|2+|B|2).
S=Ldzdydξ,
L=Ez2+Ey2-k2|E|2-γ2|E|4.
E(z, y, ξ)=[A2(ξ)exp(-ik1z)+B2(ξ)exp(ik2z)]×exp(ikyy)+[A-2(ξ)exp(ik3z)+B-2(ξ)exp(-ik4z)]exp(-ikyy).
kj=mjq,
L(ξ)=q2π02π/q ky2π02π/kyEz2+Ey2-k2|E|2-γ2|E|4dydz,
I1=qky4π202π/q02π/ky EzE*zdydz=q2m12A22+m22B22+m32A-22+m42B-22-m1m2(m1+m2)πA2B2 sin[2(m1+m2)π]-m3m4(m3+m4)πA-2B-2 sin[2(m3+m4)π]=k12A22+k22B22+k32A-22+k42B-22.
I2=qky4π202π/q02π/ky EyE*ydydz=ky2A22+B22+A-22+B-22+1(m1+m2)πA2B2 sin[2(m1+m2)π]+1(m3+m4)πA-2B-2 sin[2(m3+m4)π]=ky2(A22+B22+A-22+B-22).
I3=-k2 qky4π202π/q02π/ky|E|2dydz=-k2A22+B22+A-22+B-22+1(m1+m2)πA2B2 sin[2(m1+m2)π]+1(m3+m4)πA-2B-2 sin[2(m3+m4)π]=-k2(A22+B22+A-22+B-22).
I4=-γ2qky4π202π/q02π/ky|E|4dydz=-γ2[A24+B24+A-24+B-24+4(A22B22+A22A-22+A22B-22+B22A-22+B22B-22+A-22B-22)]+oscillatingterms.
L=I1+I2+I3+I4=k12|A2|2+k22|B2|2+k32|A-2|2+k42|B-2|2-(k2-ky2)(|A2|2+|B2|2+|A-2|2+|B-2|2)-γ2[|A2|4+|B2|4+|A-2|4+|B-2|4+4(|A2|2|B2|2+|A2|2|A-2|2+|A2|2|B-2|2+|B2|2|A-2|2+|B2|2|B-2|2+|A-2|2|B-2|2)].
L|A2|2=L|B2|2=L|A-2|2=L|B-2|2=0,
k12=k2-ky2+γ(|A2|2+2|B2|2+2|A-2|2+2|B-2|2),
k22=k2-ky2+γ(2|A2|2+|B2|2+2|A-2|2+2|B-2|2),
k32=k2-ky2+γ(2|A2|2+2|B2|2+|A-2|2+2|B-2|2),
k42=k2-ky2+γ(2|A2|2+2|B2|2+2|A-2|2+|B-2|2).
2Fz2+2Fy2+k2F+γ˜|F|2F=0,

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