Abstract

We derive a local nonlinear thin-layer theory for electromagnetic fields that propagate in layered structures of isotropic, dispersive, and spatially local Kerr media. By use of an ansatz of plane waves together with a thin-layer approximation, the two-dimensional Kerr–Maxwell equation is rigorously solved within a very thin slab, and the characteristic matrix of the nonlinear medium is determined. The theory makes use of periodicity and allows a direct calculation of the nonlinear field throughout the structure on the basis of the transmitted field. The method is applied in the two polarizations, TE and TM, and is illustrated with a numerical example. The nonlinear thin-layer technique provides a simple and accurate analytical theory that includes multiple plane-wave incident fields and takes rigorously into account all nonlinear interactions of these waves.

© 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. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  3. A. E. Kaplan, “Hysteresis reflection and refraction by a nonlinear boundary: a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).
  4. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London, 1985); J.-I. Sakai, Phase Conjugate Optics (McGraw-Hill, New York, 1992); H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
    [CrossRef]
  5. F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
    [CrossRef] [PubMed]
  6. W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987); V. M. Agranovich, S. A. Kiselev, and D. L. Mills, “Optical multistability in nonlinear superlattices with very thin layers,” Phys. Rev. B 44, 10917–10920 (1991).
    [CrossRef]
  7. S. Dutta Gupta, “Nonlinear optics of stratified media,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 1–84.
  8. G. I. Stegeman, “Comparison of guided wave approaches to optical bistability,” Appl. Phys. Lett. 41, 214–216 (1982); 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]
  9. T. A. Laine and A. T. Friberg, “Nonlinear thin-layer theory of stratified Kerr medium,” Appl. Phys. Lett. 74, 3248–3250 (1999).
    [CrossRef]
  10. 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]
  11. A system is truly periodic if all y components of the waves are fractions of the same base period, L.
  12. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sect. 1.6.
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, UK, 1987).

1999

T. A. Laine and A. T. Friberg, “Nonlinear thin-layer theory of stratified Kerr medium,” Appl. Phys. Lett. 74, 3248–3250 (1999).
[CrossRef]

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]

1986

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

1976

A. E. Kaplan, “Hysteresis reflection and refraction by a nonlinear boundary: a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

Delyon, F.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Friberg, A. T.

T. A. Laine and A. T. Friberg, “Nonlinear thin-layer theory of stratified Kerr medium,” Appl. Phys. Lett. 74, 3248–3250 (1999).
[CrossRef]

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]

Kaplan, A. E.

A. E. Kaplan, “Hysteresis reflection and refraction by a nonlinear boundary: a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

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]

T. A. Laine and A. T. Friberg, “Nonlinear thin-layer theory of stratified Kerr medium,” Appl. Phys. Lett. 74, 3248–3250 (1999).
[CrossRef]

Lévy, Y.-E.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Souillard, B.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Appl. Phys. Lett.

T. A. Laine and A. T. Friberg, “Nonlinear thin-layer theory of stratified Kerr medium,” Appl. Phys. Lett. 74, 3248–3250 (1999).
[CrossRef]

JETP Lett.

A. E. Kaplan, “Hysteresis reflection and refraction by a nonlinear boundary: a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

Opt. Commun.

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]

Phys. Rev. Lett.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Other

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987); V. M. Agranovich, S. A. Kiselev, and D. L. Mills, “Optical multistability in nonlinear superlattices with very thin layers,” Phys. Rev. B 44, 10917–10920 (1991).
[CrossRef]

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

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

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London, 1985); J.-I. Sakai, Phase Conjugate Optics (McGraw-Hill, New York, 1992); H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

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

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

A system is truly periodic if all y components of the waves are fractions of the same base period, L.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sect. 1.6.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, UK, 1987).

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Figures (3)

Fig. 1
Fig. 1

Layered nonlinear structure. In linear media the electric fields are expressed with plane waves, whose amplitudes are denoted by Fju, Gju, Fjd, and Gjd (j=-1, 0, 1).

Fig. 2
Fig. 2

Scattered intensities when nonlinearity coefficient n2 of the Kerr medium is varied. The upward- and downward-pointing triangles correspond to intensities |G0u/F0u|2 and |G1u/F1u|2, respectively.

Fig. 3
Fig. 3

Field intensity within the nonlinear medium, when n2|Fju|20.0047.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

2Ejy2+2Ejz2+0Ej+2|Ej|2Ej=0,
Ej(y, z)=Aj cos(kj,1z)+iBjkj,2sin(kj,2z),
kj,12=1Aj 0Aj+2|Aj|2Aj+2Ajy2,
kj,22=1Bj 0Bj+22|Aj|2Bj-2Aj2Bj*+2Bjy2.
kyjLmj=2π,
fˆk=1L 0Lf(y)exp-i2πkyLdy,
2AN(y)y2=k-4π2k2L3 0LEl(y, z=0)×exp-i2πkyLdyexpi2πkyL.
Aj-1Bj-1=1ihihkj,12(y)1AjBj=Mj(y)AjBj,
M(y)=j=1NMj(y).
y 1 Hy+z 1 Hz+H=0,
(y, z)=0+21 Hy2+1 Hz2,
Hj(y, z)=Cj cos(qj,1z)+ijDjqj,2 sin(qj,2z),
j3(y)-j2(y)[0+2|Dj(y)|2]-2Cj(y)y2=0.
qj,12=1Cj jCj+2Cjy2-1j jy Cjy,
qj,22=1Dj jDj+2Djy2+1j jy Djy+2jy2Dj-1j2 jy2Dj.
Cj-1Dj-1=1ihj(y)iqj,12(y)h/j(y)1CjDj=M˜j(y)CjDj.

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