Abstract

Effects of nonlinear saturation on self-trapping in planar geometry are examined analytically. The saturation is shown to have a great impact on trapping, giving rise to several interesting features that are absent in the Kerr-law nonlinearity.

© 1991 Optical Society of America

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References

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [Crossref]
  2. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  3. Y. Chen, “Vector spatial solution: trapped TM light pattern in planar geometry,” Electron. Lett. 27, 380–382 (1991).
    [Crossref]
  4. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [Crossref] [PubMed]
  5. M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
    [Crossref] [PubMed]
  6. Y. Chen, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
    [Crossref]
  7. A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
    [Crossref] [PubMed]
  8. R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
    [Crossref] [PubMed]

1991 (2)

Y. Chen, “Vector spatial solution: trapped TM light pattern in planar geometry,” Electron. Lett. 27, 380–382 (1991).
[Crossref]

Y. Chen, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
[Crossref]

1990 (1)

1987 (1)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
[Crossref] [PubMed]

1986 (1)

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
[Crossref] [PubMed]

1985 (1)

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[Crossref] [PubMed]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Aitchison, J. S.

Boyd, R. W.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
[Crossref] [PubMed]

Chen, Y.

Y. Chen, “Vector spatial solution: trapped TM light pattern in planar geometry,” Electron. Lett. 27, 380–382 (1991).
[Crossref]

Y. Chen, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Enns, R. H.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
[Crossref] [PubMed]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Jackel, J. L.

Kaplan, A. E.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
[Crossref] [PubMed]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[Crossref] [PubMed]

Kramer, M. A.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
[Crossref] [PubMed]

Leaird, D. E.

Oliver, M. K.

Rangnekar, S. S.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
[Crossref] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Silberberg, Y.

Smith, P. W.

Tompkin, W. R.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
[Crossref] [PubMed]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Vogel, E. M.

Weiner, A. M.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Electron. Lett. (1)

Y. Chen, “Vector spatial solution: trapped TM light pattern in planar geometry,” Electron. Lett. 27, 380–382 (1991).
[Crossref]

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

Opt. Lett. (1)

Phys. Rev. A (1)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, “‘Robust’ bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466–469 (1987); “Bistable-soliton pulse propagation: stability aspects,” Phys. Rev. A 36, 1270–1279 (1987).
[Crossref] [PubMed]

Phys. Rev. A. (1)

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear-optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A. 34, 2026–2031 (1986).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

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Figures (6)

Fig. 1
Fig. 1

Phase-space portrait of TM waves in the ψxψz plane for b = (β2/k2n02 − 1)1/2 = 0.7 and Δs = (nsat2/n02 − 1)1/2 = 1, where singular points (ψx, ψz) = (0, 0) and (ψx, ψz) = (±ψs, 0) with ψs = [−ln(1 − b2s2)]1/2Δs/b, which for the example here is ψs = 1.17. The solid curves identify the periodic solutions, while the dashed curve, marking the separatrix, identifies the solitary wave.

Fig. 2
Fig. 2

Evolution of the field profile and separatrix of the TM1-type solitary wave for Δs = 2; (a) b = 0.1, (b) b = 1.2, (c) b = 1.98. Note that ψx is a symmetric function and ψz is an antisymmetric function with respect to the X = 0 axis.

Fig. 3
Fig. 3

Maximum amplitude ψx(0) of the transverse-field component of the TM1 solitary wave versus the propagation constant β/kn0 for Δs = 1, 2, 3.5, 5, ∞, with Δs defined in the caption of Fig. 1.

Fig. 4
Fig. 4

Same as Fig. 3 but for the TE1 mode.

Fig. 5
Fig. 5

Normalized power Pkn2240π/n02 for trapping the TM1-type solitary wave versus the propagation constant β/kn0 for different Δs.

Fig. 6
Fig. 6

Same as Fig. 5 but for the TE1 mode.

Equations (14)

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× × E = k 2 [ n L 2 + n NL 2 ( E 2 ) ] E ,
n NL 2 = n 0 2 + ( n sat 2 - n 0 2 ) S ( E - E s ) .
n NL 2 = ( n sat 2 - n 0 2 ) [ 1 - exp ( - n 2 n sat 2 - n 0 2 E 2 ) ]
d 2 ψ y d x 2 - ψ y + Δ s 2 b 2 [ 1 - exp ( - b 2 Δ s 2 ψ y 2 ) ] ψ y = 0 ,
d ψ z d X = b ( b 2 + 1 ) 1 / 2 × { 1 - Δ s 2 b 2 + Δ s 2 b 2 exp [ - b 2 Δ s 2 ( ψ x 2 + ψ z 2 ) ] } ψ x ,
d 2 ψ z d X 2 + 1 b 2 ψ z + Δ s 2 b 2 { 1 - exp [ - b 2 Δ s 2 ( ψ x 2 + ψ z 2 ) ] } ψ z = ( b 2 + 1 ) 1 / 2 b d ψ x d X ,
ψ z 2 + { b 2 1 + b 2 [ 1 - f ( ψ x 2 + ψ z 2 ) ] 2 - 1 } b 2 ψ x 2 + Δ s 2 [ ψ x 2 + ψ z 2 - f ( ψ x 2 + ψ z 2 ) ] = Γ = const . ,
f ( η ) = [ 1 - exp ( - b 2 Δ s 2 η ) ] Δ s 2 b 2 .
X = ( b 2 + 1 ) 1 / 2 b × ψ z ( 0 ) ψ z ( 0 ) d ψ z { 1 - ( Δ s 2 / b 2 ) + ( Δ s 2 / b 2 ) exp [ - b 2 Δ s 2 ( ψ x 2 + ψ z 2 ) ] } ψ x ,
{ b 2 1 + b 2 [ 1 - f ( ψ x 0 2 ) ] 2 - 1 } b 2 ψ x 0 2 + Δ s 2 [ ψ x 0 2 - f ( ψ x 0 2 ) ] = 0 ,
ψ x 0 2 = - 1 4 ( 1 b 2 - 3 ) + 1 2 [ 1 4 ( 1 b 2 - 3 ) 2 + 4 b 2 ] 1 / 2
( d ψ y d X ) 2 - ψ y 2 + Δ s 2 b 2 { ψ y 2 - Δ s 2 b 2 [ 1 - exp ( - b 2 Δ s 2 ψ y 2 ) ] } = Γ ,
X = ± ψ y ( 0 ) ψ z ( X ) d ψ y ( Γ + ψ y 2 - ( Δ s 2 / b 2 ) { ψ y 2 - ( Δ s 2 / b 2 ) [ 1 - exp ( - b 2 ψ y 2 / Δ s 2 ) ] } ) 1 / 2 .
- ψ y 0 2 + Δ s 2 b 2 { ψ y 0 2 - Δ s 2 b 2 [ 1 - exp ( - b 2 Δ s 2 ψ y 0 2 ) ] } = 0.

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