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, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  2. V. E. Zakharov, 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, 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, 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, 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, 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, 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, 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, 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, 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, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
[CrossRef]

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

Chiao, R. Y.

R. Y. Chiao, E. Garmire, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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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