Abstract

Dispersion relations for TE modes in a planar exponentially graded-index waveguide with self-focusing nonlinear cover material have been solved numerically. It is shown that the threshold power required to pull the field maximum out of the film region into the cover is lower compared with that for the step-index waveguide and agrees well with the experimental results. Empirical relations to calculate the corresponding minimum film thickness and the minimum threshold power are given for the lowest-order mode.

© 1986 Optical Society of America

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References

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  1. N. N. Akhmediev, “Novel Class of Nonlinear Surface Waves: Asymmetric Modes in a Symmetric Layered Structure,” Sov. Phys. JETP 56, 299 (1982).
  2. G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610 (1982).
    [CrossRef]
  3. C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).
  4. U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
    [CrossRef]
  5. A. D. Boardman, P. Egan, “S-Polarized Waves in a Thin Dielectric Film asymmetrically bounded by Optically Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 1701 (1985).
    [CrossRef]
  6. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
    [CrossRef]
  7. S. M. Jensen, “The Nonlinear Coherent Coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
    [CrossRef]
  8. D. Sarid, “Analysis of Bistability in a Ring-Channel Waveguide,” Opt. Lett. 6, 552 (1981).
    [CrossRef] [PubMed]
  9. H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of Intensity-Dependent Guided Waves,” Opt. Lett. 9, 238 (1984).
    [CrossRef] [PubMed]
  10. I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
    [CrossRef]
  11. E. M. Conwell, “Modes in Optical Waveguides formed by Diffusion,” Appl. Phys. Lett. 23, 328 (1973).
    [CrossRef]
  12. R. V. Ramaswamy, R. K. Lagu, “Numerical Field Solution for an Arbitrary Asymmetrical Graded-Index Planar Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 408 (1983).
    [CrossRef]
  13. C. K. R. T. Jones, J. V. Moloney, “Stability and Instability of Nonlinear Waveguide Modes,” in Technical Digest, Topical Meeting on Optical Bistability (OB3) (Optical Society of America, Washington, DC, 1985), paper WD5.
  14. J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
    [CrossRef]

1986

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

1985

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
[CrossRef]

A. D. Boardman, P. Egan, “S-Polarized Waves in a Thin Dielectric Film asymmetrically bounded by Optically Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 1701 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
[CrossRef]

1984

1983

R. V. Ramaswamy, R. K. Lagu, “Numerical Field Solution for an Arbitrary Asymmetrical Graded-Index Planar Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 408 (1983).
[CrossRef]

1982

N. N. Akhmediev, “Novel Class of Nonlinear Surface Waves: Asymmetric Modes in a Symmetric Layered Structure,” Sov. Phys. JETP 56, 299 (1982).

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610 (1982).
[CrossRef]

S. M. Jensen, “The Nonlinear Coherent Coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

1981

1973

E. M. Conwell, “Modes in Optical Waveguides formed by Diffusion,” Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, “Novel Class of Nonlinear Surface Waves: Asymmetric Modes in a Symmetric Layered Structure,” Sov. Phys. JETP 56, 299 (1982).

Ariyasu, J.

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

Bennion, I.

I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
[CrossRef]

Boardman, A. D.

A. D. Boardman, P. Egan, “S-Polarized Waves in a Thin Dielectric Film asymmetrically bounded by Optically Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 1701 (1985).
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Conwell, E. M.

E. M. Conwell, “Modes in Optical Waveguides formed by Diffusion,” Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Egan, P.

A. D. Boardman, P. Egan, “S-Polarized Waves in a Thin Dielectric Film asymmetrically bounded by Optically Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 1701 (1985).
[CrossRef]

Goodwin, M. J.

I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The Nonlinear Coherent Coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

Jones, C. K. R. T.

C. K. R. T. Jones, J. V. Moloney, “Stability and Instability of Nonlinear Waveguide Modes,” in Technical Digest, Topical Meeting on Optical Bistability (OB3) (Optical Society of America, Washington, DC, 1985), paper WD5.

Khoo, I. C.

Lagu, R. K.

R. V. Ramaswamy, R. K. Lagu, “Numerical Field Solution for an Arbitrary Asymmetrical Graded-Index Planar Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 408 (1983).
[CrossRef]

Langbein, U.

U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
[CrossRef]

Lederer, F.

U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
[CrossRef]

Mai, Xu

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Moloney, J. V.

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

C. K. R. T. Jones, J. V. Moloney, “Stability and Instability of Nonlinear Waveguide Modes,” in Technical Digest, Topical Meeting on Optical Bistability (OB3) (Optical Society of America, Washington, DC, 1985), paper WD5.

Ponath, H. E.

U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
[CrossRef]

Ramaswamy, R. V.

R. V. Ramaswamy, R. K. Lagu, “Numerical Field Solution for an Arbitrary Asymmetrical Graded-Index Planar Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 408 (1983).
[CrossRef]

Sarid, D.

Seaton, C. T.

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of Intensity-Dependent Guided Waves,” Opt. Lett. 9, 238 (1984).
[CrossRef] [PubMed]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Stegeman, G. I.

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of Intensity-Dependent Guided Waves,” Opt. Lett. 9, 238 (1984).
[CrossRef] [PubMed]

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610 (1982).
[CrossRef]

Stewart, W. J.

I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
[CrossRef]

Vach, H.

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Winful, H. G.

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Appl. Phys. Lett.

E. M. Conwell, “Modes in Optical Waveguides formed by Diffusion,” Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

J. V. Moloney, J. Ariyasu, C. T. Seaton, G. I. Stegeman, “Stability of Nonlinear Stationary Waves guided by a Thin Film bounded by Nonlinear Media,” Appl. Phys. Lett. 43, 826 (1986).
[CrossRef]

Electron. Lett.

I. Bennion, M. J. Goodwin, W. J. Stewart, “Experimental Nonlinear Optical Waveguide Device,” Electron. Lett. 21, 41 (1985).
[CrossRef]

IEEE J. Quantum Electron.

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610 (1982).
[CrossRef]

A. D. Boardman, P. Egan, “S-Polarized Waves in a Thin Dielectric Film asymmetrically bounded by Optically Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 1701 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculations of Nonlinear TE Waves guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

S. M. Jensen, “The Nonlinear Coherent Coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

IEEE/OSA J. Lightwave Technol.

R. V. Ramaswamy, R. K. Lagu, “Numerical Field Solution for an Arbitrary Asymmetrical Graded-Index Planar Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 408 (1983).
[CrossRef]

Opt. Commun.

U. Langbein, F. Lederer, H. E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves,” Opt. Commun. 52, 417 (1985).
[CrossRef]

Opt. Eng.

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Opt. Lett.

Sov. Phys. JETP

N. N. Akhmediev, “Novel Class of Nonlinear Surface Waves: Asymmetric Modes in a Symmetric Layered Structure,” Sov. Phys. JETP 56, 299 (1982).

Other

C. K. R. T. Jones, J. V. Moloney, “Stability and Instability of Nonlinear Waveguide Modes,” in Technical Digest, Topical Meeting on Optical Bistability (OB3) (Optical Society of America, Washington, DC, 1985), paper WD5.

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Figures (5)

Fig. 1
Fig. 1

Variation of TE0 guided wave power with the effective index for d = 1.5 μm (dashed curve) and d = 3.0 μm (solid curve). Points A, B, and C represent the power level corresponding to the field profile of the TE0 mode shown in Fig. 2.

Fig. 2
Fig. 2

Field profile of the TE0 mode corresponding to three different power levels (identified by the points A, B, and C in Fig. 1) for d = 3.0 μm.

Fig. 3
Fig. 3

Minimum TE0 guided wave power as a function of (nsnc) for nb = 1.55 (curve A) and = 3.0 (curve B). Circles correspond to Ref. 6 with the homogeneous film for nb = 1.55.

Fig. 4
Fig. 4

Variation of TE1 guided wave power with effective index for d = 1.5 μm (dashed curve) and d = 3.0 μm (solid curve). Points A, B, and C correspond to the power levels for the field profile of the TE1 mode shown in Fig. 5.

Fig. 5
Fig. 5

Field profiles of the TE1 mode for three different power levels marked by points A, B, and C in Fig. 4 for d = 1.5 μm.

Equations (10)

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n 2 ( x ) = n c 2 + α E 2 for x < 0 = n b 2 [ 1 + 2 Δ exp ( - x / d ] x > 0 ,
Δ = n s 2 - n b 2 2 n b 2 .
E y ( x ) = 2 α q sech [ k 0 q ( x 1 - x ) ] for x < 0 , = A J 2 ρ [ 2 V exp ( - x / 2 d ) ] x > 0 ,
q = [ ( β / k 0 ) 2 - n c 2 ] 1 / 2 , ρ = k 0 d [ ( β / k 0 ) 2 - n b 2 ] 1 / 2 , V = k 0 d n b 2 Δ .
A = 2 α q J 2 ρ ( 2 V ) sech ( k 0 q x 1 ) ,
γ T = J 2 ρ + 1 ( 2 V ) J 2 ρ ( 2 V ) V - ρ ,
P g = ½ - ( E × H ) z * d x = P c + P f ,
P c = B 2 γ ( 1 - T ) , P f = B D ( 1 - T 2 ) , B = 2 d β q 2 k 0 n c 2 n 2 c , D = 1 J 2 ρ 2 ( 2 V ) 0 2 V 1 y J 2 ρ 2 ( y ) d y , n 2 c = α n c 2 u 0 0 . }
T = { [ ( 1 + 1 4 γ D ) 2 - P g B D ] 1 / 2 - 1 4 γ D } .
{ b = [ ( β / k 0 ) 2 - n b 2 n s 2 - n b 2 ] } 0.66 ± 0.007.

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