Abstract

We describe a quite general and accurate approach for determining the dispersion equation of TE-polarized modes propagating in planar graded-index waveguides bounded on one side by a self-focusing nonlinear cover with finite thickness. The proposed structure appears to be appropriate for wavelength converters based on a four-wave mixing process. The dependence of the guided power flow along the proposed four-layered structure (which is different from others previously published) on the refractive-index profile parameters is investigated. The validity of the reported numerical results is confirmed by the beam-propagation-method simulation. Moreover, a particular waveguide configuration is described, one especially designed to maximize the power confinement in the nonlinear medium.

© 1993 Optical Society of America

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References

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  1. R. K. Varshney, M. A. Nehme, R. Srivastava, R. V. Ramaswamy, “Guided waves in graded-index planar waveguides with nonlinear cover medium,” Appl. Opt. 25, 3899–3902 (1986).
    [CrossRef] [PubMed]
  2. K. Oguso, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
    [CrossRef]
  3. H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
    [CrossRef]
  4. N. Saiga, “Calculation of TE modes in graded-index nonlinear optical waveguides with arbitrary profile of refractive index,” J. Opt. Soc. Am. B 8, 88–94 (1991).
    [CrossRef]
  5. K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
    [CrossRef]
  6. A. E. Kaplan, “Hysteresis reflection and refraction by a boundary of a nonlinear medium—a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).
  7. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).
  8. D. Marcuse, “Reflection of a Gaussian beam from a nonlinear interface,” Appl. Opt. 19, 3130–3139 (1980).
    [CrossRef] [PubMed]
  9. A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
    [CrossRef]
  10. N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).
  11. U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
    [CrossRef]
  12. U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
    [CrossRef]
  13. U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
    [CrossRef]
  14. 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, 774–782 (1985).
    [CrossRef]
  15. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
    [CrossRef]
  16. A. D. Boardman, P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. Colloq. C5, 291–303 (1984).
  17. 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, 1701–1713 (1985).
    [CrossRef]
  18. A. D. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [CrossRef]
  19. E. Wright, G. Stegeman, “Nonlinear planar waveguide,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds. (Elsevier, New York, 1992), pp. 117–142.
  20. D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
    [CrossRef]
  21. P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
    [CrossRef]

1991 (2)

N. Saiga, “Calculation of TE modes in graded-index nonlinear optical waveguides with arbitrary profile of refractive index,” J. Opt. Soc. Am. B 8, 88–94 (1991).
[CrossRef]

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[CrossRef]

1988 (1)

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

1987 (2)

K. Oguso, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
[CrossRef]

1986 (2)

1985 (4)

U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (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, 774–782 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (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, 1701–1713 (1985).
[CrossRef]

1984 (1)

A. D. Boardman, P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. Colloq. C5, 291–303 (1984).

1983 (2)

U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[CrossRef]

U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

1982 (1)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

1981 (1)

A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[CrossRef]

1980 (1)

1978 (1)

K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
[CrossRef]

1977 (1)

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

1976 (1)

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

Akhmediev, N. N.

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
[CrossRef]

Baker, G.

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

Blau, W.

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[CrossRef]

Boardman, A. D.

A. D. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[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, 1701–1713 (1985).
[CrossRef]

A. D. Boardman, P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. Colloq. C5, 291–303 (1984).

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, 774–782 (1985).
[CrossRef]

Costa, L.

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[CrossRef]

Egan, P.

A. D. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[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, 1701–1713 (1985).
[CrossRef]

A. D. Boardman, P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. Colloq. C5, 291–303 (1984).

Etemad, S.

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

Hayata, H.

H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
[CrossRef]

Kamiya, T.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

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

Klausner, K.

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

Koshiba, M.

H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
[CrossRef]

Langbein, U.

U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[CrossRef]

Lederer, E.

U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[CrossRef]

Lederer, F.

U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Maradudin, A. A.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
[CrossRef]

A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[CrossRef]

Marcuse, D.

Nehme, M. A.

Oguso, K.

K. Oguso, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

Ponath, H. E.

U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[CrossRef]

Ramaswamy, R. V.

Saiga, N.

Schlotter, N. E.

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

Seaton, C. 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, 774–782 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
[CrossRef]

Shibayama, K.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
[CrossRef]

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, 774–782 (1985).
[CrossRef]

Skarda, V.

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[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, 774–782 (1985).
[CrossRef]

Srivastava, R.

Stegeman, G.

E. Wright, G. Stegeman, “Nonlinear planar waveguide,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds. (Elsevier, New York, 1992), pp. 117–142.

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (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, 774–782 (1985).
[CrossRef]

Suzuki, M.

H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
[CrossRef]

Townsend, P.

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

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, 774–782 (1985).
[CrossRef]

Varshney, R. K.

Wallis, R. F.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
[CrossRef]

Westland, D. J.

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[CrossRef]

Wright, E.

E. Wright, G. Stegeman, “Nonlinear planar waveguide,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds. (Elsevier, New York, 1992), pp. 117–142.

Yamanouchi, K.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

U. Langbein, F. Lederer, H. E. Ponath, “Nonlinear waves guided by a dielectric slab: 1. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

P. Townsend, G. Baker, N. E. Schlotter, K. Klausner, S. Etemad, “Waveguiding in spun films of soluble polydiacetylenes,” Appl. Phys. Lett. 53, 1782–1784 (1988).
[CrossRef]

Electron. Lett. (2)

D. J. Westland, V. Skarda, W. Blau, L. Costa, “Degenerate FWM in polydiacetylene waveguides,” Electron. Lett. 27, 1327–1328 (1991).
[CrossRef]

H. Hayata, M. Koshiba, M. Suzuki, “Finite element solution of arbitrarily nonlinear, graded index slab waveguides,” Electron. Lett. 23, 429–431 (1987).
[CrossRef]

IEEE J. Quantum Electron. (3)

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, 774–782 (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, 1701–1713 (1985).
[CrossRef]

A. D. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Yamanouchi, T. Kamiya, K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–305 (1978).
[CrossRef]

J. Appl. Phys. (1)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2559 (1985).
[CrossRef]

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

J. Phys. Colloq. (1)

A. D. Boardman, P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. Colloq. C5, 291–303 (1984).

JETP Lett. (1)

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

Opt. Commun. (3)

K. Oguso, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

U. Langbein, E. Lederer, H. E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

U. Langbein, E. Lederer, H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[CrossRef]

Sov. Phys. JETP (2)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

Z. Phys. B (1)

A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[CrossRef]

Other (1)

E. Wright, G. Stegeman, “Nonlinear planar waveguide,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds. (Elsevier, New York, 1992), pp. 117–142.

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Figures (9)

Fig. 1
Fig. 1

Sketch of the investigated nonlinear GI four-layered WG.

Fig. 2
Fig. 2

Power-dependent dispersion curves for the exponential GRIN nonlinear WG used in Ref. 1 characterized by nonlinear cladding with nc = 1.55, n2c = 10−9 m2/W, ns = 1.55, Δn = 0.02, d = 3 μm, λ = 0.515 μm. The solid curve and the filled circles are the actual and the exact results, respectively. The dashed curve refers to a four-layered structure having h = 0.515 μm.

Fig. 3
Fig. 3

Variations of guided power P (left axis) and of the electric-field value E0 at the air–nonlinear-cladding interface (right axis) with the mode index for a, an error-function; b, an exponential; and c, a semi-Gaussian RI profile with Δn = 0.118.

Fig. 4
Fig. 4

(a) Dispersion curves and (b) power amount (as a percentage) confined in the nonlinear medium versus the refractive effective index for an erfc-type WG and RI change at the surface; a, Δn = 0; b, Δn = 0.009; c, Δn = 0.05; and d, Δn = 0.118.

Fig. 5
Fig. 5

Mapping of the electric-field component Ey for neff = 1.6083 and RI change at the surface; a, Δn = 0; b, Δn = 0.009; c Δn = 0.05; and d, Δn = 0.118.

Fig. 6
Fig. 6

(a) Dispersion curves and (b) power amount (as a percentage) for the four-layered GRIN (solid curve) and SI WG (dashed curve) with Δn = 0.118.

Fig. 7
Fig. 7

Variation of the power amount (as a percentage) confined in the nonlinear medium with (a) the thickness of the nonlinear layer (curve a, h = 1.064 μm; b, h = 1.3 μm; c, h = 1.7 μm) and (b) the diffusion depth d (curve a, d = 0.5 μm; b, d = 1 μm; c, d = 1.5 μm; d, d = 2 μm) for the error-function-type WG, and RI change at the surface Δn = 0.118.

Fig. 8
Fig. 8

Comparison between the powers Pc confined in the nonlinear medium with (curve a) and without (curve b) the tapering as a function of the refractive effective index. Curve c refers to the untapered SI WG. Curve d refers to a two-step taper.

Fig. 9
Fig. 9

BPM simulation of the electric-field evolution: (a) three-dimensional representation and (b) contour lines.

Equations (13)

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

NL ( E ) = n NL 2 = n 2 + α | E | 2 ,
n ( x ) = n s + Δ n f ( x / d ) ,
d F ( u ) d u = [ A ] F ( u ) ,
d 2 F e d u 2 = ( n eff 2 n 2 ) F e ,
F e ( u ) = E 0 exp ( k a u / k 0 ) u 0 , F e ( u ) = E s exp { ( k s / k 0 ) [ u k 0 ( h + 2 d ) ] } , u k 0 ( h + 2 d ) ,
2 F c / u 2 k 2 F e + α F e 3 = 0 ,
d F e 2 d u = F h 2 , d F h 2 d u = [ ( n 3 n eff 2 ) + α | F e 2 | 2 ] F e 2 .
F e 2 ( u ) = q E 0 [ q c n ( q u / k 0 ) + k 1 s n ( q u / k 0 ) d n ( q u / k 0 ) q 2 d n 2 ( q u / k 0 ) + Λ 2 E 0 2 s n 2 ( q u / k 0 ) ] ,
d F e 3 d u = F h 3 , d F h 3 d u = { [ n 2 ( u ) n eff 2 ] } F e 3 .
F e 3 [ k 0 ( h + 2 d ) ] = E s ,
F h 3 [ k 0 ( h + 2 d ) ] = ( k s / k 0 ) E s .
P g = 1 2 s Re ( E × H * ) · z d S = P a + P c + P f + P s ,
P a = β E 0 2 4 ω μ 0 k a , P s = β E s 2 4 ω μ 0 k s , P c = β 2 ω μ 0 0 h E 2 ( x ) d x , P f = β 2 ω μ 0 h h + 2 d E 2 ( x ) d x .

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