Abstract

The propagation of TE-polarized guided waves in a nonlinear tapered waveguide is studied. The incident fields are chosen from the stable branches of the nonlinear dispersion curve in its uniform section. The propagations of the field in both directions are calculated by the beam-propagation method. Wave propagation in the tapered waveguide is discussed in terms of the nonlinear dispersion curves of the waveguide. Depending on the intensity, the field can radiate as a spatial soliton or it can have an abrupt transition between the guided mode and the surface polariton. The wave propagation of an incident field that is composed of a dominant TE0 wave and a small TE1 wave is investigated. Finally, using a property of this wave propagation, we present a new device for continuous signal routing.

© 1991 Optical Society of America

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References

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  1. L. Leine, C. Wächter, U. Langbein, and F. Lederer, “Evolution of nonlinear guided optical fields down a dielectric film with a nonlinear cladding,” J. Opt. Soc. Am. B 5, 547–558 (1988).
    [CrossRef]
  2. K. Hayata, A. Misawa, and M. Koshiba, “Split-step finite-element method applied to nonlinear integrated optics,” J. Opt. Soc. Am. B 7, 1772–1784 (1990).
    [CrossRef]
  3. M. A. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Numerical study of soliton emission from a nonlinear waveguide,” J. Opt. Soc. Am. B 4, 1837–1842 (1987).
    [CrossRef]
  4. M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
    [CrossRef]
  5. L. Leine, C. Wächter, U. Langbein, and F. Lederer, “Propagation phenomena of nonlinear film guided waves in a configuration with material losses: a numerical analysis,” Opt. Lett. 12, 747–749 (1987).
    [CrossRef] [PubMed]
  6. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [CrossRef]
  7. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  8. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  9. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [CrossRef]
  10. A. Ankiewicz, “Novel effects in nonlinear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
    [CrossRef]
  11. L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Beam-propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
    [CrossRef] [PubMed]
  12. D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
    [CrossRef]
  13. Y. Silberberg and B. G. Sfez, “All-optical phase- and power-controlled switching in nonlinear waveguide junctions,” Opt. Lett. 13, 1132–1134 (1988).
    [CrossRef] [PubMed]
  14. J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
    [CrossRef]
  15. H. Fouckhardt and Y. Silberberg, “All-optical switching in waveguide X-junctions,” J. Opt. Soc. Am. B 7, 803–809 (1990).
    [CrossRef]
  16. K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
    [CrossRef]
  17. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
    [CrossRef]
  18. S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
    [CrossRef]
  19. M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]

1990 (3)

1989 (3)

K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
[CrossRef]

J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
[CrossRef]

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

1988 (6)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

L. Leine, C. Wächter, U. Langbein, and F. Lederer, “Evolution of nonlinear guided optical fields down a dielectric film with a nonlinear cladding,” J. Opt. Soc. Am. B 5, 547–558 (1988).
[CrossRef]

A. Ankiewicz, “Novel effects in nonlinear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

Y. Silberberg and B. G. Sfez, “All-optical phase- and power-controlled switching in nonlinear waveguide junctions,” Opt. Lett. 13, 1132–1134 (1988).
[CrossRef] [PubMed]

S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
[CrossRef]

1987 (3)

1986 (1)

1985 (1)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

1980 (1)

Ankiewicz, A.

A. Ankiewicz, “Novel effects in nonlinear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Feit, M. D.

Finlayson, N.

J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Fleck, J. A.

Fouckhardt, H.

Gubbels, M.

M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
[CrossRef]

Gubbels, M. A.

Hayata, K.

K. Hayata, A. Misawa, and M. Koshiba, “Split-step finite-element method applied to nonlinear integrated optics,” J. Opt. Soc. Am. B 7, 1772–1784 (1990).
[CrossRef]

K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
[CrossRef]

Heatley, D. R.

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Koshiba, M.

K. Hayata, A. Misawa, and M. Koshiba, “Split-step finite-element method applied to nonlinear integrated optics,” J. Opt. Soc. Am. B 7, 1772–1784 (1990).
[CrossRef]

K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
[CrossRef]

Langbein, U.

Lederer, F.

Leine, L.

Misawa, A.

K. Hayata, A. Misawa, and M. Koshiba, “Split-step finite-element method applied to nonlinear integrated optics,” J. Opt. Soc. Am. B 7, 1772–1784 (1990).
[CrossRef]

K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
[CrossRef]

Moloney, J. V.

Sabini, J. P.

J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
[CrossRef]

M. A. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Numerical study of soliton emission from a nonlinear waveguide,” J. Opt. Soc. Am. B 4, 1837–1842 (1987).
[CrossRef]

L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Beam-propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
[CrossRef] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Sfez, B. G.

Shin, S.-Y.

S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
[CrossRef]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Silberberg, Y.

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
[CrossRef]

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
[CrossRef]

M. A. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Numerical study of soliton emission from a nonlinear waveguide,” J. Opt. Soc. Am. B 4, 1837–1842 (1987).
[CrossRef]

L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Beam-propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
[CrossRef] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Stolen, R. H.

Thylen, L.

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Wächter, C.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
[CrossRef]

M. A. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Numerical study of soliton emission from a nonlinear waveguide,” J. Opt. Soc. Am. B 4, 1837–1842 (1987).
[CrossRef]

M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
[CrossRef]

L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Beam-propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
[CrossRef] [PubMed]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

J. P. Sabini, N. Finlayson, and G. I. Stegeman, “All-optical switching in nonlinear X-junctions,” Appl. Phys. Lett. 55, 1176–1178 (1989).
[CrossRef]

Electron. Lett. (1)

K. Hayata, A. Misawa, and M. Koshiba, “Nonlinear beam propagation in tapered waveguides,” Electron. Lett. 25, 661–662 (1989).
[CrossRef]

IEEE J. Lightwave Technol. (2)

S.-Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” IEEE J. Lightwave Technol. 6, 977–983 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

M. Gubbels, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Effects of absorption on TE0nonlinear guided waves,” Opt. Commun. 61, 357–362 (1987).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

A. Ankiewicz, “Novel effects in nonlinear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

TE-polarized guided-wave power versus effective index for s = 2.4025, f = 2.4649, and c = 0 + α|E|2, where 0 = 2.4025 and α = 3.3776 × 10−12 m2/V2. The thickness of the film is 5 μm, and the wavelength is 1.3 μm. In the inset the z axis is the propagating axis, the linear substrate is at the left, and the nonlinear cladding is at the right.

Fig. 2
Fig. 2

(a) TE0 guided-wave power versus effective index for five different film thicknesses: 5,4, ,3, 2, and 1 μm. Points 1–7 represent the selected incident fields for the tapered waveguide. The dashed line between points 1 and 8 represents the adiabatic process of the field evolving in the tapered waveguide. (b) Detail of plots of the nonlinear dispersion curves for five different film thicknesses: 2.7,2.5,2.3,2.1, and 1.9 μm. The jump is the abrupt transition between the guided mode and the surface polariton. Path A is the forward jump, shown in Fig. 3(b), and path B is the backward jump, shown in Fig. 4. The field evolves along the solid horizontal line ab for the adiabatic process in the tapered waveguide before the jump; the dotted lines represent the jumping processes, and the dashed line represents the field evolution after the jump.

Fig. 3
Fig. 3

Field evolution for forward propagation with various input powers. The tapered waveguide is shown at the left of each panel. The tapered section is 2 mm long, and the uniform section at each end is 0.5 mm long. The thickness of the thicker end is 5 μm and that of the thinner end is 1 μm. (a) Input power, 0.08 W/mm, point 1 of Fig. 2(a). (b) Input power, 0.16 W/mm, point 2 of Fig. 2(a). (c) Input power, 0.24 W/mm point 3 of Fig. 2(a). (d) Input power, 0.32 W/mm, point 4 of Fig. 2(a). (e) Input power, 0.4 W/mm, point 5 of Fig. 2(a). (f) Input power, 0.48 W/mm, point 6 of Fig. 2(a).

Fig. 4
Fig. 4

Profile evolution for backward propagation with an input power of 0.16 W/mm, point 7 in Fig. 2(a). The tapered waveguide is shown at the left, and its dimension is the same as that in Fig. 3; the linear substrate is to the left and the nonlinear cladding is to the right of the waveguide.

Fig. 5
Fig. 5

(a) Wave propagation in the nonlinear tapered waveguide of the incident field composed of a dominant TE0 wave and a weak TE1 wave. (b) Same as (a) but for the linear case.

Fig. 6
Fig. 6

(a) TE0 wave evolving in the proposed device, which is shown schematically at the left. (b) Same as (a) but with a TE1 wave input of 3 mW/mm (i.e., 2.5% of the power of the TE0 wave). (c) Same as (a) but with a TE1 wave input of 17.4 mW/mm (i.e., 14.5% of the power of the TE0 wave). (d) Same as (b) but with a relative phase difference of 180° between the TE0 and TE1 waves.

Fig. 7
Fig. 7

Dependence of the routing angle on the input power of the TE1 wave.

Fig. 8
Fig. 8

Dependence of the routing angle on the relative phase difference between the TE0 and TE1 waves.

Equations (5)

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E y ( x , z , t ) = ½ E y ( x ) exp [ i ( k 0 β z - ω t ) ] + c . c . ,
E c y ( x ) = ( 2 α ) 1 / 2 q cosh [ k 0 q ( x - x c ) ] ,
E f y ( x ) = { A cos [ k 0 Γ ( x - x f ) ] for β 2 < f A sinh [ k 0 γ ( x - x f ) ] for β 2 > f ,
E s y ( x ) = B exp ( k 0 p x ) ,
P = 1 2 c 0 β - E y 2 ( x ) d x .

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