Abstract

In crystals that lack a center of inversion, in addition to the third-order nonlinearity a cascaded χ(2):χ(2) second-order nonlinearity contributes to the nonlinear refraction. A coupled-mode theory in the frequency domain is used to investigate theoretically the application of this new nonlinear index of refraction for realizing intensity-dependent temporal and spatial light-field evolution in waveguide structures. The significant modification and the immense enhancement of the effects of purely third-order optical nonlinearity by second-order down-mixing of the driven second-harmonic field with the incoming light at the fundamental are discussed.

© 1993 Optical Society of America

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References

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  1. S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568–1571 (1982).
    [CrossRef]
  2. Y. Chen, “Solution to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
    [CrossRef]
  3. R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of the susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
    [CrossRef]
  4. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
    [CrossRef] [PubMed]
  5. C. C. Yang, “All-optical ultrafast logic gates that use asymmetric nonlinear directional couplers,” Opt. Lett. 16, 1641–1643 (1991).
    [CrossRef] [PubMed]
  6. A. W. Snyder and D. R. Rowland, “Low power fibre coupler devices: few- versus many-period operation,” Opt. Quantum Electron. 24, 31–40 (1992).
    [CrossRef]
  7. 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]
  8. K. J. Blow, N. J. Doran, and B. K. Nayar, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756 (1989).
    [CrossRef] [PubMed]
  9. N. J. Doran and D. Wood, “Soliton processing element for all-optical switching and logic,” J. Opt. Soc. Am. B 4, 1843–1846 (1987).
    [CrossRef]
  10. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 13, 904–906 (1988).
    [CrossRef] [PubMed]
  11. R. Schiek, “Influence of second-order nonlinearity on the effects of the nonlinear index of refraction,” Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD7.
  12. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  13. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
    [CrossRef] [PubMed]
  14. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  15. R. Schiek, “Mode coupling theory in frequency domain for simulation of the dynamic of light pulse propagation in nonlinear optical waveguides,” Nonlinear Opt. (to be published).
  16. R. Schiek, “All-optical switching in the directional coupler caused by nonlinear refraction due to cascaded second-order nonlinearity,” Opt. Quantum Electron. (to be published).

1993 (1)

1992 (2)

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

A. W. Snyder and D. R. Rowland, “Low power fibre coupler devices: few- versus many-period operation,” Opt. Quantum Electron. 24, 31–40 (1992).
[CrossRef]

1991 (2)

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of the susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

C. C. Yang, “All-optical ultrafast logic gates that use asymmetric nonlinear directional couplers,” Opt. Lett. 16, 1641–1643 (1991).
[CrossRef] [PubMed]

1989 (2)

1988 (3)

1987 (1)

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568–1571 (1982).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Assanto, G.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Blow, K. J.

Chen, Y.

Y. Chen, “Solution to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

DeSalvo, R.

Doran, N. J.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Friberg, S. R.

Hagan, D. J.

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568–1571 (1982).
[CrossRef]

Nayar, B. K.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Rowland, D. R.

A. W. Snyder and D. R. Rowland, “Low power fibre coupler devices: few- versus many-period operation,” Opt. Quantum Electron. 24, 31–40 (1992).
[CrossRef]

Schiek, R.

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of the susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

R. Schiek, “Influence of second-order nonlinearity on the effects of the nonlinear index of refraction,” Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD7.

R. Schiek, “Mode coupling theory in frequency domain for simulation of the dynamic of light pulse propagation in nonlinear optical waveguides,” Nonlinear Opt. (to be published).

R. Schiek, “All-optical switching in the directional coupler caused by nonlinear refraction due to cascaded second-order nonlinearity,” Opt. Quantum Electron. (to be published).

Sfez, B. G.

Sheik-Bahae, M.

Silberberg, Y.

Smith, P. S.

Snyder, A. W.

A. W. Snyder and D. R. Rowland, “Low power fibre coupler devices: few- versus many-period operation,” Opt. Quantum Electron. 24, 31–40 (1992).
[CrossRef]

Stegeman, G. I.

Trillo, S.

Van Stryland, E.

Vanherzeele, H.

Wabnitz, S.

Weiner, A. M.

Wood, D.

Wright, E. M.

Yang, C. C.

IEEE J. Quantum Electron. (2)

Y. Chen, “Solution to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of the susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568–1571 (1982).
[CrossRef]

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

Opt. Lett. (7)

Opt. Quantum Electron. (1)

A. W. Snyder and D. R. Rowland, “Low power fibre coupler devices: few- versus many-period operation,” Opt. Quantum Electron. 24, 31–40 (1992).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Other (3)

R. Schiek, “Mode coupling theory in frequency domain for simulation of the dynamic of light pulse propagation in nonlinear optical waveguides,” Nonlinear Opt. (to be published).

R. Schiek, “All-optical switching in the directional coupler caused by nonlinear refraction due to cascaded second-order nonlinearity,” Opt. Quantum Electron. (to be published).

R. Schiek, “Influence of second-order nonlinearity on the effects of the nonlinear index of refraction,” Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD7.

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Figures (10)

Fig. 1
Fig. 1

Power of the modes and the nonlinear change of the propagation constant for input powers |Â1(z = 0)|2 equal to 1 and 5 MW/m.

Fig. 2
Fig. 2

Phase mismatch between the signal TM1 mode and several second-harmonic modes dependent on the propagation direction.

Fig. 3
Fig. 3

Typical switching states of the nonlinear coupler with auxiliary field irradiation. |Â1(z = 0)|2 is 0.001, 10, 11.3, and 14 MW/m.

Fig. 4
Fig. 4

Typical switching states of the nonlinear coupler with no auxiliary field irradiation. |Â1(z = 0)|2 is 0.001, 14, and 25 MW/m.

Fig. 5
Fig. 5

Soliton propagation with a pulse length of 1 ps. (a) Signal pulse in the time domain over one soliton period (auxiliary irradiation prevents the free-running second harmonic); (b) walk-off of the free-running second harmonic (no auxiliary second-harmonic irradiation).

Fig. 6
Fig. 6

Conversion of the signal pulse with a length of 0.1 ps into the second harmonic in the time domain.

Fig. 7
Fig. 7

Soliton propagation with a pulse duration of 0.1 ps without an auxiliary field over one soliton period.

Fig. 8
Fig. 8

Switching diagram of the nonlinear coupler with cascade process nonlinearity: left, half-beat-length coupler; right, one-beat-length coupler. Dotted–dashed curve, 100-ps pulse length; dashed curve, 0.124-ps pulse length; solid curve, 0.062-ps pulse length. (a) With auxiliary field irradiation. (b) without auxiliary field irradiation.

Fig. 9
Fig. 9

Pulse breakoff of a 100-ps pulse in the one-beat-length coupler in the time domain: top, input waveguide; bottom, second waveguide.

Fig. 10
Fig. 10

Soliton switching of a 0.062-ps pulse in the one-beat-length coupler in the time domain: top, input waveguide; bottom, second waveguide.

Equations (19)

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E t = a 1 ( ω ω L ) e 1 t + a 2 ( ω ω L ) e 2 t + a 3 ( ω ω SH ) e 3 t + a 4 ( ω ω SH ) e 4 t , H t = a 1 ( ω ω L ) h 1 t + a 2 ( ω ω L ) h 2 t + a 3 ( ω ω SH ) h 3 t + a 4 ( ω ω SH ) h 4 t .
d a i ( ω ω L ) d z + j β L ( ω ) a i ( ω ω L ) = j ω L 4 p 0 d x d y P i ( ω ω L ) · e i * , i = 1 , 2 , d a j ( ω ω SH ) d z + j β SH ( ω ) a j ( ω ω SH ) = j ω SH 4 p 0 d x d y P j ( ω ω SH ) · e j * , j = 3 , 4.
d x d y P 1 ( ω ω L ) · e 1 * = 0 [ a 1 ( ω ω L ) K SL ( 1 ) + a 2 ( ω ω L ) K CL ( 1 ) ] + 2 χ ( 2 ) [ 1 2 π d ω 1 K ( 2 ) a 3 ( ω 1 ω SH ) a 1 * ( ω 2 ω L ) ] + χ ( 3 ) [ 1 ( 2 π ) 2 d ω 1 d ω 2 K ( 3 ) a 1 ( ω 1 ω L ) + a 1 ( ω 2 ω L ) a 1 * ( ω 3 ω L ) ] , d x d y P 3 ( ω ω SH ) · e 3 * = 0 [ a 3 ( ω ω SH ) K SSH ( 1 ) + a 4 ( ω ω SH ) K CSH ( 1 ) ] + 1 χ ( 2 ) [ 1 2 π d ω 1 K ( 2 ) a 1 ( ω 1 ω L ) a 1 ( ω 2 ω L ) ] .
K SL ( 1 ) = d x d y 2 n 1 ( ω L ) Δ n 1 ( ω L ) e 1 t e 1 t * = d x d y 2 n 2 ( ω L ) Δ n 2 ( ω L ) e 2 t e 2 t * , K SSH ( 1 ) = d x d y 2 n 1 ( ω SH ) Δ n 1 ( ω SH ) e 3 t e 3 t * = d x d y 2 n 2 ( ω SH ) Δ n 2 ( ω SH ) e 4 t e 4 t * .
K CL ( 1 ) = d x d y 2 n 1 ( ω L ) Δ n 1 ( ω L ) e 2 t e 1 t * = d x d y 2 n 2 ( ω L ) Δ n 2 ( ω L ) e 1 t e 2 t * , K CSH ( 1 ) = d x d y 2 n 1 ( ω SH ) Δ n 1 ( ω SH ) e 4 t e 3 t * = d x d y 2 n 2 ( ω SH ) Δ n 2 ( ω SH ) e 3 t e 4 t *
K ( 2 ) = d x d y e 1 t e 1 t e 3 t * = d x d y e 2 t e 2 t e 4 t * .
K ( 3 ) = d x d y e 1 t e 1 t e 1 t * e 1 t * = d x d y e 2 t e 2 t e 2 t * e 2 t * .
d A 1 d z + j β L A 1 = j ω L 4 p 0 [ 2 χ ( 2 ) K ( 2 ) A 3 A 1 * + χ ( 3 ) K ( 3 ) A 1 A 1 A 1 * ] , d A 3 d z + j β SH A 3 = j ω SH 4 p 0 [ χ ( 2 ) K ( 2 ) A 1 A 1 ] .
A 1 ( z ) = A ¯ exp [ j ( β L + δ ) z ] .
δ = δ casc + δ direct = [ ω L χ ( 2 ) K ( 2 ) / ( 2 p 0 ) ] 2 2 { β L + [ ω L χ ( 3 ) K ( 3 ) / 4 p 0 ] A ¯ 1 2 } β SH ) A ¯ 1 2 + ω L χ ( 3 ) K ( 3 ) 4 p 0 A ¯ 1 2 .
χ ( 3 : casc ) = ω L [ χ ( 2 ) K ( 2 ) ] 2 K ( 3 ) [ 2 ( β L + δ ) β SH ] p 0 .
A 3 ( z = 0 ) = ω L χ ( 2 ) K ( 2 ) / 2 p 0 2 ( β L + δ ) β SH A 1 A 1 ,
p c = 4 0 K CL ( 1 ) χ ( 3 ) K ( 3 ) = 175 MW / m .
d a 1 ( ω ω L ) d ξ + j [ β L ( ω ) β L 0 β L ( ω ω L ) ] a 1 ( ω ω L ) = j ω K ( 2 ) 2 χ ( 2 ) 4 p 0 2 π d ω 1 a 3 ( ω 1 ω SH ) a 1 * ( ω + ω 1 ω L ) × exp [ + j ( 2 β L 0 β SH 0 ) ξ ] , d a 3 ( ω ω SH ) d ξ + j [ β SH ( ω ) β SH 0 β L ( ω ω SH ) ] × a 3 ( ω ω SH ) = j ω K ( 2 ) 2 χ ( 2 ) 4 p 0 2 π d ω 1 a 1 ( ω 1 ω L ) a 1 ( ω ω 1 ω L ) × exp [ j ( 2 β L 0 β SH 0 ) ξ ] , a 2 = 0 , a 4 = 0 ,
β i ( ω ) = β i 0 + β i ( ω ω i ) + 1 2 β i ( ω ω i ) 2 + .
a 1 = a ¯ 1 ( ξ ) exp { j [ β L β L 0 β L ( ω ω L ) + φ 0 ] ξ } , a 3 = a ¯ 3 ( ξ ) exp { j [ β SH β SH 0 β L ( ω ω SH ) ] ξ } ,
a 3 ( ω ω SH ) = ω SH K ( 2 ) χ ( 2 ) 4 p 0 2 π × d ω 1 a ¯ 1 ( ξ = 0 , ω 1 ω L ) a ¯ 1 ( ξ = 0 , ω ω 1 ω L ) β L ( ω 1 ) + β L ( ω ω 1 ) + 2 φ 0 β SH ( ω ) × ( exp { j [ β L ( ω 1 ) β L ( ω 1 ω L ) + β L ( ω ω 1 ) β L ( ω ω 1 ω L ) + 2 φ 0 β SH 0 ] ξ } exp { j [ β SH ( ω ) β SH 0 β L ( ω ω SH ) ] ξ } ) .
d a 1 ( ω ω L ) d ξ + j 1 2 β L ( ω ω L ) 2 a 1 ( ω ω L ) = j [ 2 ω L K ( 2 ) χ ( 2 ) 4 p 0 2 π ] 2 × d ω 1 d ω 2 a 1 ( ω 1 ω L ) a 1 ( ω ω 1 ω L ) a 1 * ( ω + ω 1 ω L ) β L ( ω 2 ) + β L ( ω 1 ω 2 ) + 2 φ 0 β SH ( ω 1 ) .
a 1 ( τ ) = [ 8 η 2 ω L | χ ( 3 : casc ) | ( K ( 3 ) / 4 p 0 ) ] 1 / 2 exp ( j 4 η 2 ξ ) cosh ( 2 η τ | 0.5 β L | 1 / 2 ) .

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