Abstract

Propagation of optical signals across a linear–nonlinear interface is investigated by using a spectral decomposition technique involving discrete sideband frequencies. The complexity of the analysis is shown to be appreciably reduced by assuming incommensurate discrete sidebands around the carrier. The efficacy of this formalism is tested for various cases, including discrete stationary modes, evolution of discrete sidebands assuming an undepleted carrier, and, finally, AM pulse propagation across the interface. Among several interesting results, the formation of a narrow-band FM pulse, spatially separated from the ubiquitous AM pulse, is demonstrated. The latter result may be interpreted as a test of the stability of the uniform plane-wave solution.

© 1990 Optical Society of America

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References

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  1. A. E. Kaplan, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).
  2. P. W. Smith and W. J. Tomlinson, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).
  3. W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, Appl. Opt. 21, 2041 (1982).
    [Crossref] [PubMed]
  4. P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
    [Crossref]
  5. A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988).
    [Crossref]
  6. A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
    [Crossref] [PubMed]
  7. G. Cao and P. P. Banerjee, J. Opt. Soc. Am. B 6, 191 (1989).
    [Crossref]
  8. See, for instance, A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985).
    [Crossref]
  9. A. Korpel and P. P. Banerjee, Proc. IEEE 72, 1109 (1984).
    [Crossref]
  10. R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).
  11. D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
    [Crossref]
  12. G. Indebetouw, K. P. Lo, and K. C. Ho, J. Opt. Soc. Am. A 5, 1030 (1988).
    [Crossref]
  13. R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1968).
  14. E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

1989 (2)

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[Crossref] [PubMed]

G. Cao and P. P. Banerjee, J. Opt. Soc. Am. B 6, 191 (1989).
[Crossref]

1988 (2)

1984 (2)

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[Crossref]

A. Korpel and P. P. Banerjee, Proc. IEEE 72, 1109 (1984).
[Crossref]

1982 (1)

1981 (1)

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

1974 (1)

R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[Crossref] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988).
[Crossref]

Banerjee, P. P.

G. Cao and P. P. Banerjee, J. Opt. Soc. Am. B 6, 191 (1989).
[Crossref]

A. Korpel and P. P. Banerjee, Proc. IEEE 72, 1109 (1984).
[Crossref]

Butkov, E.

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

Cao, G.

Gordon, J. P.

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1968).

Hermann, J.-P.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

Ho, K. C.

Indebetouw, G.

Kaplan, A. E.

W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, Appl. Opt. 21, 2041 (1982).
[Crossref] [PubMed]

A. E. Kaplan, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).

Korpel, A.

A. Korpel and P. P. Banerjee, Proc. IEEE 72, 1109 (1984).
[Crossref]

Levine, D.

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[Crossref]

Lo, K. P.

Maloney, P. J.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[Crossref] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988).
[Crossref]

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[Crossref] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988).
[Crossref]

See, for instance, A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985).
[Crossref]

Penrose, R.

R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).

Smith, P. W.

W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, Appl. Opt. 21, 2041 (1982).
[Crossref] [PubMed]

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

P. W. Smith and W. J. Tomlinson, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).

Steinhardt, P. J.

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[Crossref]

Tomlinson, W. J.

W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, Appl. Opt. 21, 2041 (1982).
[Crossref] [PubMed]

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

P. W. Smith and W. J. Tomlinson, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).

Appl. Opt. (1)

Bull. Inst. Math. Appl. (1)

R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).

IEEE J. Quantum Electron. (1)

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[Crossref]

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

J. Opt. Soc. Am. B (2)

Phys. Rev. A (1)

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[Crossref]

Proc. IEEE (1)

A. Korpel and P. P. Banerjee, Proc. IEEE 72, 1109 (1984).
[Crossref]

Other (5)

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1968).

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

See, for instance, A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985).
[Crossref]

A. E. Kaplan, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).

P. W. Smith and W. J. Tomlinson, in Optical Bistability, C. M. Bowden, N. Ciften, and H. R. Robl, eds. (Plenum, New York, 1981).

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Figures (5)

Fig. 1
Fig. 1

Problem geometry showing normal incidence at a linear–nonlinear interface.

Fig. 2
Fig. 2

Qualitative sketch of relative amplitudes of first and second sidebands on transmission and reflection.

Fig. 3
Fig. 3

Plots of (a) upper and (b) lower sideband amplitudes as a function of propagation distance and (c) composite entire real field for a single sideband pair. Undepleted carrier amplitude A0 = 1.0; sideband amplitudes A1 = A−1 = 0.004 at x = 0; c0 = 1.0, β3 = 0.1, ωc = 10.0 and Ω1 = Ω−1 = 1.0.

Fig. 4
Fig. 4

Plots of upper sideband (a) Ω1 and (b) Ω2 amplitudes as a function of propagation distance and (c) composite entire real field for double sideband pair. Undepleted carrier amplitude A0 = 1.0; sideband amplitudes A1 = A−1 = 0.004 and A2 = A−2 = 0.002 at x = 0; c0 = 1.0, β3, = 0.1, ωc = 10.0, and Ω1 = Ω−1 = 1.0, Ω2 = Ω−2 = 21/2.

Fig. 5
Fig. 5

Qualitative sketch of pulse propagation through a linear–nonlinear interface, showing formation of AM and FM pulses, and of the reflected pulse, some time after the arrival of the incident pulse at the interface.

Equations (73)

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2 ψ / t 2 c 0 2 2 ψ / x 2 = ( 2 / 3 ) β 3 2 ψ 3 / t 2 ,
c 0 c 0 ( 1 + β 3 ψ 2 ) .
ψ ( x , t ) = ( 1 / 2 ) n = N N A n ( x ) × exp { j [ ( ω c + n Ω ) t ( k c + n K ) x ] } + c . c . ,
ψ 3 : ( 3 / 8 ) n = N N m = n N n + N l = N N A l A m l A m n * × exp { j [ ( ω c + n Ω ) t ( k c + n K ) x ] } + c . c .
d A n / d x = j ( β 3 ω c n 2 / 4 c 0 2 k c n ) m = n N n + N l = N N A l A m l A m n * j ( κ ω c n 2 / 2 c 0 2 k c n ) A n .
d 2 A n / d x 2 k c n d A n / d x .
ω c n = ω c + n Ω , k c n = k c + n K , N n N ,
κ = 1 ( c 0 / c 0 ) 2 ,
ψ ( x , t ) = ( 1 / 2 ) n = N N A n ( x ) × exp { j [ ( ω c + Ω n ) t ( k c + K n ) x ] } + c . c . ,
ψ 3 : ( 3 / 8 ) n = N N { A n 2 A n * + l = N l n N 2 A n A l A l * + l = N l ± n N A n * A l A l } × exp { j [ ( ω c + Ω n ) t ( k c + K n ) x ] } + c . c .
d A n / d x = j ( β 3 ω c n 2 / 4 c 0 2 k c n ) { A n 2 A n * + l = N l n N 2 A n A l A l * + l = N l ± n N A n * A l A l } j ( κ ω c n 2 / 2 c 0 2 k c n ) A n .
ω c n = ω c + Ω n , k c n = k c + K n , N n N , n 0 , ω c 0 = ω c , k c 0 = k c ,
Ω n = Ω n , K n = K n .
( 2 κ / β 3 ) A 0 = A 0 2 A 0 * + 2 A 0 A 1 A 1 * + 2 A 0 A 1 A 1 * + 2 A 0 * A 1 A 1 ,
( 2 κ / β 3 ) A 1 = A 1 2 A 1 * + 2 A 1 A 0 A 0 * + 2 A 1 A 1 A 1 * + A 0 2 A 1 * ,
( 2 κ / β 3 ) A 1 = A 1 2 A 1 * + 2 A 1 A 0 A 0 * + 2 A 1 A 1 A 1 * + A 0 2 A 1 * .
A 0 = A R , A 1 = A 1 = a 1 R ,
a 1 2 = ( 2 / 3 ) A 2 ,
κ = ( 5 / 2 ) β 3 A 2 .
c 0 = c 0 / [ 1 ( 5 / 2 ) β 3 A 2 ] 1 / 2 c 0 [ 1 + ( 5 / 4 ) β 3 A 2 ] ,
T A t / A i = a 1 t / a 1 i = 2 η t / ( η i + η t ) = 2 c 0 / ( c 0 + c 0 )
2 c 0 [ 1 + ( 5 / 4 ) β 3 A t 2 ] / [ c 0 + c 0 ( 1 + ( 5 / 4 ) β 3 A t 2 ) ] [ using Eq . ( 3.5 ) ] .
A i A t [ ( c 0 + c 0 ) / 2 c 0 ( 5 c 0 / 8 c 0 ) β 3 A t 2 ] ,
a 1 i a 1 t [ ( c 0 + c 0 ) / 2 c 0 ( 15 c 0 / 16 c 0 ) β 3 a 1 t 2 ] .
a 1 t / A t a 1 i / A i .
A 0 = A R , A 1 , 2 = A 1 , 2 = a 1 , 2 R ,
2 κ / β 3 = A 2 + 6 a 1 2 + 6 a 2 2 + 6 a 1 2 a 2 / A ,
2 κ / β 3 = 3 A 2 + 3 a 1 2 + 6 a 2 2 + 6 A a 2 ,
2 κ / β 3 = 3 A 2 + 6 a 1 2 + 3 a 2 2 + 3 A a 1 2 / a 2 .
a 1 2 = 0.867 A 2 ,
a 2 2 = 0.563 A 2 ,
c 0 = c 0 ( 1 + 3.37 β 3 A 2 ) [ using Eq . ( 2.7 ) ] .
A i A t [ ( c 0 + c 0 ) / 2 c 0 2.07 ( c 0 / c 0 ) β 3 A t 2 ] ,
a 1 i a 1 t [ ( c 0 + c 0 ) / 2 c 0 2.38 ( c 0 / c 0 ) β 3 a 1 t 2 ] ,
a 2 i a 2 t [ ( c 0 + c 0 ) / 2 c 0 3.67 ( c 0 / c 0 ) β 3 a 2 t 2 ] .
a 1 , 2 t / A t a 1 , 2 i / A i
a 2 t / a 1 t a 2 i / a 1 i ,
[ 2 κ / β 3 2 κ / β 3 2 κ / β 3 2 κ / β 3 ] = [ 1 6 6 . 6 6 3 3 6 . 6 6 . . . . . . 3 6 6 . 3 6 3 6 6 . 6 3 ] [ A 2 a 1 2 . a N 1 2 a N 2 ] ,
a l 2 = a l 1 2 , l = 1 , 2 , N .
a l 2 = ( 2 / 3 ) A 2 , i = 1 , 2 , N
κ = ( 2 N + 1 / 2 ) β 3 A 2 .
A n = A n ( 1 ) + 2 A n ( 2 ) + , n 0 ,
κ = ( 1 / 2 ) β 3 A 0 2 .
1 : d A n ( 1 ) / d x = j ( β 3 c 0 2 k c n A 0 2 / 4 c 0 2 ) [ A n ( 1 ) + A n ( 1 ) * ] ,
2 : d A n ( 2 ) / d x = j ( β 3 c 0 2 k c n A 0 2 / 4 c 0 2 ) A n ( 2 ) ,
3 : d A n ( 3 ) / d x = j ( β 3 c 0 2 k c n / 4 c 0 2 ) [ A n ( 1 ) 2 A n ( 1 ) * + 2 A n ( 1 ) l = N l n N A l ( 1 ) A l ( 1 ) * + A n ( 1 ) * l = N l ± n N A l ( 1 ) A l ( 1 ) A 0 2 A n ( 3 ) ] ,
d A n ( 1 ) * / d x = j ( β 3 c 0 2 k c n A 0 2 / 4 c 0 2 ) [ A n ( 1 ) + A n ( 1 ) * ] .
A n ( 1 ) / d n + A n ( 1 ) * / d n = constant = A n ( 1 ) ( 0 ) / d n + A n ( 1 ) * ( 0 ) / d n ,
d n = β 3 c 0 2 k c n A 0 2 / 4 c 0 2 .
A n ( 1 ) + A n ( 1 ) * = [ A n ( 1 ) ( 0 ) + A n ( 1 ) * ( 0 ) ] exp [ j ( d n d n ) x ] .
A n ( 1 ) ( x ) = [ d n A n ( 1 ) ( 0 ) + d n A n ( 1 ) * ( 0 ) ] / ( d n d n ) [ d n / ( d n d n ) ] [ A n ( 1 ) ( 0 ) + A n ( 1 ) * ( 0 ) ] × exp [ j ( d n d n ) x ] .
d n d n = ( β 3 c 0 A 0 2 / 2 c 0 2 ) Ω n ,
Λ n 4 π c 0 / β 3 A 0 2 Ω n .
A n ( 2 ) = 0 , n 0 .
A n ( x ) A ( x ; Ω n ) .
Ω n + 1 Ω n 0 , N
Ψ e ( x ; Ω 1 , Ω 2 , , Ω N , t ) = i = B ( x ; Ω i , t ) ,
B ( x ; Ω i , t ) = A ( x ; Ω i ) exp [ j ( Ω i t K i x ) ] , Ω 0 = 0 , K 0 = 0 .
Ψ e ( x ; , Ω i , , t ) = A ( 0 ; 0 ) + n = { [ d n A ( 0 ; Ω n ) + d n A * ( 0 ; Ω n ) ] / ( d n d n ) [ d n / ( d n d n ) ] [ A ( 0 ; Ω n ) + A * ( 0 ; Ω n ) ] exp [ j ( d n d n ) x ] } exp [ j ( Ω n t K n x ) ] .
Ψ e ( x ; , Ω i , , t ) Ψ e ( x , t ) ;
a ( 0 ; t ) = Ã ( 0 ; Ω ) exp ( j Ω t ) d Ω ,
à ( 0 ; Ω ) = lim Ω n Ω [ A ( 0 ; Ω n ) / ( Ω n + 1 Ω n ) ] ;
d n / ( d n d n ) = ( ω c / Ω n ) 1 ;
ψ e ( x , t ) = j ω c t x / c 0 t x / c 0 [ a ( 0 , τ ) + a * ( 0 , τ ) ] d τ + [ a ( 0 , t x / c 0 ) a * ( 0 , t x / c 0 ) ] + [ a ( 0 , t x / c 0 ) + a * ( 0 , t x / c 0 ) ] + A ,
ψ e ( x , t ) = 2 j ω c t x / c 0 t x / c 0 a ( 0 , τ ) d τ + 2 a ( 0 , t x / c 0 ) + A .
c 0 = Ω n / [ K n ( d n d n ) ] = c 0 [ 1 ( 1 / 2 ) β 3 A 2 ] / ( 1 β 3 A 2 ) = c 0 [ 1 + ( 3 / 4 ) β 3 A 2 ] .
= a ( 0 , t x / c 0 ) ( c 0 1 c 0 1 ) x ,
c c 0 c 0 for β 3 0 ,
c 0 = c 0 [ 1 + ( 1 / 4 ) β 3 A 2 ] c 0 for β 3 0 ,
ψ t ( x , t ) = Re { [ A + j ( ω c β 3 A 2 / c 0 ) x a ( 0 , t x / c 0 ) + 2 a ( 0 , t x / c 0 ) ] exp [ j ω c ( t x / c 0 ) ] } .
A cos ( ω c { t [ 1 ( c 0 / c 0 ) β 3 A a ( 0 , t x / c 0 ) ] x / c 0 } ) ,
ψ i ( x , t ) = Re { [ A / T + ( 2 / T ) a ( 0 , t x / c 0 ) ] exp [ j ω c ( t x / c 0 ) ] } ,
c 0 = c 0 = c 0 = c 0 ,

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