Abstract

Reshaping of ultrashort pulses is predicted to occur when a pulse train from a mode-locked laser is incident upon a nonlinear Fabry–Perot cavity whose length is matched to the period of the pulse train. The temporal shape of the transmitted pulses depends on the relaxation time of the nonlinearity of the Kerr medium inserted into the Fabry– Perot cavity. When the pulse duration is shorter than the Kerr relaxation time, considerable pulse narrowing (by factors of 101–103) is predicted. If the Kerr relaxation time is longer than the period of the pulse train, the analysis shows the existence of two temporal shapes of the output pulse, leading to the possibility of bistability between these two states. A Kerr nonlinearity with an instantaneous response can be used to generate square output pulses. Both the transient and the permanent regimes are investigated, and analytical expressions for the narrowing factors are found.

© 1988 Optical Society of America

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References

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  1. For a general review of these topics, the reader is referred to H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  2. H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
    [CrossRef]
  3. M. Piché, F. Ouellette, “Compression of mode-locked pulses using nonlinear Fabry–Perot interferometers,” Opt. Lett. 11, 15–17 (1986).
    [CrossRef]
  4. T. Bishofberger, Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
    [CrossRef]
  5. F. Ouellette, M. Piché, “Picosecond pulse reshaping using a nonlinear Fabry–Perot interferometer,” in Optical Chaos, J. Chrostowski, N. B. Abraham, eds., Proc. Soc. Photo-Opt. Instrum.667, 205–211 (1986).
    [CrossRef]
  6. D. W. Rush, P. T. Ho, “The coherence time of a mode-locked pulse train,” Opt. Commun. 52, 41–45 (1984).
    [CrossRef]
  7. R. L. Fork, “Optical frequency filter for ultrashort pulses,” Opt. Lett. 11, 629–631 (1986).
    [CrossRef] [PubMed]
  8. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
    [CrossRef]
  9. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
    [CrossRef]
  10. J. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-4, 382–385 (1986).
    [CrossRef]
  11. K. J. Blow, N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. 52, 526–529 (1984).
    [CrossRef]
  12. A. M. Weiner, J. P. Heritage, R. N. Thurston, “Synthesis of phase-coherent, picosecond optical square pulses,” Opt. Lett. 11, 153–155 (1986).
    [CrossRef] [PubMed]

1986 (4)

1984 (2)

K. J. Blow, N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. 52, 526–529 (1984).
[CrossRef]

D. W. Rush, P. T. Ho, “The coherence time of a mode-locked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

1983 (1)

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

1982 (1)

H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
[CrossRef]

1979 (1)

T. Bishofberger, Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

1976 (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

Asaka, S.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Bishofberger, T.

T. Bishofberger, Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Blow, K. J.

K. J. Blow, N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. 52, 526–529 (1984).
[CrossRef]

Doran, N. J.

K. J. Blow, N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. 52, 526–529 (1984).
[CrossRef]

Eichler, H. J.

H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
[CrossRef]

Fork, R. L.

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

For a general review of these topics, the reader is referred to H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Heritage, J. P.

Ho, P. T.

D. W. Rush, P. T. Ho, “The coherence time of a mode-locked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Ikeda, K.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Itoh, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Marcuse, D.

J. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-4, 382–385 (1986).
[CrossRef]

Massman, F.

H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
[CrossRef]

Matsuoka, M.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Ouellette, F.

M. Piché, F. Ouellette, “Compression of mode-locked pulses using nonlinear Fabry–Perot interferometers,” Opt. Lett. 11, 15–17 (1986).
[CrossRef]

F. Ouellette, M. Piché, “Picosecond pulse reshaping using a nonlinear Fabry–Perot interferometer,” in Optical Chaos, J. Chrostowski, N. B. Abraham, eds., Proc. Soc. Photo-Opt. Instrum.667, 205–211 (1986).
[CrossRef]

Piché, M.

M. Piché, F. Ouellette, “Compression of mode-locked pulses using nonlinear Fabry–Perot interferometers,” Opt. Lett. 11, 15–17 (1986).
[CrossRef]

F. Ouellette, M. Piché, “Picosecond pulse reshaping using a nonlinear Fabry–Perot interferometer,” in Optical Chaos, J. Chrostowski, N. B. Abraham, eds., Proc. Soc. Photo-Opt. Instrum.667, 205–211 (1986).
[CrossRef]

Rush, D. W.

D. W. Rush, P. T. Ho, “The coherence time of a mode-locked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Shen, Y. R.

T. Bishofberger, Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Stone, J.

J. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-4, 382–385 (1986).
[CrossRef]

Thurston, R. N.

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

Weiner, A. M.

Zaki, Ch.

H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
[CrossRef]

IEEE J. Lightwave Technol. (1)

J. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-4, 382–385 (1986).
[CrossRef]

Opt. Commun. (2)

H. J. Eichler, F. Massman, Ch. Zaki, “Modulation and compression of Nd–YAG laser pulses by self-tuning of a silicon cavity,” Opt. Commun. 40, 302–306 (1982);H. M. Gibbs, T. N. C. Venkatesan, S. L. McCall, A. Passner, A. C. Gossard, W. Wiegmann, “Optical modulation by optical tuning of a cavity,” Appl. Phys. Lett. 34, 511–514 (1979).
[CrossRef]

D. W. Rush, P. T. Ho, “The coherence time of a mode-locked pulse train,” Opt. Commun. 52, 41–45 (1984).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

T. Bishofberger, Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Phys. Rev. Lett. (3)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometry,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

K. J. Blow, N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. 52, 526–529 (1984).
[CrossRef]

Other (2)

For a general review of these topics, the reader is referred to H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

F. Ouellette, M. Piché, “Picosecond pulse reshaping using a nonlinear Fabry–Perot interferometer,” in Optical Chaos, J. Chrostowski, N. B. Abraham, eds., Proc. Soc. Photo-Opt. Instrum.667, 205–211 (1986).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Two configurations of the nonlinear interferometer matched to a train of short pulses: (a) Fabry–Perot-type cavity, (b) ring-type cavity.

Fig. 2
Fig. 2

Narrowing factor versus input pulse energy for different values of ϕL: a, R = 0.8; b, R = 0.9.

Fig. 3
Fig. 3

Output pulse parameters versus reflectivity of the mirrors (for different values of initial transmittance T0 and for output pulses centered at a0 = 0): a, narrowing factors; b, pulse energy; c, K1/2/Ktr; d, Ktr/Kinc.

Fig. 4
Fig. 4

a, Input and output pulse shape for R = 0.9 and ϕL = −0.2 (C = 53); b, instantaneous frequency of the output pulse, in units of (ωω0)tp; c, output pulse before and after a cutoff filter with cutoff frequency (ωω0)tp = +0.5, showing reduction of the wings.

Fig. 5
Fig. 5

Convergence coefficients versus the number of round trips for M = 1, 30, 50 in the case R = 0.9, ϕL = −0.2: a, CV1; b, CV2; c, C.

Fig. 6
Fig. 6

Evolution of the shape of the output pulse for R = 0.8, ϕL = −0.3 during the first 40 round trips.

Fig. 7
Fig. 7

Curves of δF versus βt for a, R = 0.9, ϕL = −0.4, B = 0.5; b, R = 0.9, ϕL = −0.2, B = 0; c, R = 0.9, ϕL = −0.2, B = 0.5.

Fig. 8
Fig. 8

Curves of convergence coefficients versus the number of round trips for R = 0.9, β = 0.5: a, CV1; b, C; c, output phase-shift δF. Curve 1, ϕL = −0.4, βt = 1.43 for the first 50 round trips; curve 2, ϕL = −0.2, βt = 0.792 for the first 20 round trips. ϕL and βt are then brought back to their values −0.4 and 0.792 in 10 round trips.

Fig. 9
Fig. 9

Square output pulse shapes for R = 0.8, ϕL = −0.4: a, β = 1.0; b, β = 1.45.

Fig. 10
Fig. 10

Output pulse shape with an optical-fiber resonator for a normalized length z = 10−4 and Gaussian filters of different bandwidths Δωtp: a, infinite width; b, 100; c, 300; d, 500.

Equations (33)

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E ( z , t ) = A ( z , t ) exp [ i ( k 0 z ω 0 t ) ] ,
A r n ( 0 , t a ) = ( 1 R ) 1 / 2 A inc ( 0 , t a ) + R T A r n 1 ( 0 , t a t r ) × exp { i π [ ϕ L + ϕ NL n 1 ( t a ) ] } ,
A r ( 0 , t ) = ( 1 R ) 1 / 2 A inc ( 0 , t ) 1 R T exp [ i π ( ϕ L + ϕ NL ( t ) ) ] .
I tr ( t ) = ( 1 R ) 2 T 2 I inc ( t ) ( 1 R T ) 2 + 4 R T sin 2 { π / 2 [ ϕ L + ϕ NL ( t ) ] } .
I inc ( a ) = I inc ( a ) ( 1 R ) I 0 ( 1 R ) , I tr ( a ) = ( 1 R ) I tr ( a ) ( 1 R ) T 2 I 0 , I 0 = I 0 ( 1 R ) ( 1 R ) , R = R T , a = t / t p , l = ( 1 T 2 ) α ,
ϕ NL ( a ) = β ( 1 R ) a exp [ ( a a ) / ( τ / t p ) ] I tr ( a ) d a ,
β = n 2 I 0 ω 0 l π n 0 c ( τ / t p ) ,
I tr ( a ) = I inc ( a ) 1 + F sin 2 { π / 2 [ ϕ L + ϕ NL ( t ) ] } ,
F = 4 R ( 1 R ) 2 .
δ ( a ) = π / 2 [ ϕ L + ϕ NL ( a ) ] .
d δ d a = ( π / 2 ) β I inc ( a ) ( 1 R ) [ 1 + F sin 2 δ ( a ) ] .
δ 1 / 2 = sin 1 ( 1 / F ) .
a = ( 1 R ) [ ( 2 + F ) δ 1 / 2 ( F / 2 ) sin ( 2 δ 1 / 2 ) ] β I inc ( a 0 ) ,
a 0 = 1 ( 2 ln 2 ) 1 / 2 × erf 1 { ( 1 R ) β [ ( 2 + F ) δ i ( F / 2 ) sin ( 2 δ i ) ] 1 } ,
β t > ( 1 R ) { [ ( 1 + F / 2 ) δ i ( F / 4 ) sin ( 2 δ i ) ] + ( 1 + F / 2 ) N π } .
β t = π π 4 ln 2 β ,
T 0 = 1 1 + F sin 2 δ i .
δ ( ) = δ i .
K tr = I tr ( a ) d a = 2 ( 1 R ) δ i ( π / 2 ) β .
K tr K inc = 2 ( 1 R ) δ i β t = 2 δ i ( 2 + F ) δ i F / 2 sin ( 2 δ i ) .
K 1 / 2 K tr = δ 1 / 2 δ i 1 δ i F .
CV 1 = log | I tr n ( a ) I tr n 1 ( a ) | d a I tr n ( a ) d a .
CV 2 = log | I tr n ( a ) I tr st ( a ) | d a I tr st ( a ) d a .
I 0 n = { I 0 n / M 0 < n M I 0 n > M .
ϕ NL ( a ) = β ( 1 R ) { a exp [ ( a a ) / ( τ / t p ) ] I tr ( a ) d a + exp ( t r / τ ) × exp [ ( a a ) / ( τ / t p ) ] I tr ( a ) d a + exp ( 2 t r / τ ) × exp [ ( a a ) / ( τ / t p ) ] I tr ( a ) d a + } .
ϕ NL ( a ) = β ( 1 R ) [ a I tr ( a ) d a + exp ( t r / τ ) 1 exp ( t r / τ ) × I tr ( a ) d a ] .
δ ( a ) = δ i + ( π / 2 ) ϕ NL ( a ) + δ a ,
δ a = exp ( t r / τ ) 1 exp ( t r / τ ) ( π / 2 ) ϕ NL ( ) .
δ a = B ( π / 2 ) ϕ NL ( ) = B 1 + B ( δ F + δ i ) ,
δ F = δ ( ) .
( 1 + 2 / F ) δ F sin ( δ F ) cos [ ( 1 + 2 B ) δ F 2 δ i ] = 2 β t ( 1 R ) F .
I inc ( t ) = { 1 + F sin 2 [ ϕ L + β 1 R I out ( t ) ] } I out ( t ) .
z = | K 2 t p 2 | l ,

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