Abstract

A method for distortion-free optical pulse transmission is theoretically proposed that employs optical Fourier transformation and Fourier transform-limited (TL) pulses. With this technique, a Fourier TL pulse is used as an input signal. By noting that a Gaussian TL pulse has a Gaussian spectrum, the unchanged spectrum of the TL pulse after transmission is converted into the time domain using optical Fourier transformation. Thus the transformed waveform in the time domain has no distortion as long as the transmitted spectral envelope is not changed owing to the perturbations.

© 2005 Optical Society of America

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References

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  1. B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951-1963 (1994).
    [Crossref]
  2. M. Romagnoli, P. Franco, R. Corsini, A. Schiffini, and M. Midrio, "Time-domain Fourier optics for polarization-mode dispersion," Opt. Lett. 24, 1197-1199 (1999).
    [Crossref]
  3. L. F. Mollenauer and C. Xu, "Time-lens timing-jittercompensator in ultra-long haul DWDM dispersion managed soliton transmissions," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper CPDB1-1.
  4. L. A. Jiang, M. E. Grein, H. A. Haus, E. P. Ippen, and H. Yokoyama, "Timing jitter eater for optical pulse trains," Opt. Lett. 28, 78-80 (2003).
    [Crossref] [PubMed]
  5. T. Sakano, K. Uchiyama, I. Shake, T. Morioka, and K. Hagimoto, "Large-dispersion-tolerance optical signal transmission system based on temporal imaging," Opt. Lett. 27, 583-585 (2002).
    [Crossref]
  6. A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
    [Crossref]
  7. M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  9. M. Miyagi and S. Nishida, "Pulse spreading in a single-mode fiber due to third-order dispersion," Appl. Opt. 18, 678-682 (1979).
    [Crossref] [PubMed]
  10. C. V. Bennett and B. H. Kolner, "Aberrations in temporal imaging," IEEE J. Quantum Electron. 37, 20-32 (2001).
    [Crossref]
  11. J. van Howe, J. Hansryd, and C. Xu, "Multiwavelength pulse generator using time-lens compression," Opt. Lett. 29, 1470-1472 (2004).
    [Crossref] [PubMed]

2004 (1)

2003 (1)

2002 (1)

2001 (1)

C. V. Bennett and B. H. Kolner, "Aberrations in temporal imaging," IEEE J. Quantum Electron. 37, 20-32 (2001).
[Crossref]

1999 (1)

1994 (1)

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[Crossref]

1979 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Bennett, C. V.

C. V. Bennett and B. H. Kolner, "Aberrations in temporal imaging," IEEE J. Quantum Electron. 37, 20-32 (2001).
[Crossref]

Corsini, R.

Doi, M.

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

Franco, P.

Grein, M. E.

Hagimoto, K.

Hansryd, J.

Hasegawa, A.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

Haus, H. A.

Ippen, E. P.

Jiang, L. A.

Kolner, B. H.

C. V. Bennett and B. H. Kolner, "Aberrations in temporal imaging," IEEE J. Quantum Electron. 37, 20-32 (2001).
[Crossref]

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[Crossref]

Matsumoto, M.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

Midrio, M.

Miyagi, M.

Mollenauer, L. F.

L. F. Mollenauer and C. Xu, "Time-lens timing-jittercompensator in ultra-long haul DWDM dispersion managed soliton transmissions," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper CPDB1-1.

Morioka, T.

Nakazawa, T.

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

Nishida, S.

Onaka, H.

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

Romagnoli, M.

Sakano, T.

Schiffini, A.

Shake, I.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Sugiyama, M.

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

Taniguchi, S.

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

Uchiyama, K.

van Howe, J.

Xu, C.

J. van Howe, J. Hansryd, and C. Xu, "Multiwavelength pulse generator using time-lens compression," Opt. Lett. 29, 1470-1472 (2004).
[Crossref] [PubMed]

L. F. Mollenauer and C. Xu, "Time-lens timing-jittercompensator in ultra-long haul DWDM dispersion managed soliton transmissions," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper CPDB1-1.

Yokoyama, H.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

C. V. Bennett and B. H. Kolner, "Aberrations in temporal imaging," IEEE J. Quantum Electron. 37, 20-32 (2001).
[Crossref]

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[Crossref]

Opt. Lett. (4)

Other (4)

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

M. Sugiyama, M. Doi, S. Taniguchi, T. Nakazawa, and H. Onaka, "Driver-less 40 Gb/sLiNbO3 modulator with sub-1 V drive voltage," in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper FB6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

L. F. Mollenauer and C. Xu, "Time-lens timing-jittercompensator in ultra-long haul DWDM dispersion managed soliton transmissions," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postdeadline paper CPDB1-1.

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Figures (13)

Fig. 1
Fig. 1

Principle of distortion-free pulse transmission. A TL pulse is used at the input, and a FT pulse is used at the output.

Fig. 2
Fig. 2

Example of the distortion-free transmission.

Fig. 3
Fig. 3

Elimination of second-order dispersion. Transmission fiber length is 150 m with a GVD of 17 ps km nm . (a-1) is a broadened-pulse waveform throughout transmission and (a-2) is the corresponding spectrum. (b-1) and (b-2) are the recovered waveform and corresponding FT spectral profile for ideal OFTC, respectively. (c-1) and (c-2) are the recovered waveform and corresponding FT spectral profile with sinusoidal OFT, respectively.

Fig. 4
Fig. 4

Elimination of second-order dispersion. Transmission fiber length is 300 m with a GVD of 17 ps km nm . (a-1) is a broadened-pulse waveform throughout transmission and (a-2) is the corresponding spectrum. (b-1) and (b-2) are the recovered waveform and corresponding FT spectral profile with ideal OFTC, respectively. (c-1) and (c-2) are the recovered waveform and corresponding FT spectral profile with sinusoidal OFT, respectively.

Fig. 5
Fig. 5

Elimination of second-order dispersion. Transmission fiber length is 500 m with a GVD of 17 ps km nm . (a-1) is a broadened-pulse waveform throughout transmission and (a-2) is the corresponding spectrum. (b-1) and (b-2) are the recovered waveform and corresponding FT spectral profile with ideal OFTC, respectively. (c-1) and (c-2) are the recovered waveform and corresponding FT spectral profile with sinusoidal OFT, respectively.

Fig. 6
Fig. 6

Elimination of third-order dispersion. Transmission length is 40 km with a TOD of 0.07 ps km nm 2 . (a-1) is a distorted pulse waveform throughout transmission and (a-2) is the corresponding spectrum. (b-1) and (b-2) are the recovered waveform and corresponding FT spectral profile with ideal OFTC, respectively. (c-1) and (c-2) are the recovered waveform and corresponding FT spectral profile with sinusoidal OFT, respectively.

Fig. 7
Fig. 7

Elimination of third-order dispersion. Transmission length is 120 km with a TOD of 0.07 ps km nm 2 . (a-1) is a distorted pulse waveform throughout transmission and (a-2) is the corresponding spectrum. The waveform is extended into an adjacent time slot. (b-1) and (b-2) are the incompletely recovered waveform and corresponding FT spectral profile with ideal OFTC, respectively. The waveform is not ideally recovered. (c-1) and (c-2) are the incompletely recovered waveform and corresponding FT spectral profile with sinusoidal OFTC. The distortion is larger than those with ideal OFTC.

Fig. 8
Fig. 8

Jitter elimination with TOD. (a) is the pulse after the transmission, and this changes into (b) through the ideal OFT. When we use an OFT with sinusoidal modulation, it should be noted, as shown in (c), that the jittered pulse does not become a single pulse, since there is nonlinear chirp on both wings of the modulation.

Fig. 9
Fig. 9

FT output waveform of the transmitted pulse in a fiber with a GVD of 17 ps nm km and a length of 300 m , computed for several values of the second-order dispersion used in OFT. The second-order dispersion D was varied between D = 0.5 K 1.5 K . When we reduce the dispersion from D = 1 K (top) to 0.8 K (b) and 0.5 K (a), the pulse gradually broadens. When we increase the dispersion to 1.2 K (c) 1.5 K (d), the pulse is narrower than with ideal FT.

Fig. 10
Fig. 10

Output pulse width T out after OFTC as a function of the normalized GVD in the OFTC. The normalized GVD of 1.0 corresponds to the ideal OFT condition, in which the pulse width recovers to the original pulse width of 2 ps regardless of the value of D z , namely, the length and the dispersion of the transmission fiber.

Fig. 11
Fig. 11

FT output waveform of the transmitted pulse in a fiber with a TOD of 0.07 ps km nm 2 and a length of 40 km , computed for several values of the second-order dispersion used in OFT. The second-order dispersion D was varied between D = 0.5 K 1.5 K . When the dispersion is reduced from D = 1 K to 0.8 K (b) and 0.5 K (a), there is still a small amount of TOD in the trailing edge area. When the dispersion is increased to 1.2 K (c) 1.5 K (d), the pulses are asymmetrically broadened.

Fig. 12
Fig. 12

The same result as Fig. 8, but with phase modulation given by Eqs. (26) and (28). (a) shows ϕ ( t ) in Eq. (26) and (28) (thick solid curve), compared with the ideal parabola (thin solid curve) and a single sinusoidal function (dashed curve). (b) shows the output waveform after OFT.

Fig. 13
Fig. 13

The same result as Fig. 8, but with phase modulation given by Eq. (29), (a) shows ϕ ( t ) in Eq. (29) (thick solid curve), compared with the ideal parabola (thin solid curve) and a single sinusoidal function (dashed curve). (b) shows the output waveform after OFT.

Equations (32)

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( i ) u ( z , t ) z = n D n ( i t ) n u ( z , t ) ,
U ( z , ω ) = U ( 0 , ω ) exp ( i n D n ω n z ) ,
u chirp ( t ) = u ( z , t ) exp ( i 1 2 K t 2 ) .
v ( t ) = ( i 2 π D ) 1 2 u chirp ( t ) exp [ i 2 D ( t t ) 2 ] d t ,
v ( t ) = ( i 2 π D ) 1 2 exp ( i 1 2 K t 2 ) u ( z , t ) exp [ i ( t D ) t ] d t = ( i 2 π D ) 1 2 exp ( i 1 2 K t 2 ) U ( z , t D ) .
v ( t ) = ( i 2 π D ) 1 2 U ( 0 , t D ) exp [ i 1 2 K t 2 + i ϕ ( t D ) ]
ϕ ( t D ) = ϕ ( ω ) = β ( ω ) z .
v ( t ) = ( i T 0 2 D ) 1 2 A exp [ t 2 ( 2 D 2 T 0 2 ) ] exp [ i 1 2 K t 2 + i ϕ ( t D ) ] .
D = T 0 2 ,
v ( t ) = [ i sign ( k ) ] 1 2 A exp ( 1 2 t 2 T 0 2 ) exp [ i 1 2 K t 2 + i ϕ ( t D ) ] .
v ( t ) 2 = A 2 exp ( t 2 T 0 2 ) .
Δ ϕ ( t ) M ( 1 1 2 ω m 2 t 2 ) .
u ( z , t ) = 1 2 π U ( 0 , ω ω 0 ) exp [ i β 2 2 ( ω ω 0 ) 2 z + i β 3 6 ( ω ω 0 ) 3 z i ( ω ω 0 ) t ] d ( ω ω 0 ) .
u ( z , t ) = A T 0 2 π exp [ i S 6 ( ω ω 0 ) 3 T 0 2 2 ( ω ω 0 ) 2 i ( ω ω 0 ) t ] d ω ,
u ( z , t ) = 2 π A T 0 ( 1 2 S ) 1 3 exp ( T 0 6 3 S 2 T 0 2 S t ) × Ai [ ( 1 2 S ) 1 3 ( T 0 4 2 S 2 t ) ] ,
Ai ( t ) = 1 2 π exp ( i x 3 3 + i x t ) d x , x = ω ω 0 .
v ( t ) = ( i 2 π D ) 1 2 exp ( i K t 2 2 ) u ( z , t ) exp [ i 2 D ( t t ) 2 ] d t = ( i 2 π D ) 1 2 exp ( i K t 2 2 ) 2 π A T 0 ( 1 2 S ) 1 3 exp ( T 0 6 3 S 2 ) exp ( T 0 2 S t ) Ai [ ( 1 2 S ) 1 3 ( T 0 4 2 S 2 t ) ] exp ( i t D t ) d t .
1 2 π exp { i [ t D ( 2 ) 2 3 S 1 3 x ] t } d t = δ [ t D ( 2 ) 2 3 S 1 3 x ] ,
v ( t ) = i D 2 exp ( i K t 2 2 ) A T 0 ( 1 2 S ) 1 3 exp ( T 0 6 6 S 2 ) × exp { i 3 [ x + i T 0 2 ( 2 S ) 2 3 ] 3 + i ( 1 2 S ) 4 3 T 0 4 x } × δ [ t D ( 2 ) 2 3 S 1 3 x ] d x .
v ( t ) = i D 2 exp ( i K t 2 2 ) A T 0 ( 1 2 S ) 1 3 exp ( T 0 6 6 S 2 ) S 1 3 ( 2 ) 2 3 exp { i 3 [ ( S 2 ) 1 3 t D + i T 0 2 ( 2 S ) 2 3 ] 3 + i ( 1 2 S ) 4 3 T 0 4 ( S 2 ) 1 3 t D } = i D A T 0 exp ( i K t 2 2 ) exp ( i 6 S t 3 D 3 T 0 2 2 D 2 t 2 ) .
v ( t ) 2 = A 2 exp ( t 2 T 0 2 ) ,
U ( z , ω ) = u ( 0 , t Z v g ) exp ( i ω t ) d t = U ( 0 , ω ) exp ( i ω z v g ) .
v ( t ) = [ i sign ( k ) ] 1 2 A exp [ 1 2 T 0 2 ( 1 + i C ) t 2 ] ,
C = sign ( k ) T 0 2 + D z T 0 2 .
δ D = C 1 + C 2 T 0 2 .
T out = [ ( 1 + C δ D T 0 2 ) 2 + ( δ D T 0 2 ) 2 ] 1 2 T 0 .
2 C 1 + C 2 T 0 2 < δ D < 0 for C > 0 ,
0 < δ D < 2 C 1 + C 2 T 0 2 for C < 0 .
ϕ ( t ) = A cos ω m t + B cos 2 ω m t .
ϕ ( t ) = A ( 1 ω m 2 t 2 2 + ω m 4 t 4 24 + ) + B ( 1 4 ω m 2 t 2 2 + 16 ω m 4 t 4 24 + ) = ( A + B ) ω m 2 t 2 2 ( A + 4 B ) + ω m 4 t 4 24 ( A + 16 B ) +
A = 4 K 3 ω m 2 , B = K 12 ω m 2 .
ϕ ( t ) = K π 2 6 ω m 2 2 K ω m 2 n = 1 ( 1 ) n n 2 cos ( n ω m t ) .

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