Abstract

We propose an optical time-to-two-dimensional (2-D)-space-to-time-to-2-D-space conversion technique for ultrafast image transmission with ultrashort-pulse lasers. The proposed technique is based on the concepts of the time–space transform, the spatial time–frequency transform, and their inverses. We describe and analyze the proposed technique for ultrafast all-optical processors that can convert an input 2-D spatial object into a modulated ultrafast optical pulse sequence and can retrieve the original 2-D spatial image from the temporal signals transmitted through the optical fiber channel with ultrahigh bandwidth. To verify the proposed technique, we report preliminary experimental results.

© 1999 Optical Society of America

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  1. C. A. Brackett, “Is there an emerging consensus on WDM networking?” J. Lightwave Technol. 14, 936–941 (1996).
  2. H. Taga, “Long distance transmission experiments using the WDM technology,” J. Lightwave Technol. 14, 1287–1298 (1996).
    [CrossRef]
  3. V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
    [CrossRef]
  4. H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
    [CrossRef]
  5. S. Baroni, P. Bayvel, “Wavelength requirements in arbitrarily connected wavelength-routed optical networks,” J. Lightwave Technol. 15, 242–251 (1997).
    [CrossRef]
  6. Y. T. Mazurenko, “Reconstruction of a nonstationary wave field by holography in a 3-D medium,” Opt. Spectrosc. (USSR) 57, 343–344 (1984).
  7. P. C. Sun, Y. T. Mazurenko, W. S. C. Chang, P. K. L. Yu, Y. Fainman, “All-optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. 20, 1728–1730 (1995).
    [CrossRef] [PubMed]
  8. P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
    [CrossRef]
  9. R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
    [CrossRef]
  10. A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
    [CrossRef]
  11. M. M. Wefers, K. A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light-modulators,” J. Opt. Soc. Am. B 12, 1343–1362 (1995).
    [CrossRef]
  12. M. M. Wefers, K. A. Nelson, A. M. Weiner, “Multidimensional shaping of ultrafast optical waveforms,” Opt. Lett. 21, 746–748 (1996).
    [CrossRef] [PubMed]
  13. H. M. Ozaktas, M. C. Nuss, “Time-variant linear pulse processing,” Opt. Commun. 131, 114–118 (1996).
    [CrossRef]
  14. L. Cohen, “Time–frequency distribution,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  15. L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  16. A. Cohen, J. Kovacevic, “Wavelets: the mathematical background,” Proc. IEEE 84, 514–522 (1996).
    [CrossRef]
  17. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
    [CrossRef] [PubMed]
  18. K. G. Purchase, D. J. Brady, K. Wagner, “Time-of-flight cross correlation on a detector array for ultrafast packet detection,” Opt. Lett. 18, 2129–2131 (1993).
    [CrossRef] [PubMed]
  19. D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
    [CrossRef]

1997 (3)

H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
[CrossRef]

S. Baroni, P. Bayvel, “Wavelength requirements in arbitrarily connected wavelength-routed optical networks,” J. Lightwave Technol. 15, 242–251 (1997).
[CrossRef]

P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
[CrossRef]

1996 (6)

A. Cohen, J. Kovacevic, “Wavelets: the mathematical background,” Proc. IEEE 84, 514–522 (1996).
[CrossRef]

M. M. Wefers, K. A. Nelson, A. M. Weiner, “Multidimensional shaping of ultrafast optical waveforms,” Opt. Lett. 21, 746–748 (1996).
[CrossRef] [PubMed]

C. A. Brackett, “Is there an emerging consensus on WDM networking?” J. Lightwave Technol. 14, 936–941 (1996).

H. Taga, “Long distance transmission experiments using the WDM technology,” J. Lightwave Technol. 14, 1287–1298 (1996).
[CrossRef]

V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
[CrossRef]

H. M. Ozaktas, M. C. Nuss, “Time-variant linear pulse processing,” Opt. Commun. 131, 114–118 (1996).
[CrossRef]

1995 (2)

1993 (2)

K. G. Purchase, D. J. Brady, K. Wagner, “Time-of-flight cross correlation on a detector array for ultrafast packet detection,” Opt. Lett. 18, 2129–2131 (1993).
[CrossRef] [PubMed]

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

1992 (2)

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

1989 (1)

L. Cohen, “Time–frequency distribution,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1986 (1)

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

1984 (1)

Y. T. Mazurenko, “Reconstruction of a nonstationary wave field by holography in a 3-D medium,” Opt. Spectrosc. (USSR) 57, 343–344 (1984).

Baroni, S.

S. Baroni, P. Bayvel, “Wavelength requirements in arbitrarily connected wavelength-routed optical networks,” J. Lightwave Technol. 15, 242–251 (1997).
[CrossRef]

Bayvel, P.

S. Baroni, P. Bayvel, “Wavelength requirements in arbitrarily connected wavelength-routed optical networks,” J. Lightwave Technol. 15, 242–251 (1997).
[CrossRef]

Brackett, C. A.

C. A. Brackett, “Is there an emerging consensus on WDM networking?” J. Lightwave Technol. 14, 936–941 (1996).

Brady, D. J.

Caviris, N. P.

Chang, W. S. C.

Chu, P. L.

H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
[CrossRef]

Cohen, A.

A. Cohen, J. Kovacevic, “Wavelets: the mathematical background,” Proc. IEEE 84, 514–522 (1996).
[CrossRef]

Cohen, L.

L. Cohen, “Time–frequency distribution,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Fainman, Y.

Hatami-Hanza, H.

H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
[CrossRef]

Heritage, J. P.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Heuring, V. P.

V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
[CrossRef]

Jordan, H. F.

V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
[CrossRef]

Kane, D. J.

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

Kanterakis, E. G.

Katz, A.

Kovacevic, J.

A. Cohen, J. Kovacevic, “Wavelets: the mathematical background,” Proc. IEEE 84, 514–522 (1996).
[CrossRef]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

Li, Y.

Lu, X. J.

Mazurenko, Y. T.

Mostofi, A.

H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
[CrossRef]

Nelson, K. A.

Nuss, M. C.

H. M. Ozaktas, M. C. Nuss, “Time-variant linear pulse processing,” Opt. Commun. 131, 114–118 (1996).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, M. C. Nuss, “Time-variant linear pulse processing,” Opt. Commun. 131, 114–118 (1996).
[CrossRef]

Patel, J. S.

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

Pratt, J. P.

V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
[CrossRef]

Purchase, K. G.

Sun, P. C.

Taga, H.

H. Taga, “Long distance transmission experiments using the WDM technology,” J. Lightwave Technol. 14, 1287–1298 (1996).
[CrossRef]

Thurston, R. N.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Tolimieri, R.

Tomlonson, W. J.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Trebino, R.

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

Wagner, K.

Wefers, M. M.

Weiner, A. M.

M. M. Wefers, K. A. Nelson, A. M. Weiner, “Multidimensional shaping of ultrafast optical waveforms,” Opt. Lett. 21, 746–748 (1996).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Wullert, J. R.

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

Yu, P. K. L.

Zhang, Y.

IEEE J. Quantum Electron. (3)

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlonson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–920 (1992).
[CrossRef]

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

J. Lightwave Technol. (5)

C. A. Brackett, “Is there an emerging consensus on WDM networking?” J. Lightwave Technol. 14, 936–941 (1996).

H. Taga, “Long distance transmission experiments using the WDM technology,” J. Lightwave Technol. 14, 1287–1298 (1996).
[CrossRef]

V. P. Heuring, H. F. Jordan, J. P. Pratt, “Energy transfers and frequency shifts from three soliton collisions in a multiplexed transmission line with periodic amplification,” J. Lightwave Technol. 14, 1639–1643 (1996).
[CrossRef]

H. Hatami-Hanza, A. Mostofi, P. L. Chu, “A multilevel soliton communication system,” J. Lightwave Technol. 15, 6–19 (1997).
[CrossRef]

S. Baroni, P. Bayvel, “Wavelength requirements in arbitrarily connected wavelength-routed optical networks,” J. Lightwave Technol. 15, 242–251 (1997).
[CrossRef]

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

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

Opt. Commun. (1)

H. M. Ozaktas, M. C. Nuss, “Time-variant linear pulse processing,” Opt. Commun. 131, 114–118 (1996).
[CrossRef]

Opt. Lett. (4)

Opt. Spectrosc. (USSR) (1)

Y. T. Mazurenko, “Reconstruction of a nonstationary wave field by holography in a 3-D medium,” Opt. Spectrosc. (USSR) 57, 343–344 (1984).

Proc. IEEE (2)

L. Cohen, “Time–frequency distribution,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

A. Cohen, J. Kovacevic, “Wavelets: the mathematical background,” Proc. IEEE 84, 514–522 (1996).
[CrossRef]

Other (1)

L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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Figures (12)

Fig. 1
Fig. 1

Conceptual diagram of the time-to-2-D-space-to-time conversion.

Fig. 2
Fig. 2

Illustrating diagram of the spatial wavelet transform.

Fig. 3
Fig. 3

Block diagram of the proposed time-to-2-D-space-to-time-to-2-D-space conversion.

Fig. 4
Fig. 4

Optical system for optical time-to-pseudo-2-D-space conversion.

Fig. 5
Fig. 5

Optical system for optical time-to-2-D-space-to-time conversion.

Fig. 6
Fig. 6

Illustrating diagram of encoding a 2-D spatial signal into a temporal signal.

Fig. 7
Fig. 7

(a) Autocorrelation measurement of an ultrashort pulse, (b) spectral measurement of the ultrashort pulse.

Fig. 8
Fig. 8

(a) Light field obtained by time-to-pseudo-2-D-space conversion that is photographed by a 2-D CCD detector, (b) experimental results of interferometric time-of-flight cross correlation in the plane P2 (bounded portions around interference patterns are contrast enhanced by an image processing technique).

Fig. 8
Fig. 8

(c) Schematic diagram of time-of-flight cross correlation in the plane P2 (interference between the field obtained and the collimated reference pulse beam from a mode-locked Ti:sapphire laser with different time delays), (d) series of spectra of the field obtained in the vertical direction.

Fig. 9
Fig. 9

Input 2-D object for optical 2-D-space-to-time conversion.

Fig. 10
Fig. 10

(a) Experimental results of time-of-flight cross correlation in the plane P4 (bounded portions around interference patterns are contrast enhanced by an image processing technique.)

Fig. 10
Fig. 10

(b) Schematic diagram of interferometric time-of-flight cross correlation in the plane P4 (interference between the output distribution and the collimated reference pulse beam from a mode-locked Ti:sapphire laser with different time delays), (c) series of spectral distributions of the light field at different positions in the horizontal direction of the plane P4 obtained by optical 2-D-space-to-time conversion.

Equations (38)

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WT(x, νx)=-f (x)ψx-xνxdx,
f(x)=-W T(x, νx)ψ *x-xνxdνxd x,
ri(x0, y0; t)=r˜0(t)exp(+jωct),
G0(x0, y0; ω-ωc)=w(x0, y0)exp+j ω-ωccαx0,
α=λd cos θ,
Ri(x0, y0; ω, t)=R˜0(ω-ωc)exp(+jωt),
R˜0(ω-ωc)=-r˜0(t)exp(+jωct)exp(-jωt)d t.
RiG(x0, y0; ω, t)=R˜0(ω-ωc)exp(+jωt)w(x0, y0)×exp+j ω-ωccα x0.
R1(x1, y1; ω, t)=R˜0(ω-ωc)exp(+jωt)Wx×ωx12πcf+(ω-ωc)α2πc,y1,
F(x1, y1)=n=1NFn(x1)recty1-nΔyΔy,
rect(y)=1,|y|1/20,otherwise ,
Fn(x1)H(x1nΔx)=-hxnΔxexp-j ω x1cfxdx.
ω-ωc=-x1ωcx1+fα.
F(x1, y1)F [β(ω-ωc), y1],
β=-fαωc.
R1F(x1, y1; ω, t)
=R1(x1, y1; ω, t)F(x1, y1)=R˜0(ω-ωc)F[β(ω-ωc), y1]exp(+jωt)×Wxωx12πcf+(ω-ωc)α2πc, y1.
R2(x2, y2; ω, t)
=-R1F(x1, y1; ω, t)exp-j ωx1cfx2d x1
=R˜0(ω-ωc)F [β(ω-ωc),y1]×exp(+jωt)w(x2, y2)×exp+j ω-ωccα x2.
r2(x2, y2; t)=-R2(x2, y2: ω, t)dω=n=1Nrit-αx2c,y2t htnβΔxrecty2-nΔyΔyw(x2, y2),
r2s(x2, y2; t)=r2(x2, y2; t)s(x2, y2).
R2s(x2, y2; ω, t)=R˜0(ω-ωc)F[β(ω-ωc, y2)]×exp(+jω t)w(x2, y2)s(x2, y2)×exp+j ω-ωccα x4.
R3(x3, y3; ω, t)
=-R2s(x2, y2; ω, t)exp-j ωx3cfx2dx2=R˜0(ω-ωc)F[β(ω-ωc), y3]exp(+jωt)×Wxωx3cf+(ω-ωc)α2πc, y3x Sx(x3, y3),
F(x3, y3)=n=1NFn(x3)recty3-nΔ yΔy,
Fn(x3)H(x3 nΔx)=-hxnΔxexp-j ω x3cfxd x.
F(x3, y3)F[β(ω-ωc),y3].
R3F(x3, y3; ω, t)
=R3(x3, y3: ω, t)F(x3, y3)=R˜0(ω-ωc)F[β(ω-ωc), y3]×F[β(ω-ωc), y3]exp(+jωt)×Wxωx3cf+(ω-ωc)α2πc, y3x Sx(x3, y3).
R4(x4, y4; ω, t)=-R3F(x3, y3; ω, t)×exp-jωx3cfx4+ωy3cfy4dx3dy3=R˜0(ω-ωc)Wy(x4, y4)×exp+j ω-ωccαx4Sy(x4, y4)×exp(+jωt)n=1NFn[β(ω-ωc)]×Fn[β(ω-ωc)]sinc(Δyy4)exp(-jnΔyy4),
sinc(y)=sin(πy)πy.
r4(x4, y4; t)
=Sy(x4, y4)Wy(x4, y4)sinc(Δy y4)exp(+jωc t)×n=1Nrit-αx4cthtnβΔxt htnβΔxexp(-jnΔyy4),
s(x2, y2)=n=1Nm=1Msnm rectx2-mΔxΔxrecty2-nΔyΔy.
h(χ)=cos(2πf0χ) 12πσexp-χ22σ2,
snm=1,n=am+b1n=am+b20,otherwise,
ΔtΔωconst.

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