Abstract

We use the well-known duality between paraxial diffraction in space and dispersion in time to propose a time-domain analog to spatial Fraunhofer diffraction. This analog permits the design of real-time optical Fourier-transformer systems. These systems are shown to be realizable by use of linearly chirped fiber gratings as dispersive media.

© 1999 Optical Society of America

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References

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  1. B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
    [CrossRef]
  2. A. Papoulis, J. Opt. Soc. Am. A 11, 3 (1994).
    [CrossRef]
  3. T. Jannson, Opt. Lett. 8, 232 (1983).
    [CrossRef] [PubMed]
  4. B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630 (1989).
    [CrossRef] [PubMed]
  5. A. W. Lohmann and D. Mendlovic, Appl. Opt. 31, 6212 (1992).
    [CrossRef] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  7. F. Ouellette, Opt. Lett. 12, 847 (1987).
    [CrossRef] [PubMed]
  8. J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
    [CrossRef]
  9. J. Capmany and M. A. Muriel, IEEE J. Lightwave Technol. 9, 27 (1991).
    [CrossRef]
  10. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  11. M. A. Muriel and A. Carballar, IEEE Photon. Technol. Lett. 9, 955 (1997).
    [CrossRef]
  12. K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
    [CrossRef]

1998 (1)

K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
[CrossRef]

1997 (1)

M. A. Muriel and A. Carballar, IEEE Photon. Technol. Lett. 9, 955 (1997).
[CrossRef]

1995 (1)

J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
[CrossRef]

1994 (2)

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

A. Papoulis, J. Opt. Soc. Am. A 11, 3 (1994).
[CrossRef]

1992 (1)

1991 (1)

J. Capmany and M. A. Muriel, IEEE J. Lightwave Technol. 9, 27 (1991).
[CrossRef]

1989 (1)

1987 (1)

1983 (1)

Bennion, I.

J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
[CrossRef]

Capmany, J.

J. Capmany and M. A. Muriel, IEEE J. Lightwave Technol. 9, 27 (1991).
[CrossRef]

Carballar, A.

M. A. Muriel and A. Carballar, IEEE Photon. Technol. Lett. 9, 955 (1997).
[CrossRef]

Ennser, K.

K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Jannson, T.

Kolner, B. H.

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630 (1989).
[CrossRef] [PubMed]

Laming, R.

K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Muriel, M. A.

M. A. Muriel and A. Carballar, IEEE Photon. Technol. Lett. 9, 955 (1997).
[CrossRef]

J. Capmany and M. A. Muriel, IEEE J. Lightwave Technol. 9, 27 (1991).
[CrossRef]

Nazarathy, M.

Ouellette, F.

Papoulis, A.

Williams, J. A. R.

J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zervas, M. N.

K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
[CrossRef]

Zhang, L.

J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
[CrossRef]

Appl. Opt. (1)

IEEE J. Lightwave Technol. (1)

J. Capmany and M. A. Muriel, IEEE J. Lightwave Technol. 9, 27 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Ennser, M. N. Zervas, and R. Laming, IEEE J. Quantum Electron. 34, 770 (1998).
[CrossRef]

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. A. R. Williams, I. Bennion, and L. Zhang, IEEE Photon. Technol. Lett. 7, 491 (1995).
[CrossRef]

M. A. Muriel and A. Carballar, IEEE Photon. Technol. Lett. 9, 955 (1997).
[CrossRef]

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

Opt. Lett. (3)

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (4)

Fig. 1
Fig. 1

Diagram of the real-time optical Fourier transformer.

Fig. 2
Fig. 2

Reflection spectral characteristics of the LCFG designed to process pulses with time widths from 10 to 100  ps. (a) LCFG reflectivity versus optical frequency. The grating has a 200-GHz bandwidth centered at 193.1  THz λ=1552.524 nm. (b) LCFG reflection group delay. The grating provides a linear group delay with a slope (dispersion coefficient) of 104 ps2.

Fig. 3
Fig. 3

First simulation of the real-time Fourier transformer. (a) Normalized average power of the squared pulse (80-ps width) entering the system. (b) Output normalized average power (solid curve). Dashed curve, the squared magnitude of the Fourier transform of the input signal envelope with tps=fTHz×104.

Fig. 4
Fig. 4

Second simulation of the real-time Fourier transformer. (a) Input normalized average power, which is composed of two 10-ps-width Gaussian pulses separated by 80  ps. (b) Output normalized average power (solid curve). Dashed curve, the squared magnitude of the Fourier transform of the input signal envelope with tps=fTHz×104.

Equations (6)

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hzx,yexp-jπλzx2+y2,
Φ¨=-2Φωω2=τgωω
htexpjπ2Φ¨t2,
art=C-+aiτexpj2Φ¨t-τ2dτ=Cexpj2Φ¨t2-+aiτexpj2Φ¨τ2×exp-jΦ¨tτdτ,
Δt022πΦ¨1,
art=Cexpj2Φ¨t2-+aiτexp-jΦ¨tτdτ=Cexpj2Φ¨t2Faitw=t/Φ¨,

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