Abstract

The Fourier transform of a coherent optical image can be evaluated physically by use of a single lens plus free-space propagation, thereby providing the basis for the field of Fourier optics. I point out that one can similarly evaluate the discrete Fourier transform of a sampled or pixelated optical array physically by passing the discrete array amplitudes through a network of single-mode fibers or optical waveguides. A passive optical network that evaluates the fast Fourier transform of a coherent array can be fabricated by use of N/2log2N optical 3-dB couplers plus small added phase shifts. Implementing such networks in fiber or integrated optical form could provide the basis for a possible technology of fiber Fourier optics.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  3. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  4. J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Comput. Math. 19, 297–301 (1965).
    [CrossRef]
  5. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).
  6. L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  7. J. W. Goodman, A. R. Dias, and L. M. Woody, “Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms,” Opt. Lett. 2, 1–3 (1978).
    [CrossRef] [PubMed]

1978

1965

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Comput. Math. 19, 297–301 (1965).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Comput. Math. 19, 297–301 (1965).
[CrossRef]

Dias, A. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gold, B.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Goodman, J. W.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Rabiner, L. R.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Comput. Math. 19, 297–301 (1965).
[CrossRef]

Woody, L. M.

Comput. Math.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Comput. Math. 19, 297–301 (1965).
[CrossRef]

Opt. Lett.

Other

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

A 2×2 Fourier transform is equivalent physically to an asymmetric 3-dB fiber-optic coupler or similar 50/50 beam splitter.

Fig. 2
Fig. 2

Fiber-optic network that performs an eight-order DFT, using the decimation-in-time algorithm. The circles represent asymmetrical 3-dB couplers; the boxes represent added phase shifts of value -q2π/N, where q is the integer shown in the box.

Fig. 3
Fig. 3

Fiber-optic network that evaluates the decimation-in-frequency FFT algorithm.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

bm=1Nn=0N-1anexp-jmn2π/N,0m,nN-1.
bm=q=1Np=0N/2-1a2pexp-jqpπ/N+exp-jq2π/Np=0N/2-1a2p+1exp-jqpπ/N,
bm=q+N/2=1Np=0N/2-1a2pexp-jqpπ/N-exp-jq2π/Np=0N/2-1a2p+1exp-jqp2π/N,
b0=1/2a0+a1,  b1=1/2a0-a1,
S=12111-1,

Metrics