Abstract

We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space–bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy.

© 2006 Optical Society of America

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References

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    [CrossRef]

2005 (1)

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

1997 (3)

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

1996 (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

1994 (1)

1979 (1)

Abe, S.

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

Barshan, B.

B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

Bastiaans, M. J.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall, 1995).

Erden, M. F.

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Hennelly, B. M.

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Kutay, M. A.

B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Ozaktas, H. M.

B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Sheridan, J. T.

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

IEEE Trans. Signal Process. (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 42, 2141 (1996).
[CrossRef]

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

Opt. Lett. (1)

Other (3)

L. Cohen, Time-Frequency Analysis (Prentice-Hall, 1995).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (2)

Fig. 1
Fig. 1

Decomposition of an arbitrary quadratic-phase system: (a) region to which the signal energy is initially confined, (b) energy distribution after rotation, (c) after scaling, (d) final distribution of signal energy after shearing.

Fig. 2
Fig. 2

rms errors versus N. Inset, rms error percentages.

Equations (7)

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

g ( u ) = β exp ( i π 4 ) exp [ i π ( α u 2 2 β u u + γ u 2 ) ] f ( u ) d u ,
g ( u ) = exp ( i a π 4 ) exp ( i π q u 2 ) f sc ( u ) ,
f sc ( u ) = 1 M f a ( u M ) ,
a = ( 2 π ) arccot γ ,
M = { 1 + γ 2 β γ 0 1 + γ 2 β γ < 0 } ,
q = γ β 2 ( 1 + γ 2 ) α .
k = 1 + γ α ( 1 + γ 2 ) β 2 .

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