Abstract

Sampling a function periodically replicates its spectrum. As a bilinear function of the signal, the associated Wigner distribution function contains cross terms between the replicas. Often neglected, these cross terms affect numerical simulations of paraxial optical systems. We develop expressions for these cross terms and show their effect on an example calculation.

© 2010 Optical Society of America

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References

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  1. L. Cohen, Proc. IEEE 77, 941 (1989).
    [CrossRef]
  2. T. A. C. M. Claasen and W. F. G. Mecklenbrauker, IEEE Trans. Acoust., Speech, Signal Process. 31, 1067 (1983).
    [CrossRef]
  3. L. Onural and M. T. Ozgen, J. Opt. Soc. Am. A 9, 252 (1992).
    [CrossRef]
  4. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
    [CrossRef]
  5. B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 917 (2005).
    [CrossRef] [PubMed]
  6. J. J. Healy and J. T. Sheridan, J. Opt. Soc. Am. A 27, 21 (2010).
    [CrossRef]
  7. R. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).
  8. J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
    [CrossRef]

2010 (1)

2005 (1)

B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 917 (2005).
[CrossRef] [PubMed]

2000 (1)

R. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

1999 (1)

J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
[CrossRef]

1996 (1)

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

1992 (1)

1989 (1)

L. Cohen, Proc. IEEE 77, 941 (1989).
[CrossRef]

1983 (1)

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, IEEE Trans. Acoust., Speech, Signal Process. 31, 1067 (1983).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

Claasen, T. A. C. M.

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, IEEE Trans. Acoust., Speech, Signal Process. 31, 1067 (1983).
[CrossRef]

Cohen, L.

L. Cohen, Proc. IEEE 77, 941 (1989).
[CrossRef]

Dorsch, R. G.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

Flandrin, P.

J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
[CrossRef]

Healy, J. J.

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 917 (2005).
[CrossRef] [PubMed]

Lohmann, A. W.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

Mecklenbrauker, W. F. G.

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, IEEE Trans. Acoust., Speech, Signal Process. 31, 1067 (1983).
[CrossRef]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

O'Neill, J. C.

J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
[CrossRef]

Onural, L.

Ozgen, M. T.

Sheridan, J. T.

J. J. Healy and J. T. Sheridan, J. Opt. Soc. Am. A 27, 21 (2010).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 917 (2005).
[CrossRef] [PubMed]

Williams, W. J.

J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
[CrossRef]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

IEEE Signal Process. Lett. (1)

J. C. O'Neill, P. Flandrin, and W. J. Williams, IEEE Signal Process. Lett. 6, 304 (1999).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, IEEE Trans. Acoust., Speech, Signal Process. 31, 1067 (1983).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. A Opt. Image Sci. Vis (1)

B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 917 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[CrossRef]

Proc. IEEE (1)

L. Cohen, Proc. IEEE 77, 941 (1989).
[CrossRef]

Other (1)

R. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

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Figures (3)

Fig. 1
Fig. 1

Modified phase space diagram (PSD) of a sampled Gaussian, indicating the cross terms. This example’s WDF is circularly symmetric and so may be represented by a circle, diameter chosen as in [5]. The sampling rate is twice the Nyquist rate. At this rate, the PSD terms lie at intervals of twice their diameter. The even cross terms, indicated by fine, dashed circles, lie atop the PSD terms. The odd cross terms, represented by circles of alternating thick dashes and dots, lie between the PSD terms. For a lower sampling rate, the odd cross terms overlap the PSD terms. The figure is infinitely periodic in k.

Fig. 2
Fig. 2

Plots of the integral over k of the three basic units making up the WDF of a sampled Gaussian. (a) The PSD term (2); (b) The even cross-term Eq. (5a) (truncating the sum to 1 q 20 ); (c) The odd cross-term Eq. (5b) (truncating the sum to 1 q 20 ); (d) The sum of the previous three terms [(a)–(c)]. The periodically sampled structure appears to emerge.

Fig. 3
Fig. 3

(a) Marginal of the order 1 2 FRT of a Gaussian of variance 2, sampled at unit intervals. Circles indicate where the even (thin dash) and odd (thick dash dot) cross terms are located, even terms on the peaks, and odd terms in the troughs. (b) Marginal of an even term, Eq. (7a). (c) Marginal of an odd term, Eq. (7b), with a shift to center it on π / 2 .

Equations (9)

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W { f s ( x ) } ( x , k ) = 2 π T x 2 [ n = F ( k 2 π n T x ε 2 ) ] [ m = F ( k 2 π m T x + ε 2 ) ] exp ( j ε x ) d ε .
2 π T x 2 W { f ( x ) } ( x , k 2 π n T x ) .
2 π T x exp [ j 2 π x ( m n ) T x ] W { f ( x ) } ( x , k 2 π T x n + m 2 ) .
4 π T x cos [ 2 π x ( m n ) T x ] W { f ( x ) } ( x , k 2 π T x n + m 2 ) .
4 π T x cos [ 2 π x 2 q T x ] W { f ( x ) } ( x , k + 2 π [ 2 r ] T x ) ,
4 π T x cos [ 2 π x 2 q 1 T x ] W { f ( x ) } ( x , k + 2 π [ 2 r 1 ] T x ) ,
W { f ( x ) } ( x , k ) = exp ( x 2 k 2 ) .
4 π T x q = 1 π   exp ( x 2 [ 2 π q   sin ( θ ) T x ] 2 ) cos ( 4 π q x   cos ( θ ) T x ) ,
4 π T x q = 1 π   exp ( x 2 [ π ( 2 q 1 ) sin ( θ ) T x ] 2 ) cos ( 2 π ( 2 q 1 ) x   cos ( θ ) T x ) .

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