## Two-dimensional Fourier transform of scaled Dirac delta curves

JOSA A, Vol. 21, Issue 9, pp. 1682-1688 (2004)

http://dx.doi.org/10.1364/JOSAA.21.001682

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### Abstract

We obtain a Fourier transform scaling relation to find analytically, numerically, or experimentally the spectrum of an arbitrary scaled two-dimensional Dirac delta curve from the spectrum of the nonscaled curve. An amplitude factor is derived and given explicitly in terms of the scaling factors and the angle of the forward tangent at each point of the curve about the positive *x* axis. With the scaling relation we determine the spectrum of an elliptic curve by a circular geometry instead of an elliptical one. The generalization to *N*-dimensional Dirac delta curves is also included.

© 2004 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1960) Diffraction and gratings : Diffraction theory

(070.2590) Fourier optics and signal processing : ABCD transforms

(350.5500) Other areas of optics : Propagation

**History**

Original Manuscript: February 3, 2004

Revised Manuscript: March 30, 2004

Manuscript Accepted: March 30, 2004

Published: September 1, 2004

**Citation**

Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Two-dimensional Fourier transform of scaled Dirac delta curves," J. Opt. Soc. Am. A **21**, 1682-1688 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-9-1682

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### References

- J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).
- K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]
- Strictly speaking, the curves where spectrum WEvanishes are not ellipses. We would need to solve the equation WE(u, v)=0to determine these nodal lines.
- The proposed scaling theorem of Eq. (7) provides a very efficient way to draw an elliptic Dirac delta curve with constant amplitude in a square grid matrix. By solving for δ(f(αx, βy))we obtain explicitly δ(f(αx, βy))=H(x, y)F-1{G(u/α, v/β)}/|αβ|.It is known that the spectrum Gof a circular delta is a zero-order Bessel function; thus the inverse fast Fourier transform of the scaled spectrum G(u/α, v/β)yields an elliptic Dirac delta curve whose amplitude is modulated by the factor h(x, y).The effect of multiplying by H(x, y)is to demodulate the Dirac delta curve such that now it has a constant amplitude.
- It is worth noting that the experimental setup shown in Fig. 5is indeed the same arrangement commonly used to generate nondiffracting beams (Refs. 3, 4, and 9). We can then take the image of the spectrum not only in the Fourier plane but at any plane within the well-known propagation distance of the nondiffracting beam.
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). [CrossRef]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001). [CrossRef]
- I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

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