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

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).
  2. K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  4. J. Durnin, J. J. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  5. 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.
  6. 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.
  7. 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.
  8. 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]
  9. 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]
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

2001 (1)

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]

2000 (1)

1987 (2)

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]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).

Chávez-Cerda, S.

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]

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]

Durnin, J.

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]

Eberly, J. H.

J. Durnin, J. J. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Gutiérrez-Vega, J. C.

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]

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]

Iturbe-Castillo, M. D.

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]

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]

Micely, J. J.

J. Durnin, J. J. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

New, G. H. C.

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]

Rami´rez, G. A.

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]

Rodri´guez-Dagnino, R. M.

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]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Tepichi´n, E.

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]

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

Opt. Commun. (1)

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]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (6)

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.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

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).

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Figures (7)

Fig. 1
Fig. 1

(a) FT of an arbitrary two-dimensional curve δ(f(x, y)) is given by the spectrum G(u, v). (b) FT of the scaled curve δ(f(αx, βy)) is given by the spectrum W(u, v), which is structurally different from G(u, v). (c) FT of the properly modulated curve h(x, y)δ(f(αx, βy)) yields the scaled spectrum G(u/α, v/β)/|αβ|.

Fig. 2
Fig. 2

(a) Effect of scaling a straight segment of length L originally placed at (x0, y0) and oriented at an angle ϕ about the positive x axis. (b) At each point, the factor H(ϕ) depends on the angle ϕ of the forward tangent measured from the positive x axis.

Fig. 3
Fig. 3

(a) Absolute value of the transverse shape of the spectrum of an elliptic contour. (b) Amplitude of each ring is not constant but depends on the angular position φ.

Fig. 4
Fig. 4

(a) Absolute value of the transverse shape of the spectrum of an elliptic contour computed with the fast Fourier transform algorithm. (b) Behavior along the v axis of the numerically calculated spectrum (dotted curve) and that computed from the analytical solution of Eq. (18) (solid curve).

Fig. 5
Fig. 5

Experimental setup to measure the scaled power spectrum of an elliptic contour by using a circular contour.

Fig. 6
Fig. 6

(a), (b) Theoretical scaled power spectrum of an elliptic contour |WE(u, βv)|2. (c), (d) Image of the transverse intensity pattern of |WE(u, βv)|2 obtained experimentally at the Fourier plane of the setup shown in Fig. 5: (e), (f) Behavior of the power spectrum along the axes u and v for the theoretical solution (solid curves) and the experimental measurement (dotted curves).

Fig. 7
Fig. 7

(a) Scaling changes the rectangular shape of the straight Dirac delta segment. (b) To correct the segment it is necessary to multiply by a suitable amplitude factor.

Equations (41)

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F{g(x, y)}=G(u, v)=-g(x, y)exp[i(ux+vy)]dxdy,
F{g(αx, βy)}=1|αβ| Guα, vβ,
F{δ(f(αx, βy))}=W(u, v)=-δ(f(αx, βy))exp[i(ux+vy)]dxdy,
δ(f(x, y))=rectx2aδ(y-b)+recty2aδ(x-b),
G(u, v)=2a exp(ibv)sinc(au)+2a exp(ibu)sinc(av),
W(u, v)=2aαexpi bvβsincauα+2aβexpi buαsincavβ.
F{h(x, y)δ(f(αx, βy))}=1|αβ| Guα, vβ.
F{δ(f(αx, βy))}=W(u, v)=1|αβ|FF-1Guα, vβH(x, y),
G(u, v)=L exp(iux0+ivy0)×sincL2 (ucosϕ+vsinϕ).
1|αβ| Guα, vβ=L|αβ|expiux0α+ivy0β×sincL2ucosϕα+vsinϕβ.
F{δ(fL(αx, βy))}=W(u, v)=Lexp(iux0+ivy0)×sincL2 (ucosϕ+vsinϕ),
H(x, y)=H(ϕ)=(β2cos2 ϕ+α2sin2 ϕ)1/2.
W(u, v)=1|αβ|-H(x, y)δ(f(x, y))×expiuα x+vβ ydxdy,
WE(u, v)=1β02π0H(x, y)δ(r-a)×expiurcosθ+vβ rsinθrdrdθ,
WE(u, v)=aβ02π(β2sin2 θ+cos2 θ)1/2×expiaucosθ+avsinθβdθ.
WE(u, v)=a02π(1-e2cos2 θ)1/2×exp[iR cos(θ-φ)]dθ,
R(κx2+κy2)1/2=a(u2+v2/β2)1/2,
φtan-1(κy/κx)=tan-1(v/βu).
WE(u, v)=πaA0J0(R)+2πam=1(-1)mA2m(cos 2mφ)J2m(R),
A2m=1π02π(cos 2mφ)(1-e2cos2 φ)1/2dφ.
W(u)=1|α1α2αN|H(x)δ(f(x))×expiu1x1α1+u2x2α2+dxN,
H(φ)=|α1α2αN|j=1Ncos2 φjαj21/2,
A=L[xˆcos ϕ+yˆsin ϕ],
B=τ[-xˆsin ϕ+yˆcos ϕ],
C=(1/τ)zˆ.
A=Lcos ϕαxˆ+sin ϕβyˆ,
B=τ-sin ϕαxˆ+cos ϕβyˆ.
limτ0(A×B)C=Lαβ.
limτ0(A×B)C=Lαβ (β2cos2 ϕ+α2sin2 ϕ)1/2.
H(x, y)=H(ϕ)=(β2cos2 ϕ+α2sin2 ϕ)1/2,
x=f cosh ξcosη,
y=f sinh ξsinη,
G(u, v)=02π0g(ξ, η)exp[i(uf cosh ξcosη+vf sinh ξsinη)]dxdy,
g(ξ, η)=δ(ξ-ξ0)f(cosh2 ξ-cos2 η)1/2.
G(u, v)=a02π(1-e2cos2 η)1/2exp[i(aucosη+bvsinη)]dη.
G(u, v)=a02π(1-e2cos2 η)1/2exp[iR cos(η-φ)]dη,
(1-e2cos2 η)1/2=A02+n=1Ancos(nη).
An1π02π(1-e2cos2 η)1/2cos(nη)dη,
02πexp{i[R cos(η-φ)]}cos(nη)dη=in2π[cos(nφ)]Jn(R),
G(u, v)=πaA0J0(R)+2πam=1(-1)mA2m[cos(2mφ)]J2m(R),
A2m(e)=1π02π(1-e2cos2 η)1/2cos(2mη)dη.

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