Abstract

A fractional cross-spectral density is defined by use of an optical fractional Fourier transform based on the rotation of a Wigner distribution function within the accuracy of the paraxial approximation. The properties of the planar Gaussian Schell-model (GSM) source field in a fractional Fourier plane are analyzed theoretically. As a result, it becomes clear that the extent, the wave-front-curvature radius, and the spectral shift of the fractional Fourier field depend strongly on the fractional order of the Fourier transform, the Fresnel number associated with the GSM source size, and the scaled spatial-coherence length of the GSM source.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  2. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  3. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  4. Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
    [CrossRef] [PubMed]
  5. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  6. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
  7. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  8. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  12. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
    [CrossRef]
  13. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
    [CrossRef]
  14. A. Gamliel, G. P. Agrawal, “Wolf effect in homogeneous and inhomogeneous media,” J. Opt. Soc. Am. A 7, 2184–2192 (1990).
    [CrossRef]
  15. A constant 1/iλf1 sin ϕ is added to the definition of Eq. (6.7) in Ref. 10 for accuracy. The constant is derived from the transfer matrix, or ABCD matrix, of the optical systems depicted in Fig. 1. See, for example, Z. Jiang, “Scaling law and simultaneous optical implementation of various order fractional Fourier transforms,” Opt. Lett. 20, 2408–2410 (1995).
    [CrossRef] [PubMed]
  16. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995). The averaging procedure for stationary fields in the frequency domain is described in Section 4.7.
  17. The spectral density of the partially coherent light field inside a nondispersive graded-index optical fiber has already been derived as Eq. (6) in Ref. 14 under the paraxial approximation. The equation becomes equivalent to our result under the condition that αz=ϕ,α=1/f1,k=2π/λ,r=x,ρ1=ξ1=ξ-Δξ/2, and ρ2=ξ2=ξ+Δξ/2.
  18. B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
    [CrossRef]
  19. Eq. (9) in Ref. 14 becomes equivalent to our result under the condition that S(0)(ω)=A exp[-(ω-ω0)2/2Γ02],σI=w/2,σg=σw,αz=ϕ, and α=1/f1.
  20. In general, the normalized FRSD may be defined as sp(x, ω)=Sp(x, ω)/∫0∞Sp(x, ω)dω. In our study, however, the normalized FRSD is defined as the normalization of FRSD Sp(x, ω) by its maximum value, because Eq. (16) cannot be integrated analytically with respect to ω.
  21. H. Yoshimura, T. Iwai, “Fractional cross-spectral density on a Gaussian Schell-model source,” in Seventeenth Congress of the International Commission for Optics: Optics for Science and New Technology, J. Chang, J. Lee, S. Lee, C. Nam, eds., Proc. SPIE2778, 355–356 (1996).

1996 (5)

1995 (4)

1994 (2)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

1993 (2)

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1990 (1)

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Agrawal, G. P.

Bitran, Y.

Cai, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Gamliel, A.

Iwai, T.

H. Yoshimura, T. Iwai, “Fractional cross-spectral density on a Gaussian Schell-model source,” in Seventeenth Congress of the International Commission for Optics: Optics for Science and New Technology, J. Chang, J. Lee, S. Lee, C. Nam, eds., Proc. SPIE2778, 355–356 (1996).

Jiang, Z.

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Lohmann, A. W.

Lü, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995). The averaging procedure for stationary fields in the frequency domain is described in Section 4.7.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Ozaktas, H. M.

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995). The averaging procedure for stationary fields in the frequency domain is described in Section 4.7.

Yoshimura, H.

H. Yoshimura, T. Iwai, “Fractional cross-spectral density on a Gaussian Schell-model source,” in Seventeenth Congress of the International Commission for Optics: Optics for Science and New Technology, J. Chang, J. Lee, S. Lee, C. Nam, eds., Proc. SPIE2778, 355–356 (1996).

Zalevsky, Z.

Zhang, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Appl. Opt. (4)

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Mod. Opt. (1)

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Opt. Eng. (1)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994), and references therein.
[CrossRef]

Opt. Lett. (2)

Other (5)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995). The averaging procedure for stationary fields in the frequency domain is described in Section 4.7.

The spectral density of the partially coherent light field inside a nondispersive graded-index optical fiber has already been derived as Eq. (6) in Ref. 14 under the paraxial approximation. The equation becomes equivalent to our result under the condition that αz=ϕ,α=1/f1,k=2π/λ,r=x,ρ1=ξ1=ξ-Δξ/2, and ρ2=ξ2=ξ+Δξ/2.

Eq. (9) in Ref. 14 becomes equivalent to our result under the condition that S(0)(ω)=A exp[-(ω-ω0)2/2Γ02],σI=w/2,σg=σw,αz=ϕ, and α=1/f1.

In general, the normalized FRSD may be defined as sp(x, ω)=Sp(x, ω)/∫0∞Sp(x, ω)dω. In our study, however, the normalized FRSD is defined as the normalization of FRSD Sp(x, ω) by its maximum value, because Eq. (16) cannot be integrated analytically with respect to ω.

H. Yoshimura, T. Iwai, “Fractional cross-spectral density on a Gaussian Schell-model source,” in Seventeenth Congress of the International Commission for Optics: Optics for Science and New Technology, J. Chang, J. Lee, S. Lee, C. Nam, eds., Proc. SPIE2778, 355–356 (1996).

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 (5)

Fig. 1
Fig. 1

Optical systems for performing the FRT defined on the basis of the idea of the rotation of a WDF. A lens having a focal length f1/sin ϕ is used in type I, and two lenses having a focal length f1/tan(ϕ/2) are used in type II. up(x; λ) corresponds to the fractional Fourier field of a stationary source field u0(ξ; λ). ϕ represents a rotation angle of the WDF of up(x; λ) and is given by pπ/2, in which p denotes a fractional order of the Fourier transform.

Fig. 2
Fig. 2

Dependence of the extent wp (in units of w) of the fractional average intensity on the fractional order p under the conditions that the Fresnel numbers associated with the GSM source size Nw equal (a) 0.1, (b) 1, (c) 10. Results for three scaled spatial-coherence lengths of the GSM source, σ: solid curve, σ=0.1; dashed curve, σ=1; dotted curve, σ=10.

Fig. 3
Fig. 3

Dependence of the wave-front-curvature radius Rp (in units of πw2/λ0) of the fractional Fourier field on p under the conditions that Nw equal (a) 0.1, (b) 1, and (c) 10. Results for, solid curve, σ=0.1; dashed curve, σ=1; dotted curve, σ=10.

Fig. 4
Fig. 4

Solid curve, normalized FRSD at the center of the fractional Fourier plane (i.e., x=o) when p=0.5, Nw=0.1, σ=1.0, and Γ/ω0=0.05. Dotted curve, normalized spectral density (in units of 1/Γ02π) of the GSM source.

Fig. 5
Fig. 5

Dependence of the relative frequency shift zp at x=o on p under the conditions that Nw=(a) 0.1, (b) 1, and (c) 10 for Γ0/ω0=0.05. Results for solid curve, σ=0.1; dashed curve, σ=1; dotted curve, σ=10.

Equations (18)

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

up(x; λ)=1iλf1 sin ϕu0(ξ; λ)expiπ(|x|2+|ξ|2)λf1 tan ϕ×exp-2πix·ξλf1 sin ϕd2ξ,
ϕ=pπ/2,
Wp(x, Δx; λ)=up(x+Δx/2; λ)up*(x-Δx/2; λ)=1(λf1 sin ϕ)2exp2πix·Δxλf1 tan ϕ×W0(ξ, Δξ; λ)×exp-2πi(x·Δξ+ξ·Δx)λf1 sin ϕ×exp2πiξ·Δξλf1 tan ϕd2ξd2Δξ,
W0(ξ, Δξ; λ)=u0(ξ+Δξ/2; λ)u0*(ξ-Δξ/2; λ).
Sp(x; λ)=up(x; λ)up*(x; λ)=1(λf1 sin ϕ)2W0(ξ, Δξ; λ)×exp-2πix·Δξλf1 sin ϕexp2πiξ·Δξλf1 tan ϕd2ξd2Δξ.
W0(ξ, Δξ; ω)=[I0(ξ+Δξ/2)I0(ξ-Δξ/2)]1/2γ0(Δξ)s0(ω)=A exp-2|ξ|2w2exp-12w21+1σ2|Δξ|2×exp-(ω-ω0)22Γ02,
I0(ξ)=AΓ02π exp[-2|ξ|2/w2],
γ0(Δξ)=exp[-|Δξ|2/2(wσ)2],
s0(ω)=1Γ02πexp[-(ω-ω0)2/2Γ02].
Wp(x, Δx; ω)=Aw2wp2(ω)exp-2|x|2wp2(ω)×exp-12wp2(ω)1+1σ2|Δx|2×exp-iωx·ΔxcRp(ω)exp-(ω-ω0)22Γ02,
wp(ω)=wcos2(pπ/2)+sin(pπ/2)πNw2×1+1σ2ω0ω21/2.
Rp(ω)=f1 tan(pπ/2)1[1-(1/πNw)2(1+1/σ2)(ω0/ω)2]sin2(pπ/2)-1.
Nw=w2ω02πcf1=w2λ0f1,
2(1/πNw)2(1+1/σ2)1/2(1/πNw)2(1+1/σ2)-1,
p=1πcos-1(1/πNw)2(1+1/σ2)-1(1/πNw)2(1+1/σ2)+1.
Sp(x; ω)=Bw2wp2(ω)exp-2|x|2wp2(ω)s0(ω),
Sp(o; ω)=Bw2wp2(ω)s0(ω).
zp=ω0-ωpωp=λp-λ0λ0,

Metrics