Abstract

On the basis of the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, approximate analytical expressions for the output field distribution of a Laguerre-Gaussian beam and an elegant Laguerre-Gaussian beam passing through apertured fractional Hankel transform systems are derived. Some numerical simulation comparisons are done, by using the approximate analytical formulas and diffraction integral formulas, and it is shown that our method can significantly improve the numerical calculation efficiency.

© 2004 Optical Society of America

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  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  3. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
  7. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  8. A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. Elsevier, Amsterdam, 2002).
  9. V. Namias, “Fractionalization of Hankel transforms,” J. Inst. Math. Appl. 26, 187–197 (1980).
    [CrossRef]
  10. L. Yu, X. Lu, Y. Zeng, M. Huang, M. Chen, W. Huang, Z. Zhu, “Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform,” Opt. Lett. 23, 1158–1160 (1998).
    [CrossRef]
  11. D. Zhao, “Collins formula in frequency-domain described by fractional Fourier transforms or fractional Hankel transforms,” Optik (Stuttgart) 111, 9–12 (2000).
  12. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  13. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1998).
    [CrossRef]
  14. D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
    [CrossRef]
  15. D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
    [CrossRef]
  16. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  17. B. Lu, R. Peng, “Relative phase shift in Laguerre-Gaussian beams propagating through an apertured paraxial ABCD system,” J. Mod. Opt. 50, 857–865 (2003).
    [CrossRef]
  18. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
  19. S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
    [CrossRef]

2004 (1)

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

2003 (2)

B. Lu, R. Peng, “Relative phase shift in Laguerre-Gaussian beams propagating through an apertured paraxial ABCD system,” J. Mod. Opt. 50, 857–865 (2003).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

2000 (1)

D. Zhao, “Collins formula in frequency-domain described by fractional Fourier transforms or fractional Hankel transforms,” Optik (Stuttgart) 111, 9–12 (2000).

1998 (3)

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

L. Yu, X. Lu, Y. Zeng, M. Huang, M. Chen, W. Huang, Z. Zhu, “Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform,” Opt. Lett. 23, 1158–1160 (1998).
[CrossRef]

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1998).
[CrossRef]

1994 (1)

1993 (3)

1987 (1)

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

1980 (2)

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

V. Namias, “Fractionalization of Hankel transforms,” J. Inst. Math. Appl. 26, 187–197 (1980).
[CrossRef]

1970 (1)

Breazeale, M. A.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1998).
[CrossRef]

Chen, M.

Collins, S. A.

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Huang, M.

Huang, W.

Jing, F.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

Kerr, F. H.

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

Kutay, M. A.

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

Liu, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

Lohmann, A. W.

Lu, B.

B. Lu, R. Peng, “Relative phase shift in Laguerre-Gaussian beams propagating through an apertured paraxial ABCD system,” J. Mod. Opt. 50, 857–865 (2003).
[CrossRef]

Lu, X.

Mao, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

McBride, A. C.

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

Mendlovic, D.

Namias, V.

V. Namias, “Fractionalization of Hankel transforms,” J. Inst. Math. Appl. 26, 187–197 (1980).
[CrossRef]

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

Ozaktas, H. M.

Peng, R.

B. Lu, R. Peng, “Relative phase shift in Laguerre-Gaussian beams propagating through an apertured paraxial ABCD system,” J. Mod. Opt. 50, 857–865 (2003).
[CrossRef]

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Shen, M.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Torre, A.

A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. Elsevier, Amsterdam, 2002).

Wang, S.

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

Wen, J. J.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1998).
[CrossRef]

Yu, L.

Zalevsky, Z.

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

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).

Zeng, Y.

Zhang, W.

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

D. Zhao, “Collins formula in frequency-domain described by fractional Fourier transforms or fractional Hankel transforms,” Optik (Stuttgart) 111, 9–12 (2000).

Zhu, Q.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

Zhu, Z.

Appl. Opt. (1)

IMA J. Appl. Math. (1)

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

J. Acoust. Soc. Am. (1)

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1998).
[CrossRef]

J. Inst. Math. Appl. (2)

V. Namias, “Fractionalization of Hankel transforms,” J. Inst. Math. Appl. 26, 187–197 (1980).
[CrossRef]

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

J. Mod. Opt. (2)

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

B. Lu, R. Peng, “Relative phase shift in Laguerre-Gaussian beams propagating through an apertured paraxial ABCD system,” J. Mod. Opt. 50, 857–865 (2003).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148–154 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

D. Zhao, “Collins formula in frequency-domain described by fractional Fourier transforms or fractional Hankel transforms,” Optik (Stuttgart) 111, 9–12 (2000).

Other (4)

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).

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

A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. Elsevier, Amsterdam, 2002).

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

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Figures (4)

Fig. 1
Fig. 1

Two optical setups for implementing FRHT. (a) Type I, (b) type II.

Fig. 2
Fig. 2

(a) Real and imaginary parts of the Gaussian expansion [Eq. (4)] for the aperture function. (b) Magnitude and phase of the Gaussian expansion.

Fig. 3
Fig. 3

Intensity distributions in the FRHT output plane of the type I setup with incidence of a Laguerre–Gaussian beam with l=2, m=3 and a/w0=0.5 by using two methods. Solid curves, results with the approximate analytical formula; dotted curves, results with the diffraction integral formulas. (a) p=0.1, (b) p=0.3, (c) p=0.5, (d) p=1.

Fig. 4
Fig. 4

Same as Fig. 3 but for type II.

Tables (2)

Tables Icon

Table 1 Ratio of Numerical Calculation Time with the Diffraction Integral Formulas to Calculation Time with Approximate Analytical Expressions

Tables Icon

Table 2 Comparison of Standard Error S of the Results Obtained by Using the Approximate Analytical Expressions and the Diffraction Integral Formulas

Equations (29)

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E(r, θ, z)=ik2πB02π0aE(r0, θ0, 0)exp-ik2B [Ar02-2rr0cos(θ-θ0)+Dr2]r0dr0dθ0,
Ap(r)=1,ra0,r>a,
E(r, θ, z)=ik2πB02π0E(r0, θ0, 0)Ap(r)×exp-ik2B [Ar02-2rr0cos(θ-θ0)+Dr2]r0dr0dθ0.
Ap(r)=h=1NAhexp-Bha2 r2,
E(r1, θ1, z1)=ik2πd02π0E(r0, θ0, 0)×exp-ik2d [r02-2r1r0cos(θ1-θ0)+r12]r0dr0dθ0,
E(r2, θ2, z)=ik2πd02π0E(r1, θ1, z1)Ap(r)×exp-ik2d1-dfr12-2r2r1cos(θ2-θ1)+r22r1dr1dθ1.
E(r2, θ2, z)=ik2πd02π0E(r0, θ0, 0)Ap(r)×exp-ik2d1-dfr02-2r2r0cos(θ2-θ0)+1-dfr22r0dr0dθ0.
E(r0, θ0, 0)=2r0w0mLlm2r02w02×exp-r02w02exp(-imθ0),
02πexpikB rr0cos(θ-θ0)exp(-imθ0)dθ0
=im2πJmkB rr0exp(-imθ),
0xv+1/2exp(-βx2)Lnv(αx2)Jv(xy)(xy)1/2dx=2-v-1β-v-n-1(β-α)nyv+1/2×exp-y24βLnvαy24β(α-β),
(Re β>0, Re >0),
E(r2, θ2, z)=i2m+22-2m-2exp(-imθ2)exp-ikr222dym×h=1NAh(β1β2)-m-l-1[(β1-1)×(β2-α)]l×exp-y24β2Llmαy24β2(α-β2),
y=2r2w0,
β1=12+ikw024d,
α=14β1(1-β1),
β2=14β1+idkw02+id cos ϕkw02+2Bhd2k2w02a2.
E(r2, θ2, z)=1d im+12-m-2kw02ymexp(-imθ2)×exp-ik cos ϕ2d r22×h=1NAhβ-m-l-1(β-1)lexp-y24βLlmy24β(1-β),
y=kw02d r2,
β=Bhw022a2+ikw02cos ϕ4d+12.
E(r0, θ0, 0)=r0w0mLlmr02w02exp-r02w02exp(-imθ0).
E(r2, θ2, z)=i2m+22-2m-2exp(-imθ2)exp-ikr222dym×h=1NAh(β1β2)-m-l-1[(β1-1)(β2-α)]lexp-y24β2Llmαy24β2(α-β2),
y=r2w0,
β1=1+ikw022d,
α=14β1(1-β1),
β2=14β1+id2kw02+id cos ϕ2kw02+Bhd2k2w02a2.
E(r2, θ2, z)=1d im+12-m-1kw02ym×exp(-imθ2)exp-ik cos ϕ2d r22×h=1NAhβ-m-l-1×(β-1)lexp-y24βLlmy24β(1-β),
y=kw0d r2,
β=Bhw02a2+ikw02cos ϕ2d+1.

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