Abstract

A semianalytical method is described for calculating the diffraction integral for paraxial propagation through an optical system. The field at the input plane is represented by a linear superposition of nearly Gaussian basis functions that keep a simple analytical form under ABCD propagation. The coefficients of the superposition are obtained by a numerical fit. The flexibility of the basis functions makes the method well suited to dealing with sharp local variations of the input field.

© 2000 Optical Society of America

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References

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  1. P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
    [CrossRef]
  2. P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
    [CrossRef]
  3. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  5. The definition of the transfer matrix adopted here follows that given in Ref. 4.
  6. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
  7. Since the coefficients of the expansion are not obtained by projection of the input field on the basis functions, the only practical constraint on their normalization is that their maximum value should not be so large, or so small, as to hinder the numerical computations.
  8. A mathematically rigorous discussion of the completeness of pseudowavelet bases is outside the scope of this work. It is clear, however, that any smoothly varying field distribution can be approached at any desirable level of accuracy by a linear superposition of such functions, as is the case with spline bases. A basis containing only pseudowavelets with the same parameter a and with n varying from 0 to N by a step of one is equivalent to a basis of N Laguerre–Gaussian modes. Bases containing pseudowavelets with different values of a, on the other hand, are subsets of overcomplete bases formed by the union of several Laguerre–Gaussian bases.
  9. Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
    [CrossRef] [PubMed]
  10. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  11. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  12. D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).
  13. B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
    [CrossRef]
  14. B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
    [CrossRef]
  15. The calculations reported in Refs. 13 and 14 are based on Eq. (B5).

1999 (2)

B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
[CrossRef]

B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

1998 (1)

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

1996 (1)

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1987 (1)

Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
[CrossRef] [PubMed]

1970 (1)

1969 (2)

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

Aiello, D.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

Ambrosini, D.

Bagini, V.

Baues, P.

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

Borghi, R.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Collins, S. A.

Durnin, J.

Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
[CrossRef] [PubMed]

Eberly, J. H.

Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
[CrossRef] [PubMed]

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Lü, B.

B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
[CrossRef]

Luo, S.

B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
[CrossRef]

Luo, S. R.

Miceli, J. J.

Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
[CrossRef] [PubMed]

Pacileo, A. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Santarsiero, M.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Schirripa Spagnolo, G.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Vicalvi, S.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

Zhang, B.

B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Optik (1)

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

Opto-Electronics (2)

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

Phys. Rev. Lett. (1)

Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)].
[CrossRef] [PubMed]

Other (6)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

The definition of the transfer matrix adopted here follows that given in Ref. 4.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Since the coefficients of the expansion are not obtained by projection of the input field on the basis functions, the only practical constraint on their normalization is that their maximum value should not be so large, or so small, as to hinder the numerical computations.

A mathematically rigorous discussion of the completeness of pseudowavelet bases is outside the scope of this work. It is clear, however, that any smoothly varying field distribution can be approached at any desirable level of accuracy by a linear superposition of such functions, as is the case with spline bases. A basis containing only pseudowavelets with the same parameter a and with n varying from 0 to N by a step of one is equivalent to a basis of N Laguerre–Gaussian modes. Bases containing pseudowavelets with different values of a, on the other hand, are subsets of overcomplete bases formed by the union of several Laguerre–Gaussian bases.

The calculations reported in Refs. 13 and 14 are based on Eq. (B5).

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Figures (4)

Fig. 1
Fig. 1

Functions Φ1n(r1; a) for a=1 mm and n=5, 10, and 20 (solid curves) and for a=0.5 mm and n=20 (dotted–dashed curve).

Fig. 2
Fig. 2

Intensity profile of the field defined by Eq. (8): (a) in the input plane, (b) in a plane at distance d=1.2 f from the lens, and (c) in a plane at distance d=2.0f. Dotted curves, exact results; solid curves, results for a basis of seven pseudowavelets; dashed curves, results for a basis of six pseudowavelets.

Fig. 3
Fig. 3

Intensity profile of the flattened Gaussian beam field defined by Eq. (9): (a) in the input plane and (b) at 200 mm of the input plane, after free propagation. The wavelength is 1064 nm. Dotted curves, exact results; solid curves, results for the basis of nine pseudowavelets defined in the text. (c) The variation in the input plane of the nine functions Φ1nj(r1; aj) forming the basis. (The pseudowavelet with n=0 is larger than 0.6 for r1<0.36 mm.)

Fig. 4
Fig. 4

Intensity profile of the beam whose profile in the input plane is defined by Eq. (11), after free propagation over a distance of either 35 or 350 mm. Dotted curves, exact results; solid curves, results for a basis of 18 pseudowavelets.

Equations (31)

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E2(x2, y2)
=exp(-ikL0) ik2πB--
×exp-i k2B (Ar12-2x1x2-2y1y2+Dr22)
×E1(x1, y1)dx1dy1.
E2(r2)=exp(-ikL0) ikB 0 exp-i k2B (Ar12+Dr22)×J0kB r1r2E1(r1)r1dr1.
Φ1n(r1; a)=1n! r1a2n exp-r12a2.
Φ2n(r2; a)=exp(-ikL0) ik2B exp-i kDr222B×1α2 1α2a2n exp-k2r224B2α2Lnk2r224B2α2,
α2=1a2+i kA2B.
E1(r1)E1(approx)(r1)=j=0NcjΦ1nj(r1; aj),
E2(r2)E2(approx)(r2)=j=0NcjΦ2nj(r2; aj).
E1(r1)=19! r1R18 exp-r12R2,
E1(r1)=G(r1, N, w0)
exp-N+1r1w02n=0N 1n! N+1r1w02n,
E1(r1)=n=0NΦ1n(r1; a),
E1(r1)=G(r1, N, w0,1)-G(r1, N, w0,2),
j=118cjΦ1nj(r1, i; a)=E1(r1, i)i=1,, 18,
Φ1n(r1; a)
=1n! exp-r12a2-n ln n-n ln1+r12-rmax2rmax2
Φ1n(r1rmax; a)nn exp(n)n! exp-12na4 (r12-rmax2)2.
Φ1n(r1rmax; a)nn exp(n)n! exp[-2(r1-rmax)2/a2],
Φ1n,m(r1; a)=1n! r1a2n+m exp-r12a2
0 exp-i kA2B r12JmkB r1r2Φ1n,m(r1; a)r1dr1
=12α2 1α2a2nkr22Bα2am
× exp-k2r224B2α2Lnmk2r224B2α2,
Φ1n,ev(x1; a)=1n! x1a2n exp-x12a2
Φ1n,odd(x1; a)=1n! x1a2n+1 exp-x12a2.
- exp-i kA2B x12expi kB x1x2Φ1n,ev(x1; a)dx1
=(-1)nπn! 1α 12αa2n exp-k2x224B2α2H2nkx22Bα
- exp-i kA2B x12expi kB x1x2Φ1n,odd(x1; a)dx1
=(-1)niπn! 1α 12αa2n+1 exp-k2x224B2α2
×H2n+1kx22Bα

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