Abstract

The concept of ray parameters for decentered Gaussian beams is developed on a formal basis. When the beam propagates through first-order optical systems, these parameters are transformed as the ray parameters of geometrical optics. It is shown how this feature helps one to understand the behavior of more sophisticated beams that can be considered as bundles of decentered Gaussian beams. In particular, the case of Bessel–Gauss beams and their recently introduced generalizations is analyzed, and simple transformation formulas are obtained.

© 1997 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. XV, pp. 581–625, and Chap. XX, pp. 777–814.
  2. A. Yariv, Optical Electronics (Saunders, Bristol, U.K., 1991), Chap. 2, pp. 35–74.
  3. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  4. A. R. Al-Rashed, B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
    [CrossRef] [PubMed]
  5. J. Keller, W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  6. S. Chandrasekar, Radiative Transfer (Dover, New York, 1950).
  7. E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96, 315–320 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1982).
    [CrossRef]
  8. R. Simon, “Generalized rays: ray dispersion, dark rays and statistical inhomogeneity,” Opt. Commun. 64, 94–98 (1987).
    [CrossRef]
  9. A. N. Norris, “Complex point-source representation of real point sources and the Gaussian beam summation method,” J. Opt. Soc. Am. A 3, 2005–2010 (1986).
    [CrossRef]
  10. Y. Nishiyama, “Trajectory in the optical wave,” J. Opt. Soc. Am. A 12, 1390–1397 (1995).
    [CrossRef]
  11. L. W. Casperson, “Gaussian beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973);“Mode stability of lasers and periodical optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef] [PubMed]
  12. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  13. F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  14. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  15. F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 139–148.
  16. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  17. K. B. Wolf, “Diffraction-free beams remain diffraction free under all paraxial optical transformations,” Phys. Rev. Lett. 60, 757–759 (1988).
    [CrossRef] [PubMed]
  18. A. T. Friberg, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
    [CrossRef] [PubMed]
  19. C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
    [CrossRef]

1996

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

1995

1991

A. T. Friberg, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

1988

K. B. Wolf, “Diffraction-free beams remain diffraction free under all paraxial optical transformations,” Phys. Rev. Lett. 60, 757–759 (1988).
[CrossRef] [PubMed]

1987

R. Simon, “Generalized rays: ray dispersion, dark rays and statistical inhomogeneity,” Opt. Commun. 64, 94–98 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1986

1979

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96, 315–320 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1982).
[CrossRef]

1973

1971

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

J. Keller, W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
[CrossRef]

1970

Al-Rashed, A. R.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Casperson, L. W.

Chandrasekar, S.

S. Chandrasekar, Radiative Transfer (Dover, New York, 1950).

Cincotti, G.

C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Friberg, A. T.

A. T. Friberg, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 139–148.

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Keller, J.

Nishiyama, Y.

Norris, A. N.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Palma, C.

C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Saleh, B. E. A.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. XV, pp. 581–625, and Chap. XX, pp. 777–814.

Simon, R.

R. Simon, “Generalized rays: ray dispersion, dark rays and statistical inhomogeneity,” Opt. Commun. 64, 94–98 (1987).
[CrossRef]

Streifer, W.

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96, 315–320 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1982).
[CrossRef]

Turunen, J.

A. T. Friberg, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Wolf, K. B.

K. B. Wolf, “Diffraction-free beams remain diffraction free under all paraxial optical transformations,” Phys. Rev. Lett. 60, 757–759 (1988).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Bristol, U.K., 1991), Chap. 2, pp. 35–74.

Appl. Opt.

Electron. Lett.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

J. Mod. Opt.

C. Palma, G. Cincotti, M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

R. Simon, “Generalized rays: ray dispersion, dark rays and statistical inhomogeneity,” Opt. Commun. 64, 94–98 (1987).
[CrossRef]

Phys. Rev. A

A. T. Friberg, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett.

K. B. Wolf, “Diffraction-free beams remain diffraction free under all paraxial optical transformations,” Phys. Rev. Lett. 60, 757–759 (1988).
[CrossRef] [PubMed]

Physica

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96, 315–320 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1982).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. XV, pp. 581–625, and Chap. XX, pp. 777–814.

A. Yariv, Optical Electronics (Saunders, Bristol, U.K., 1991), Chap. 2, pp. 35–74.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

S. Chandrasekar, Radiative Transfer (Dover, New York, 1950).

F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 139–148.

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Figures (1)

Fig. 1
Fig. 1

Geometry for the superposition of DGB’s. The two parallel vectors ε0 and rd (whose lengths amount to the radius of the broken line circumference) have the same anomaly g. The direction of the peak intensity line is indicated by the vector k, whose inclination on the z axis is φ.

Equations (23)

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

Ux, y, 0=1q0expik2q0ix02+y-yd-iy02,
q0=-ikw022=-iz0,
Ux, y, 0=1q0expi k2x02+y02z0-ixdx0z0+ydy0z0+ik2q0x-xd2+y-yd2+ik×xx0z0+yy0z0=U0 expik2q0r-rd2+ikε·r,
U0=i/z0expk2x02z0-2ixdx0z0+ydy0z0=i/z0expkz0ε02/2-ikε0·rd, rd=xd, yd, ε0=ε0x,ε0y=x0z0,y0z0.
Eαfx=-iα-fξexpπiαx-ξ2dξ,
E1/λzfξexpiεξx=-iλz-fξexpiεξexpik2zx-ξ2dξ=expiεx-εz2kE1/λzfξx-εzk.
Uξ, η, z=-iλBexpikLUx, y, 0expik2BAx2+y2-2xξ+yη+Dξ2+η2dxdy,
Ux, y, 0=X0xY0y
Xzξ=U0x-iλBexpikL2expik2q0x-xd2-2ixx0+ik2BAx2-2xξ+Dξ2dx,
U0x=i/z0expkz0ε0x2/2-ikxdε0x.
Xzξ=U0x1/A+B/q0expikL/2exp×ik2q1zξ-x1d2+ikε1xξ+iφx
q1z=Aq0+BCq0+D; x1dε1x=ABCDxdε0x; φx=k2Cxdx1d+Bε0xε1x.
φ=φx+φy=k2Cxdx1d+ydy1d+Bε0xε1x+ε0yε1y
Ur, θ=R0 expik2q0r2+rd2-2rrd cosγ-θ+ikε0r cosγ-θ,
R0=expkz0ε02/2-ikrdε0/q0.
Vr, θ=2πR0 expik2q0r2+rd2J0krε0-rdq0.
Vr, θ=A0 exp-r2w02J0kε0r,
q1=Aq0+BCq0+D,ε1=ε1x2+ε1y2=Crd+Dε0, r1d=x1d2+y1d2=Ard+Bε0, φ=k2Crdr1d+Bε0ε1.
q1=q0+z,r1d=rd+ε0z,ε1=ε0, φ=k2zε02.
U1r, θ=R0 expik2q1r2+r1d2-2rr1d×cosγ-θ+ikε1r cosγ-θ,
R0=R0q0/q1 expikzε02/2.
Vzr, θ=2πR0 expik2q0r2+r1d2×J0kε1-rd1/q1r.
Vzr, θ=2πR0 expik2q1r2+ε02z2J0kε0q0r/q1.

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