Abstract

Beams of a high angle of convergence and divergence are called high-aperture beams. Different ways of defining high-aperture generalizations to paraxial beams are reviewed for both scalar beams and electromagnetic beams. The different approaches are divided into three types. The particular examples of Gaussian beams and Bessel beams are discussed. For Gaussian beams, beams that exhibit a Gaussian variation in the waist necessarily include evanescent components, which rules out their use in describing propagation over all space. Generalizations of the definitions of beam width and the beam-propagation factor M2 for high-aperture beams are described. The similarities among the three types of high-aperture beams and the different models of ultrashort-pulsed beams are discussed.

© 2001 Optical Society of America

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2000 (2)

1999 (8)

1998 (8)

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

1997 (5)

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

X. D. Zeng, C. H. Liang, Y. Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042–2047 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (1)

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1994 (4)

M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A 27, L391–L398 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

1993 (1)

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1992 (1)

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

1990 (2)

S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
[CrossRef] [PubMed]

L. Cicchitelli, H. Hora, R. Postle, “Longitudinal components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732 (1990).
[CrossRef] [PubMed]

1989 (1)

1987 (3)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

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

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

1986 (2)

1985 (1)

1984 (1)

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1982 (1)

L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams,’” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

1981 (2)

M. Couture, P.-A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

L. W. Davis, G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981).
[CrossRef] [PubMed]

1980 (1)

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

1979 (3)

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

G. P. Agrawal, D. N. Pattanayck, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

1978 (3)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisel, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1974 (4)

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik 41, 281–291 (1974).

J. F. Nye, M. Berry, “Dislocations of wave fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[CrossRef]

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. A 64, 779–788 (1974).
[CrossRef]

1972 (1)

1971 (1)

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

1969 (1)

1966 (2)

G. Toraldo di Francia, “Optical resonators,” Opt. Acta 13, 323–342 (1966).
[CrossRef]

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

1965 (2)

1964 (1)

A. G. van Nie, “Rigorous calculation of the electromagnetic field of a wave beam,” Philips Res. Rep. 19, 378–394 (1964).

1961 (1)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

1960 (2)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength lasers,” Bell Syst. Tech. J. 40, 489–508 (1960).
[CrossRef]

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (Leipzig) 9, 24–75 (1960).
[CrossRef]

1959 (3)

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems 1. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

1955 (1)

A. Nisbet, “Hertzian electromagnetic potentials and associated gauge transformations,” Proc. R. Soc. London Ser. A 231, 250–263 (1955).
[CrossRef]

1954 (1)

1953 (1)

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London Ser. B 66, 145–149 (1953).
[CrossRef]

1950 (1)

H. G. Booker, P. C. Clemmow, “The concept of an angular spectrum of plane waves and its relation to that of polar diagrams and aperture distributions,” Proc. Inst. Electr. Eng. 97 III, 11–17 (1950).

1928 (1)

G. C. Steward, “IV Aberration diffraction effects,” Proc. R. Soc. London Ser. A 225, 131 (1928).

1904 (1)

E. T. Whittaker, “On an expression of the electromagnetic field due to electrons by means of two scalar potential functions,” Proc. London Math. Soc. 1, 367–372 (1904).
[CrossRef]

1872 (1)

Rayleigh, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).

1841 (1)

G. B. Airy, “The diffraction of an annular aperture,” Philos. Mag. Ser. 3 18, 1–10 (1841).

Agrawal, G.

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayck, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Airy, G. B.

G. B. Airy, “The diffraction of an annular aperture,” Philos. Mag. Ser. 3 18, 1–10 (1841).

Alonso, M. A.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

M. A. Alonso, G. W. Forbes are preparing a paper titled “Uncertainty products for nonparaxial wave fields” for publication.

An, Y. Y.

Asakura, T.

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik 41, 281–291 (1974).

Belanger, P.-A.

M. Couture, P.-A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Berry, M.

J. F. Nye, M. Berry, “Dislocations of wave fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Berry, M. V.

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A 27, L391–L398 (1994).
[CrossRef]

Booker, H. G.

H. G. Booker, P. C. Clemmow, “The concept of an angular spectrum of plane waves and its relation to that of polar diagrams and aperture distributions,” Proc. Inst. Electr. Eng. 97 III, 11–17 (1950).

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
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C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
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C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
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K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
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C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
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C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

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E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London Ser. B 66, 145–149 (1953).
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A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

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F. Gori, G. Guatteri, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
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C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
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C. J. R. Sheppard, S. Saghafi, “Electric and magnetic dipole beam modes beyond the paraxial approximation,” Optik 110, 487–491 (1999).

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
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C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
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G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
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C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
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C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
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C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
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C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
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K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
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C. J. R. Sheppard, S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
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C. J. R. Sheppard, S. Saghafi, “Electric and magnetic dipole beam modes beyond the paraxial approximation,” Optik 110, 487–491 (1999).

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Appl. Opt. (4)

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C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
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C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
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IEEE J. Quantum Electron. (1)

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
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J. Mod. Opt. (5)

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
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C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 45, 1881–1892 (1999).
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Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
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C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
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C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
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P. S. Carney, G. Gbur, “Optimal apodizations for finite apertures,” J. Opt. Soc. Am. A 16, 1638–1640 (1999).
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K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
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K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
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Rayleigh, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).

Opt. Acta (2)

G. Toraldo di Francia, “Optical resonators,” Opt. Acta 13, 323–342 (1966).
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W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
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Opt. Commun. (9)

C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

P. Varga, P. Török, “Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

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

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

M. A. Porras, “The best optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
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Opt. Lett. (5)

Optik (5)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik 41, 281–291 (1974).

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

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Figures (8)

Fig. 1
Fig. 1

Intensity in the waist region of the scalar complex-source–sink model of a Gaussian beam for a value of kz0=2.

Fig. 2
Fig. 2

Contours of constant phase for the scalar complex-source–sink model of a Gaussian beam for the case of kz0=2. Reproduced from Sheppard and Saghafi (Ref. 51).

Fig. 3
Fig. 3

Contours of time-averaged electric energy density for a vectorial Bessel beam of high aperture. The polarization is that produced from the focusing of a plane-polarized wave, which is equivalent to that of crossed electric and magnetic dipoles placed at the focus. The semiangle of convergence is (a) π/6, (b) π/3, (c) π/2, (d) 2π/3. Reproduced from Sheppard (Ref. 10).

Fig. 4
Fig. 4

Contours of equal intensity for a scalar Bessel–Gauss beam with (a) a=0.1, (b) a=0.05. Reproduced from Sheppard and Wilson (Ref. 63).

Fig. 5
Fig. 5

Simple model of the beam that consists of Gaussian beams convergent on the waist from around a cone. Thus we could generate a Bessel–Gauss beam in a toroidal resonator. Reproduced from Sheppard and Wilson (Ref. 63).

Fig. 6
Fig. 6

Intensity in a far-field plane for the hemisphere of convergent radiation that produces the focal spot with the smallest second moment.

Fig. 7
Fig. 7

Intensity in the waist for the hemisphere of convergent radiation that produces the focal spot with the smallest second moment.

Fig. 8
Fig. 8

Value of the high-aperture generalized beam-propagation factor M2 for the complex-source–sink Gaussian beam.

Equations (29)

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U(ρ, ϕ, z)
=ifλexp(ikf)02π0πa(θ, ϕ)exp{ik[(ρ cos(ϕ-ϕ)
×sin θ+z cos θ)]}sin θdθdϕ,
U=sin krkr.
U=exp(ikf)-exp(-ikf)2ikf,
U=kz02 sinh kz0[exp(ikR)-exp(-ikR)]kR,
R=[x2+y2+(z-iz0)2]1/2.
ρ=z0cosh ζ sin η,
z=z0sinh ζ cos η,
ikR=kz0[(1+i sinh ζ cos η)2-cosh2 ζ sin2 η]1/2.
ikRkz0(1+i sinh ζ)-cosh2 ζ sin2 η2(1+i sinh ζ),
U=exp(ikz0sinh ζ)exp[-kz0cosh2 ζ sin2 η/2(1+i sinh ζ)]1+i sinh ζ
exp(ikz0sinh ζ)exp(-1/2kz0sin2 η)1+i sinh ζ,
ikR=kz0(cos η+i sinh ζ),
U=i1-exp(-2kz0)exp(-2kz0sin2 η/2)exp(ikz0sinh ζ)-exp(-2kz0cos2 η/2)exp(-ikz0sinh ζ)cos η+i sinh ζ.
Usource=iexp(ikz0sinh ζ)exp(-2kz0sin2 η/2)cos η+i sinh ζ
kz0sinh ζ=(q+2)π2,
U=J0(v)exp(ikz cos α),
v=kρ sin α
WE=J0(v)2+2 tan2α2[J1(v)]2+tan4α2[J2(v)]2+2 cos 2ϕ tan2α2×{[J1(v)]2+J0(v)J2(v)}.
U=exp(ikz)1+i tan ψexp-v2a22(1+i tan ψ)×exp-i2a2tan ψ1+i tan ψJ0v1+i tan ψ,
tan ψ=zz0,
a(θ)=exp-12a2sin θsin α-12.
I=1(1+t2)2exp-k2w02t24(1+t2),
t=tan θ
I=1(1+t2)2J02j01t1+t2,
i=J0(kρ)1-(kρ/j01)22,
M2=[1-L2(2kz0)][1+kz0L(2kz0)],
L(x)=coth x-1/x.

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