P. S. Carney, G. Gbur, “Optimal apodizations for finite apertures,” J. Opt. Soc. Am. A 16, 1638–1640 (1999).

[CrossRef]

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).

[CrossRef]

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).

[CrossRef]

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]

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).

[CrossRef]

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, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).

[CrossRef]

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

[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]

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

[CrossRef]

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

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]

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

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

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

[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]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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]

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

[CrossRef]
[PubMed]

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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]

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]

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

[CrossRef]

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

[CrossRef]

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

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

[CrossRef]

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]

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

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]

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

[CrossRef]

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

[CrossRef]

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).

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

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]

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

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

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

[CrossRef]

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

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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).

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

[CrossRef]

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

[CrossRef]

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).

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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).

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Frankfurt, Germany, 1981).

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

P. Jacquinot, M. B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.

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

[CrossRef]

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

[CrossRef]

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).

[CrossRef]

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).

[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]

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

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

[CrossRef]

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

[CrossRef]

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

[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]

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

[CrossRef]

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]
[PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[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]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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).

[CrossRef]

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

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]

P. Jacquinot, M. B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.

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

[CrossRef]

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Frankfurt, Germany, 1981).

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).

[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).

[CrossRef]

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, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).

[CrossRef]

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

[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).

[CrossRef]

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).

[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).

[CrossRef]

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).

[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).

[CrossRef]

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, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).

[CrossRef]

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

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

[CrossRef]

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]

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

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

[CrossRef]

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

[CrossRef]

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]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).

[CrossRef]

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

[CrossRef]
[PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).

[CrossRef]

W. H. Steel, “Axicons with spherical surfaces,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 181–193.

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

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 360.

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

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

[CrossRef]
[PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).

[CrossRef]

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

[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]

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]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).

[CrossRef]
[PubMed]

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

[CrossRef]

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

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]

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

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]

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

[CrossRef]

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]

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

[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]

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

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]

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]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

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]

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

[CrossRef]

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

[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]

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]

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]

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

[CrossRef]

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

[CrossRef]

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]

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

[CrossRef]

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

[CrossRef]

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).

[CrossRef]

D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).

[CrossRef]

J. C. Heurtley, W. Streifer, “Optical resonator modes—circular reflectors of spherical curvature,” J. Opt. Soc. Am. 65, 1472–1479 (1965).

[CrossRef]

C. W. McCutchen, “Two families of apodization problems,” J. Opt. Soc. Am. 59, 1163–1171 (1969).

[CrossRef]

A. Papoulis, “Apodization for optimum imaging of smooth objects,” J. Opt. Soc. Am. 62, 1423–1429 (1972).

[CrossRef]

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).

[CrossRef]

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

[CrossRef]

A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).

[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]

T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).

[CrossRef]

E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).

[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).

[CrossRef]

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

[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).

[CrossRef]

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]

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).

[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximations,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).

[CrossRef]

E. Heyman, L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).

[CrossRef]

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

[CrossRef]

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).

[CrossRef]

P. S. Carney, G. Gbur, “Optimal apodizations for finite apertures,” J. Opt. Soc. Am. A 16, 1638–1640 (1999).

[CrossRef]

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

[CrossRef]

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[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]

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]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).

[CrossRef]
[PubMed]

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, S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).

[CrossRef]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

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

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).

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

C. J. R. Sheppard, S. Saghafi, “Electric and magnetic dipole beam modes beyond the paraxial approximation,” Optik 110, 487–491 (1999).

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

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

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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).

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]

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

[CrossRef]

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

[CrossRef]

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

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 360.

W. H. Steel, “Axicons with spherical surfaces,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 181–193.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Frankfurt, Germany, 1981).

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

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).

[CrossRef]

P. Jacquinot, M. B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.