Abstract

In paraxial optics, the power carried by an optical beam can be accurately calculated by means of the integral of the squared modulus of its electric field over a plane transverse to the propagation axis. However, for nonparaxial electromagnetic beams, it is more appropriate to define the power carried by the beam by the integral of the longitudinal component of its time-averaged Poynting vector over a plane transverse to the propagation axis. In this paper, the expression of the power carried by a high-aperture transverse magnetic (TM) beam of any order is determined. The general expression of the power carried by a TM beam, which also applies for a transverse electric (TE) beam, is given in terms of a modified Struve function of order equal to an integer plus one-half.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  2. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71, 026603 (2005).
    [CrossRef]
  3. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 36, 681-683 (2008).
    [CrossRef]
  4. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34, 3601-3603 (2009).
    [CrossRef] [PubMed]
  5. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]
  6. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
    [CrossRef]
  7. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31, 2619-2621 (2006).
    [CrossRef] [PubMed]
  8. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793-1798 (2007).
    [CrossRef]
  9. Q. Cao and X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212-216 (1998).
    [CrossRef]
  10. J. Lekner, “TM, TE and 'TEM' beam modes: exact solutions and their problems,” J. Opt. A, Pure Appl. Opt. 3, 407-412 (2001).
    [CrossRef]
  11. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  12. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543-1545 (1999).
    [CrossRef]
  13. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77-87 (2000).
    [CrossRef] [PubMed]
  14. I. J. Cooper, M. Roy, and C. J. R. Sheppard, “Focusing of pseudoradial polarized beams,” Opt. Express 13, 1066-1071 (2005).
    [CrossRef] [PubMed]
  15. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228-1234 (2006).
    [CrossRef]
  16. C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the longitudinal electric field resulting from the tight focusing of an ultrafast transverse-magnetic laser beam,” J. Opt. Soc. Am. A 23, 2027-2038 (2006).
    [CrossRef]
  17. A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33, 1563-1565 (2008).
    [CrossRef] [PubMed]
  18. S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. 45, 5335-5345 (2006).
    [CrossRef] [PubMed]
  19. H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33, 1984-1986 (2008).
    [CrossRef] [PubMed]
  20. A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D: Appl. Phys. 32, 2871-2875 (1999).
    [CrossRef]
  21. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234-2239 (1990).
    [CrossRef] [PubMed]
  22. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  23. V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393-8399 (2006).
    [CrossRef] [PubMed]
  24. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
    [CrossRef]
  25. Y. Kozawa and S. Sato, “Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler,” Opt. Lett. 33, 2278-2280 (2008).
    [CrossRef] [PubMed]
  26. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43, 2397-2399 (2004).
    [CrossRef] [PubMed]
  27. M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929-1931 (2002).
    [CrossRef]
  28. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A. 253, 358-379 (1959).
    [CrossRef]
  29. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
    [CrossRef]
  30. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  31. M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
    [CrossRef]
  32. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
    [CrossRef]
  33. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. 25, 1792-1794 (2000).
    [CrossRef]
  34. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33, 1392-1394 (2008).
    [CrossRef] [PubMed]
  35. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
  36. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

2009 (1)

2008 (5)

2007 (1)

2006 (5)

2005 (3)

I. J. Cooper, M. Roy, and C. J. R. Sheppard, “Focusing of pseudoradial polarized beams,” Opt. Express 13, 1066-1071 (2005).
[CrossRef] [PubMed]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71, 026603 (2005).
[CrossRef]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

2004 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (2)

J. Lekner, “TM, TE and 'TEM' beam modes: exact solutions and their problems,” J. Opt. A, Pure Appl. Opt. 3, 407-412 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

2000 (3)

1999 (3)

1998 (2)

Q. Cao and X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212-216 (1998).
[CrossRef]

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

1990 (1)

1981 (1)

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

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
[CrossRef]

1971 (1)

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

1959 (1)

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

Ahmed, M. A.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

April, A.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Bélanger, P.-A.

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

Brown, T. G.

Cao, Q.

Q. Cao and X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212-216 (1998).
[CrossRef]

Chang, R. S.

Cooper, I. J.

Couture, M.

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

De Koninck, Y.

Dehez, H.

Deng, D.

Deng, X.

Q. Cao and X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212-216 (1998).
[CrossRef]

Deschamps, G. A.

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

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Ford, D. H.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Glur, H.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Gradshteyn, I. S.

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

Graf, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Hershcovitz, O.

Juskaitis, R.

Kawauchi, H.

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Kimura, W. D.

Kozawa, Y.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
[CrossRef]

Lekner, J.

J. Lekner, “TM, TE and 'TEM' beam modes: exact solutions and their problems,” J. Opt. A, Pure Appl. Opt. 3, 407-412 (2001).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Lipson, S. G.

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
[CrossRef]

Ludlow, I. K.

Massoumian, F.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
[CrossRef]

Meixner, A. J.

Moed, S.

Moser, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Neil, M. A. A.

Nesterov, A. V.

V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393-8399 (2006).
[CrossRef] [PubMed]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D: Appl. Phys. 32, 2871-2875 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393-8399 (2006).
[CrossRef] [PubMed]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D: Appl. Phys. 32, 2871-2875 (1999).
[CrossRef]

Parriaux, O.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Piché, M.

Pigeon, F.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the longitudinal electric field resulting from the tight focusing of an ultrafast transverse-magnetic laser beam,” J. Opt. Soc. Am. A 23, 2027-2038 (2006).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71, 026603 (2005).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Richards, B.

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

Romano, V.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

Rotschild, C.

Roy, M.

Ryzhik, I. M.

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

Saghafi, S.

Salamin, Y. I.

Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31, 2619-2621 (2006).
[CrossRef] [PubMed]

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Sato, S.

Seshadri, S. R.

Sheppard,

Sheppard, C. J. R.

Stadler, J.

Stanciu, C.

Stupperich, C.

Tidwell, S. C.

Ulanowski, Z.

Varin, C.

C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the longitudinal electric field resulting from the tight focusing of an ultrafast transverse-magnetic laser beam,” J. Opt. Soc. Am. A 23, 2027-2038 (2006).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71, 026603 (2005).
[CrossRef]

Wilson, T.

Wolf, E.

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

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D: Appl. Phys. 32, 2871-2875 (1999).
[CrossRef]

Youngworth, K. S.

Zommer, S.

Appl. Opt. (4)

Appl. Phys. B (2)

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707-713 (2005).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Electron. Lett. (1)

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

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

J. Lekner, “TM, TE and 'TEM' beam modes: exact solutions and their problems,” J. Opt. A, Pure Appl. Opt. 3, 407-412 (2001).
[CrossRef]

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

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

J. Phys. D: Appl. Phys. (1)

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D: Appl. Phys. 32, 2871-2875 (1999).
[CrossRef]

Opt. Commun. (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Q. Cao and X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212-216 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (10)

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

A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33, 1563-1565 (2008).
[CrossRef] [PubMed]

Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31, 2619-2621 (2006).
[CrossRef] [PubMed]

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 36, 681-683 (2008).
[CrossRef]

H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34, 3601-3603 (2009).
[CrossRef] [PubMed]

Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. 25, 1792-1794 (2000).
[CrossRef]

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33, 1392-1394 (2008).
[CrossRef] [PubMed]

Y. Kozawa and S. Sato, “Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler,” Opt. Lett. 33, 2278-2280 (2008).
[CrossRef] [PubMed]

M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929-1931 (2002).
[CrossRef]

H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33, 1984-1986 (2008).
[CrossRef] [PubMed]

Phys. Rev. A (3)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1369 (1975).
[CrossRef]

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

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

Phys. Rev. E (1)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71, 026603 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. Roy. Soc. A. (1)

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

Other (2)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Power carried by a TM p , m + 1 beam normalized with the power of its paraxial counterpart as a function of the parameter k a for the first values of p and m.

Equations (25)

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

E x = E o K p , 0 K p , 1 V ̃ p , 1 e ,
E y = E o K p , 0 K p , 1 V ̃ p , 1 o ,
H x = H o K p , 0 K p , 1 U ̃ p , 1 o ,
H y = H o K p , 0 K p , 1 U ̃ p , 1 e ,
U ̃ p , m e = K p , m 2 a p ! ( a 2 k ) p + m 2 exp ( k a ) cos ( m φ ) 0 k r 2 p + m cos [ k z ( z + j a ) ] k z J m ( k r r ) k r d k r ,
V ̃ p , m e = j K p , m 2 a p ! ( a 2 k ) p + m 2 exp ( k a ) k cos ( m φ ) 0 k r 2 p + m sin [ k z ( z + j a ) ] J m ( k r r ) k r d k r ,
P 0 2 π 0 S a ̂ z r d r d φ = 1 2 0 2 π 0 Re { E x H y * E y H x * } r d r d φ .
P p , 1 = | E o | 2 2 η o | K p , 0 | 2 | K p , 1 | 2 0 2 π 0 Re { V ̃ p , 1 e U ̃ p , 1 * e + V ̃ p , 1 o U ̃ p , 1 * o } r d r d φ .
P p , 1 = a | E o | 2 | K p , 0 | 2 2 2 p ( p ! ) 2 η o ( a k ) 2 p + 2 exp ( 2 k a ) Re { 0 2 π 0 0 k r 2 p + 1 k r 2 p + 1 × cos [ k z * ( z j a ) ] sin [ k z ( z + j a ) ] j k z * 0 J 1 ( k r r ) J 1 ( k r r ) r d r k r k r d k r d k r d φ } ,
P p , 1 = a | E o | 2 | K p , 0 | 2 2 2 p ( p ! ) 2 η o ( a k ) 2 p + 2 exp ( 2 k a ) Re { 0 2 π 0 k r 4 p + 2 cos [ k z * ( z j a ) ] sin [ k z ( z + j a ) ] j k z * k r d k r d φ } .
P p , 1 = a | E o | 2 | K p , 0 | 2 2 2 p + 1 ( p ! ) 2 η o ( a k ) 2 p + 2 exp ( 2 k a ) 0 2 π 0 k k r 4 p + 2 sinh ( 2 k z a ) k z k r d k r d φ .
P p , 1 = a | E o | 2 | K p , 0 | 2 ( k a ) 2 p + 2 2 2 p + 1 ( p ! ) 2 k η o exp ( 2 k a ) 0 2 π 0 π 2 sin 4 p + 3 θ sinh ( 2 k a cos θ ) d θ d φ .
Φ p , 1 ( θ , φ ) a | E o | 2 | K p , 0 | 2 ( k a ) 2 p + 2 2 2 p + 1 ( p ! ) 2 k η o exp ( 2 k a ) sin 4 p + 2 θ sinh ( 2 k a cos θ )
P p , 1 = π | E o | 2 | K p , 0 | 2 ( 2 p + 1 ) ! 2 2 p + 1 ( p ! ) 2 η o ( π k a 3 ) 1 2 exp ( 2 k a ) k L 2 p + 3 2 ( 2 k a ) .
P p , m + 1 = ( 1 + δ 0 , m ) π | E o | 2 | K p , m | 2 ( 2 p + m + 1 ) ! 2 2 p + m + 2 ( p ! ) 2 η o ( π k a 3 ) 1 2 exp ( 2 k a ) k L 2 p + m + 3 2 ( 2 k a ) ,
P 0 , 1 = π | E o | 2 | K 0 , 0 | 2 4 η o k 2 exp ( 2 k a ) [ 2 k a sinh ( 2 k a ) cosh ( 2 k a ) + 1 2 ( k a ) 2 ] .
P p , m + 1 ( 0 ) lim k a 1 P p , m + 1 = ( 1 + δ 0 , m ) ( π a 2 k ) | E o | 2 | K p , m | 2 ( 2 p + m + 1 ) ! 2 2 p + m + 2 ( p ! ) 2 η o .
P p , m + 1 P p , m + 1 ( 0 ) = 2 ( π k a ) 1 2 exp ( 2 k a ) L 2 p + m + 3 2 ( 2 k a ) .
L ν ( x ) = s = 0 1 Γ ( s + 3 2 ) Γ ( ν + s + 3 2 ) ( x 2 ) 2 s + ν + 1 ,
L ν ( x ) = 2 ( x 2 ) ν π Γ ( ν + 1 2 ) 0 π 2 sin 2 ν θ sinh ( x cos θ ) d θ .
L n + 1 2 ( x ) = I n 1 2 ( x ) ( 2 π x ) 1 2 m = 0 n ( 1 ) m ( 2 m 1 ) !! ( 2 n 2 m ) !! x n 2 m ,
L 1 2 ( x ) = ( 2 π x ) 1 2 ( 2 cosh x 2 ) ,
L 3 2 ( x ) = ( 2 π x 3 ) 1 2 ( 2 x sinh x 2 cosh x + 2 x 2 ) ,
L 5 2 ( x ) = ( 2 π x 5 ) 1 2 ( 6 + x 2 1 4 x 4 + 2 ( x 2 + 3 ) cosh x 6 x sinh x ) .
lim x 1 L ν ( x ) = exp ( x ) ( 2 π x ) 1 2 1 π 1 2 Γ ( ν + 1 2 ) ( x 2 ) ν 1 .

Metrics