Abstract

A spherical Hankel function series solution for the vector components of a general flattened Gaussian laser field is derived, based on the angular spectrum of plane waves. This perturbative series is valid for spot sizes greater than ten wavelengths, creating a complete vector solution for a general flattened Gaussian laser profile for all focusing conditions when coupled to the model developed in Part I of this investigation [J. Opt. Soc. Am. B 23, 2157 (2006) ]. The focusing and propagation properties of these fields are then explored numerically. Finally, the exact solution is compared to the perturbative Hermite–Gaussian (0,0) laser mode by comparing the focal plane boundary conditions imposed in each and is found to be a separate and distinct solution under tight focusing conditions.

© 2006 Optical Society of America

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  1. S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. I. Small F-number laser focusing," J. Opt. Soc. Am. B 23, 2157-2165 (2006).
    [CrossRef]
  2. L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
    [CrossRef] [PubMed]
  3. B. W. Boreham and B. Luther-Davies, "High-energy electron acceleration by ponderomotive forces in tenuous plasmas," J. Appl. Phys. 50, 2533-2538 (1979).
    [CrossRef]
  4. S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
    [CrossRef] [PubMed]
  5. B. W. Boreham and H. Hora, "Debye length discrimination of nonlinear laser forces acting on electrons in tenuous plasmas," Phys. Rev. Lett. 42, 776-779 (1979).
    [CrossRef]
  6. B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
    [CrossRef]
  7. A. Maltsev and T. Ditmire, "Above threshold ionization and in tightly focused, strongly relativistic laser fields," Phys. Rev. Lett. 90, 053002 (2003).
    [CrossRef] [PubMed]
  8. H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
    [CrossRef]
  9. S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
    [CrossRef]
  10. H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).
  11. H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).
  12. H. Hora, Laser Plasma Physics (SPIE, 2000).
  13. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  14. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  15. J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  16. J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
    [CrossRef]
  17. Y. I. Salamin and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  18. R. Borghi, A. Ciattoni, and M. Santarisiero, "Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions," J. Opt. Soc. Am. A 19, 1207-1211 (2002).
    [CrossRef]
  19. P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
    [CrossRef]
  20. S. Sepke and D. Umstadter, "Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes," Opt. Lett. 31, 1447-1449 (2006).
    [CrossRef] [PubMed]

2006 (2)

2005 (2)

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

2003 (1)

A. Maltsev and T. Ditmire, "Above threshold ionization and in tightly focused, strongly relativistic laser fields," Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (1)

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

1998 (2)

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[CrossRef]

1990 (1)

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

1989 (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1979 (3)

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

B. W. Boreham and B. Luther-Davies, "High-energy electron acceleration by ponderomotive forces in tenuous plasmas," J. Appl. Phys. 50, 2533-2538 (1979).
[CrossRef]

B. W. Boreham and H. Hora, "Debye length discrimination of nonlinear laser forces acting on electrons in tenuous plasmas," Phys. Rev. Lett. 42, 776-779 (1979).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Banerjee, S.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Barton, J. P.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Boreham, B. W.

B. W. Boreham and H. Hora, "Debye length discrimination of nonlinear laser forces acting on electrons in tenuous plasmas," Phys. Rev. Lett. 42, 776-779 (1979).
[CrossRef]

B. W. Boreham and B. Luther-Davies, "High-energy electron acceleration by ponderomotive forces in tenuous plasmas," J. Appl. Phys. 50, 2533-2538 (1979).
[CrossRef]

Borghi, R.

Castillo, R.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

Ciattoni, A.

Cicchitelli, L.

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

Davis, L. W.

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

Ditmire, T.

A. Maltsev and T. Ditmire, "Above threshold ionization and in tightly focused, strongly relativistic laser fields," Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

Hauser, T.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Ho, Y. K.

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

Hoelss, M.

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

Hora, H.

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

B. W. Boreham and H. Hora, "Debye length discrimination of nonlinear laser forces acting on electrons in tenuous plasmas," Phys. Rev. Lett. 42, 776-779 (1979).
[CrossRef]

H. Hora, Laser Plasma Physics (SPIE, 2000).

H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Kando, M.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Kato, Y.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

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]

Kitagawa, Y.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Kotaki, H.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Luther-Davies, B.

B. W. Boreham and B. Luther-Davies, "High-energy electron acceleration by ponderomotive forces in tenuous plasmas," J. Appl. Phys. 50, 2533-2538 (1979).
[CrossRef]

Maksimchuk, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Maltsev, A.

A. Maltsev and T. Ditmire, "Above threshold ionization and in tightly focused, strongly relativistic laser fields," Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

Masuda, S.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Mima, K.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Mora, P.

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[CrossRef]

Nakajima, K.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Osman, F.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

Postle, R.

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

Quesnel, B.

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[CrossRef]

Salamin, Y. I.

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

Santarisiero, M.

Scheid, W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Sepke, S.

Shah, R.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Sheid, W.

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

Török, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Umstadter, D.

Valenzuela, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Varga, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Wang, J. W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

Wang, J. X.

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

Yamanaka, T.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

J. Appl. Phys. (2)

B. W. Boreham and B. Luther-Davies, "High-energy electron acceleration by ponderomotive forces in tenuous plasmas," J. Appl. Phys. 50, 2533-2538 (1979).
[CrossRef]

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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

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

Opt. Commun. (1)

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Plasmas (1)

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, "Suppression of electron scattering by the longitudinal components of tightly focused laser fields," Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Phys. Rev. A (3)

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

M. Lax, W. H. Louisell, and 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]

Phys. Rev. E (2)

J. X. Wang, W. Sheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[CrossRef]

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[CrossRef]

Phys. Rev. Lett. (4)

A. Maltsev and T. Ditmire, "Above threshold ionization and in tightly focused, strongly relativistic laser fields," Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

B. W. Boreham and H. Hora, "Debye length discrimination of nonlinear laser forces acting on electrons in tenuous plasmas," Phys. Rev. Lett. 42, 776-779 (1979).
[CrossRef]

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

Other (4)

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, "Free wave laser acceleration of electrons and consequences for the Umstadter experiment," in Proceedings of the 13th International Conference on Laser Interaction and Related Plasma Phenomena, AIP Conf. Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Acceleration of electrons in vacuum by lasers and the accuracy principle of nonlinearity," in High-Power Lasers in Energy Engineering, K.Mima, G.L.Kulcinski, and W.Hogan, eds., Proc. SPIE 3886, 145-156 (2000).

H. Hora, Laser Plasma Physics (SPIE, 2000).

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Figures (14)

Fig. 1
Fig. 1

Relative error of the series expansion of E x compared to the exact integral solution for w 0 = 5 μ m along the line ( x , y ) = ( 1 , 1 ) μ m to the order of ϵ 0 (dotted–dashed curve), ϵ 8 (dotted curve), ϵ 18 (dashed curve), and ϵ 38 (solid curve) for λ 0 = 1 μ m .

Fig. 2
Fig. 2

Flattened Gaussian boundary condition for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 3
Fig. 3

Angular spectrum solution for E x in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 4
Fig. 4

Angular spectrum solution for E y in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 5
Fig. 5

Angular spectrum solution for E z in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 6
Fig. 6

Angular spectrum solution for E z in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) . Notice the region of zero field intensity around the laser axis.

Fig. 7
Fig. 7

Angular spectrum solution for E z in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 ) , a pure Gaussian. Notice the region of zero field intensity around the laser axis is now only a narrow line along x = 0 .

Fig. 8
Fig. 8

Angular spectrum solution for E x along x = y versus z for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 9
Fig. 9

Angular spectrum solution for E y along x = y versus z for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 10
Fig. 10

Angular spectrum solution for E z along x = y versus z for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 1 , 1 , 0.5 ) .

Fig. 11
Fig. 11

Hollow core angular spectrum solution for E x in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 0 , 1 ) .

Fig. 12
Fig. 12

Hollow core angular spectrum solution for E y in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 0 , 1 ) .

Fig. 13
Fig. 13

Hollow core angular spectrum solution for E z in the focal plane ( z = 0 ) for a wavelength λ 0 = 1 μ m , waist w 0 = 5 μ m , E 0 = 1 , and A N = ( 0 , 1 ) .

Fig. 14
Fig. 14

Absolute difference between the symmetric TEM 00 and angular spectrum solutions for E x along y = z = 0 for a wavelength λ 0 = 1 μ m and waist w 0 = 5 μ m corresponding to ϵ 2 = 4.1 × 10 3 and E 0 normalized to 1. The dots have been calculated from the field models and the solid curve is the result predicted by Eq. (10).

Tables (1)

Tables Icon

Table 1 Electric Field Components of the Symmetric TEM 00 Mode to the Order 2

Equations (32)

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E x ( x , y , z = 0 ) = N = 0 A N ( r w 0 ) 2 N e ( r 2 w 0 2 ) ,
E x a = 0 1 e ( b 2 ϵ 2 ) e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
E z a = i x ¯ 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
B x a = x ¯ y ¯ 2 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
B y a = [ 0 1 e ( b 2 ϵ 2 ) m e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b x ¯ 2 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ] ,
B z a = i y ¯ 0 1 e ( b 2 ϵ 2 ) e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
e i m z = n = 0 1 n ! ( b 2 2 ) n z n + 1 h n 1 ( 1 ) ( z ) ,
m e i m z = i n = 0 1 n ! ( b 2 2 ) n z n [ ( z ) h n ( 1 ) ( z ) 2 n h n 1 ( 1 ) ( z ) ] ,
e i m z m = i n = 0 1 n ! ( b 2 2 ) n z n + 1 h n ( 1 ) ( z ) ,
E x = i E 0 N = 0 A N n = 0 ϵ 2 n ( k 0 z ) n + 1 h n ( 1 ) ( k 0 z ) m = 0 N Δ m n N y ¯ 2 G m + n ( ξ ) + E 0 N = 0 A N n = 0 ϵ 2 n ( k 0 z ) n { [ ( k 0 z ) 2 i n ] h n 1 ( 1 ) ( k 0 z ) + i ( k 0 z ) h n ( 1 ) ( k 0 z ) } m = 0 N Δ m n N G m + n ( ξ ) ,
E y = i E 0 N = 0 A N n = 0 ϵ 2 n ( k 0 z ) n + 1 h n ( 1 ) ( k 0 z ) m = 0 N Δ m n N x ¯ y ¯ 2 G m + n ( ξ ) ,
E z = E 0 N = 0 A N n = 0 ϵ 2 n ( k 0 z ) n + 1 [ i h n 1 ( 1 ) ( k 0 z ) h n ( 1 ) ( k 0 z ) ] m = 0 N Δ m n N x ¯ G m + n ( ξ )
Δ m n N ( 1 ) m [ N ! N ! ( m + n ) ! ( N m ) ! n ! m ! m ! 2 n + 1 ] .
E x = i E 0 n = 0 ϵ 2 n ( k 0 z 2 ) n + 1 h n ( 1 ) ( k 0 z ) y ¯ 2 G n ( r 2 w 0 2 ) + E 0 2 n = 0 ϵ 2 n ( k 0 z 2 ) n F x ( k 0 z ) G n ( r 2 w 0 2 ) ,
E y = i E 0 n = 0 ϵ 2 n ( k 0 z 2 ) n + 1 h n ( 1 ) ( k 0 z ) x ¯ y ¯ 2 G n ( r 2 w 0 2 ) ,
E z = E 0 n = 0 ϵ 2 n ( k 0 z 2 ) n + 1 F z ( k 0 z ) x ¯ G n ( r 2 w 0 2 ) ,
E x = 2 E 0 sin ( η ) e ( r 2 w 0 2 ) N = 0 m = 0 N A N Δ m 0 N L m ( r 2 w 0 2 ) ,
E y = ϵ 2 E 0 sin ( η ) e ( r 2 w 0 2 ) ( x y r 2 ) N = 0 m = 0 N A N Δ m 0 N Θ m ,
E z = 2 ϵ E 0 cos ( η ) e ( r 2 w 0 2 ) ( x r ) N = 0 m = 0 N A N Δ m 0 N ψ m ,
ψ m ( r ) = m ξ 1 [ L m ( ξ ) L m 1 ( ξ ) ] ξ L m ( ξ ) ,
Θ m ( r ) = { ξ L m ( ξ ) 2 m [ L m ( ξ ) L m 1 ( ξ ) ] + m ( m 1 ) ξ 1 [ L m ( ξ ) 2 L m 1 ( ξ ) + L m 2 ( ξ ) ] } ,
E z z x E x d z i k 0 E x x ,
2 A 1 c 2 2 A t 2 = 0 .
A = x ̂ A 0 g ( η ) Ψ ( r ) e i η
Ψ 0 = σ e σ ρ 2 = ( w 0 w ) e ( r 2 w 2 ) exp { i [ η + arctan ( z z R ) z r 2 z R w 2 ] } ,
1 ϵ 2 t = 1 1 2 n = 1 Γ n 1 2 n ! ϵ 2 n t n ,
1 1 ϵ 2 t = n = 0 Γ n + 1 2 n ! ϵ 2 n t n
I 1 = ϵ 2 2 0 e t ( 1 + 1 ϵ 2 t ) J 0 ( 2 ρ 2 t ) d t ϵ 2 2 { 1 1 2 n = 1 Γ n 1 2 ϵ 2 n L n ( r 2 w 0 2 ) } e ( r 2 w 0 2 ) ,
E x = E 0 e ( r 2 w 0 2 ) [ 1 + 1 2 n = 1 Γ n 1 2 ( n y 2 r 2 1 2 ) ϵ 2 n L n 0 ( r 2 w 0 2 ) + 1 4 ϵ 2 ( x 2 y 2 r 2 ) n = 0 Γ n + 1 2 ϵ 2 n L n 1 ( r 2 w 0 2 ) ] ,
E x ( 2 ) ( x , y , z = 0 ) = E 0 4 ( 2 x 2 r 2 w 0 2 ) e ( r 2 w 0 2 ) .
E x ( 2 ) = E 0 4 { ( i C 1 + 2 ) ( r w 0 ) 4 + [ 2 ( x 2 w 0 2 ) + ( i C 1 + 3 ) ( r w 0 ) 2 ] } e ( r 2 w 0 2 ) ,
Δ E x = 1 4 E 0 ϵ 2 ( r 2 w 0 2 ) ( 2 r 2 w 0 2 ) e ( r 2 w 0 2 ) ,

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