Abstract

Many laser interaction models assume that incident focused laser fields are Gaussian and use either the approximate TEM00 series model or the exact integral Gaussian angular-spectrum solution. Many practical laser systems, however, produce flat-top transverse intensity profiles, and indeed, such profiles are often desired. Here, an exact, integral solution is derived for all of the vector components having a general flattened Gaussian profile using the angular-spectrum method. This solution includes the pure and annular Gaussian modes as special cases. The resulting integrals are solved for tight focusing conditions exactly by making use of a Fourier–Gegenbauer expansion. This technique follows closely that of Sepke and Umstadter [Opt. Lett. 31, 1447 (2006)] but, by redefining the expansion coefficients, the simplicity of the model is greatly enhanced and the computation time reduced by roughly a factor of 2 beyond the 2 orders of magnitude improvement obtained previously. This series solution is stable at all points and converges after S20w0 terms, where w0 is the 1e waist normalized to the laser wavelength.

© 2006 Optical Society of America

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2005

H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
[CrossRef]

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]

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[CrossRef]

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[CrossRef]

2004

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, "Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal," Opt. Commun. 232, 77-82 (2004).
[CrossRef]

S. P. D. Mangles, C. D. Murphy, Z. Najmudin, A. G. R. Thomas, J. L. Collier, A. E. Dangor, E. J. Divall, P. S. Foster, J. G. Gallacher, C. J. Hooker, D. A. Jaroszynski, A. J. Langley, W. B. Mori, P. A. Norreys, F. S. Tsung, R. Viskup, B. R. Walton, and K. Krushelnick, "Monoenergetic beams of relativistic electrons from intense laser-plasma interactions," Nature 431, 535-538 (2004).
[CrossRef] [PubMed]

C. G. R. Geddes, C. Toth, J. van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. P. Leemans, "High-quality electron beams from a laser wakefield accelerator using plasma channel guiding," Nature 431, 538-541 (2004).
[CrossRef] [PubMed]

J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J.-P. Rousseau, F. Burgy, and V. Malka, "A laser-plasma accelerator producing monoenergetic electron beams," Nature 431, 541-544 (2004).
[CrossRef] [PubMed]

2003

X. H. Lu, X. M. Chen, L. Zhang, and D. J. Xue, "High-order Bessel-Gaussian beam and its propagation properties," Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

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

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, "Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams," Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).
[CrossRef]

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

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]

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

2001

R. Borghi, "Elegant Laguerre-Gaussian beams as a new tool for describing axisymmetric flattened Gaussian beams," J. Opt. Soc. Am. A 18, 1627-1633 (2001).
[CrossRef]

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]

2000

1999

1998

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]

T. Y. Cherezova, S. S. Chesnokov, L. N. Kaptsov, and A. V. Kudryashov, "Super-Gaussian laser intensity output formation by means of adaptive optics," Opt. Commun. 155, 99-106 (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]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axicon," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

1997

G. Malka, E. Lefebvre, and J. L. Miquel, "Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser," Phys. Rev. Lett. 78, 3314-3317 (1997).
[CrossRef]

W. C. Yen, C. T. Huang, H. P. Liu, and L. P. Lee, "A Nd:YAG laser with a flat-top beam profile and constant divergence," Opt. Laser Technol. 29, 57-61 (1997).
[CrossRef]

1996

1995

T. Lehecka, R. H. Lehmberg, A. V. Deniz, K. A. Gerber,S. P. Obenschain, C. J. Pawley, M. S. Pronko, and C. A. Sullivan, "Production of high energy, uniform focal profiles with the Nike laser," Opt. Commun. 117, 485-491 (1995).
[CrossRef]

V. V. Goloviznin, P. W. van Amersfoort, N. E. Andreev, and V. I. Kirsanov, "Self-resonant plasma wake-field excitation by a laser pulse with a steep leading edge for particle acceleration," Phys. Rev. E 52, 5327-5332 (1995).
[CrossRef]

1994

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, "Optical smoothing techniques for shock wave generation in laser-produced plasmas," Phys. Rev. E 50, R3314-R3317 (1994).
[CrossRef]

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1993

R. Kant, "An analytical solution of vector diffraction forfocusing optical systems," J. Mod. Opt. 40, 337-347 (1993).
[CrossRef]

U. Mohideen, M. H. Sher, H. W. K. Tom, G. D. Aumiller, O. R. Wood, R. R. Freeman, J. Boker, and P. H. Bucksbaum, "High intensity above-threshold ionization of He," Phys. Rev. Lett. 71, 509-512 (1993).
[CrossRef] [PubMed]

1990

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

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]

1988

A. Pertovaara, T. J. Morrow, and K. L. Casey, "Cutaneous pain and detection thresholds to short CO2 laser pulses in humans: evidence on afferent mechanisms and the influence of varying stimulus conditions," Pain 34, 261-269 (1988).
[CrossRef] [PubMed]

S. DeSilvestri, P. Laporta, V. Magni, and O. Svelto, "Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach," IEEE J. Quantum Electron. QE-24, 1172-1177 (1988).
[CrossRef]

1983

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, "A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam," Opt. Commun. 48, 44-46 (1983).
[CrossRef]

1982

D. Shafer, "Gaussian to flat-top intensity distributing lens," Opt. Laser Technol. 14, 159-160 (1982).
[CrossRef]

1979

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]

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

G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979).
[CrossRef]

1975

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

Agrawal, G. P.

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]

Ambrosini, D.

Andreev, N. E.

V. V. Goloviznin, P. W. van Amersfoort, N. E. Andreev, and V. I. Kirsanov, "Self-resonant plasma wake-field excitation by a laser pulse with a steep leading edge for particle acceleration," Phys. Rev. E 52, 5327-5332 (1995).
[CrossRef]

Aumiller, G. D.

U. Mohideen, M. H. Sher, H. W. K. Tom, G. D. Aumiller, O. R. Wood, R. R. Freeman, J. Boker, and P. H. Bucksbaum, "High intensity above-threshold ionization of He," Phys. Rev. Lett. 71, 509-512 (1993).
[CrossRef] [PubMed]

Bagini, V.

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]

Batani, D.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, "Optical smoothing techniques for shock wave generation in laser-produced plasmas," Phys. Rev. E 50, R3314-R3317 (1994).
[CrossRef]

Benuzzi, A.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, "Optical smoothing techniques for shock wave generation in laser-produced plasmas," Phys. Rev. E 50, R3314-R3317 (1994).
[CrossRef]

Boker, J.

U. Mohideen, M. H. Sher, H. W. K. Tom, G. D. Aumiller, O. R. Wood, R. R. Freeman, J. Boker, and P. H. Bucksbaum, "High intensity above-threshold ionization of He," Phys. Rev. Lett. 71, 509-512 (1993).
[CrossRef] [PubMed]

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.

Bossi, S.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, "Optical smoothing techniques for shock wave generation in laser-produced plasmas," Phys. Rev. E 50, R3314-R3317 (1994).
[CrossRef]

Boudenne, J. M.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, "Optical smoothing techniques for shock wave generation in laser-produced plasmas," Phys. Rev. E 50, R3314-R3317 (1994).
[CrossRef]

Bourdet, G.

A. Fulop, G. Bourdet, J. C. Chanteloup, C. Dambrine, S. Ferre, S. L. Moal, A. Pichot, G. L. Touze, H. Yu, and Z. Zhao, "Diode pumped Yb:YAG V-shape unstable super-Gaussian laser resonators for 10 Hzto100 Joule clase laser," Proc. SPIE 5708, 20-31 (2005).
[CrossRef]

Bruhwiler, D.

C. G. R. Geddes, C. Toth, J. van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. P. Leemans, "High-quality electron beams from a laser wakefield accelerator using plasma channel guiding," Nature 431, 538-541 (2004).
[CrossRef] [PubMed]

Buchter, S. C.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, "Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal," Opt. Commun. 232, 77-82 (2004).
[CrossRef]

Bucksbaum, P. H.

U. Mohideen, M. H. Sher, H. W. K. Tom, G. D. Aumiller, O. R. Wood, R. R. Freeman, J. Boker, and P. H. Bucksbaum, "High intensity above-threshold ionization of He," Phys. Rev. Lett. 71, 509-512 (1993).
[CrossRef] [PubMed]

Burgy, F.

J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J.-P. Rousseau, F. Burgy, and V. Malka, "A laser-plasma accelerator producing monoenergetic electron beams," Nature 431, 541-544 (2004).
[CrossRef] [PubMed]

Cai, Y.

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).
[CrossRef]

Cary, J.

C. G. R. Geddes, C. Toth, J. van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. P. Leemans, "High-quality electron beams from a laser wakefield accelerator using plasma channel guiding," Nature 431, 538-541 (2004).
[CrossRef] [PubMed]

Casey, K. L.

A. Pertovaara, T. J. Morrow, and K. L. Casey, "Cutaneous pain and detection thresholds to short CO2 laser pulses in humans: evidence on afferent mechanisms and the influence of varying stimulus conditions," Pain 34, 261-269 (1988).
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Castillo, R.

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Figures (3)

Fig. 1
Fig. 1

Four sets of expansion coefficients, a ̂ s , 0 , b ̂ s , 0 , c ̂ s , 0 , and d ̂ s , 0 , for the case of w 0 = λ 0 ( ϵ = 1 π 0.3183 ) .

Fig. 2
Fig. 2

Transform function f 2 ( θ ) = exp [ ( cos 2 θ 1 ) ϵ 2 ] from I 2 for several values of ϵ = λ 0 π w 0 . Note that as the focusing loosens, the laser beam becomes directed only along z ̂ , corresponding here to θ = 0 . That is, the diffraction lessens as expected.

Fig. 3
Fig. 3

Value of E x E 0 at the focus as a function of the spot size, w 0 . This follows as ( 2 a 0 , 0 0 + b 0 , 0 0 ) ϵ 2 from Eq. (22).

Equations (52)

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A x ( p , q ) = 1 λ 0 2 R 2 E x ( x , y , z = 0 ) e i k 0 ( p x + q y ) d x d y ,
E x a ( x , y , z ) = R 2 A x ( p , q ) e i k 0 ( p x + q y + m z ) d p d q ,
E z a = i x ¯ R 2 A x ( p , q ) e i k 0 ( p x + q y + m z ) m d p d q ,
B x a = i x y ¯ 2 R 2 A x ( p , q ) e i k 0 ( p x + q y + m z ) m d p d q ,
B y a = R 2 A x ( p , q ) m e i k 0 ( p x + q y + m z ) d p d q x ¯ 2 R 2 A x ( p , q ) e i k 0 ( p x + q y + m z ) m d p d q ,
B z a = i y ¯ R 2 A x ( p , q ) e i k 0 ( p x + q y + m z ) d p d q ,
E x ( x , y , z = 0 ) = E 0 N = 0 A N ( r 2 w 0 2 ) N e ( r 2 w 0 2 ) ,
A x ( p , q ) = ( E 0 π ϵ 2 ) e ( b 2 ϵ 2 ) N = 0 A N N ! L N ( b 2 ϵ 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 x ( x , y , z ) = E 0 ϵ 2 ( I 1 + x 2 y 2 k 0 r 3 I 2 + y 2 r 2 I 3 ) ,
E y ( x , y , z ) = E 0 ϵ 2 x y k 0 r 3 ( 2 I 2 k 0 r I 3 ) ,
E z ( x , y , z ) = E 0 ϵ 2 x r I 4 .
I 1 = N = 0 A ̂ N 0 1 e ζ 2 ( m + m 2 ) sin ( ϕ m ) J 0 ( Λ ) L N ( ζ 2 ) d m ,
I 2 = N = 0 A ̂ N 0 1 e ζ 2 sin ( ϕ m ) J 1 ( Λ ) L N ( ζ 2 ) 1 m 2 d m ,
I 3 = N = 0 A ̂ N 0 1 e ζ 2 sin ( ϕ m ) J 0 ( Λ ) L N ( ζ 2 ) ( 1 m 2 ) d m ,
I 4 = N = 0 A ̂ N 0 1 e ζ 2 κ ( m ) cos ( ϕ m ) J 1 ( Λ ) L N ( ζ 2 ) d m
lim r 0 E x = 1 4 0 1 e ζ 2 sin ϕ m L N ( ζ 2 ) ( m + 1 ) 2 d m ,
0 π 2 f n ( θ ) sin μ + 1 2 θ J μ 1 2 ( k 0 r sin θ ) e i k 0 z cos θ d θ .
f n ( θ ) = s = 0 a s , 0 μ 1 2 C s μ ( cos θ ) ,
a s , d μ 1 2 = N s μ 1 2 0 π 2 f n ( θ ) cos d θ C s μ ( cos θ ) sin μ + 1 2 θ d θ ,
N s μ 1 2 = [ 2 2 μ ( s + μ ) Γ 2 ( μ ) Γ ( s + 1 ) 2 π Γ ( s + 2 μ ) ] ,
2 s = 0 a s , 0 μ 1 2 i s ( r ρ ) μ 1 2 C s μ ( z ρ ) j μ 1 2 + s ( k 0 ρ ) ,
s C s μ ( t ) = 2 ( μ + s 1 ) t C s 1 μ ( t ) ( 2 μ + s 2 ) C s 2 μ ( t )
a 1 , d μ 1 2 = 2 μ ( N 1 μ 1 2 N 0 μ 1 2 ) a 0 , d + 1 μ 1 2 ,
a s , d μ 1 2 = [ 2 ( μ + s 1 ) s ] ( N s μ 1 2 N s 1 μ 1 2 ) a s 1 , d + 1 μ 1 2 ( 2 μ + s 2 s ) ( N s μ 1 2 N s 2 μ 1 2 ) a s 2 , d μ 1 2
a 0 , d μ 1 2 = N 0 μ 1 2 0 π 2 f n ( θ ) cos d θ sin μ + 1 2 θ d θ
I 2 = 1 k 0 r 0 1 e ζ 2 ( e i ϕ m e i ϕ m 2 i ) J 0 ( Λ ) d m ,
I 4 = 1 k 0 r 0 1 e ζ 2 ( m + 1 ) ( e i ϕ m + e i ϕ m 2 ) J 0 ( Λ ) d m ,
f 2 ( cos θ ) = e ζ 2 = e ( cos 2 θ 1 ) ϵ 2 ,
b ̂ 0 , d = 1 d ( i ϵ ) d + 1 γ ( d + 1 2 , 1 ϵ 2 ) e 1 ϵ 2 ,
γ ( a , x ) = 0 x t a 1 e t d t x Γ ( a ) ,
b ̂ 1 , d = 3 b ̂ 0 , d + 1 ,
b ̂ s , d = ( 2 s 1 s ) [ b ̂ s 1 , d + 1 ( s 1 2 s 3 ) b ̂ s 2 , d ] .
N s 0 1 f 2 ( θ ) ( cos d + 1 θ + cos d + 2 θ ) C s 1 2 ( cos θ ) d ( cos θ ) = N s 0 1 f 1 ( θ ) cos d θ C s 1 2 ( cos θ ) d ( cos θ ) ,
a ̂ s , d = b ̂ s , d + 1 + b ̂ s , d + 2 .
c ̂ s , d = b ̂ s , d b ̂ s , d + 2 ,
d ̂ s , d = b ̂ s , d + b ̂ s , d + 1 ,
E x ( x , t ) = E 0 ( 2 ϵ 2 ) s = 0 sin φ s { ( a ̂ s , 0 + y 2 r 2 c ̂ s , 0 ) C s 1 2 ( z ρ ) j s ( k 0 ρ ) ( x 2 y 2 k 0 r 3 ) b ̂ s , 0 r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] } ,
E y ( x , t ) = E 0 ( 2 ϵ 2 ) ( x y k 0 r 3 ) s = 0 sin φ s { 2 b ̂ s , 0 r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] + ( k 0 r ) c ̂ s , 0 C s 1 2 ( z ρ ) j s ( k 0 ρ ) } ,
E z ( x , t ) = E 0 ( 2 ϵ 2 ) ( x r ) s = 0 cos φ s { d ̂ s , 0 r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] } ,
r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] = ( s z k 0 r ρ ) [ ( z ρ ) C s 1 2 ( z ρ ) C s 1 1 2 ( z ρ ) ] j s ( k 0 ρ ) + ( r ρ ) C s 1 2 ( z ρ ) [ s j s 1 ( k 0 ρ ) ( s + 1 ) j s + 1 ( k 0 ρ ) 2 s + 1 ] ,
lim r 0 E x = ϵ 2 E 0 s = 0 ( 2 a ̂ s , 0 + c ̂ s , 0 ) j s ( k 0 z ) sin φ s .
lim ρ 0 E x = ( 2 a ̂ 0 , 0 + c ̂ 0 , 0 ϵ 2 ) E 0 = [ 1 4 ( 3 2 e ( 1 ϵ 2 ) ) i ( 2 ϵ 2 8 ϵ ) e ( 1 ϵ 2 ) γ ( 1 2 , 1 ϵ 2 ) ] E 0 .
f n  N ( m ) = f n ( m ) L N ( ζ 2 ) = l = 0 N k = 0 l ( 1 ) l + N k N ! ϵ 2 l ( N l ) ! ( l k ) ! k ! [ f n ( m ) m 2 N 2 k ] .
b ̂ s , d N = l = 0 N k = 0 l ( 1 ) l + N k N ! ϵ 2 l b ̂ s , d + 2 N 2 k ( N l ) ! ( l k ) ! k ! ,
E x ( x , t ) = E 0 ( 2 ϵ 2 ) N = 0 A ̂ N s = 0 sin φ s { ( a ̂ s , 0 N + y 2 r 2 c ̂ s , 0 N ) C s 1 2 ( z ρ ) j s ( k 0 ρ ) ( x 2 y 2 k 0 r 3 ) b ̂ s , 0 N r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] } ,
E y ( x , y ) = E 0 ( 2 ϵ 2 ) ( x y k 0 r 3 ) N = 0 A ̂ N s = 0 sin φ s { 2 b ̂ s , 0 N r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] + ( k 0 r ) c ̂ s , 0 N C s 1 2 ( z ρ ) j s ( k 0 ρ ) } ,
E z ( x , t ) = E 0 ( 2 ϵ 2 ) ( x r ) N = 0 A ̂ N s = 0 cos φ s { d ̂ s , 0 N r ¯ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ] } .

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