Abstract

In the first two parts of this study, the electromagnetic field components were derived for infinitely long, flattened Gaussian laser beams [J. Opt. Soc. Am. B 23, 2157 and J. Opt. Soc. Am. B 23, 2166 (2006) ]. These results are now extended without approximation to allow for finite laser pulses having an arbitrary duration beginning with the standard Gaussian beam profile and then generalizing these results to a flattened Gaussian. The resulting models thus allow for all pulse durations and spot sizes from infinite, paraxial beams to single-cycle, wavelength-size spots, with a savings of more than 2 orders of magnitude in computation time. Pulses having fewer than ten cycles exhibit significant modification from the monochromatic fields as a result of the finite bandwidth. Specifically, the energy in the focus is shown to decrease from the theoretical value of 86.5% to as low as 72.2% for a single-cycle pulse.

© 2006 Optical Society of America

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2006

S. Stagira, G. Sansone, C. Vozzi, and M. Nisoli, "Classical trajectories of molecules exposed to few-optical-cycle light pulses," Phys. Rev. A 73, 043403 (2006).
[CrossRef]

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Controlling vibrational wave packets with intense, few-cycle laser pulses," Phys. Rev. A 73, 021401(R) (2006).
[CrossRef]

S. X. Hu and L. A. Collins, "Attosecond pump probe: exploring ultrafast electron motion inside an atom," Phys. Rev. Lett. 96, 073004 (2006).
[CrossRef] [PubMed]

M. Wickenhauser, X. M. Tong, and C. D. Lin, "Laser-induced substructures in above-threshold-ionization spectra from intense few-cycle laser pulses," Phys. Rev. A 73, 011401(R) (2006).
[CrossRef]

S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of tightly focused laser beams of arbitrary pulse length," Opt. Lett. 31, 2589-2591 (2006).
[CrossRef] [PubMed]

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]

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]

S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. II. Large F-number laser focusing," J. Opt. Soc. Am. B 23, 2166-2173 (2006).
[CrossRef]

2005

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

J. Faure, Y. Glinec, J. J. Santos, F. Ewald, J.-P. Rousseau, S. Kiselev, A. Pukhov, T. Hosokai, and V. Malka, "Observation of laser-pulse shortening in nonlinear plasma waves," Phys. Rev. Lett. 95, 205003 (2005).
[CrossRef] [PubMed]

P. Johnsson, R. Lopez-Martens, S. Kazamias, J. Mauritsson, C. Valentin, T. Remetter, K. Varju, M. B. Gaarde, Y. Mairesse, H. Wabnitz, P. Salieres, P. Balcou, K. J. Schafer, and A. L'Huillier, "Attosecond electron wave packet dynamics in strong laser fields," Phys. Rev. Lett. 95, 013001 (2005).
[CrossRef] [PubMed]

P. J. Ho and J. H. Eberly, "Classical effects of laser pulse duration on strong-field double ionization," Phys. Rev. Lett. 95, 193002 (2005).
[CrossRef] [PubMed]

A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, K. J. Gaffney, C. Blome, O. Synnergren, J. Sheppard, C. Caleman, A. G. MacPhee, D. Weinstein, D. P. Lowney, T. K. Allison, T. Matthews, R. W. Falcone, A. L. Cavalieri, D. M. Fritz, S. H. Lee, P. H. Bucksbaum, D. A. Reis, J. Rudati, P. H. Fuoss, C. C. Kao, D. P. Siddons, R. Pahl, J. Als-Nielsen, S. Duesterer, R. Ischebeck, H. Schlarb, H. Schulte-Schrepping, T. Tschentscher, J. Schneider, D. von der Linde, O. Hignette, F. Sette, H. N. Chapman, R. W. Lee, T. N. Hansen, S. Techert, J. S. Wark, M. Bergh, G. Huldt, D. van der Spoel, N. Timneanu, J. Hajdu, R. A. Akre, E. Bong, P. Krejcik, J. Arthur, S. Brennan, K. Luening, and J. B. Hastings, "Atomic-scale visualization of inertial dynamics," Science 308, 392-395 (2005).
[CrossRef] [PubMed]

A. A. Balakin, G. M. Fraiman, N. J. Fisch, and S. Suckewer, "Backward Raman amplification in a partially ionized gas," Phys. Rev. E 72, 036401 (2005).
[CrossRef]

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, "Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient," Opt. Lett. 30, 2657-2659 (2005).
[CrossRef] [PubMed]

A. J. Waddie, M. J. Thomson, and M. R. Taghizadeh, "Comparison of one- and two-dimensional dielectric reflector geometries for high-energy laser pulse compression," Opt. Lett. 30, 991-993 (2005).
[CrossRef] [PubMed]

2004

N. L. Wagner, E. A. Gibson, T. Popmintchev, and I. P. Christov, "Self-compression of ultrashort pulses through ionization-induced spatiotemproal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

N. M. Naumova, J. A. Nees, I. V. Sokolov, B. Houl, and G. A. Mourou, "Relativistic generation of isolated attosecond pulses in a lambda-cubed focal volume," Phys. Rev. Lett. 92, 063902 (2004).
[CrossRef] [PubMed]

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[CrossRef]

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, "High-order corrected fields of ultrashort, tightly-focused laser pulses," Appl. Phys. Lett. 85, 3705-3707 (2004).
[CrossRef]

2003

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

M. Spanner, M. Y. Ivanov, V. Kalosha, J. Hermann, D. A. Wiersma, and M. Pshenichnikov, "Tunable optimal compression of ultrabroadband pulses by cross-phase modulation," Opt. Lett. 28, 749-751 (2003).
[CrossRef] [PubMed]

V. P. Kalosha and J. Herrmann, "Ultrawide spectral broadening and compression of single extremely short pulses in the visible, uv-vuv, and middle infrared by high-order stimulated Raman scattering," Phys. Rev. A 68, 023812 (2003).
[CrossRef]

J. Seres, A. Müller, E. Seres, K. O'Keeffe, M. Lenner, R. Herzog, D. Kaplan, C. Spielmann, and F. Krausz, "Sub-10-fs, terawatt-scale Ti:sapphire laser system," Opt. Lett. 28, 1832-1834 (2003).
[CrossRef] [PubMed]

2002

N. Zhavoronkov and G. Korn, "Generation of single intense short optical pulses by ultrafast molecular phase modulation," Phys. Rev. Lett. 88, 203901 (2002).
[CrossRef] [PubMed]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2002).
[CrossRef]

P. X. Wang and J. X. Wang, "Classical field description for ultrashort tightly-focused laser pulses," Appl. Phys. Lett. 81, 4473-4475 (2002).
[CrossRef]

A. Ludu and R. F. O'Connell, "Laplace transform of spherical Bessel functions," Phys. Scr. 65, 369-372 (2002).
[CrossRef]

2001

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]

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. DeSilvestri, "Absolute phase phenomona in photoionization with few-cycle laser pulses," Nature 414, 182-184 (2001).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, "X-ray pulses approaching the attosecond frontier," Science 291, 1923-1927 (2001).
[CrossRef] [PubMed]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, "Attosecond metrology," Nature 414, 509-513 (2001).
[CrossRef] [PubMed]

H. Ihee, V. A. Lobastov, U. M. Gomez, B. M. Goodson, R. Srinivasan, C.-Y. Ruan, and A. H. Zewail, "Direct imaging of transient molecular structures with ultrafast diffraction," Science 291, 458-462 (2001).
[CrossRef] [PubMed]

1999

1998

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]

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]

1997

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]

1979

1975

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

Agostini, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Agrawal, G. P.

Akre, R. A.

A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, K. J. Gaffney, C. Blome, O. Synnergren, J. Sheppard, C. Caleman, A. G. MacPhee, D. Weinstein, D. P. Lowney, T. K. Allison, T. Matthews, R. W. Falcone, A. L. Cavalieri, D. M. Fritz, S. H. Lee, P. H. Bucksbaum, D. A. Reis, J. Rudati, P. H. Fuoss, C. C. Kao, D. P. Siddons, R. Pahl, J. Als-Nielsen, S. Duesterer, R. Ischebeck, H. Schlarb, H. Schulte-Schrepping, T. Tschentscher, J. Schneider, D. von der Linde, O. Hignette, F. Sette, H. N. Chapman, R. W. Lee, T. N. Hansen, S. Techert, J. S. Wark, M. Bergh, G. Huldt, D. van der Spoel, N. Timneanu, J. Hajdu, R. A. Akre, E. Bong, P. Krejcik, J. Arthur, S. Brennan, K. Luening, and J. B. Hastings, "Atomic-scale visualization of inertial dynamics," Science 308, 392-395 (2005).
[CrossRef] [PubMed]

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]

Allison, T. K.

A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, K. J. Gaffney, C. Blome, O. Synnergren, J. Sheppard, C. Caleman, A. G. MacPhee, D. Weinstein, D. P. Lowney, T. K. Allison, T. Matthews, R. W. Falcone, A. L. Cavalieri, D. M. Fritz, S. H. Lee, P. H. Bucksbaum, D. A. Reis, J. Rudati, P. H. Fuoss, C. C. Kao, D. P. Siddons, R. Pahl, J. Als-Nielsen, S. Duesterer, R. Ischebeck, H. Schlarb, H. Schulte-Schrepping, T. Tschentscher, J. Schneider, D. von der Linde, O. Hignette, F. Sette, H. N. Chapman, R. W. Lee, T. N. Hansen, S. Techert, J. S. Wark, M. Bergh, G. Huldt, D. van der Spoel, N. Timneanu, J. Hajdu, R. A. Akre, E. Bong, P. Krejcik, J. Arthur, S. Brennan, K. Luening, and J. B. Hastings, "Atomic-scale visualization of inertial dynamics," Science 308, 392-395 (2005).
[CrossRef] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

Fraction of the total energy in the focus at the peak of the pulse as a function of the pulse duration for w 0 = λ 0 2 (blue circles), w 0 = ( 2 π ) λ 0 (red triangles), and w 0 = 10 λ 0 (black diamonds). The limit of a perfect Gaussian [ 1 exp ( 2 ) 0.86466 ] is included as the blue line for reference.

Tables (1)

Tables Icon

Table 1 Electric Field Components for a Loosely Focused Gaussian Beam a

Equations (57)

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E x = E 0 ϵ 2 [ I 1 + x 2 y 2 r 3 I 2 + y 2 r 2 I 3 ] e i ϕ 0 ,
E y = E 0 ϵ 2 x y r 2 [ 2 r I 2 I 3 ] e i ϕ 0 ,
E z = E 0 ϵ 2 x r I 4 e i ( ϕ 0 π 2 ) ,
f ̃ ( ω ω 0 ) = 1 2 π f ( t ) e i ( ω ω 0 ) t d t ,
I 1 = f ̃ ( ω ω 0 ) e i ω t 0 1 e μ ( m + m 2 ) J 0 ( Λ ) d m d ω ,
I 2 = f ̃ ( ω ω 0 ) e i ω t k 0 1 e μ J 1 ( Λ ) 1 m 2 d m d ω ,
I 3 = f ̃ ( ω ω 0 ) e i ω t 0 1 e μ J 0 ( Λ ) ( 1 m 2 ) d m d ω ,
I 4 = f ̃ ( ω ω 0 ) e i ω t 0 1 e μ κ ( m ) J 1 ( Λ ) d m d ω ,
I 1 = 2 e i ω 0 t s = 0 a ̂ s , 0 i s C s 1 2 ( z ρ ) j s ( ω ρ ¯ ) ,
I 2 = 2 c 2 ω 0 2 e i ω 0 t s = 0 b ̃ s , 0 i s r [ C s 1 2 ( z ρ ) j s ( ω ρ ¯ ) ] ,
I 3 = 2 e i ω 0 t s = 0 c ̂ s , 0 i s C s 1 2 ( z ρ ) j s ( ω ρ ¯ ) ,
I 4 = i 2 c ω 0 e i ω 0 t s = 0 d ̂ s , 0 i s r [ C s 1 2 ( z ρ ) j s ( ω ρ ¯ ) ] ,
b ̂ 0 , d = 1 4 ( i ϵ ) d + 1 γ ( d + 1 2 , 1 ϵ 2 ) e ( 1 ϵ 2 ) ,
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 ] ,
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 .
I s , n = 2 i s f ̃ ( ω ω 0 ) e i ( ω ω 0 ) t ω n j s ( ω ρ ¯ ) d ω
I 1 = e i ω 0 t s = 0 a ̂ s , 0 C s 1 2 ( z ρ ) I s , 0 ,
I 2 = c r ρ e i ω 0 t s = 0 b ̂ s , 0 { ( c z ρ 2 ) C s 1 3 2 ( z ρ ) I s , 2 i C s 1 2 ( z ρ ) [ s I s 1 , 1 + ( s + 1 ) I s + 1 , 1 2 s + 1 ] } ,
I 3 = e i ω 0 t s = 0 c ̂ s , 0 C s 1 2 ( z ρ ) I s , 0 ,
I 4 = r ρ e i ω 0 t s = 0 d ̂ s , 0 { ( i c z ρ 2 ) C s 1 3 2 ( z ρ ) I s , 1 + C s 1 2 ( z ρ ) [ s I s 1 , 0 + ( s + 1 ) I s + 1 , 0 2 s + 1 ] } ,
I s , n = i s π f ( t T ) H n ( T ) e i ω 0 T d T ,
H n ( T ) = j s ( ω ρ ¯ ) ω n e i ω T d ω ,
H 0 ( T ) = π i s rect ( T ρ ¯ ) k = 0 2 k s C k ( s ) ( 1 ) k ρ ¯ 2 k s 1 T s 2 k ,
( l + 1 ) C k ( l + 1 ) = l C k 1 ( l 1 ) ( 2 l + 1 ) C k ( l )
d H 1 d T = i H 0 ( T ) , d 2 H 2 d T 2 = H 0 ( T ) .
H 1 ( T ) = π i 1 s k = 0 2 k s [ C k ( s ) ( 1 ) k s 2 k + 1 ] { ( T ρ ¯ ) s 2 k + 1 rect ( T ρ ¯ ) + ( 1 ) s 2 k [ H ( T + ρ ¯ ) + ( 1 ) s 2 k H ( T ρ ¯ ) ] } ,
H 2 ( T ) = π i s k = 0 2 k s C k ( s ) ( 1 ) k × { ρ ¯ ( s 2 k + 1 ) ( s 2 k + 2 ) rect ( T ρ ¯ ) ( T ρ ¯ ) s 2 k + 2 + ρ ¯ ( 1 ) s 2 k s 2 k + 2 [ H ( T + ρ ¯ ) ( 1 ) s 2 k H ( T ρ ¯ ) ] + ( 1 ) s 2 k s 2 k + 1 [ H ( T + ρ ¯ ) + ( 1 ) s 2 k H ( T ρ ¯ ) ] T } ,
I s , 0 = k = 0 2 k s C k ( s ) ( 1 ) s + k ρ ¯ 2 k s 1 F s 2 k ( ρ ¯ , ρ ¯ ) ,
I s , 1 = i k = 0 2 k s C k ( s ) ( 1 ) s + k s 2 k + 1 ( F s 2 k + 1 ( ρ ¯ , ρ ¯ ) ρ ¯ s 2 k + 1 + ( 1 ) s 2 k { F 0 ( ρ ¯ , ρ ) + [ 1 + ( 1 ) s 2 k ] F 0 ( ρ ¯ , ) } ) ,
I s , 2 = k = 0 2 k s C k ( s ) ( 1 ) s + k ( F s 2 k + 2 ( ρ ¯ , ρ ¯ ) ρ ¯ 2 k s 1 ( s 2 k + 1 ) ( s 2 k + 2 ) + ρ ¯ ( 1 ) s 2 k s 2 k + 2 { F 0 ( ρ ¯ , ρ ¯ ) + [ 1 ( 1 ) s 2 k ] F 0 ( ρ ¯ , ) } + ( 1 ) s 2 k s 2 k + 1 { F 1 ( ρ ¯ , ρ ¯ ) + [ 1 + ( 1 ) s 2 k ] F 1 ( ρ ¯ , ) } ) ,
F n ( α , β ) = 1 4 { ( 2 C n ( ω 0 ) + cos ( Ω 0 t ) [ C n ( Ω + ) + C n ( Ω ) ] + sin ( Ω 0 t ) [ S n ( Ω + ) + S n ( Ω ) ] ) + i ( 2 S n ( ω 0 ) + cos ( Ω 0 t ) [ S n ( Ω + ) S n ( Ω ) ] sin ( Ω 0 t ) [ C n ( Ω + ) C n ( Ω ) ] ) } α β ,
S p ( ω ) x p sin ( ω x ) d x = ( 1 ) p 2 + 1 ( p ! ) [ cos ( ω x ) k = 0 p 2 ( 1 ) k x 2 k ( 2 k ) ! ω p 2 k + 1 + sin ( ω x ) k = 1 p 2 ( 1 ) k + 1 x 2 k 1 ( 2 k 1 ) ! ω p 2 k + 2 ] , p even ,
= ( x p ω ) cos ( ω x ) + ( p ω ) C p 1 ( ω ) , p odd ,
C p ( ω ) x p cos ( ω x ) d x = ( 1 ) p 2 ( p ! ) [ sin ( ω x ) k = 0 p 2 ( 1 ) k x 2 k ( 2 k ) ! ω p 2 k + 1 cos ( ω x ) k = 1 p 2 ( 1 ) k + 1 x 2 k 1 ( 2 k 1 ) ! ω p 2 k + 2 ] , p even
= ( x p ω ) sin ( ω x ) ( p ω ) S p 1 ( ω ) , p odd
lim r 0 E x = E 0 ϵ 2 s = 0 ( 2 a ̂ s , 0 + c ̂ s , 0 ) j s ( k 0 z ) i s e i ( ω 0 t ϕ 0 ) ,
lim ρ 0 E x = E 0 ϵ 2 ( 2 a ̂ 0 , 0 + c ̂ 0 , 0 ) e i ( ω 0 t ϕ 0 ) ,
lim r 0 E x = E 0 2 ϵ 2 e i ( ω 0 t ϕ 0 ) s = 0 ( 2 a ̂ s , 0 + c ̂ s , 0 ) I s , 0 r = 0 ,
lim ρ 0 I s , 0 = k = 0 2 k s C k ( s ) ( 1 ) s + k [ T 1 s 2 k + 1 T 0 s 2 k + 1 s 2 k + 1 ] f ( t ) ,
lim ρ 0 E x = E 0 ϵ 2 e i ( ω 0 t ϕ 0 ) cos 2 ( π t 2 Δ τ ) ( 2 a ̂ 0 , 0 + c ̂ 0 , 0 ) × k = 0 2 k s C k ( s ) ( 1 ) s + k [ T 1 s 2 k + 1 T 0 s 2 k + 1 s 2 k + 1 ] .
lim ρ 0 I s , 0 = 2 cos 2 ( π t 2 Δ τ )
E 0 N = 0 A N ( r w 0 ) 2 N exp ( r 2 w 0 2 ) ,
b ̂ s , n N = ν = 0 N k = 0 ν ( 1 ) k ϵ 2 ν [ N ! b ̂ s , 2 ν 2 k + n ( N ν ) ! ν ! ( ν k ) ! k ! ] ,
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 ) ,
L n , m k e i ω 0 ( t z c ) e r 2 w 0 2 ω 0 m L n k ( r 2 w 0 2 ) .
L n , m k ν = 0 n ( 1 ) ν ν ! ( n + k ν + k ) ( r ϵ 2 c ) 2 ν κ 2 ν + m ,
κ n = f ̃ ( ω ω 0 ) e i ω ( t z c ) e r 2 ϵ 2 ω 2 4 c 2 ω n d ω = ( ω 0 n 2 π ) exp ( ω 0 2 Δ τ 2 4 [ δ 1 ] δ η 2 ω 0 2 Δ τ 2 ) e i δ η × k = 0 n m = 0 n k γ n , k , m δ n k 2 + 1 2 η m ( ω 0 Δ τ ) 2 m + k ,
γ n , k , m = n ! 2 k + m Γ ( k + 1 2 ) [ 1 + ( 1 ) k ] i m ( n k m ) ! k ! m !
E x E 0 e i ( η ϕ 0 ) exp [ ( t z c ) 2 Δ τ 2 ] ,
L n , m k , N l = 0 N Δ l n N ν = 0 n + l ( 1 ) ν ν ! ( n + l + k ν + k ) ( r ϵ 2 c ) 2 ν κ 2 ν + m ,
Δ l n N ( 1 ) l [ N ! ( l + n ) ! ( N l ) ! n ! l ! l ! ] ,
lim ρ 0 E x 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 ) .

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