Abstract

An analytical formula is derived for calculating the flux of radiation from a Gaussian source irradiating a spherical object. The formula was derived for the radiant intensity function represented by a paraxial approximate solution of the Halmholtz scalar wave equation. All calculations are presented in the Cartesian 0xyz coordinate system, where the coordinates, x, y and z, determine the center of the spherical object. The center of the source was located at the point P(0,0,0) in the plane z = 0. Some computer simulation results were illustrated graphically and analyzed.

© 2005 Optical Society of America

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    [CrossRef]
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Acta Metal. Mater. (1)

C. Hu and T.N. Baker, �??Prediction of laser transformation hardening depth using a line source model.�?? Acta Metal. Mater. 43, 3563-3569 (1995).
[CrossRef]

Appl. Opt. (3)

Handbook of Optical Engineering (1)

M. Strojnik and G Paez, �??Radiometry�?? in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) 649-699.

J. Food Eng. (1)

G. Brescia, R. Moreira, L. Braby and E. Castell-Perez, �??Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,�?? J. Food Eng. 60, 31-39 (2003).
[CrossRef]

J. Quantum Spectrosc. Radiat. Transfer (1)

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quantum Spectrosc. Radiat. Transfer 55, 673-681 (1996).
[CrossRef]

J. Res. Natl. Bur. Stand. (1)

J.H. Habbell, R.L. Bach and R.J. Herbold, �??Radiation field from a circular disk source,�?? J. Res. Natl. Bur. Stand. 65C, 249-264 (1961).

Laser Focus (1)

W.H.A. Wilde, W.H. Parr and D.W. McPeak, �??Seeds bask in laser light,�?? Laser Focus 5, 41-42 (1969).

Nucl. Instrum. Methods (1)

R.P. Gardner and K. Verghese, �??On the solid angle subtended by a circular disc,�?? Nucl. Instrum. Methods, 93, 163-167 (1971).
[CrossRef]

Opt. Commun. (4)

Y.P. Han, L. Méès, K.F. Ren, G. Gouesbet, S.Z. Wu and G. Gréhan, �??Scattering of light by spheroids: the far field case,�?? Opt. Commun. 210, 1-9 (2002).
[CrossRef]

K. Shinozaki, C. Xu, H. Sasaki and T. Kamijoh, �??A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,�?? Opt. Commun. 133, 300-304 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan and Y Qiu, �??Propagation of paraxial circular symmetric beams in a general optical system,�?? Opt. Commun. 140, 226-230 (1997).
[CrossRef]

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

Opt. Express (1)

Opt. Laser Technol. (1)

R.C. Gauthier, �??Optical trapping: a tool to assist optical machining,�?? Opt. Laser Technol. 29, 389-399 (1997).
[CrossRef]

Opt. Lasers. Eng. (2)

L. Zeni, S. Campopiano, A. Cutolo and G. D�??Angelo, �??Power semiconductor laser diode characterization,�?? Opt. Lasers Eng. 39, 203-217 (2003).
[CrossRef]

M.R. Taghizadeh, P. Blair, K. Ballüder, N.J. Waddie, P. Rudman and N. Ross, �??Design and fabrication of diffractive elements for laser material processing applications,�?? Opt. Lasers Eng. 34, 289-307 (2000).
[CrossRef]

Phys. Lett. (1)

S. Liu, H. Guo, M. Liu and G. Wu, �??A comparison of propagation characteristics of focused Gaussian beam and fundamental Gaussian beam in vacuum,�?? Phys. Lett. A, 327, 254-262 (2004).
[CrossRef]

Phys. Plantarum (1)

S.R. Govil, D.C. Agrawal, K.P. Rai and S.N. Thakur, �??Growth responses of Vigna radiata seeds to laser irradiation in the UV-A region,�?? Phys. Plantarum 63, 133-134 (1985).
[CrossRef]

Phys. Rep. (1)

G. Grynberg and C Robilliard, �??Cold atom in dissipative optical lattices,�?? Phys. Rep. 355, 335-451 (2001).
[CrossRef]

Prog. Energy Combust. Sci. (1)

A.R. Jones, �??Light scattering for particle characterization,�?? Prog. Energy Combust. Sci., 25, 1-53 (1999).
[CrossRef]

Prog. Quantum Electron. (1)

C.S. Adams and E. Riis, �??Laser cooling and trapping of neutral atoms,�?? Prog. Quantum Electron. 21, 1-79 (1997).
[CrossRef]

Radiat. Phys. Chem. (2)

J.F. Diehl, �??Food irradiation-past, present and future,�?? Radiat. Phys. Chem. 63, 211-215 (2002).
[CrossRef]

G. W. Gould, �??Potential of irradiation as a component of mild combination preservation procedures,�??Radiat. Phys. Chem. 48, 366 (1996).
[CrossRef]

Other (3)

F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) p. 37.

E.J. Rethwell and M.J. Cloud, Electromagnetics, (CRC, Boca Raton, 2001) Chap. IV, 6-51.

S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), 44-186.

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Figures (6)

Fig. 1.
Fig. 1.

A spherical object with the center 01 at the distance r from the origin 0 of the main coordinate system 0xyz erected at the point P or at the center of the emitting Gaussian aperture G. Additional coordinate systems 01x1y1z1 and 01 x 11 y 11 z 11 are erected at the center 01 of the sphere. The parameter a 2, given by Eq. (4e), is always greater than the radius rob when the geometry is illustrated on the 0̀x 11z1 plane. Similar dependencies between geometrical variables used in Eq. (3) are shown in Ref. [22].

Fig. 2.
Fig. 2.

Graphical illustrations of perpendicular projection of the radiative flux with the effective radius ρef at the distance z from the origin 0 of the main coordinate system 0xyz and the surface subtended by the spherical object when 0 ≤ rxyρef - a 1 (a) and 0 < ρef - a1rxy (b). The first function in Eqs. (3) and (5) was calculated for the surface S shown in the part (a). The second function in Eq. (3) was calculated for the surfaces S 1 + S 2 + S 3 shown in the part (b). The radiative flux related to the complete black surface S 3 illustrated in the part (b) was included erroneously into the second function of Eq. (3) in Ref. [22]. The first term of the second function in Eq. (5) was obtained at 0 < ρef - a1rxy for the surface S 1, while the second term of this function for the surface S 2.

Fig. 3.
Fig. 3.

The transverse distribution of the radiant intensity I(x, y, z; I 0, ρ 0, λ) and spot sizes ρ0, ρ1 and ρ2 given by Eq. (12a) at distances z 0 = 0 [m], z 1 = 100 [m] and z 2 = 200 [m] from the emitting Gaussian aperture. For the dependence of I(x, y, z; I 0, ρ0, λ) on the variable I 0, ρ0 and λ see Eqs. (12a), (13) and (19). The data were obtained for ρ0 = 2.5×10-3 [m] and λ = 632.8×10-9 [m] corresponding to the wavelength of He-Ne laser emission. Solid lines represent the influence of the distance z on the radiant intensity I(x, y, z; I 0, ρ0, λ) equal to 2I 0/e, I 0/e, and I 0/(16 e), where I 0 denotes the maximal intensity at rxy = 0. The beam divergence in the far field (for z » zR ) is illustrated by the dashed straight lines and is equal to the half-angle θ = λ/(π ρ0) [24]. Here zR = 31.03 [m] and θ = 8.06×10-5 [rad]. For the distance z » zR the extended Gaussian source may be considered as a point Gaussian source. In these plots the radius rxy is expressed in millimeters and z in meters, so it is clearly seen that the extended Gaussian source will be well approximated by the point Gaussian source at z » zR . When the Gaussian beam is focused by a thin lens of focal length, f, then the focused spot size, ρ0f = ρ0/[1+(zR /f)2]1/2, and the far field approximation may be used at z » zRf = π0f )2/λ [24].

Fig. 4.
Fig. 4.

The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of x and y at rob = 0.0001 [m] (a), rob = 0.0025 [m] (b), rob = 0.0050 [m] (c), and rob = 0.0500 [m] (d). The data were calculated for I 0 = 1 [W sr-1], z = 0.2 [m], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Fig. 5.
Fig. 5.

The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated for I 0 = 1 [W sr-1], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Fig. 6.
Fig. 6.

The absolute values of the total radiative flux Φp (x, y, z; I 0, ρef , rob ), denoted by ΦP , from the isotropic point source P as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated from Eq. (5) at I 0 = 1 [W sr-1] and ρef = 0.005 [m] for the radiant intensity function I(x, y, z; x 11, y 11) = I 0 z 2/[(x 11 + rxy )2 + z 2 + y112], where I 0 = I(x=0, y=0, z).

Equations (70)

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d Φ P ( x , y , z ; x 11 , y 11 ) = I ( x , y , z ; x 11 , y 11 ) z d x 11 d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 ,
r xy = x 2 + y 2 ,
r = x 2 + y 2 + z 2 .
Φ P ( x , y , z ; ρ ef , r ob ) = { a 2 a 1 d x 11 y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 r xy ρ ef a 1 , a 2 ρ ef r xy d x 11 y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 < ρ ef a 1 r xy , 0 , if 0 < ρ ef + a 2 < r xy ,
y 11 / = b a 2 [ x 11 ( a 1 a ) ] 2 a ,
a = a 1 + a 2 = r ob z r 2 r ob 2 ( z 2 r ob 2 ) ,
a 1 = r 2 r ob ( z r 2 r ob 2 r ob r 2 z 2 ) ,
a 2 = r 2 r ob ( z r 2 r ob 2 + r ob r 2 z 2 ) ,
b = { r r ob ( r xy + a 1 a ) ( r xy r 2 r ob 2 ) , if r xy > 0 , r ob z z 2 r ob 2 , if r xy = 0 ,
Φ G ( x , y , z ; ρ ef , r ob ) = { a 2 a 1 d x 11 y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 if 0 r xy ρ ef a 1 , a 2 x 11 / d x 11 y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 + x 11 / ρ ef r xy y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 < ρ ef a 1 < r xy , 0 , if 0 < ρ ef + a 2 < r xy ,
ρ ef b 0 = r ob z z 2 r ob 2 ,
x 11 / = b 2 ( a a 1 ) a 2 r xy + a a 2 ρ ef 2 2 a b 2 ( a 1 + r xy ) + b 2 [ ( b 2 + ρ ef 2 ) + ( a 1 + r xy ) 2 ] a 2 b 2
y 11 // = ρ ef 2 ( x 11 + r xy ) 2 ,
[ x 11 ( a 1 a ) ] 2 a 2 + y 11 2 b 2 = 1 ,
( x 11 + r xy ) 2 + y 11 2 = ρ ef 2 ,
ρ ef < b 0 = r ob z z 2 r ob 2 ,
Φ P ( x , y , z ; ρ ef , r ob ) = { ρ ef r xy ρ ef r xy d x 11 y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 if 0 r xy a 2 ρ ef , a 2 x 11 / d x 11 y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 + x 11 / ρ ef r xy d x 11 y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 a 2 ρ ef < r xy a 2 + ρ ef , 0 , if 0 < ρ ef + a 2 < r xy .
E ( x , y , z ; A , ρ 0 , λ ) = A ρ 0 ρ exp ( r xy 2 ρ 2 ) exp { i [ k r xy 2 2 R + k z arctan ( z z R ) ] } ,
ρ = ρ 0 1 + ( z z R ) 2 ,
R = z [ 1 + ( z R z ) 2 ] .
z R = π ρ 0 2 λ ,
S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 Re [ S c ( x , y , z ; A , ρ 0 , λ ) ] ,
S c ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = E ( x , y , z ; A , ρ 0 , λ ) × H * ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) ,
μ r μ 0 H 2 ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = ε r ε 0 E 2 ( x , y , z ; A , ρ 0 , λ ) ,
S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 ε r ε 0 μ r μ 0 E ( x , y , z ; A , ρ 0 , λ ) E * ( x , y , z ; A , ρ 0 , λ ) .
L ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 π S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r )
= c ε 0 4 π ε r μ r E ( x , y , z ; A , ρ 0 , λ ) E * ( x , y , z ; A , ρ 0 , λ ) ,
L ( x , y , z ; A , ρ 0 , λ , ε r , n ) = A 2 c ε r ρ 0 2 4 π n ρ 2 exp ( 2 r xy 2 ρ 2 ) ,
I ( x , y , z ; A , ρ 0 , λ , ε r , n ) = I 0 ( A , ρ 0 , ε r , n ) exp ( 2 r xy 2 ρ 2 ) ,
I 0 ( A , ρ 0 , ε r , n ) = [ L ( x , y , z ; A , ρ 0 , λ , ε r , n ) dy ] d x = A 2 c ε r ε 0 ρ 0 2 ( 8 n ρ 2 ) .
I ( x , y , z ; A , ρ 0 , λ , ε r , n ) = I 0 ( A , ρ 0 , ε r , n ) exp { 2 [ ( x + x 1 ) 2 + ( y + y 1 ) 2 ] ρ 2 } ,
I ( x , y , z ; I 0 , ρ 0 , λ ) = I 0 exp { 2 [ ( x 11 + r xy ) 2 + y 11 2 ρ 2 } ,
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z a 2 a 1 e 2 ( x 11 + r xy ) 2 ρ 2 d x 11 y 11 / y 11 / e 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 ,
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z { a 2 x 11 / e 2 ( x 11 + r xy ) 2 ρ 2 d x 11 y 11 / y 11 / e 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2
+ x 11 / ρ ef r xy e 2 ( x 11 + r xy ) 2 ρ 2 d x 11 y 11 // y 11 // e 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 } ,
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = { F 1 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) , if 0 r xy ρ ef a 1 , F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) + F 3 // ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) , if 0 < ρ ef a 1 r xy < ρ ef + a 2 , 0 , if 0 < ρ ef + a 2 r xy ,
F 1 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z a 2 a 1 f / ( x , y , z ; ρ 0 , λ ) d x 11
= I 0 z { i = 1 k = 1 ( 1 ) i + k ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ρ 2 ( i k ) ( 2 i 2 ) ! ! ( 2 k 1 ) ( k 1 ) ! 2 2 i k 2
× a 2 a 1 ( y 11 / ) 2 k 1 e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11
+ i = 1 j = 1 i l ( 1 ) i 2 ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ( 2 i 2 ) ! ! [ 2 ( i j ) 1 ] ! ! ( ρ 2 ) 2 j
× a 2 a 1 ( y 11 / ) 2 ( i j ) 1 e [ 2 ( x 11 + r xy ) 2 + ( y 11 / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } ,
F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z a 2 x 11 / f / ( x , y , z ; ρ 0 , λ ) d x 11
= I 0 z { i = 1 k = 1 ( 1 ) i + k ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ρ 2 ( i k ) ( 2 i 2 ) ! ! ( 2 k 1 ) ( k 1 ) ! 2 2 i k 2
× a 2 x 11 / ( y 11 / ) 2 k 1 e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11
+ i = 1 j = 1 i l ( 1 ) i 2 ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ( 2 i 2 ) ! ! [ 2 ( i j ) 1 ] ! ! ( ρ 2 ) 2 j
× a 2 x 11 / ( y 11 / ) 2 ( i j ) 1 e 2 [ ( x 11 + r xy ) 2 + ( y 11 / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 , }
F 3 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z x 11 / ρ ef r x y f / / ( x , y , z ; ρ 0 , λ ) d x 11
= I 0 z { i = 1 k = 1 ( 1 ) i + k ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ρ 2 ( i k ) ( 2 i 2 ) ! ! ( 2 k 1 ) ( k 1 ) ! 2 2 i k 2
× x 11 / ρ e f r x y ( y 11 / / ) 2 k 1 e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11
+ i = 1 j = 1 i l ( 1 ) i 2 ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ( 2 i 2 ) ! ! [ 2 ( i j ) 1 ] ! ! ( ρ 2 ) 2 j
× x 11 / ρ e f r x y ( y 11 / / ) 2 ( i j ) 1 e 2 [ ( x 11 + r xy ) 2 + ( y 11 / / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } .
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = { F 4 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) if 0 r xy a ρ ef , F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) + F 3 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) if 0 a 2 ρ ef r xy < a 2 + ρ ef , 0 , if 0 < ρ ef + a r xy ,
F 4 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ρ e f r x y ρ ef r x y f / / ( x , y , z ; ρ 0 , λ ) d x 11
I 0 z { i = 1 k = 1 ( 1 ) i + k ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ρ 2 ( i k ) ( 2 i 2 ) ! ! ( 2 k 1 ) ( k 1 ) ! 2 2 i k 2
× ρ ef r xy ρ e f r x y ( y 11 / / ) 2 k 1 e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11
+ i = 1 j = 1 i l ( 1 ) i 2 ( 2 i 1 ) ! ! ( 2 i 3 ) ! ! ( 2 i 2 ) ! ! [ 2 ( i j ) 1 ] ! ! ( ρ 2 ) 2 j
× ρ ef r xy ρ e f r x y ( y 11 / / ) 2 ( i j ) 1 e 2 [ ( x 11 + r xy ) 2 + ( y 11 / / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } .
1 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 = 1 [ ( x 11 + r xy ) 2 + z 2 ] 3 2 { 1 + y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] } 3 2 ,
y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] < 1 ,
1 { 1 + y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] } 3 2 = i = 1 ( 1 ) i 1 ( 2 i 1 ) ! ! ( 2 i 2 ) ! ! ( y 11 ) 2 ( i 1 ) [ ( x 11 + r xy ) 2 + z 2 ] i 1 .
y 11 2 ( i 1 ) e 2 ( y 11 ) 2 ρ 2 d y 11 = ( 2 i 3 ) ! ! 2 2 i 2 π ρ ( 2 i 1 ) Erf ( 2 y 11 ρ )
j = 1 i 1 ( 2 i 3 ) ! ! y 11 ( 2 i 2 j 1 ) ( 2 i 2 j 1 ) ! ! ( ρ 2 ) 2 j e 2 ( y 11 ρ ) 2 + C ,
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z a 2 a 1 f / ( x , y , z ; ρ 0 , λ ) d x 11 ,
Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z [ a 2 x 11 / f / ( x , y , z ; ρ 0 , λ ) d x 11
+ x 11 / ρ ef r xy f / / ( x , y , z ; ρ 0 , λ ) d x 11 ] ,
f / ( x , y , z ; ρ 0 , λ ) = i = 1 ( 1 ) i 1 ( 2 i 1 ) ! ! ( 2 i 3 ) !! ( 2 i 2 ) ! ! 2 ( ρ 2 ) 2 i 1 { 2 y 11 / ρ 2 y 11 / ρ e ( y 11 ) 2 d y 11
j = 1 i 1 ( ρ 2 ) 2 j 1 2 ( y 11 / ) 2 i 2 j 1 e 2 ( y 11 / ρ ) 2 [ 2 ( i j ) 1 ] ! ! } e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 ,
f / / ( x , y , z ; ρ 0 , λ ) = i = 1 ( 1 ) i 1 ( 2 i 1 ) ! ! ( 2 i 3 ) !! ( 2 i 2 ) ! ! 2 ( ρ 2 ) 2 i 1 { 2 y 11 // ρ 2 y 11 // ρ e ( y 11 ) 2 d y 11
j = 1 i 1 ( ρ 2 ) 2 j 1 2 ( y 11 // ) 2 i 2 j 1 e 2 ( y 11 // ρ ) 2 [ 2 ( i j ) 1 ] ! ! } e 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 .
e ( y 11 ) 2 d y 11 = k = 1 ( 1 ) k + 1 1 2 k 1 ( y 11 ) 2 k 1 ( k 1 ) ! ,

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