Abstract

Deterministic mode representation (DMR) is introduced for a three-dimensional random medium with a statistically stationary refractive index distribution. The DMR allows for the designing and fine tuning of novel random media by adjusting the weights of individual deterministic modes. To illustrate its usefulness, we have applied the decomposition to the problem of weak light scattering from a Gaussian Schell-model medium. In particular, we have shown how individual deterministic modes of the medium contribute to the scattered far-field spectral density distribution.

© 2017 Optical Society of America

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References

  • View by:

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    [Crossref]
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    [Crossref]
  31. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [Crossref]
  32. C. Schwartz and A. Dogariu, “Mode coupling approach to beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 329–338 (2006).
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    [Crossref]
  34. G. Gbur, “Nonradiating sources and the inverse source problem,” Ph.D. dissertation (University of Rochester, 2002).
  35. M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercontinuum,” Opt. Lett. 37, 169–171 (2012).
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2016 (1)

2015 (5)

2013 (1)

2012 (2)

2010 (1)

2009 (1)

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (2)

C. Schwartz and A. Dogariu, “Mode coupling approach to beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 329–338 (2006).

K. Kim and E. Wolf, “A scalar-mode representation of stochastic planar, electromagnetic sources,” Opt. Commun. 261, 19–22 (2006).
[Crossref]

2005 (1)

T. Setälä, J. Lindberg, and K. Blomstedt, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume,” Phys. Rev. E 71, 036618 (2005).
[Crossref]

2004 (2)

2003 (2)

2002 (1)

C. C. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

2001 (1)

1995 (2)

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[Crossref]

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[Crossref]

1993 (1)

1989 (1)

F. Gori, C. Palma, and C. Padovani, “Modal expansion of blackbody cross spectral density,” Atti Fondaz. G. Ronchi 44, 383 (1989).

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

1986 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

1982 (2)

1981 (1)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[Crossref]

1980 (1)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

1979 (1)

R. Martinez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).
[Crossref]

1909 (1)

J. Mercer, “Functions of positive and negative type and their connection with the theory of integral equations,” Philos. Trans. R. Soc. A 209, 415–446 (1909).
[Crossref]

Blomstedt, K.

T. Voipio, K. Blomstedt, T. Setala, and A. T. Friberg, “Conservation of electromagnetic coherent mode structure on propagation,” Opt. Commun. 340, 93–101 (2015).
[Crossref]

T. Setälä, J. Lindberg, and K. Blomstedt, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume,” Phys. Rev. E 71, 036618 (2005).
[Crossref]

Borghi, R.

Dogariu, A.

Erkintalo, M.

Friberg, A. T.

Gbur, G.

G. Gbur, “Nonradiating sources and the inverse source problem,” Ph.D. dissertation (University of Rochester, 2002).

Genty, G.

Gilchrest, Y. V.

C. C. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Gori, F.

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref]

R. Borghi, F. Gori, O. Korotkova, and M. Santarsiero, “Propagation of cross-spectral densities from spherical sources,” Opt. Lett. 37, 3183–3185 (2012).
[Crossref]

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33, 1857–1859 (2008).
[Crossref]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[Crossref]

F. Gori, C. Palma, and C. Padovani, “Modal expansion of blackbody cross spectral density,” Atti Fondaz. G. Ronchi 44, 383 (1989).

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Guattari, G.

Huttunen, J.

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[Crossref]

Kim, K.

K. Kim and E. Wolf, “A scalar-mode representation of stochastic planar, electromagnetic sources,” Opt. Commun. 261, 19–22 (2006).
[Crossref]

Korotkova, O.

Lajunen, H.

Li, J.

Lindberg, J.

T. Setälä, J. Lindberg, and K. Blomstedt, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume,” Phys. Rev. E 71, 036618 (2005).
[Crossref]

Macon, B. R.

C. C. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Majeed, H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez-Herrero, R.

R. Martinez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).
[Crossref]

Mercer, J.

J. Mercer, “Functions of positive and negative type and their connection with the theory of integral equations,” Philos. Trans. R. Soc. A 209, 415–446 (1909).
[Crossref]

Mukunda, N.

Nguyen, T. H.

Ostrovsky, A. S.

A. S. Ostrovsky, Coherent-Mode Representation in Optics (SPIE, 2006).

Padovani, C.

F. Gori, C. Palma, and C. Padovani, “Modal expansion of blackbody cross spectral density,” Atti Fondaz. G. Ronchi 44, 383 (1989).

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Palma, C.

F. Gori, C. Palma, and C. Padovani, “Modal expansion of blackbody cross spectral density,” Atti Fondaz. G. Ronchi 44, 383 (1989).

Piquero, G.

Ponomarenko, S. A.

Popescu, G.

Santarsiero, M.

Schwartz, C.

Setala, T.

T. Voipio, K. Blomstedt, T. Setala, and A. T. Friberg, “Conservation of electromagnetic coherent mode structure on propagation,” Opt. Commun. 340, 93–101 (2015).
[Crossref]

Setälä, T.

Shirai, T.

Simon, R.

Starikov, A.

Sundar, K.

Surakka, M.

Tervo, J.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Voipio, T.

T. Voipio, K. Blomstedt, T. Setala, and A. T. Friberg, “Conservation of electromagnetic coherent mode structure on propagation,” Opt. Commun. 340, 93–101 (2015).
[Crossref]

T. Voipio, T. Setälä, and A. T. Friberg, “Coherent-mode representation of partially polarized pulsed electromagnetic beams,” J. Opt. Soc. Am. A 30, 2433–2443 (2013).
[Crossref]

Wolf, E.

Wyrowski, F.

Young, C. C.

C. C. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Atti Fondaz. G. Ronchi (1)

F. Gori, C. Palma, and C. Padovani, “Modal expansion of blackbody cross spectral density,” Atti Fondaz. G. Ronchi 44, 383 (1989).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (11)

E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
[Crossref]

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[Crossref]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
[Crossref]

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[Crossref]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Shifted-elementary-mode representation for partially coherent vectorial fields,” J. Opt. Soc. Am. A 27, 2004–2014 (2010).
[Crossref]

T. Voipio, T. Setälä, and A. T. Friberg, “Coherent-mode representation of partially polarized pulsed electromagnetic beams,” J. Opt. Soc. Am. A 30, 2433–2443 (2013).
[Crossref]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[Crossref]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

C. Schwartz and A. Dogariu, “Mode coupling approach to beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 329–338 (2006).

Nuovo Cimento B (1)

R. Martinez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).
[Crossref]

Opt. Commun. (7)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

T. Voipio, K. Blomstedt, T. Setala, and A. T. Friberg, “Conservation of electromagnetic coherent mode structure on propagation,” Opt. Commun. 340, 93–101 (2015).
[Crossref]

K. Kim and E. Wolf, “A scalar-mode representation of stochastic planar, electromagnetic sources,” Opt. Commun. 261, 19–22 (2006).
[Crossref]

Opt. Eng. (1)

C. C. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Opt. Express (1)

Opt. Lett. (8)

Philos. Trans. R. Soc. A (1)

J. Mercer, “Functions of positive and negative type and their connection with the theory of integral equations,” Philos. Trans. R. Soc. A 209, 415–446 (1909).
[Crossref]

Phys. Rev. E (2)

T. Setälä, J. Lindberg, and K. Blomstedt, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume,” Phys. Rev. E 71, 036618 (2005).
[Crossref]

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[Crossref]

Other (3)

G. Gbur, “Nonradiating sources and the inverse source problem,” Ph.D. dissertation (University of Rochester, 2002).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

A. S. Ostrovsky, Coherent-Mode Representation in Optics (SPIE, 2006).

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Figures (7)

Fig. 1.
Fig. 1. Logarithmic curves of ratio μ l m n ( ω ) / μ 000 ( ω ) versus the DGC ( l g ) of the medium for modes l = m = n = 0 , 1, 2, 3, and 4, respectively. The figure is plotted from Eq. (26).
Fig. 2.
Fig. 2. Eigenvalues of the medium versus the mode indexes ( l = m = n ) for (a) different potential strength rms widths: σ s = λ , 2 λ , and 3 λ , (b) different rms widths of the potential’s correlation: δ s = 0.5 λ , 0.3 λ , and 0.1 λ . The figures are plotted from Eqs. (20) and (23).
Fig. 3.
Fig. 3. 3D profiles of individual eigenmodes of the medium. The scattering potential parameters are chosen as σ s = λ and δ s = 0.2 λ . The figures are plotted from Eqs. (21)–(23).
Fig. 4.
Fig. 4. 3D profiles of superposed eigenmodes of the GSM medium for the same parameters as in Fig. 3, plotted from Eqs. (21)–(23).
Fig. 5.
Fig. 5. 3D plots of the spectral density scattered by individual eigenmodes of the medium for the same parameters as in Fig. 3. The figures are plotted from Eq. (37).
Fig. 6.
Fig. 6. 3D plots of the spectral density scattered from superpositions of eigenmodes. The medium parameters are the same as in Fig. 3. The figures are plotted from Eq. (37).
Fig. 7.
Fig. 7. Normalized rms widths of the scattered intensity from the medium with eigenmodes l = m = n = 0 , 1, 2, 3, and 4, respectively. The asterisk (red) and rectangle (blue) marks represent the cases where l g = 0.2 and 0.5, respectively.

Equations (39)

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F n ( r , ω ) = k 2 4 π 2 [ n 2 ( r , ω ) 1 ] ,
C F ( r 1 , r 2 ; ω ) = F n * ( r 1 ; ω ) F n ( r 2 ; ω ) M ,
D D | C F ( r 1 , r 2 ; ω ) | 2 d 3 r 1 d 3 r 2 < ,
C F ( r 2 , r 1 ; ω ) = C F * ( r 1 , r 2 ; ω ) ,
D D C F ( r 1 , r 2 ; ω ) f * ( r 1 ; ω ) f ( r 2 ; ω ) d 3 r 1 d 3 r 2 0 .
C F ( r 1 , r 2 ; ω ) = n = 0 μ n ( ω ) ϕ n * ( r 1 , ω ) ϕ n ( r 2 , ω ) ,
D C F ( r 1 , r 2 ; ω ) ϕ n ( r 1 , ω ) d 3 r 1 = μ n ( ω ) ϕ n ( r 2 , ω ) .
μ n ( ω ) 0 ,
D ϕ m * ( r , ω ) ϕ n ( r , ω ) d 3 r = δ m n ( ω ) ,
C F ( r 1 , r 2 ; ω ) = n = 0 μ n ( ω ) C n ( r 1 , r 2 ; ω ) ,
C n ( r 1 , r 2 ; ω ) = ϕ n * ( r 1 , ω ) ϕ n ( r 2 , ω )
| η n ( r 1 , r 2 ; ω ) | = | C n ( r 1 , r 2 ; ω ) | C n ( r 1 , r 1 ; ω ) C n ( r 2 , r 2 ; ω ) = 1 .
I F ( r ; ω ) = C F ( r , r ; ω ) .
I F ( r ; ω ) = n = 0 μ n ( ω ) I n ( r ; ω ) ,
I n ( r ; ω ) = C n ( r , r ; ω ) = | ϕ n ( r , ω ) | 2
D I n ( r ; ω ) d 3 r = 1 ,
D I F ( r ; ω ) d 3 r = n = 0 μ n ( ω ) ,
C F ( r 1 , r 2 ; ω ) = l = 0 m = 0 n = 0 μ l m n ( ω ) × ϕ l m n * ( x 1 , y 1 , z 1 ; ω ) ϕ l m n ( x 2 , y 2 , z 2 ; ω ) ,
C F ( r 1 , r 2 ; ω ) = A exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] ,
μ l m n ( ω ) = A μ 0 3 / 2 q 0 l + m + n = A π 3 / 2 ( a 0 + b 0 ) 3 ( b 0 a 0 b 0 + a 0 ) l + m + n
ϕ l m n ( x , y , z ) = ( 2 π ) 3 / 4 ( 2 l + m + n l ! m ! n ! ) 1 / 2 p 0 3 / 2 H l ( 2 x p ) × H m ( 2 y p ) H n ( 2 z p 0 ) exp ( x 2 + y 2 + z 2 p 0 2 ) ,
p 0 = 1 ( 2 a 0 b 0 ) 2 , q 0 = b 0 a 0 b 0 + a 0 , μ 0 = π ( a 0 + b 0 ) 2 ,
a 0 2 = 1 8 σ 0 2 , b 0 2 = 1 8 σ 0 2 + 1 2 δ 0 2 ,
C F ( r 1 , r 2 ; ω ) = A ( 2 μ 0 π p 0 2 ) 3 / 2 exp ( x 1 2 + y 1 2 + z 1 2 + x 2 2 + y 2 2 + z 2 2 p 0 2 ) × l = 0 m = 0 n = 0 ( q 0 / 2 ) l + m + n l ! m ! n ! H l ( 2 x 1 p 0 ) H m ( 2 y 1 p 0 ) H n ( 2 z 1 p 0 ) × H l ( 2 x 2 p 0 ) H m ( 2 y 2 p 0 ) H n ( 2 z 2 p 0 ) .
l g = δ 0 / σ 0 ,
μ l m n ( ω ) μ 000 ( ω ) = [ 1 l g 2 / 2 + 1 + l g ( l g / 2 ) 2 + 1 ] l + m + n .
U ( i ) ( r ; ω ) = a ( i ) ( ω ) exp ( i k s 0 · r ) ,
W ( s ) ( r s 1 , r s 2 ; ω ) = D D U ( i ) * ( r 1 ; ω ) U ( i ) ( r 2 ; ω ) C F ( r 1 , r 2 ; ω ) × G * ( r s 1 , r 1 ; ω ) G ( r s 2 , r 2 ; ω ) d 3 r 1 d 3 r 2 ,
G ( r s , r ; ω ) = exp ( i k r ) r exp ( i k s · r ) , as k r .
W ( s ) ( r s 1 , r s 2 ; ω ) = | a ( i ) ( ω ) | 2 r 2 l = 0 m = 0 n = 0 μ l m n ( ω ) × [ ϕ l m n ˜ ( K 1 ; ω ) ] * · ϕ l m n ˜ ( K 2 ; ω ) ,
ϕ l m n ˜ ( K j ; ω ) = D ϕ l m n ( x , y , z ) exp ( i K j · r ) d 3 r , ( j = 1 , 2 ) .
S ( s ) ( r s ; ω ) = | a ( i ) ( ω ) | 2 r 2 l = 0 m = 0 n = 0 μ l m n ( ω ) | ϕ l m n ˜ ( K ; ω ) | 2 .
W ( s ) ( r s 1 , r s 2 ; ω ) = | a ( i ) ( ω ) | 2 r 2 l = 0 m = 0 n = 0 μ l m n ( ω ) × { ϕ l m n ˜ [ k s 1 x , k s 1 y , k ( 1 s 1 z ) ] } * · ϕ l m n ˜ [ k s 2 x , k s 2 y , k ( 1 s 2 z ) ] ,
ϕ l m n ˜ [ k s x , k s y , k ( 1 s z ) ] = D ϕ l m n ( x , y , z ) × exp { i k [ s x x + s y y + ( 1 s z ) z ] } d x d y d z .
ψ l m n ( r s ; ω ) = a ( i ) ( ω ) r ϕ l m n ˜ ( K ; ω ) ,
W ( s ) ( r s 1 , r s 2 ; ω ) = l = 0 m = 0 n = 0 μ l m n ( ω ) ψ l m n * ( r s 1 ; ω ) ψ l m n ( r s 2 ; ω ) .
S ( s ) ( r s ) = A r 2 ( 2 π p 0 a 0 + b 0 ) 3 exp [ k 2 p 0 2 ( 1 s z ) ] × l = 0 m = 0 n = 0 1 l ! m ! n ! ( b 0 a 0 2 b 0 + 2 a 0 ) l + m + n H l 2 ( k p 0 s x 2 ) × H m 2 ( k p 0 s y 2 ) H n 2 [ k p 0 2 ( s z 1 ) ] .
W l m n = [ 0 2 π d ϕ π / 2 π / 2 θ 2 S l m n ( s ) ( θ , ϕ ) d θ 0 2 π d ϕ π / 2 π / 2 S l m n ( s ) ( θ , ϕ ) d θ ] 1 / 2 ,
S l m n ( s ) ( θ , ϕ ) = A r 2 ( 2 π p 0 a 0 + b 0 ) 3 exp [ k 2 p 0 2 ( 1 cos θ ) ] × 1 l ! m ! n ! ( b 0 a 0 2 b 0 + 2 a 0 ) l + m + n H l 2 ( k p 0 sin θ cos ϕ 2 ) × H m 2 ( k p 0 sin θ sin ϕ 2 ) H n 2 [ k p 0 2 ( cos θ 1 ) ]

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