Scattering of multi-Gaussian Schell-model beams on a random medium Yuanyuan Zhang and Daomu Zhao* Department of Physics, Zhejiang University, Hangzhou 310027, China * [email protected]
Abstract: Using the angular spectrum representation of plane waves, we investigate the scattering of multi-Gaussian Schell-model (MGSM) beams from a random medium within the accuracy of the first-order Born approximation. The far-zone properties, including the normalized spectral density and the spectral degree of coherence, are discussed. It is shown that the normalized spectral density and the spectral degree of coherence are influenced by the boundary of the beam profile (i.e., M ), the transverse beam width, the correlation width of the source, and the properties of the scatterer. ©2013 Optical Society of America OCIS codes: (290.0290) Scattering; (290.5825) Scattering theory; (140.3295) Laser beam characterization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007). D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt. 57, 262–308 (2012). D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007). T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35(3), 318–320 (2010). X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010). C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011). Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasihomogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13(12), 1060– 1062 (1988). P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155(1–3), 1–6 (1998). T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasihomogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006). T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun. 300, 38–44 (2013). S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24781
1. Introduction Due to its potential applications in remote sensing, medical diagnosis and so on, light scattering is always a subject of considerable importance. And the researches for the light scattering have been mainly devoted to the scattering of plane waves [1–8], whether monochromatic or polychromatic plane waves. But since the development of the laser in the 1960s, lots of scattering experiments have been and are being performed with various laser beams rather than with plane waves. So it is necessary to extend the theory of scattering of plane waves to the scattering of various laser beams. In 1988, Jannson et al. discussed the influences of the degree of spatial coherence of the incident radiation on the scattered intensity [9], and Carney and Wolf derived an energy theorem for scattering of partially coherent beams on a deterministic scattering object [10]. In 2006, two reciprocity relations of the far field generated by scattering of light from a quasi-homogeneous source on a quasihomogeneous random medium were presented [11]. After that, in the book written by Wolf in 2007 [1], the general theory of scattering of partially coherent waves on random media was presented in detail. Besides, Dijk et al. illustrated numerical examples show how the effective spectral coherence length of the incident Gaussian Schell-model beam affects the angular distribution of the radiant intensity of the scattered field generated by scattering on a homogeneous spherical scatterer [12]. More recently, Zhang and Zhao extended the analysis of weak scattering to the Hermite-Gaussian incident beam and found some new features [13]. On the other hand, a new class of source, i.e. the multi-Gaussian Schell-model (MGSM) source, which can generate far fields with flat intensity profiles, was introduced [14], and the behaviors of fields generated by such sources on propagation in free space, in linear isotropic random media and in turbulent atmosphere [15,16] were examined. Besides that, Mei et al. extended the propagation of the MGSM beam to the electromagnetic domain and investigated the behavior of the polarization properties of such beams [17]. However, to the best of our knowledge, all these researches are only limited to the propagation properties of the MGSM beams. In this paper, we study the scattering of the MGSM beams on a random medium by using the angular spectrum representation of plane waves and investigate the statistic properties of the far-zone scattered field within the accuracy of the first-order Born approximation. We show the influences of the boundary of the beam profile (i.e., M ), the transverse beam width, the correlation width of the source, and the properties of the scatterer on the spectral density and the spectral degree of coherence of the scattered field. 2. Scattering of a partially coherent stochastic beam on a random medium We assume that a scalar field with time dependence exp( −iωt ) (not explicitly shown in the subsequent analysis), propagating in a direction specified by a unit vector s0 ( s0 = p, q, 1 − p 2 − q 2 ), is incident on a linear, isotropic medium occupying a finite domain D. Such a field may be of an arbitrary form and can be represented as an angular spectrum of plane waves propagating into the z ≥ 0 half-space as [18]
U (i ) (r, ω ) =
a( p, q, ω ) exp(iks0 ⋅ r )dpdq,
(1)
p 2 + q 2 ≤1
where r denotes the position vector of a point in space, ω denotes the angular frequency, k = 2π / λ is the wave number with λ being the wavelength, and the evanescent waves have been omitted. Let F (r ′, ω ) be the scattering potential of the medium and assume that the medium is so weak that the scattering can be analyzed within the accuracy of the first-order Born approximation [19]. Then the scattered field in the far-zone of the scatterer, at a point specified by a position vector r = rs ( s2 = 1 ) is expressed as [12]
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24782
U ( s ) (r, ω ) =
eikr r
a( p, q, ω ) f (s, s0 , ω )dpdq,
(2)
f (s, s0 , ω ) = F (r ′, ω ) exp −ik ( s − s0 ) ⋅ r ′d 3 r ′,
(3)
p 2 + q 2 ≤1
with D
where f (s, s0 , ω ) is the scattering amplitude. Next consider the situation when the incident field is partially coherent and the scatterer is a random medium, the cross-spectral density of the scattered field at a pair of points specified by the position vectors r1 = rs1 and r2 = rs2 can be given by the formula [1]
W ( s ) (rs1 , rs 2 , ω ) = U ( s )∗ (rs1 , ω )U ( s ) (rs 2 , ω ) ,
(4)
where the asterisk denotes the complex conjugate and the angular brackets denote ensemble average. On substituting from Eq. (2) into Eq. (4), the cross-spectral density function can be expressed as W ( s ) ( rs1 , r s 2 , ω ) =
1 r2
p12 + q12 ≤1
p22 + q22 ≤1
A(s01 , s02 , ω )C F − k ( s1 − s01 ) , k ( s 2 − s02 ) , ω dp1dq1dp2 dq2 ,
(5) where
A(s01 , s02 ,ω ) = a∗ ( p1 , q1 , ω )a( p2 , q2 , ω )
(6)
is the so-called angular correlation function of the stochastic field [18], and C F −k ( s1 − s 01 ) , k ( s 2 − s 02 ) , ω = f ∗ (s1 , s 01 , ω ) f (s 2 , s 02 , ω ) = CF (r1′, r2′ , ω ) exp ik ( s1 − s 01 ) ⋅ r1′ − ik ( s 2 − s 02 ) ⋅ r2′ d 3 r1′d 3 r2′ DD
(7) is
the
six-dimensional
spatial
Fourier
transform
of
CF (r1′, r2′ , ω )
with
CF (r1′, r2′ , ω ) = F (r1′, ω ) F (r2′ , ω ) being the correlation function of the scattering potential of ∗
the medium specified by position vectors r1′ and r2′ within the scatterer. 3. Scattering of multi-Gaussian Schell-model beams on a Gaussian Schell-model random medium We assume the incident field is of a MGSM form, and the second-order statistical properties of such field can be characterized by the cross-spectral density function, which is expressed as [14]
W
(0)
ρ2 + ρ2 (ρ1 , ρ2 , ω ) = exp − 1 2 2 4σ
1 C0
m −1
M ( −1) m m =1 m M
ρ −ρ 2 exp − 2 21 2mδ
. (8)
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24783
In the above formula ρ1 = ( x1 , y1 ) and ρ2 = ( x2 , y2 ) are two-dimensional position vectors in M M M ( −1) the source plane, C0 = is the normalization factor, stand for binomial m m =1 m m coefficients, σ is the transverse beam width of the source, and δ is the correlation width. The angular correlation function of such a beam may be expressed as a four-dimensional Fourier transform of its cross-spectral density function in the source plane [18] m −1
k A(s01 , s02 , ω ) = 2π
4
+∞
−∞
W (0) (ρ1 , ρ 2 , ω ) exp −ik ( s02⊥ ⋅ ρ 2 − s01⊥ ⋅ ρ1 ) d 2 ρ1d 2 ρ 2 . (9)
On substituting from Eq. (8) into Eq. (9), one gets the angular correlation function of a MGSM beam in the following form A(s 01 , s 02 , ω ) =
M ( −1) m m =1 m
k 4σ 2 1 4π 2 C0
m −1
M
σ eff2
2 k 2 2 2 σ eff 2 × exp − ( s 01⊥ − s 02 ⊥ ) σ + ( s 01⊥ + s 02 ⊥ ) 4 2
(10)
,
where
1
σ
2 eff
=
1 4σ
2
1 . mδ 2
+
(11)
Next we suppose the correlation function of the scattering potential of the medium has a Gaussian Schell-model (GSM) form, i.e. r ′2 + r ′2 CF (r1′, r2′ , ω ) = A0 exp − 1 2 2 4σ I
r′ − r′ 2 2 1 exp − 2 2 σ μ
,
(12)
where A0 , σ I , σ μ are positive constants, and σ I , σ μ denote the effective radius and correlation length of the scatterer respectively. On substituting from Eq. (12) into Eq. (7), one obtains the expression
(
3 6 3 2 2 C F [ − k ( s1 − s 01 ) , k ( s 2 − s 02 ) , ω ] = 64π A0σ I σ μ σ μ + 4σ I
k 2σ I2
× exp −
[( s
1
k σIσμ 2
3 2
2
2 2
−
− s 01 ) − ( s 2 − s 02 )]
× exp −
)
(13)
2
[( s
− s 01 ) + ( s 2 − s 02 )] . 2
2 (σ μ + 4σ ) On substituting from Eqs. (10) and (13) into Eq. (5) and after some calculations, we obtain the cross-spectral density function of the far-zone scattered field that is valid to within the accuracy of the first-order Born approximation expressed in the following three-dimensional form 2
2 I
1
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24784
W
(
)
16π k σ A0σ I σ μ σ μ + 4σ I 4
(s)
( rs 1 , r s 2 , ω ) =
2
6
3
2
kσ
exp[ −
k σ eff 2
2
M ( −1) m
M
m
m −1
σ eff 2
( p2 − 2 p2 p1 + p1 + q2 − 2 q2 q1 + q1 )] 2
2
2
2
2
( p2 + 2 p2 p1 + p1 + q2 + 2 q2 q1 + q1 )] 2
8 k σI 2
× exp{−
3
2
2
2
p1 + q1 ≤ 1 p 2 + q2 ≤ 1
× exp[ −
−
m =1
2
2
2
2
C0 r
2
2
2
2
2
2
[( s1 x − s 2 x + p2 ) − 2 ( s1 x − s2 x + p2 ) p1 2
2
(
2
)
+ ( s1 y + s2 y − q2 ) − 2 ( s1 y + s2 y − q2 ) q1 + s1 z + s 2 z − 1 − p2 − q2
)
+ ( s1 y − s2 y + q2 ) − 2 ( s1 y − s2 y + q2 ) q1 + s1 z − s 2 z + 1 − p2 − q2 2
(
)
−2 s1 z − s2 z + 1 − p2 − q2 2
k σIσμ 2
× exp{−
(
2
2
2
2
(14)
1 − p1 − q1 + 1]} 2
2
2
2 σ μ + 4σ I 2
2
[( s1 x + s2 x − p2 ) − 2 ( s1 x + s 2 x − p2 ) p1 2
)
(
2
(
)
−2 s1 z + s2 z − 1 − p2 − q2 2
2
2
2
2
1 − p1 − q1 + 1]} dp1 dq1 dp2 dq2 . 2
2
Consider the values of the angular correlation function of the MGSM beams in Eq. (10) decay exponentially as the values of p , q increase, we assume that only values with
p 2 + q 2
Abstract: Using the angular spectrum representation of plane waves, we investigate the scattering of multi-Gaussian Schell-model (MGSM) beams from a random medium within the accuracy of the first-order Born approximation. The far-zone properties, including the normalized spectral density and the spectral degree of coherence, are discussed. It is shown that the normalized spectral density and the spectral degree of coherence are influenced by the boundary of the beam profile (i.e., M ), the transverse beam width, the correlation width of the source, and the properties of the scatterer. ©2013 Optical Society of America OCIS codes: (290.0290) Scattering; (290.5825) Scattering theory; (140.3295) Laser beam characterization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007). D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt. 57, 262–308 (2012). D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007). T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35(3), 318–320 (2010). X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010). C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011). Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasihomogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13(12), 1060– 1062 (1988). P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155(1–3), 1–6 (1998). T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasihomogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006). T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun. 300, 38–44 (2013). S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24781
1. Introduction Due to its potential applications in remote sensing, medical diagnosis and so on, light scattering is always a subject of considerable importance. And the researches for the light scattering have been mainly devoted to the scattering of plane waves [1–8], whether monochromatic or polychromatic plane waves. But since the development of the laser in the 1960s, lots of scattering experiments have been and are being performed with various laser beams rather than with plane waves. So it is necessary to extend the theory of scattering of plane waves to the scattering of various laser beams. In 1988, Jannson et al. discussed the influences of the degree of spatial coherence of the incident radiation on the scattered intensity [9], and Carney and Wolf derived an energy theorem for scattering of partially coherent beams on a deterministic scattering object [10]. In 2006, two reciprocity relations of the far field generated by scattering of light from a quasi-homogeneous source on a quasihomogeneous random medium were presented [11]. After that, in the book written by Wolf in 2007 [1], the general theory of scattering of partially coherent waves on random media was presented in detail. Besides, Dijk et al. illustrated numerical examples show how the effective spectral coherence length of the incident Gaussian Schell-model beam affects the angular distribution of the radiant intensity of the scattered field generated by scattering on a homogeneous spherical scatterer [12]. More recently, Zhang and Zhao extended the analysis of weak scattering to the Hermite-Gaussian incident beam and found some new features [13]. On the other hand, a new class of source, i.e. the multi-Gaussian Schell-model (MGSM) source, which can generate far fields with flat intensity profiles, was introduced [14], and the behaviors of fields generated by such sources on propagation in free space, in linear isotropic random media and in turbulent atmosphere [15,16] were examined. Besides that, Mei et al. extended the propagation of the MGSM beam to the electromagnetic domain and investigated the behavior of the polarization properties of such beams [17]. However, to the best of our knowledge, all these researches are only limited to the propagation properties of the MGSM beams. In this paper, we study the scattering of the MGSM beams on a random medium by using the angular spectrum representation of plane waves and investigate the statistic properties of the far-zone scattered field within the accuracy of the first-order Born approximation. We show the influences of the boundary of the beam profile (i.e., M ), the transverse beam width, the correlation width of the source, and the properties of the scatterer on the spectral density and the spectral degree of coherence of the scattered field. 2. Scattering of a partially coherent stochastic beam on a random medium We assume that a scalar field with time dependence exp( −iωt ) (not explicitly shown in the subsequent analysis), propagating in a direction specified by a unit vector s0 ( s0 = p, q, 1 − p 2 − q 2 ), is incident on a linear, isotropic medium occupying a finite domain D. Such a field may be of an arbitrary form and can be represented as an angular spectrum of plane waves propagating into the z ≥ 0 half-space as [18]
U (i ) (r, ω ) =
a( p, q, ω ) exp(iks0 ⋅ r )dpdq,
(1)
p 2 + q 2 ≤1
where r denotes the position vector of a point in space, ω denotes the angular frequency, k = 2π / λ is the wave number with λ being the wavelength, and the evanescent waves have been omitted. Let F (r ′, ω ) be the scattering potential of the medium and assume that the medium is so weak that the scattering can be analyzed within the accuracy of the first-order Born approximation [19]. Then the scattered field in the far-zone of the scatterer, at a point specified by a position vector r = rs ( s2 = 1 ) is expressed as [12]
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24782
U ( s ) (r, ω ) =
eikr r
a( p, q, ω ) f (s, s0 , ω )dpdq,
(2)
f (s, s0 , ω ) = F (r ′, ω ) exp −ik ( s − s0 ) ⋅ r ′d 3 r ′,
(3)
p 2 + q 2 ≤1
with D
where f (s, s0 , ω ) is the scattering amplitude. Next consider the situation when the incident field is partially coherent and the scatterer is a random medium, the cross-spectral density of the scattered field at a pair of points specified by the position vectors r1 = rs1 and r2 = rs2 can be given by the formula [1]
W ( s ) (rs1 , rs 2 , ω ) = U ( s )∗ (rs1 , ω )U ( s ) (rs 2 , ω ) ,
(4)
where the asterisk denotes the complex conjugate and the angular brackets denote ensemble average. On substituting from Eq. (2) into Eq. (4), the cross-spectral density function can be expressed as W ( s ) ( rs1 , r s 2 , ω ) =
1 r2
p12 + q12 ≤1
p22 + q22 ≤1
A(s01 , s02 , ω )C F − k ( s1 − s01 ) , k ( s 2 − s02 ) , ω dp1dq1dp2 dq2 ,
(5) where
A(s01 , s02 ,ω ) = a∗ ( p1 , q1 , ω )a( p2 , q2 , ω )
(6)
is the so-called angular correlation function of the stochastic field [18], and C F −k ( s1 − s 01 ) , k ( s 2 − s 02 ) , ω = f ∗ (s1 , s 01 , ω ) f (s 2 , s 02 , ω ) = CF (r1′, r2′ , ω ) exp ik ( s1 − s 01 ) ⋅ r1′ − ik ( s 2 − s 02 ) ⋅ r2′ d 3 r1′d 3 r2′ DD
(7) is
the
six-dimensional
spatial
Fourier
transform
of
CF (r1′, r2′ , ω )
with
CF (r1′, r2′ , ω ) = F (r1′, ω ) F (r2′ , ω ) being the correlation function of the scattering potential of ∗
the medium specified by position vectors r1′ and r2′ within the scatterer. 3. Scattering of multi-Gaussian Schell-model beams on a Gaussian Schell-model random medium We assume the incident field is of a MGSM form, and the second-order statistical properties of such field can be characterized by the cross-spectral density function, which is expressed as [14]
W
(0)
ρ2 + ρ2 (ρ1 , ρ2 , ω ) = exp − 1 2 2 4σ
1 C0
m −1
M ( −1) m m =1 m M
ρ −ρ 2 exp − 2 21 2mδ
. (8)
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24783
In the above formula ρ1 = ( x1 , y1 ) and ρ2 = ( x2 , y2 ) are two-dimensional position vectors in M M M ( −1) the source plane, C0 = is the normalization factor, stand for binomial m m =1 m m coefficients, σ is the transverse beam width of the source, and δ is the correlation width. The angular correlation function of such a beam may be expressed as a four-dimensional Fourier transform of its cross-spectral density function in the source plane [18] m −1
k A(s01 , s02 , ω ) = 2π
4
+∞
−∞
W (0) (ρ1 , ρ 2 , ω ) exp −ik ( s02⊥ ⋅ ρ 2 − s01⊥ ⋅ ρ1 ) d 2 ρ1d 2 ρ 2 . (9)
On substituting from Eq. (8) into Eq. (9), one gets the angular correlation function of a MGSM beam in the following form A(s 01 , s 02 , ω ) =
M ( −1) m m =1 m
k 4σ 2 1 4π 2 C0
m −1
M
σ eff2
2 k 2 2 2 σ eff 2 × exp − ( s 01⊥ − s 02 ⊥ ) σ + ( s 01⊥ + s 02 ⊥ ) 4 2
(10)
,
where
1
σ
2 eff
=
1 4σ
2
1 . mδ 2
+
(11)
Next we suppose the correlation function of the scattering potential of the medium has a Gaussian Schell-model (GSM) form, i.e. r ′2 + r ′2 CF (r1′, r2′ , ω ) = A0 exp − 1 2 2 4σ I
r′ − r′ 2 2 1 exp − 2 2 σ μ
,
(12)
where A0 , σ I , σ μ are positive constants, and σ I , σ μ denote the effective radius and correlation length of the scatterer respectively. On substituting from Eq. (12) into Eq. (7), one obtains the expression
(
3 6 3 2 2 C F [ − k ( s1 − s 01 ) , k ( s 2 − s 02 ) , ω ] = 64π A0σ I σ μ σ μ + 4σ I
k 2σ I2
× exp −
[( s
1
k σIσμ 2
3 2
2
2 2
−
− s 01 ) − ( s 2 − s 02 )]
× exp −
)
(13)
2
[( s
− s 01 ) + ( s 2 − s 02 )] . 2
2 (σ μ + 4σ ) On substituting from Eqs. (10) and (13) into Eq. (5) and after some calculations, we obtain the cross-spectral density function of the far-zone scattered field that is valid to within the accuracy of the first-order Born approximation expressed in the following three-dimensional form 2
2 I
1
#194796 - $15.00 USD Received 2 Aug 2013; accepted 26 Sep 2013; published 9 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024781 | OPTICS EXPRESS 24784
W
(
)
16π k σ A0σ I σ μ σ μ + 4σ I 4
(s)
( rs 1 , r s 2 , ω ) =
2
6
3
2
kσ
exp[ −
k σ eff 2
2
M ( −1) m
M
m
m −1
σ eff 2
( p2 − 2 p2 p1 + p1 + q2 − 2 q2 q1 + q1 )] 2
2
2
2
2
( p2 + 2 p2 p1 + p1 + q2 + 2 q2 q1 + q1 )] 2
8 k σI 2
× exp{−
3
2
2
2
p1 + q1 ≤ 1 p 2 + q2 ≤ 1
× exp[ −
−
m =1
2
2
2
2
C0 r
2
2
2
2
2
2
[( s1 x − s 2 x + p2 ) − 2 ( s1 x − s2 x + p2 ) p1 2
2
(
2
)
+ ( s1 y + s2 y − q2 ) − 2 ( s1 y + s2 y − q2 ) q1 + s1 z + s 2 z − 1 − p2 − q2
)
+ ( s1 y − s2 y + q2 ) − 2 ( s1 y − s2 y + q2 ) q1 + s1 z − s 2 z + 1 − p2 − q2 2
(
)
−2 s1 z − s2 z + 1 − p2 − q2 2
k σIσμ 2
× exp{−
(
2
2
2
2
(14)
1 − p1 − q1 + 1]} 2
2
2
2 σ μ + 4σ I 2
2
[( s1 x + s2 x − p2 ) − 2 ( s1 x + s 2 x − p2 ) p1 2
)
(
2
(
)
−2 s1 z + s2 z − 1 − p2 − q2 2
2
2
2
2
1 − p1 − q1 + 1]} dp1 dq1 dp2 dq2 . 2
2
Consider the values of the angular correlation function of the MGSM beams in Eq. (10) decay exponentially as the values of p , q increase, we assume that only values with
p 2 + q 2
