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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 24781–24792
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Scattering of multi-Gaussian Schell-model beams on a random medium

Yuanyuan Zhang and Daomu Zhao  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 24781-24792 (2013)
http://dx.doi.org/10.1364/OE.21.024781


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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

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

1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

8

8. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). [CrossRef] [PubMed]

], 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

9. J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13(12), 1060–1062 (1988). [CrossRef] [PubMed]

], and Carney and Wolf derived an energy theorem for scattering of partially coherent beams on a deterministic scattering object [10

10. P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155(1–3), 1–6 (1998). [CrossRef]

]. In 2006, two reciprocity relations of the far field generated by scattering of light from a quasi-homogeneous source on a quasi-homogeneous random medium were presented [11

11. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006). [CrossRef] [PubMed]

]. After that, in the book written by Wolf in 2007 [1

1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

], 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

12. 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). [CrossRef] [PubMed]

]. More recently, Zhang and Zhao extended the analysis of weak scattering to the Hermite-Gaussian incident beam and found some new features [13

13. Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun. 300, 38–44 (2013). [CrossRef]

].

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

14. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

], and the behaviors of fields generated by such sources on propagation in free space, in linear isotropic random media and in turbulent atmosphere [15

15. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]

,16

16. 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). [CrossRef]

] 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

17. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]

]. 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,1p2q2), 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 z0 half-space as [18

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

]
U(i)(r,ω)=p2+q21a(p,q,ω)exp(iks0r)dpdq,
(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

19. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

]. 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

12. 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). [CrossRef] [PubMed]

]
U(s)(r,ω)=eikrrp2+q21a(p,q,ω)f(s,s0,ω)dpdq,
(2)
with
f(s,s0,ω)=DF(r,ω)exp[ik(ss0)r]d3r,
(3)
where f(s,s0,ω) is the scattering amplitude.

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

14. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

]
W(0)(ρ1,ρ2,ω)=exp(ρ12+ρ224σ2)1C0m=1M(Mm)(1)m1mexp(|ρ2ρ1|22mδ2).
(8)
In the above formula ρ1=(x1,y1) and ρ2=(x2,y2) are two-dimensional position vectors in the source plane, C0=m=1M(Mm)(1)m1m is the normalization factor, (Mm) stand for binomial 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

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

]
A(s01,s02,ω)=(k2π)4+W(0)(ρ1,ρ2,ω)exp[ik(s02ρ2s01ρ1)]d2ρ1d2ρ2.
(9)
On substituting from Eq. (8) into Eq. (9), one gets the angular correlation function of a MGSM beam in the following form
A(s01,s02,ω)=k4σ24π21C0m=1M(Mm)(1)m1mσeff2×exp{k22[(s01s02)2σ2+(s01+s02)2σeff24]},
(10)
where

1σeff2=14σ2+1mδ2.
(11)

C˜F[k(s1s01),k(s2s02),ω]=64π3A0σI6σμ3(σμ2+4σI2)32×exp{k2σI22[(s1s01)(s2s02)]2}×exp{k2σI2σμ22(σμ2+4σI2)[(s1s01)+(s2s02)]2}.
(13)

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

W(s)(rs1,rs2,ω)=16πk4σ2A0σI6σμ3(σμ2+4σI2)32C0r2m=1M(Mm)(1)m1mσeff2p12+q121p22+q221exp[k2σ22(p222p2p1+p12+q222q2q1+q12)]×exp[k2σeff28(p22+2p2p1+p12+q22+2q2q1+q12)]×exp{k2σI22[(s1xs2x+p2)22(s1xs2x+p2)p1+(s1ys2y+q2)22(s1ys2y+q2)q1+(s1zs2z+1p22q22)22(s1zs2z+1p22q22)1p12q12+1]}×exp{k2σI2σμ22(σμ2+4σI2)[(s1x+s2xp2)22(s1x+s2xp2)p1+(s1y+s2yq2)22(s1y+s2yq2)q1+(s1z+s2z1p22q22)22(s1z+s2z1p22q22)1p12q12+1]}dp1dq1dp2dq2.
(14)

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 p2+q2<<1 can contribute significantly to the integral in Eq. (14). Thus within the domain, 1p12q12 in Eq. (14) may be approximated as 1p12+q122, and Eq. (14) may be rewritten as

W(s)(rs1,rs2,ω)==16πk4σ2A0σI6σμ3(σμ2+4σI2)32C0r2m=1M(Mm)(1)m1mσeff2p12+q12<<1p22+q221exp[k2σ22(p222p2p1+p12+q222q2q1+q12)]×exp[k2σeff28(p22+2p2p1+p12+q22+2q2q1+q12)]×exp{k2σI22[(s1xs2x+p2)22(s1xs2x+p2)p1+(s1ys2y+q2)22(s1ys2y+q2)q1+(s1zs2z+1p22q22)22(s1zs2z+1p22q22)+(s1zs2z+1p22q22)(p12+q12)+1]}×exp{k2σI2σμ22(σμ2+4σI2)[(s1x+s2xp2)22(s1x+s2xp2)p1+(s1y+s2yq2)22(s1y+s2yq2)q1+(s1z+s2z1p22q22)22(s1z+s2z1p22q22)+(s1z+s2z1p22q22)(p12+q12)+1]}dp1dq1dp2dq2.
(15)

For carrying out the integral in Eq. (15) analytically we suppose the values of the angular correlation function of the MGSM beams in Eq. (10) might not be zero only within the domain p12+q12<<1, so the integral limits in Eq. (15) may be extended to ± and Eq. (15) can be rewritten as

W(s)(rs1,rs2,ω)=16πk4σ2A0σI6σμ3(σμ2+4σI2)32C0r2m=1M(Mm)(1)m1mσeff2p22+q221exp[k2σ22(p222p2p1+p12+q222q2q1+q12)]×exp[k2σeff28(p22+2p2p1+p12+q22+2q2q1+q12)]×exp{k2σI22[(s1xs2x+p2)22(s1xs2x+p2)p1+(s1ys2y+q2)22(s1ys2y+q2)q1+(s1zs2z+1p22q22)22(s1zs2z+1p22q22)+(s1zs2z+1p22q22)(p12+q12)+1]}×exp{k2σI2σμ22(σμ2+4σI2)[(s1x+s2xp2)22(s1x+s2xp2)p1+(s1y+s2yq2)22(s1y+s2yq2)q1+(s1z+s2z1p22q22)22(s1z+s2z1p22q22)+(s1z+s2z1p22q22)(p12+q12)+1]}dp1dq1dp2dq2.
(16)

After integrating on p1 and q1 separately, we finally obtain the cross-spectral density function of the far-zone scattered field from Eq. (16) in the following form
W(s)(rs1,rs2,ω)=16π2k4σ2A0σI6σμ3(σμ2+4σI2)32C0r2m=1M(Mm)(1)m1mσeff2p22+q2211aexp(b2+c24a)exp[k22(σ2+σeff24)(p22+q22)]×exp{k2σI22[(s1xs2x+p2)2+(s1ys2y+q2)2+(s1zs2z+1p22q22)22(s1zs2z+1p22q22)+1]}×exp{k2σI2σμ22(σμ2+4σI2)[(s1x+s2xp2)2+(s1y+s2yq2)2+(s1z+s2z1p22q22)22(s1z+s2z1p22q22)+1]}dp2dq2,
(17)
where

a=k2σ22+k2σeff28+k2σI22(s1zs2z+1p22q22)+k2σI2σμ22(σμ2+4σI2)(s1z+s2z1p22q22),
(18)
b=k2σ2p2k2σeff2p24+k2σI2(s1xs2x+p2)+k2σI2σμ2σμ2+4σI2(s1x+s2xp2),
(19)
c=k2σ2q2k2σeff2q24+k2σI2(s1ys2y+q2)+k2σI2σμ2σμ2+4σI2(s1y+s2yq2).
(20)

From the expressions for the cross-spectral density of the far-zone scattered field, the spectral density is calculated by setting s1=s2=s and is expressed as follows
S(s)(rs,ω)=16π2k4σ2A0σI6σμ3(σμ2+4σI2)32C0r2m=1M(Mm)(1)m1mσeff2p22+q2211aexp(b2+c24a)exp[k22(σ2+σeff24)(p22+q22)]×exp[k2σI2(11p22q22)]×exp[k2σI2σμ22(σμ2+4σI2)(64sxp24syq24sz1p22q224sz+21p22q22)]dp2dq2,
(21)
where

a=k2σ22+k2σeff28+k2σI221p22q22+k2σI2σμ22(σμ2+4σI2)(2sz1p22q22),
(22)
b=k2σ2p2k2σeff2p24+k2σI2p2+k2σI2σμ2σμ2+4σI2(2sxp2),
(23)
c=k2σ2q2k2σeff2q24+k2σI2q2+k2σI2σμ2σμ2+4σI2(2syq2).
(24)

The spectral degree of coherence is derived from Eqs. (17) and (21) by use of the expression [1

1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

]
μ(s)(rs1,rs2,ω)=W(s)(rs1,rs2,ω)S(s)(rs1,ω)S(s)(rs2,ω),
(25)
and is expressed as
μ(s)(rs1,rs2,ω)=m=1M(Mm)(1)m1mσeff2p22+q2211aexp(b2+c24a)×exp[k22(σ2+σeff24)(p22+q22)]×exp{k2σI22[(s1xs2x+p2)2+(s1ys2y+q2)2+(s1zs2z+1p22q22)22(s1zs2z+1p22q22)+1]}×exp{k2σI2σμ22(σμ2+4σI2)[(s1x+s2xp2)2+(s1y+s2yq2)2+(s1z+s2z1p22q22)22(s1z+s2z1p22q22)+1]}dp2dq2/{m=1M(Mm)(1)m1mσeff2p22+q2211a1exp(b12+c124a1)×exp[k22(σ2+σeff24)(p22+q22)]exp[k2σI2(11p22q22)]×exp[k2σI2σμ22(σμ2+4σI2)(64s1xp24s1yq24s1z1p22q224s1z+21p22q22)]dp2dq2}12/{m=1M(Mm)(1)m1mσeff2p22+q2211a2exp(b22+c224a2)×exp[k22(σ2+σeff24)(p22+q22)]exp[k2σI2(11p22q22)]×exp[k2σI2σμ22(σμ2+4σI2)(64s2xp24s2yq24s2z1p22q224s2z+21p22q22)]dp2dq2}12,
(26)
where a, b and c are expressed in Eqs. (18), (19),and (20) respectively, and

a1=k2σ22+k2σeff28+k2σI221p22q22+k2σI2σμ22(σμ2+4σI2)(2s1z1p22q22),
(27)
b1=k2σ2p2k2σeff2p24+k2σI2p2+k2σI2σμ2σμ2+4σI2(2s1xp2),
(28)
c1=k2σ2q2k2σeff2q24+k2σI2q2+k2σI2σμ2σμ2+4σI2(2s1yq2),
(29)
a2=k2σ22+k2σeff28+k2σI221p22q22+k2σI2σμ22(σμ2+4σI2)(2s2z1p22q22),
(30)
b2=k2σ2p2k2σeff2p24+k2σI2p2+k2σI2σμ2σμ2+4σI2(2s2xp2),
(31)
c2=k2σ2q2k2σeff2q24+k2σI2q2+k2σI2σμ2σμ2+4σI2(2s2yq2).
(32)

4. Numerical simulations and discussions

According to Eq. (21) and Eq. (26) we can 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 contours of the normalized spectral density and the spectral degree of coherence of a MGSM beam scattered by a random medium (as shown in Figs. 1-5).
Fig. 1 The far-zone scattered field for selected values of M as a function of sx. The other parameters are chosen as follows: σ=15λ, δ=5λ, σI=25λ, σμ=10λ. (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s0.

Figure 1(a) illustrates the behavior of the normalized spectral density of the far-zone scattered field as a function of the x component of the scattering direction s for several values of M. It is shown that when M=1 (now the beam reduces to the ordinary GSM beam), the normalized spectral density in the far-zone preserves Gaussian shape, but as the index M increases, the contours of the normalized spectral density begin to grow in width and present flat profiles around the beam axis. From Fig. 1(b) showing the distributions of the spectral degree of coherence, one finds that there are no obvious changes with different values of M and all curves are Gaussian-profile, which means that the distributions of the spectral degree of coherence are almost not affected by the index M.

Fig. 2 The far-zone scattered field for selected values of the transverse beam width of the source σ as a function of sx. The other parameters are chosen as follows:δ=5λ, σI=25λ, σμ=10λ, M=4. (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s0.
In Fig. 2(a)and Fig. 2(b), we present the influences of the transverse beam width of the source σ on the distributions of the normalized spectral density and the spectral degree of coherence of the far-zone scattered field. It is shown that the transverse beam width of the source has almost no effects on the distribution of the normalized spectral density (see Fig. 2(a)), but has significant impact on the distribution of the spectral degree of coherence (see Fig. 2(b)), and the larger the value of σ is, the faster the spectral degree of coherence decays.

Figure 3(a) reveals that the correlation width of the source δ may markedly affect the profiles of the normalized spectral density in the far-zone. Specifically, the shape of the normalized spectral density turns wider and is more likely to be flat-topped profile with a lower δ. In Fig. 3(b), it is clearly displayed that the influence of δ on the distribution of the spectral degree of coherence is not as important as that on the distribution of the normalized spectral density, and there is only a slight difference with different values of δ.
Fig. 3 The far-zone scattered field for selected values of the correlation width of the source δ as a function of sx. The other parameters are chosen as follows: σ=15λ, σI=25λ, σμ=10λ, M=4. (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s0.

From Figs. 4(a) and 4(b) one finds that the influences of the effective radius of the scatterer σI on the distributions of the normalized spectral density and the spectral degree of coherence are almost the same as the influences of the transverse beam width of the source σ on the distributions of the normalized spectral density and the spectral degree of coherence in Figs. 2(a) and 2(b). There are nearly no changes of the normalized spectral density with different values of σI, but with a larger σI there is a more quick reduce of the spectral degree of coherence.
Fig. 4 The far-zone scattered field for selected values of the effective radius of the scatterer σI as a function of sx. The other parameters are chosen as follows: σ=15λ, δ=5λ, σμ=10λ, M=4. (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s0.

Figure 5(a) exhibits the profiles of the normalized spectral density for different values of the correlation length of the scatterer σμ. It indicates that as the value of σμ increases, the shape of the normalized intensity grows in width and becomes a flat intensity profile around the beam axis. Besides, along with the direction of scattering from on-axis to off-axis, the normalized intensity firstly decreases more slowly and then drops faster with an increasing value of σμ. Figure 5(b) shows that there is no change of the spectral degree of coherence with an increasing value of the correlation length of the scatterer σμ, which indicates that σμ has almost no impact on the distribution of the spectral degree of coherence in the far-zone.

Fig. 5 The far-zone scattered field for selected values of the correlation length of the scatterer σμ as a function of sx. The other parameters are chosen as follows: σ=15λ, δ=5λ, σI=25λ, M=4. (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s0.

5. Conclusion

We summarize our work by saying that we have analyzed the case when a MGSM beam is scattered by a random medium within the accuracy of the first-order Born approximation. The expressions for the spectral density and the spectral degree of coherence of the far-zone scattered field have been derived. The numerical examples are given to show that the contours of the normalized spectral density are more likely to present far fields with flat profiles around the beam axis with a larger index M, a smaller correlation width of the source or a greater correlation length of the scatterer. Besides, the numerical results demonstrate that a greater transverse beam width of the source or a larger effective radius of the scatterer can result in a faster decline of the spectral degree of coherence in the far zone. The ability of the MGSM beams scattered by random media to generate far fields with flat intensity profiles around the beam axis may have potential use in the theory of weak scattering.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11074219, 11274273, 10874150, and J1210046), and the Zhejiang Provincial Natural Science Foundation of China (R1090168).

References and links

1.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

2.

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3.

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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). [CrossRef] [PubMed]

8.

Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). [CrossRef] [PubMed]

9.

J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13(12), 1060–1062 (1988). [CrossRef] [PubMed]

10.

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155(1–3), 1–6 (1998). [CrossRef]

11.

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006). [CrossRef] [PubMed]

12.

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). [CrossRef] [PubMed]

13.

Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun. 300, 38–44 (2013). [CrossRef]

14.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

15.

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]

16.

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). [CrossRef]

17.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]

18.

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

19.

M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

OCIS Codes
(290.0290) Scattering : Scattering
(140.3295) Lasers and laser optics : Laser beam characterization
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: August 2, 2013
Manuscript Accepted: September 26, 2013
Published: October 9, 2013

Citation
Yuanyuan Zhang and Daomu Zhao, "Scattering of multi-Gaussian Schell-model beams on a random medium," Opt. Express 21, 24781-24792 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24781


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References

  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  2. D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt.57, 262–308 (2012).
  3. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007). [CrossRef] [PubMed]
  4. 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). [CrossRef] [PubMed]
  5. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett.35(3), 384–386 (2010). [CrossRef] [PubMed]
  6. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010). [CrossRef]
  7. 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). [CrossRef] [PubMed]
  8. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010). [CrossRef] [PubMed]
  9. J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett.13(12), 1060–1062 (1988). [CrossRef] [PubMed]
  10. P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun.155(1–3), 1–6 (1998). [CrossRef]
  11. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A23(7), 1631–1638 (2006). [CrossRef] [PubMed]
  12. 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). [CrossRef] [PubMed]
  13. Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun.300, 38–44 (2013). [CrossRef]
  14. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett.37(14), 2970–2972 (2012). [CrossRef] [PubMed]
  15. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A29(10), 2159–2164 (2012). [CrossRef] [PubMed]
  16. 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). [CrossRef]
  17. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013). [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  19. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

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