Abstract

A generalized random medium is introduced of which the effective radius and the correlation length are assumed to depend on the wavelength of the incident light. The generalized Stokes parameters of a random electromagnetic beam are employed to characterize the statistical properties of the far-zone scattered field generated by the introduced random medium. It is shown that the dependence of the effective radius and the correlation length of a medium on the wavelength can affect the behaviors of the intensity, coherence and polarization of the scattered field. Besides, the distributions of the scattered field are found to have many differences between the situations of medium-led modulation and beam-led modulation.

© 2017 Optical Society of America

1. Introduction

The light scattering, which studies the interaction between light waves and media, is a subject of great importance due to its potential applications in the fields like target detection, remote sensing, medical diagnosis and so on. During the past few decades, many efforts have been devoted to this subject, both for continuous media and particle collections [1–6]. Recently, proposing new models for scattering media has attracted a lot of attention, and there are two starting points for this work. On the one hand, the structural information about an unknown object can be obtained from the knowledge of the scattered field, so it makes more sense to explore the scattered field generated by more realistic media. Based on this starting point, some generalized models for relatively realistic media have been introduced. For example, the isotropic media have been extended to anisotropic ones [7, 8]; the soft-edge media and the hard-edge media, which are both idealizations being mathematically convenient, have been generalized to the media with adjustable boundaries [9,10]; besides, the stationary nonuniformly correlated media have been proposed as a generalization of the uniformly correlated media [11]. On the other hand, the modulation of the scattered light consists of two parts: one is from the incident light, and the other is from the scattering medium, which indicates that the desired scattered light can be produced by designing the medium. In order to obtain expected scattered lights, some novel models for designed media have been developed. For instance, the circular flat and ring-like intensity distributions can be formed by media with well-chosen correlation properties [12]; a model proposed in [13] has been shown to produce square or rectangular intensity patterns; in addition, two examples of media have been introduced to generate frame-like intensity profiles and intensity patterns with azimuthal dependence [14].

Besides scattering media, incident light beams in the scattering theory have also been a focus of research in recent years. According to the actual needs for scattering experiments, the scattering of idealized plane waves have been extended to that of various laser beams [15–20]. In particular, because of containing polarization properties in addition to intensity and coherence properties, the scattering of electromagnetic laser beams are more important. In [21], the generalized Stokes parameters, treated as a two-point extension of the usual Stokes parameters, have been proposed to characterize the statistical properties of random electromagnetic beams. Subsequently, much work has employed them to investigate the behaviors of electromagnetic beams on propagation [22–24]. In addition, Singh et al. have extended the generalized Stokes parameters to the scattered field and the spatial polarization fluctuations of the scattered field have been studied in detail [25].

In this paper, we first introduce a generalized model for a random medium based on the former staring point. Up to now, for all the media in the scattering theory, their properties are assumed to be irrelevant to the frequency/wavelength of the incident light for simplicity. But in practice, the frequency/wavelength of the incident light can affect the refractive index of a medium with dispersion. The reason is that the electric vector of light can lead to the polarization of a medium; for light waves with different frequencies, their energy flux densities and electric vectors are different, which causes the change in the dielectric susceptibility of a medium as well as its refractive index. The original idea of the spectral dependence belongs to [26] which proposed a spectral Gaussian Schell-model source. Here we take the spectral dependence into consideration to model a more realistic random medium. Then, with the help of the generalized Stokes parameters, we derive the expressions for the spectral density, the spectral degree of coherence and the spectral degree of polarization of the scattered field generated by the introduced medium. Finally, the influences of the spectral dependence of the medium on the scattered field are illustrated by numerical examples, and some important results are concluded.

2. Spectral random media

In the weak potential scattering theory, the properties of a medium can be characterized by the scattering potential function, which is defined as [27]

F(r;ω)=k2[n2(r;ω)1]4π,
where n(r;ω) is the refractive index of the medium, with r being the position vector within the medium and ω being the frequency of the incident light. k=2π/λ is the wave number and λ is the wavelength.

For a random medium, the scattering potential varies stochastically as a function of position. In this case, the second-order statistical properties of the medium can be described by the correlation function of the scattering potential as [28]

CF(r1,r2;ω)=F(r1;ω)F(r2;ω)m,
where the asterisk denotes the complex conjugate, the angular brackets denote the statistical ensemble average in the space-frequency domain and m represents the whole system of the medium.

Analogous to the description method of a random source, the correlation function of a random medium can also be expressed as [29]

CF(r1,r2;ω)=IF(r1;ω)IF(r2;ω)μF(r1,r2;ω),
where
IF(r;ω)=CF(r,r;ω)
is the strength of the scattering potential function and
μF(r1,r2;ω)=CF(r1,r2;ω)IF(r1;ω)IF(r2;ω)
is the correlation coefficient of the scattering potential function.

It is worth noting that, for almost all kinds of media, the scattering potential functions or the correlation functions are influenced by the frequency of the incident light in addition to positions. The reason is that the refractive index of a medium changes with the variation of the incident light frequency, which results in the corresponding change in the scattering potential function or the correlation function. However, the dependence on the frequency has always been omitted in previous studies for the sake of simplicity.

To introduce a generalization of random media, we first review a classic random model, the quasi-homogeneous (QH) model, which provides a physically realistic description of many types of random media. The QH media have some special properties as follows: the correlation coefficient of the scattering potential μF(r1,r2;ω) depends on the two points r1 and r2 only through their difference r2r1; besides, the strength of the scattering potential IF(r;ω) varies much more slowly with the point r than the correlation coefficient μF(r1,r2;ω) does with r2r1. Therefore, the correlation function of the scattering potential of QH media can be given by [1]

CF(r1,r2;ω)=IF(r1+r22;ω)μF(r2r1;ω).
Let us assume that the strength and the correlation coefficient of QH media satisfy the Gaussian distribution, then the correlation function in Eq. (6) has the form
CF(r1,r2;ω)=C0exp[(r1+r2)28σI2]exp[(r2r1)22σμ2],
where C0 is a positive constant, σI denotes the effective radius and σμ denotes the correlation length of the medium. In previous researches, the parameters σI and σμ have always been assumed to be constants, but in general, they are both related to the frequency of the incident light.

In this paper, we take the spectral dependence into consideration for a generalization of random media. Then the correlation function in Eq. (7) can be rewritten as

CF(r1,r2;ω)=C0exp[(r1+r2)28σI2(ω)]exp[(r2r1)22σμ2(ω)],
where σI(ω) and σμ(ω) can have quite arbitrary dependence on the frequency. The frequency and the wavelength are both basic properties of the incident light, and they have the relation as λ=2πc/ω with c being the speed of light in vacuum. It can be found that the numerator of the relation is a constant, so the dependence on the frequency can also be regarded as relying on the wavelength in the same manner
CF(r1,r2;λ)=C0exp[(r1+r2)28σI2(λ)]exp[(r2r1)22σμ2(λ)].
We assume that σI(λ) and σμ(λ) take the forms of Gaussian functions, i.e.,
σI(λ)=σI0exp[(λλσI)22ΛσI2],
σμ(λ)=σμ0exp[(λλσμ)22Λσμ2],
where λσI and λσμ are the central wavelengths of the spectral distributions of the effective radius and the correlation length of the medium, respectively. ΛσI and Λσμ are the rms widths of these distributions, σI0 and σμ0 are the maximum values of the two functions at the central wavelengths.

The family of the generalized QH media which involve such spectral dependence will be termed as the spectral quasi-homogeneous (SQH) media. It can be seen from Eqs. (9)-(11) that besides the two parameters of the classic QH model, σI0 and σμ0, the SQH model includes four other parameters, making it a very rich class of media.

In order to be physically realizable, any correlation function needs to obey some conditions, of which non-negative definiteness is the most difficult one to verify. The sufficient condition for a legitimate two-dimensional (2D) correlation function has been extended to a 3D case [12], and it is found that a genuine CF must have the following integral representation:

CF(r1,r2;λ)=p(v;λ)H0(r1,v;λ)H0(r2,v;λ)d3v,
where v is a 3D vector, H0(r,v;λ) is an arbitrary function, and p(v;λ), which can define a family of media with different degrees of correlation function, must be non-negative and Fourier-transformable. For a classic QH medium,
p(v;λ)μ˜F(r2r1;λ),
where μ˜F(r2r1;λ) denotes the 3D Fourier transform of the correlation coefficient. The approximation also applies to the SQH medium. For a SQH medium, the correlation coefficient has the form
μF(r2r1;λ)=exp[(r2r1)22σμ2(λ)].
On substituting from Eq. (14) into Eq. (13), one can find that

p(v;λ)(2π)3/2σμ3(λ)exp[σμ2(λ)v32]0.

3. Generalized Stokes parameters of random electromagnetic beams on scattering

Consider a stochastic, statistically stationary electromagnetic beam, propagating in a direction specified by a unit vector s0, that is incident on a medium occupying a finite domain D. The second-order properties of the incident field can be characterized by a 2×2 cross-spectral density matrix, which has the definition [28]

W(i)(r1,r2,s0;λ)[Wij(i)(r1,r2,s0;λ)]=[Ei(i)(r1,s0;λ)Ej(i)(r2,s0;λ)],(i,j=x,y),
where r1 and r2 are a pair of points in the incident field, Ex(i)(r,s0;λ) and Ey(i)(r,s0;λ) are the electric-field components of the incident beam along the x and y axes in Cartesian coordinates, respectively.

Using the angular spectrum representation of plane waves, Ei(i)(r,s0;λ) can be expressed as the integral form [29]

Ei(i)(r,s0;λ)=|s0|21ai(s0;λ)exp(iks0r)d2s0,
where s0 is a 2D vector and ai(s0;λ) is the Cartesian component of the amplitude.

On substituting from Eq. (17) into Eq. (16), we obtain the element of the cross-spectral density matrix of the stochastic incident field

Wij(i)(r1,r2,s01,s02;λ)=|s01|21|s02|21Aij(s01,s02;λ)×exp[ik(s01r1s02r2)]d2s01d2s02,
where Aij(s01,s02;λ)=ai(s01;λ)aj(s02;λ) denotes the angular correlation function and it can be determined by [29]
Aij(s01,s02;λ)=(k2π)4+Wij(0)(ρ1,ρ2;λ)×exp[ik(s02ρ2s01ρ1)]d2ρ1d2ρ2,
where Wij(0)(ρ1,ρ2;λ) is the element of the cross-spectral density matrix at the source plane, ρ1 and ρ2 are 2D position vectors in the plane.

In the weak potential scattering theory, the scattering is considered within the accuracy of the first-order Born approximation. In this case, the relation between the scattered and the incident electric field can be expressed as [30]

Eθ(s)(rs;λ)=DF(r;λ)G(rs,r;λ)×[cosθcosφEx(i)(r,s0;λ)+cosθsinφEy(i)(r,s0;λ)]d3r,
Eφ(s)(rs;λ)=DF(r;λ)G(rs,r;λ)×[sinφEx(i)(r,s0;λ)+cosφEy(i)(r,s0;λ)]d3r,
Er(s)(rs;λ)=0,
where Eθ(s)(rs;λ), Eφ(s)(rs;λ) and Er(s)(rs;λ) are the three components of the scattered electric field in spherical coordinates, with s=(sinθcosφ,sinθsinφ,cosθ) being the scattering direction. G(rs,r;λ) is the free-space Green function. In practical situation, the measurements of the scattered field are made far away from media, so the Green function can be approximated in the far zone as [27]

G(rs,r;λ)exp(ikr)rexp(iksr).

The generalized Stokes parameters of random electromagnetic beams have been introduced in [21]. By analogy, the generalized Stokes parameters of the scattered field at a pair of points rs1 and rs2 in spherical coordinates can be defined as the following formulas:

S0(s)(rs1,rs2;λ)=Eθ(s)(rs1;λ)Eθ(s)(rs2;λ)+Eφ(s)(rs1;λ)Eφ(s)(rs2;λ),
S1(s)(rs1,rs2;λ)=Eθ(s)(rs1;λ)Eθ(s)(rs2;λ)Eφ(s)(rs1;λ)Eφ(s)(rs2;λ),
S2(s)(rs1,rs2;λ)=Eθ(s)(rs1;λ)Eφ(s)(rs2;λ)+Eφ(s)(rs1;λ)Eθ(s)(rs2;λ),
S3(s)(rs1,rs2;λ)=i[Eφ(s)(rs1;λ)Eθ(s)(rs2;λ)Eθ(s)(rs1;λ)Eφ(s)(rs2;λ)].

On substituting from Eqs. (20) and (21) into Eqs. (24)-(27), after some calculation, one can find that

S0(s)(rs1,rs2;λ)=1r2[(m3n4+n1)Hxx(s)(rs1,rs2;λ)+(m3n3n2)Hxy(s)(rs1,rs2;λ)+(m3n2n3)Hyx(s)(rs1,rs2;λ)+(m3n1+n4)Hyy(s)(rs1,rs2;λ)],
S1(s)(rs1,rs2;λ)=1r2[(m3n4n1)Hxx(s)(rs1,rs2;λ)+(m3n3+n2)Hxy(s)(rs1,rs2;λ)+(m3n2+n3)Hyx(s)(rs1,rs2;λ)+(m3n1n4)Hyy(s)(rs1,rs2;λ)],
S2(s)(rs1,rs2;λ)=1r2[(m1n3m2n2)Hxx(s)(rs1,rs2;λ)+(m1n4m2n1)Hxy(s)(rs1,rs2;λ)+(m1n1+m2n4)Hyx(s)(rs1,rs2;λ)+(m1n2+m2n3)Hyy(s)(rs1,rs2;λ)],
S3(s)(rs1,rs2;λ)=1r2[i(m2n2+m1n3)Hxx(s)(rs1,rs2;λ)+i(m2n1m1n4)Hxy(s)(rs1,rs2;λ)+i(m2n4+m1n1)Hyx(s)(rs1,rs2;λ)+i(m2n3m1n2)Hyy(s)(rs1,rs2;λ)],
where m1=cosθ1, m2=cosθ2, m3=cosθ1cosθ2, n1=sinφ1sinφ2, n2=sinφ1cosφ2, n3=cosφ1sinφ2, n4=cosφ1cosφ2,
Hij(s)(rs1,rs2;λ)=|s01|21|s02|21Aij(s01,s02;λ)C˜F(K1,K2;λ)d2s01d2s02,
and
C˜F(K1,K2;λ)=DDCF(r1,r2;λ)exp[i(K2r2K1r1)]d3r1d3r2
is the 6D spatial Fourier transform of the correlation function of a random medium, with K1=k(s1s01) and K2=k(s2s02) being the momentum transfer vectors.

Based on the generalized Stokes parameters of the far-zone scattered field, the spectral density, spectral degree of coherence and spectral degree of polarization can be obtained by the formulas [29]:

S(s)(rs;λ)=S0(s)(rs,rs;λ),
μ(s)(rs1,rs2;λ)=S0(s)(rs1,rs2;λ)S0(s)(rs1,rs1;λ)S0(s)(rs2,rs2;λ),
P(s)(rs;λ)=α=13Sα(s)2(rs,rs;λ)S0(s)(rs,rs;λ).

Assume that the scatterer is a SQH medium and the illumination is a classic electromagnetic Gaussian Schell-model beam. In the previous section, we have introduced the correlation function of the scattering potential of a SQH medium in Eq. (9). On substituting from Eq. (9) into Eq. (33), we get

C˜F(K1,K2;λ)=C0(2π)3σI3(λ)σμ3(λ)exp[σI2(λ)2(K1Κ2)2]×exp[σμ2(λ)8(K1+Κ2)2].

For a classic electromagnetic Gaussian Schell-model beam, the element of the cross-spectral density matrix at the source plane has the form [28]

Wij(0)(ρ1,ρ2;λ)=AiAjBijexp(ρ12+ρ224σ2)exp[(ρ2ρ1)22δij2],
where σ and δij are the beam size and correlation width of the source, respectively. Ai is the amplitude and Bij is the single-point correlation coefficient which has the following properties [28]:
Bij=1,wheni=j,|Bij|1,whenij,Bij=Bji.
For simplicity, we suppose that the two electric-field components of the source plane are uncorrelated, that is, Bij=0 (when ij).

Besides, in order for the source to be physically realizable, some constrains must be satisfied to guarantee that the source generates a beam-like field and the cross-spectral density matrix is non-negative, viz., [28, 31]

14σ2+1δij22π2λ2,(i,j=x,y),
and

δxx2+δyy22δxyδxxδyy|Bxy|.

The angular correlation function of the incident field can be given by substituting from Eq. (38) into Eq. (19), and it is expressed as

Aij(s01,s02;λ)=AiAjBijk4σ2σij2(2π)2exp[k2σ22(s01s02)2]×exp[k2σij28(s01+s02)2],
where

1σij2=14σ2+1δij2.

On substituting from Eqs. (37) and (42) into Eq. (32), after some calculation, we obtain the expression

Hij(s)(rs1,rs2;λ)=AiAjBijC0k2(2π)2σ2σij2σI3(λ)σμ3(λ)×|s02|211aexp[k2(b2+c2)2a]exp[k2(σ2+σij2/4)2s022]×exp{k2σI2(λ)2[(s1s2+s02)2+(m1m2+1s0221)2]}×exp{k2σμ2(λ)8[(s1+s2s02)2+(m1+m21s0221)2]}d2s02,
where

a=σ2+σij24+σI2(λ)(m1m2+1s022)+σμ2(λ)4(m1+m21s022),
b=(σ2σij24)sinθ02cosφ02+σI2(λ)(sinθ1cosφ1sinθ2cosφ2+sinθ02cosφ02)+σμ2(λ)4(sinθ1cosφ1+sinθ2cosφ2sinθ02cosφ02),
c=(σ2σij24)sinθ02sinφ02+σI2(λ)(sinθ1sinφ1sinθ2sinφ2+sinθ02sinφ02)+σμ2(λ)4(sinθ1sinφ1+sinθ2sinφ2sinθ02sinφ02).

Then the generalized Stokes parameters of the far-zone scattered field can be obtained by substituting from Eq. (44) into Eqs. (28)-(31). With the help of the generalized Stokes parameters, we finally get the expressions for the spectral density, spectral degree of coherence and spectral degree of polarization of the scattered field.

4. Numerical simulations

In this section, we will investigate the scattering from a SQH medium by numerical examples. Specifically, the behaviors of the spectral density, spectral degree of coherence and spectral degree of polarization of the scattered field will be discussed in detail. Unless specified in figure captions, the parameters are chosen as follows: φ=0, λ=0.5μm, Λσ=Λδ=λ/4, Ax=2, Ay=1, σ=20λ, δxx=10λ and δyy=5λ.

When a laser beam is scattered by a medium, the modulation of the scattered field consists of two parts: one is from the medium, and the other is from the beam. The proportion of each part changes under different circumstances. Figures 1 and 2 show the distributions of the normalized spectral density as a function of the scattering direction θ with the variation of a medium parameter and a beam parameter, respectively. The influences of these parameters on the scattered field are illustrated under two opposite conditions: in the left columns of the figures, the values of σμ(λ) are set to be on the order of the wavelength; while in the right columns, the values are set to be much larger than λ.

 figure: Fig. 1

Fig. 1 The influence of the central wavelength of the correlation length λσμ of the medium on the far-zone intensity distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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 figure: Fig. 2

Fig. 2 The influence of the correlation width δ of the beam on the far-zone intensity distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, λσμ=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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Through the comparison of Figs. 1 and 2, we can find that σμ(λ) is a key factor in determining the proportions of the two parts in the modulation. When σμ(λ) is close to λ [see Figs. 1(a) and 2(a)], the modulation of the scattered field is dominated by the medium. In this situation, the effect of the medium parameter is obvious, while the effect of the beam parameter is negligible. However, when σμ(λ) becomes much larger than λ [see Figs. 1(b) and 2(b)], the medium-led modulation is transformed into the beam-led modulation, and the effect of the beam parameter plays a major role in the scattered field. Besides, it can be found that compared with the situation of beam-led modulation, when the scattered field is mainly modulated by the medium, the spectral density has a much broader distribution. The explanation for this phenomenon is that in the medium-led modulation mode the scattering strength approaches a peak value, so the incident light is scattered to a larger range.

Figure 1 illustrates the influence of the central wavelength of the correlation length of the medium on the spectral density profile. For a SQH medium of which the statistical properties depend on the wavelength of the incident light, the parameter λσμ is very important for the modulation of the scattered field. The reason is that the deviation of λσμ from the wavelength can affect the value of the correlation length of the medium, which may lead to changes in the scattered field. It is shown that as the deviation becomes larger, the distribution area of the spectral density turns wider. Figure 2 presents the effect of the correlation width of the beam on the intensity profile. It can be seen that a wider intensity distribution can also be made with a smaller correlation width of the beam.

In Figs. 3 and 4, the spectral degrees of coherence of the scattered field are investigated under the two situations consistent with Figs. 1 and 2. It is found that due to the stronger scattering strength, the distribution range of the coherence in the situation of medium-led modulation is much broader than that in the situation of beam-led modulation. The influence of the central wavelength of the effective radius of the medium on the coherence is shown in Fig. 3. Just like λσμ, parameter λσI also plays a significant role in the modulation of the scattered field, because the effective radius of the medium, which is a direct acting factor in the modulation, is associated with the deviation between λσI and the wavelength. It can be seen that for a smaller value of the deviation there is a stronger attenuation of the coherence. From Fig. 4 we can find that the coherence is also affected by the beam width. The larger σ is, the faster the degree of coherence decays.

 figure: Fig. 3

Fig. 3 The influence of the central wavelength of the effective radius λσI of the medium on the far-zone coherence distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσμ=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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 figure: Fig. 4

Fig. 4 The influence of the beam size σ on the far-zone coherence distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, λσμ=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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Next we pay attention to the changes in the spectral degree of polarization of the scattered field. Likewise, the two opposite situations and their placements in Figs. 1 and 2 apply to the rest of figures in this paper. For the incident beams with different degrees of polarization, the distributions of polarization of the scattered field are illustrated in Fig. 5. It is shown that if the incident beam is linearly polarized, i.e., Ax=1,Ay=0, the polarization of the scattered field keeps invariant as unity. However, if the incident beam is uniformly unpolarized, i.e., Ax=1,Ay=1, or partially polarized, i.e., Ax=2,Ay=1, the polarization of the scattered field experiences more complicated changes with increasing scattering angle θ. In general, when θ=0, the polarization has a value less than unity; as θ increases, the polarization first declines to a minimum value around zero and then rises to a maximum value as unity. Besides, compared with a partially polarized incident beam, for a uniformly unpolarized beam, the degree of polarization at θ=0 is smaller, and with the growth of θ it will reduce to zero at a smaller θ.

 figure: Fig. 5

Fig. 5 The far-zone polarization distributions for different degrees of polarization of the incident beam under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, λσμ=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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From Fig. 5 it can also be found that even in the situation of medium-led modulation, for different degrees of polarization of the incident beam, the far-zone polarization distributions are different. Compared with the beam-led modulation mode, the polarization of the central point in the far-zone has a smaller value in the medium-led modulation mode, but it takes a much longer distance for the polarization to experience changes and finally reach to unity. Besides, the differences between the cases of a partially polarized incident beam and a uniformly unpolarized beam are greater in the medium-led modulation mode.

We now take the case of a partially polarized incident beam for example to study the influences of the parameters of the medium and the beam on the far-zone polarization distribution. As we have mentioned before, the central wavelength of the correlation length of the medium plays an important role in the modulation of the scattered field. It is shown in Fig. 6 that as the deviation of λσμ from the wavelength decreases, the polarization first declines at a faster rate and then grows at a slower rate. Figure 7 illustrates the effect of the correlation width of the beam on the polarization of the scattered field. As can be seen in Fig. 7(b), if the beam is anisotropic, i.e., δxxδyy, for different values of δxx or δyy the degrees of polarization at θ=0 are different, so are the decreasing rate and the increasing rate. However, if the beam is isotropic, i.e., δxx=δyy, the polarization of the scattered field remains invariant and it is equal to the polarization of the incident beam. In addition, the major differences between the cases of isotropic beams and anisotropic beams are not demonstrated in Fig. 7(a), which also indicates that the scattered field is mainly modulated by the medium in the situation of σμ(λ) being on the order of the wavelength.

 figure: Fig. 6

Fig. 6 The influence of the central wavelength of the correlation length λσμ of the medium on the far-zone polarization distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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 figure: Fig. 7

Fig. 7 The influence of the correlation width δ of the beam on the far-zone polarization distribution under two situations: σμ(λ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λσI=0.5μm, λσμ=0.5μm, σI0=10λ (the left column), σI0=30λ (the right column).

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5. Conclusions

In summary, we have introduced a generalization of a classic random medium. The dependence of a medium on the wavelength of an incident light has been taken into consideration, and we term the generalized medium as a spectral random medium. Based on the particular model of Gaussian-correlated QH medium, the effective radius and the correlation length are assumed to be arbitrary functions of the wavelength, and we take the forms of Gaussian functions as an example in this paper. With the help of the generalized Stokes parameters of random electromagnetic beams, we have derived the expressions for the spectral density, the spectral degree of coherence and the spectral degree of polarization of the far-zone scattered field generated by a spectral random medium. Numerical examples reveal that the dependence of the effective radius and correlation length on the wavelength has important influences on the behaviors of the scattered field. In particular, the distributions of the intensity, coherence and polarization can be modulated by the central wavelengths at which the effective radius and correlation length reach their maximum values.

In addition, it is also found that these distributions have many differences between the situations of medium-led modulation and beam-led modulation. To be specific, the intensity and coherence have much broader distributions in the medium-led modulation mode. The reason is that the scattering strength approaches its peak value in this mode, which causes the scattered light to spread to a larger range. As for the distribution of the polarization, if the incident beam is linearly polarized, the polarization keeps invariant as unity in the two modes; if the incident beam is not linearly polarized, with the growth of θ it takes a much longer distance for the polarization to experience changes and finally reach to unity in the medium-led modulation mode.

Our results have potential applications in two aspects. One is the modulation and control of random electromagnetic beams; the other is the inverse scattering problem to determine the structural information about a more realistic medium with spectral dependence.

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) (11474253 and 11274273) and the Fundamental Research Funds for the Central Universities (2017FZA3005).

References and links

1. 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]  

2. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008). [CrossRef]  

3. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009). [CrossRef]   [PubMed]  

4. 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]  

5. D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012). [CrossRef]  

6. Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012). [CrossRef]   [PubMed]  

7. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). [CrossRef]   [PubMed]  

8. X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35(10), 1518–1520 (2010). [CrossRef]   [PubMed]  

9. S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011). [CrossRef]   [PubMed]  

10. O. Korotkova, S. Sahin, and E. Shchepakina, “Light scattering by three-dimensional objects with semi-hard boundaries,” J. Opt. Soc. Am. A 31(8), 1782–1787 (2014). [CrossRef]   [PubMed]  

11. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016). [CrossRef]   [PubMed]  

12. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015). [CrossRef]   [PubMed]  

13. O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015). [CrossRef]   [PubMed]  

14. G. Zheng, D. Ye, X. Peng, M. Song, and Q. Zhao, “Tunable scattering intensity with prescribed weak media,” Opt. Express 24(21), 24169–24178 (2016). [CrossRef]   [PubMed]  

15. 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]  

16. J. Li, “Determination of correlation function of scattering potential of random medium by Gaussian vortex beam,” Opt. Commun. 308, 164–168 (2013). [CrossRef]  

17. Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013). [CrossRef]   [PubMed]  

18. Y. Zhang and D. Zhao, “The coherence and polarization properties of electromagnetic rectangular Gaussian Schell-model sources scattered by a deterministic medium,” J. Opt. 16(12), 125709 (2014). [CrossRef]  

19. X. Wang, Z. Liu, K. Huang, and D. Zhu, “Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium,” J. Opt. Soc. Am. A 33(10), 1955–1960 (2016). [CrossRef]   [PubMed]  

20. X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017). [CrossRef]  

21. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef]   [PubMed]  

22. Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008). [CrossRef]   [PubMed]  

23. Z. Mei, “Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams,” Opt. Express 18(22), 22826–22832 (2010). [CrossRef]   [PubMed]  

24. Z. Mei, “Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams,” Opt. Express 18(26), 27105–27111 (2010). [CrossRef]   [PubMed]  

25. R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014). [CrossRef]  

26. E. Shchepakina and O. Korotkova, “Spectral Gaussian Schell-model beams,” Opt. Lett. 38(13), 2233–2236 (2013). [CrossRef]   [PubMed]  

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

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

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

30. T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012). [CrossRef]  

31. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]   [PubMed]  

References

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  1. 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]
  2. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
    [Crossref]
  3. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
    [Crossref] [PubMed]
  4. 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]
  5. D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
    [Crossref]
  6. Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
    [Crossref] [PubMed]
  7. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010).
    [Crossref] [PubMed]
  8. X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35(10), 1518–1520 (2010).
    [Crossref] [PubMed]
  9. S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
    [Crossref] [PubMed]
  10. O. Korotkova, S. Sahin, and E. Shchepakina, “Light scattering by three-dimensional objects with semi-hard boundaries,” J. Opt. Soc. Am. A 31(8), 1782–1787 (2014).
    [Crossref] [PubMed]
  11. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
    [Crossref] [PubMed]
  12. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015).
    [Crossref] [PubMed]
  13. O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015).
    [Crossref] [PubMed]
  14. G. Zheng, D. Ye, X. Peng, M. Song, and Q. Zhao, “Tunable scattering intensity with prescribed weak media,” Opt. Express 24(21), 24169–24178 (2016).
    [Crossref] [PubMed]
  15. 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]
  16. J. Li, “Determination of correlation function of scattering potential of random medium by Gaussian vortex beam,” Opt. Commun. 308, 164–168 (2013).
    [Crossref]
  17. Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013).
    [Crossref] [PubMed]
  18. Y. Zhang and D. Zhao, “The coherence and polarization properties of electromagnetic rectangular Gaussian Schell-model sources scattered by a deterministic medium,” J. Opt. 16(12), 125709 (2014).
    [Crossref]
  19. X. Wang, Z. Liu, K. Huang, and D. Zhu, “Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium,” J. Opt. Soc. Am. A 33(10), 1955–1960 (2016).
    [Crossref] [PubMed]
  20. X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
    [Crossref]
  21. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
    [Crossref] [PubMed]
  22. Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
    [Crossref] [PubMed]
  23. Z. Mei, “Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams,” Opt. Express 18(22), 22826–22832 (2010).
    [Crossref] [PubMed]
  24. Z. Mei, “Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams,” Opt. Express 18(26), 27105–27111 (2010).
    [Crossref] [PubMed]
  25. R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
    [Crossref]
  26. E. Shchepakina and O. Korotkova, “Spectral Gaussian Schell-model beams,” Opt. Lett. 38(13), 2233–2236 (2013).
    [Crossref] [PubMed]
  27. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University, 1999).
  28. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  29. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  30. T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012).
    [Crossref]
  31. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
    [Crossref] [PubMed]

2017 (1)

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

2016 (3)

2015 (2)

2014 (3)

O. Korotkova, S. Sahin, and E. Shchepakina, “Light scattering by three-dimensional objects with semi-hard boundaries,” J. Opt. Soc. Am. A 31(8), 1782–1787 (2014).
[Crossref] [PubMed]

Y. Zhang and D. Zhao, “The coherence and polarization properties of electromagnetic rectangular Gaussian Schell-model sources scattered by a deterministic medium,” J. Opt. 16(12), 125709 (2014).
[Crossref]

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

2013 (3)

2012 (3)

D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
[Crossref]

Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
[Crossref] [PubMed]

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012).
[Crossref]

2011 (2)

2010 (5)

2009 (1)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

2008 (3)

2006 (1)

2005 (1)

Borghi, R.

Cai, Y.

Ding, C.

Du, X.

Fischer, D. G.

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]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

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]

Gbur, G.

Gori, F.

Huang, K.

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

X. Wang, Z. Liu, K. Huang, and D. Zhu, “Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium,” J. Opt. Soc. Am. A 33(10), 1955–1960 (2016).
[Crossref] [PubMed]

Itou, H.

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

Korotkova, O.

J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
[Crossref] [PubMed]

O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015).
[Crossref] [PubMed]

O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Light scattering by three-dimensional objects with semi-hard boundaries,” J. Opt. Soc. Am. A 31(8), 1782–1787 (2014).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Spectral Gaussian Schell-model beams,” Opt. Lett. 38(13), 2233–2236 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
[Crossref] [PubMed]

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]

S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
[Crossref]

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[Crossref] [PubMed]

Lahiri, M.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

Li, J.

J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
[Crossref] [PubMed]

J. Li, “Determination of correlation function of scattering potential of random medium by Gaussian vortex beam,” Opt. Commun. 308, 164–168 (2013).
[Crossref]

Liu, Z.

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

X. Wang, Z. Liu, K. Huang, and D. Zhu, “Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium,” J. Opt. Soc. Am. A 33(10), 1955–1960 (2016).
[Crossref] [PubMed]

Mei, Z.

Miyamoto, Y.

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

Naik, D. N.

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

Pan, L.

Peng, X.

Ramírez-Sánchez, V.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Shirai, T.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

Singh, R. K.

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

Song, M.

Sun, J.

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Takeda, M.

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

van Dijk, T.

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]

Visser, T. D.

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]

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]

Wang, T.

D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
[Crossref]

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012).
[Crossref]

Wang, X.

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

X. Wang, Z. Liu, K. Huang, and D. Zhu, “Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium,” J. Opt. Soc. Am. A 33(10), 1955–1960 (2016).
[Crossref] [PubMed]

Wolf, E.

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]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

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]

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[Crossref] [PubMed]

Ye, D.

Zhang, Y.

Zhao, D.

Zhao, Q.

Zheng, G.

Zhu, D.

Zhu, Y.

J. Opt. (2)

Y. Zhang and D. Zhao, “The coherence and polarization properties of electromagnetic rectangular Gaussian Schell-model sources scattered by a deterministic medium,” J. Opt. 16(12), 125709 (2014).
[Crossref]

R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Characterization of spatial polarization fluctuations in scattered field,” J. Opt. 16(10), 105010 (2014).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

J. Li, “Determination of correlation function of scattering potential of random medium by Gaussian vortex beam,” Opt. Commun. 308, 164–168 (2013).
[Crossref]

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012).
[Crossref]

X. Wang, Z. Liu, K. Huang, and J. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Opt. Express (5)

Opt. Lett. (9)

Phys. Rev. A (1)

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
[Crossref]

Phys. Rev. Lett. (2)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

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]

Prog. Opt. (1)

D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
[Crossref]

Other (3)

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

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

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

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Figures (7)

Fig. 1
Fig. 1 The influence of the central wavelength of the correlation length λ σ μ of the medium on the far-zone intensity distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 2
Fig. 2 The influence of the correlation width δ of the beam on the far-zone intensity distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , λ σ μ = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 3
Fig. 3 The influence of the central wavelength of the effective radius λ σ I of the medium on the far-zone coherence distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ μ = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 4
Fig. 4 The influence of the beam size σ on the far-zone coherence distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , λ σ μ = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 5
Fig. 5 The far-zone polarization distributions for different degrees of polarization of the incident beam under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , λ σ μ = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 6
Fig. 6 The influence of the central wavelength of the correlation length λ σ μ of the medium on the far-zone polarization distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).
Fig. 7
Fig. 7 The influence of the correlation width δ of the beam on the far-zone polarization distribution under two situations: σ μ ( λ ) being on the order of λ (the left column) and being much larger than λ (the right column). The other parameters are chosen as follows: λ σ I = 0.5 μ m , λ σ μ = 0.5 μ m , σ I 0 = 10 λ (the left column), σ I 0 = 30 λ (the right column).

Equations (47)

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F ( r ; ω ) = k 2 [ n 2 ( r ; ω ) 1 ] 4 π ,
C F ( r 1 , r 2 ; ω ) = F ( r 1 ; ω ) F ( r 2 ; ω ) m ,
C F ( r 1 , r 2 ; ω ) = I F ( r 1 ; ω ) I F ( r 2 ; ω ) μ F ( r 1 , r 2 ; ω ) ,
I F ( r ; ω ) = C F ( r , r ; ω )
μ F ( r 1 , r 2 ; ω ) = C F ( r 1 , r 2 ; ω ) I F ( r 1 ; ω ) I F ( r 2 ; ω )
C F ( r 1 , r 2 ; ω ) = I F ( r 1 + r 2 2 ; ω ) μ F ( r 2 r 1 ; ω ) .
C F ( r 1 , r 2 ; ω ) = C 0 exp [ ( r 1 + r 2 ) 2 8 σ I 2 ] exp [ ( r 2 r 1 ) 2 2 σ μ 2 ] ,
C F ( r 1 , r 2 ; ω ) = C 0 exp [ ( r 1 + r 2 ) 2 8 σ I 2 ( ω ) ] exp [ ( r 2 r 1 ) 2 2 σ μ 2 ( ω ) ] ,
C F ( r 1 , r 2 ; λ ) = C 0 exp [ ( r 1 + r 2 ) 2 8 σ I 2 ( λ ) ] exp [ ( r 2 r 1 ) 2 2 σ μ 2 ( λ ) ] .
σ I ( λ ) = σ I 0 exp [ ( λ λ σ I ) 2 2 Λ σ I 2 ] ,
σ μ ( λ ) = σ μ 0 exp [ ( λ λ σ μ ) 2 2 Λ σ μ 2 ] ,
C F ( r 1 , r 2 ; λ ) = p ( v ; λ ) H 0 ( r 1 , v ; λ ) H 0 ( r 2 , v ; λ ) d 3 v ,
p ( v ; λ ) μ ˜ F ( r 2 r 1 ; λ ) ,
μ F ( r 2 r 1 ; λ ) = exp [ ( r 2 r 1 ) 2 2 σ μ 2 ( λ ) ] .
p ( v ; λ ) ( 2 π ) 3 / 2 σ μ 3 ( λ ) e x p [ σ μ 2 ( λ ) v 3 2 ] 0.
W ( i ) ( r 1 , r 2 , s 0 ; λ ) [ W i j ( i ) ( r 1 , r 2 , s 0 ; λ ) ] = [ E i ( i ) ( r 1 , s 0 ; λ ) E j ( i ) ( r 2 , s 0 ; λ ) ] , ( i , j = x , y ) ,
E i ( i ) ( r , s 0 ; λ ) = | s 0 | 2 1 a i ( s 0 ; λ ) exp ( i k s 0 r ) d 2 s 0 ,
W i j ( i ) ( r 1 , r 2 , s 01 , s 02 ; λ ) = | s 01 | 2 1 | s 02 | 2 1 A i j ( s 01 , s 02 ; λ ) × exp [ i k ( s 01 r 1 s 02 r 2 ) ] d 2 s 01 d 2 s 02 ,
A i j ( s 01 , s 02 ; λ ) = ( k 2 π ) 4 + W i j ( 0 ) ( ρ 1 , ρ 2 ; λ ) × exp [ i k ( s 02 ρ 2 s 01 ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 ,
E θ ( s ) ( r s ; λ ) = D F ( r ; λ ) G ( r s , r ; λ ) × [ cos θ cos φ E x ( i ) ( r , s 0 ; λ ) + cos θ s i n φ E y ( i ) ( r , s 0 ; λ ) ] d 3 r ,
E φ ( s ) ( r s ; λ ) = D F ( r ; λ ) G ( r s , r ; λ ) × [ sin φ E x ( i ) ( r , s 0 ; λ ) + cos φ E y ( i ) ( r , s 0 ; λ ) ] d 3 r ,
E r ( s ) ( r s ; λ ) = 0 ,
G ( r s , r ; λ ) exp ( i k r ) r exp ( i k s r ) .
S 0 ( s ) ( r s 1 , r s 2 ; λ ) = E θ ( s ) ( r s 1 ; λ ) E θ ( s ) ( r s 2 ; λ ) + E φ ( s ) ( r s 1 ; λ ) E φ ( s ) ( r s 2 ; λ ) ,
S 1 ( s ) ( r s 1 , r s 2 ; λ ) = E θ ( s ) ( r s 1 ; λ ) E θ ( s ) ( r s 2 ; λ ) E φ ( s ) ( r s 1 ; λ ) E φ ( s ) ( r s 2 ; λ ) ,
S 2 ( s ) ( r s 1 , r s 2 ; λ ) = E θ ( s ) ( r s 1 ; λ ) E φ ( s ) ( r s 2 ; λ ) + E φ ( s ) ( r s 1 ; λ ) E θ ( s ) ( r s 2 ; λ ) ,
S 3 ( s ) ( r s 1 , r s 2 ; λ ) = i [ E φ ( s ) ( r s 1 ; λ ) E θ ( s ) ( r s 2 ; λ ) E θ ( s ) ( r s 1 ; λ ) E φ ( s ) ( r s 2 ; λ ) ] .
S 0 ( s ) ( r s 1 , r s 2 ; λ ) = 1 r 2 [ ( m 3 n 4 + n 1 ) H x x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 3 n 2 ) H x y ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 2 n 3 ) H y x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 1 + n 4 ) H y y ( s ) ( r s 1 , r s 2 ; λ ) ] ,
S 1 ( s ) ( r s 1 , r s 2 ; λ ) = 1 r 2 [ ( m 3 n 4 n 1 ) H x x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 3 + n 2 ) H x y ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 2 + n 3 ) H y x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 3 n 1 n 4 ) H y y ( s ) ( r s 1 , r s 2 ; λ ) ] ,
S 2 ( s ) ( r s 1 , r s 2 ; λ ) = 1 r 2 [ ( m 1 n 3 m 2 n 2 ) H x x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 1 n 4 m 2 n 1 ) H x y ( s ) ( r s 1 , r s 2 ; λ ) + ( m 1 n 1 + m 2 n 4 ) H y x ( s ) ( r s 1 , r s 2 ; λ ) + ( m 1 n 2 + m 2 n 3 ) H y y ( s ) ( r s 1 , r s 2 ; λ ) ] ,
S 3 ( s ) ( r s 1 , r s 2 ; λ ) = 1 r 2 [ i ( m 2 n 2 + m 1 n 3 ) H x x ( s ) ( r s 1 , r s 2 ; λ ) + i ( m 2 n 1 m 1 n 4 ) H x y ( s ) ( r s 1 , r s 2 ; λ ) + i ( m 2 n 4 + m 1 n 1 ) H y x ( s ) ( r s 1 , r s 2 ; λ ) + i ( m 2 n 3 m 1 n 2 ) H y y ( s ) ( r s 1 , r s 2 ; λ ) ] ,
H i j ( s ) ( r s 1 , r s 2 ; λ ) = | s 01 | 2 1 | s 02 | 2 1 A i j ( s 01 , s 02 ; λ ) C ˜ F ( K 1 , K 2 ; λ )d 2 s 01 d 2 s 02 ,
C ˜ F ( K 1 , K 2 ; λ )= D D C F ( r 1 , r 2 ; λ ) exp [ i ( K 2 r 2 K 1 r 1 ) ] d 3 r 1 d 3 r 2
S ( s ) ( r s ; λ ) = S 0 ( s ) ( r s , r s ; λ ) ,
μ ( s ) ( r s 1 , r s 2 ; λ ) = S 0 ( s ) ( r s 1 , r s 2 ; λ ) S 0 ( s ) ( r s 1 , r s 1 ; λ ) S 0 ( s ) ( r s 2 , r s 2 ; λ ) ,
P ( s ) ( r s ; λ ) = α = 1 3 S α ( s ) 2 ( r s , r s ; λ ) S 0 ( s ) ( r s , r s ; λ ) .
C ˜ F ( K 1 , K 2 ; λ )= C 0 ( 2 π ) 3 σ I 3 ( λ ) σ μ 3 ( λ ) exp [ σ I 2 ( λ ) 2 ( K 1 Κ 2 ) 2 ] × exp [ σ μ 2 ( λ ) 8 ( K 1 + Κ 2 ) 2 ] .
W i j ( 0 ) ( ρ 1 , ρ 2 ; λ ) = A i A j B i j exp ( ρ 1 2 + ρ 2 2 4 σ 2 ) exp [ ( ρ 2 ρ 1 ) 2 2 δ i j 2 ] ,
B i j = 1 , when i = j , | B i j | 1 , when i j , B i j = B j i .
1 4 σ 2 + 1 δ i j 2 2 π 2 λ 2 , ( i , j = x , y ) ,
δ x x 2 + δ y y 2 2 δ x y δ x x δ y y | B x y | .
A i j ( s 01 , s 02 ; λ ) = A i A j B i j k 4 σ 2 σ i j 2 ( 2 π ) 2 exp [ k 2 σ 2 2 ( s 01 s 02 ) 2 ] × exp [ k 2 σ i j 2 8 ( s 01 + s 02 ) 2 ] ,
1 σ i j 2 = 1 4 σ 2 + 1 δ i j 2 .
H i j ( s ) ( r s 1 , r s 2 ; λ ) = A i A j B i j C 0 k 2 ( 2 π ) 2 σ 2 σ i j 2 σ I 3 ( λ ) σ μ 3 ( λ ) × | s 02 | 2 1 1 a exp [ k 2 ( b 2 + c 2 ) 2 a ] exp [ k 2 ( σ 2 + σ i j 2 / 4 ) 2 s 02 2 ] × exp { k 2 σ I 2 ( λ ) 2 [ ( s 1 s 2 + s 02 ) 2 + ( m 1 m 2 + 1 s 02 2 1 ) 2 ] } × exp { k 2 σ μ 2 ( λ ) 8 [ ( s 1 + s 2 s 02 ) 2 + ( m 1 + m 2 1 s 02 2 1 ) 2 ] } d 2 s 02 ,
a = σ 2 + σ i j 2 4 + σ I 2 ( λ ) ( m 1 m 2 + 1 s 02 2 ) + σ μ 2 ( λ ) 4 ( m 1 + m 2 1 s 02 2 ) ,
b = ( σ 2 σ i j 2 4 ) sin θ 02 cos φ 02 + σ I 2 ( λ ) ( sin θ 1 cos φ 1 sin θ 2 cos φ 2 + sin θ 02 cos φ 02 ) + σ μ 2 ( λ ) 4 ( sin θ 1 cos φ 1 + sin θ 2 cos φ 2 sin θ 02 cos φ 02 ) ,
c = ( σ 2 σ i j 2 4 ) sin θ 02 sin φ 02 + σ I 2 ( λ ) ( sin θ 1 sin φ 1 sin θ 2 sin φ 2 + sin θ 02 sin φ 02 ) + σ μ 2 ( λ ) 4 ( sin θ 1 sin φ 1 + sin θ 2 sin φ 2 sin θ 02 sin φ 02 ) .

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