## 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]

where $n(r;\omega )$ is the refractive index of the medium, with $r$ being the position vector within the medium and $\omega $ being the frequency of the incident light. $k=2\pi /\lambda $ is the wave number and $\lambda $ 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]

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

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 ${\mu}_{F}({r}_{1},{r}_{2};\omega )$ depends on the two points ${r}_{1}$ and ${r}_{2}$ only through their difference ${r}_{2}-{r}_{1}$; besides, the strength of the scattering potential ${I}_{F}(r;\omega )$ varies much more slowly with the point $r$ than the correlation coefficient ${\mu}_{F}({r}_{1},{r}_{2};\omega )$ does with ${r}_{2}-{r}_{1}$. Therefore, the correlation function of the scattering potential of QH media can be given by [1]

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

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, ${\sigma}_{I0}$ and ${\sigma}_{\mu 0}$, 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 ${C}_{F}$ must have the following integral representation:

## 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 ${s}_{0}$, that is incident on a medium occupying a finite domain *D*. The second-order properties of the incident field can be characterized by a $2\times 2$ cross-spectral density matrix, which has the definition [28]

Using the angular spectrum representation of plane waves, ${E}_{i}^{(i)}({r}^{\prime},{s}_{0};\lambda )$ can be expressed as the integral form [29]

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

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]

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 $r{s}_{1}$ and $r{s}_{2}$ in spherical coordinates can be defined as the following formulas:

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

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]:

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

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

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]

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

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

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: $\phi =0$, $\lambda =0.5\mu m$, ${\Lambda}_{\sigma}={\Lambda}_{\delta}=\lambda /4$, ${A}_{x}=2$, ${A}_{y}=1$, $\sigma =20\lambda $, ${\delta}_{xx}=10\lambda $ and ${\delta}_{yy}=5\lambda $.

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 $\theta $ 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 ${\sigma}_{\mu}(\lambda )$ are set to be on the order of the wavelength; while in the right columns, the values are set to be much larger than $\lambda $.

Through the comparison of Figs. 1 and 2, we can find that ${\sigma}_{\mu}(\lambda )$ is a key factor in determining the proportions of the two parts in the modulation. When ${\sigma}_{\mu}(\lambda )$ is close to $\lambda $ [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 ${\sigma}_{\mu}(\lambda )$ becomes much larger than $\lambda $ [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 ${\lambda}_{\sigma \mu}$ is very important for the modulation of the scattered field. The reason is that the deviation of ${\lambda}_{\sigma \mu}$ 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 ${\lambda}_{\sigma \mu}$, parameter ${\lambda}_{\sigma I}$ 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 ${\lambda}_{\sigma I}$ 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 $\sigma $ is, the faster the degree of coherence decays.

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., ${A}_{x}=1,{A}_{y}=0$, the polarization of the scattered field keeps invariant as unity. However, if the incident beam is uniformly unpolarized, i.e., ${A}_{x}=1,{A}_{y}=1$, or partially polarized, i.e., ${A}_{x}=2,{A}_{y}=1$, the polarization of the scattered field experiences more complicated changes with increasing scattering angle $\theta $. In general, when $\theta =0$, the polarization has a value less than unity; as $\theta $ 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 $\theta =0$ is smaller, and with the growth of $\theta $ it will reduce to zero at a smaller $\theta $.

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 ${\lambda}_{\sigma \mu}$ 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., ${\delta}_{xx}\ne {\delta}_{yy}$, for different values of ${\delta}_{xx}$ or ${\delta}_{yy}$ the degrees of polarization at $\theta =0$ are different, so are the decreasing rate and the increasing rate. However, if the beam is isotropic, i.e., ${\delta}_{xx}={\delta}_{yy}$, 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 ${\sigma}_{\mu}(\lambda )$ being on the order of the wavelength.

## 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 $\theta $ 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]