1. Introduction

Light as an electromagnetic wave carries a certain momentum and therefore can exert the optical forces on material particles which stand in its propagation path. These optical forces have been used experimentally to manipulate atoms, molecules and even micrometer-sized particles, which gives rise to the field of optical micromanipulation, including optical trapping [1], optical binding [2] and optical pushing [3] and so on. In optical micromanipulation, the realization of the optimization of optical forces usually requires not only to determine and control the momentum flow of the light field falling on the particles, but also to explore the distribution characteristics of the momentum flow of light scattering from the particles. In this regard, Kim and Gbur considered momentum flow in a general scattering formalism [4]. The authors pointed out that the net momentum flow into a finite spatial volume containing the scatterer not only relies on the scattered pattern generated by the scatterer but also on the spatial coherence between the incident and scattered wave. Subsequently, Tong and Korotkova, within the accuracy of the first-order Born approximation, systematically discussed the angular distribution of the momentum flow of the far field produced by scattering of a polychromatic electromagnetic plane wave on a system of isotropic particles [5]. It has been shown that the polarization properties of the incident electromagnetic wave, the size of particles, the separation between particles and their position distribution characteristics (deterministic or random) are important factors that affect the angular distribution of the scattered momentum flow in the far zone. Very recently, the momentum flow of an electromagnetic light wave on scattering from a homogeneous, isotropic, and Gaussian-correlated sphere has been discussed. The results have shown that the angular distribution of the scattered momentum flow behaves predominantly in the forward direction, but it becomes more diffuse as the spatial coherence radius of the sphere decreases [6]. These findings provide useful guidance for regulating and utilizing the momentum flow of light scattering from a system of isotropic particles or a homogeneous sphere.

On the other hand, as an important extension of the traditional isotropic scatterer, the spatially anisotropic scatterer recently has attracted considerable interest (e.g., see Refs. [7–22] and references therein) due to its elegant ability to mimic many practical scatterers, such as ellipsoids. The anisotropic scatterer model was introduced into the classical potential scattering theory due to the classic paper by Du and Zhao on light scattering from a Gaussian-correlated, quasi-homogeneous, anisotropic medium [7]. In this work, the differences in light scattering between isotropic and anisotropic media were compared in detail. It was found, unlike the distribution of light scattered from an isotropic scatterer only relies on the angle between the directions of incidence and scattering [23–26], that due to the spatial anisotropy of the scatterer itself the scattered pattern generated by an anisotropic scatterer behaves a strong azimuthal angular dependence, that is, the scattered pattern lacks rotational symmetry. Later, Du and Zhao further found that the distribution of the far field even produced by scattering of a plane wave on an anisotropic scatterer can be rotationally symmetric provided that the structural parameters of the scattering medium meet certain restrictive conditions [10]. However, such a cognition about light scattering from spatially anisotropic media is completely confined to the scalar treatment of the scattered field. To date, the corresponding analysis has not been performed yet within the framework of electromagnetic scattering. Therefore, many conceptual questions on the process of electromagnetic light scattering from an anisotropic medium still remain open. For example, is it possible to obtain a rotationally symmetric scattered momentum flow in the far zone, when a polychromatic electromagnetic plane wave is scattered by an anisotropic medium? If possible, what are the necessary and sufficient conditions for producing such a distribution?

In the present work, we aim to examine these questions. We will first derive the tensor form of the analytic expression for the angular distribution of the momentum flow of the far field generated by scattering of a polychromatic electromagnetic plane wave on an anisotropic, Gaussian, Schell-model medium (i.e., the strength distribution and the normalized correlation coefficient of the scattering potential of the medium both obey Gaussian distributions. Examples of practical media with such characteristics include troposphere, confined plasmas, elliptocyte and so on [27]). Based on this, we will formulate the necessary and sufficient conditions for producing a rotationally symmetric scattered momentum flow in the far zone, and further elaborate the relationship between the rotationally symmetric distribution of the scattered momentum flow and the structural properties of the scattering medium, the polarization characteristics of the incident light source. Finally, numerical examples will be presented to confirm our results. Our research not only provides a deeper insight into the far-zone distribution of the process of electromagnetic light scattering from an anisotropic scatterer, but also has implication for optical micromanipulation.

2. Momentum flow of a ploychromatic electromagnetic plane wave on scattering from a spatially anisotropic random medium

Let us consider that a polychromatic electromagnetic plane wave propagating in the direction of an unit vector $\mathbf {s}_{0}$ is scattered by a statistically stationary anisotropic random scatterer with a finite volume D, as shown in Fig. 1. The incident electric field at a point, specified by a vector $\mathbf {r}^{\prime }$, is represented by a statistical ensemble $\bigl \{E_{i}(\mathbf {r}^{\prime },\omega )\bigr \}$ $(i=x,y,z)$ of monochromatic realizations oscillating at the frequency $\omega$, with forms of

 

Fig. 1. Illustration of notations.

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The second-order correlation properties of the incident electric field at a point $\mathbf {r}^{\prime }$ can be characterized by its spectral polarization matrix [27,28]. With the help of Eq. (1), this matrix can be formulated as

If the refractive index of the medium is close to unity, the scattering is weak and therefore can be addressed within the accuracy of the first-order Born approximation ([27], Sec. 6.1). As is well-known that the scattered wave in the far zone behaves globally like an outgoing spherical wave, which implies that the analysis will be more convenient if both the scattered electric field and the scattered magnetic field are expressed in the spherical coordinate system, where their radial components vanish. Within this description the scattered electric field and the scattered magnetic field at a point $r\mathbf {s}$ $(\mathbf {s}=\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta )$ in the far zone have been given by the following expressions [29]

Similarly, based on Eqs. (3)–(5), the second-order correlation properties of the scattered electric field and of the scattered magnetic field at a point $r\mathbf {s}$ in the far zone can also be found from their individual spectral polarization matrices, which are given by the following expressions

On substituting from Eqs. (13) and (14) into Eq. (12), after some trivial mathematics, the resulting expression for the correlation function of the scattering potential of an anisotropic, Gaussian, Schell-model medium can be written in the following compact tensor form, viz.,

The momentum flow of the light field at any spatial position and any time instant can be readily calculated from the Maxwell stress tensor of the field [31], and the same is true of the scattered field outside the scattering object. Recently, the expression for the Maxwell stress tensor of the scattered field at a point $r\mathbf {s}$ in the far zone in the space-frequency representation has been derived, which is given as [4,5]

On substituting from Eq. (15) into Eq. (10) first, after manipulating Fourier transform, and then inserting the result into Eqs. (8) and (9), and finally into Eq. (20), after pretty tedious calculations, the expression for the momentum flow of the scattered field in the far zone, as a function of the direction of scattering $\mathbf {s}$, is formulated as

Equation (21) is one of the main results of this work. It gives the tensor form of the analytic expression for the angular distribution of the momentum flow of the far field generated by scattering of a polychromatic electromagnetic plane wave on an anisotropic, Gaussian, Schell-model medium. From Eq. (21), we see that the scattered momentum flow in the far zone relies heavily on the azimuthal angle of scattering $\phi$, which implies that the distribution of the scattered momentum flow in the far zone is rotationally asymmetric. What’s more, we see that the dependence of the scattered momentum flow on the azimuthal angle of scattering $\phi$ is contained in two factors: one is the exponential term, i.e., $\exp {\bigl [-\frac {1}{4}\widehat {\mathbf {K}}^{\top }\mathbf {U}^{-1}\widehat {\mathbf {K}}\bigr ]}$, and the other is the trace term, i.e., $\text {Tr}\bigl [\mathbf {W}(\mathbf {r}^{\prime },\omega )\mathbf {Y}(\theta,\phi )\bigr ]$. The former refers to the scattering medium, while the latter involves the incident light source. In other words, only by imposing certain constraints on the physical properties of both the scattering medium and the incident light source can the rotationally symmetric distribution of the scattered momentum flow in the far zone be achieved. To see these constraints more clearly, we now rewrite Eq. (21) without tensor form, which is expressed as

3. Necessary and sufficient conditions for producing a rotationally symmetric scattered momentum flow in the far zone

Equation (24) suggests that the far-zone distribution of the momentum flow of the scattered field will be the same in every azimuthal angle of scattering $\phi$, if the following three conditions hold simultaneously

Equation (25) gives the constraints to the effective width $\sigma$ and the effective correlation width $\mu$ of the scatterer along the $x$ and $y$ axes. We see that if the values of ($\sigma _{x},\sigma _{y}$) are fixed but those of ($\mu _{x},\mu _{y}$) are allowed to change. For a given value of $\mu _{x}$, there always exists a value of $\mu _{y}$ which makes Eq. (25) hold. This implies that the media have the same effective widths ($\sigma _{x},\sigma _{y}$) but different effective correlation widths ($\mu _{x},\mu _{y}$), yet all of them are capable of producing rotationally symmetric distributions of the scattered momentum flow in the far zone provided that Eqs. (26) and (27) have been met in advance. The same is true of the complementary situation–the media have the same effective correlation widths ($\mu _{x},\mu _{y}$) but different effective widths ($\sigma _{x},\sigma _{y}$), and all of them likewise have the ability to produce a rotationally symmetric scattered momentum flow in the far zone. Notably, since Eq. (25) itself does not impose any constraints on $\sigma _{z}$ and $\mu _{z}$, the scatterers have the same effective widths ($\sigma _{x},\sigma _{y}$), which does not mean that they have the same strength distribution of the scattering potential $I_{F}(\mathbf {r}^{\prime },\omega )$. The fact that the media have the same effective correlation widths ($\mu _{x},\mu _{y}$) also does not mean that they have the same normalized correlation coefficient of the scattering potential $\mu _{F}(\mathbf {r}^{\prime }_{2}-\mathbf {r}^{\prime }_{1},\omega )$. Alternatively, in the special case of a quasi-homogeneous anisotropic medium ($\sigma \gg \mu$) which is an important subclass of anisotropic Gaussian Schell-model media, Eq. (25) reduces to a pretty simple form: $\mu _{x}=\mu _{y}$. In this case, only the effective correlation widths of the scatterer along the $x$ and $y$ axes are limited. This is because the reciprocity theorems relating to scattering from quasi-homogeneous anisotropic scatterers lead the scattered momentum flow in the far zone to be proportional to the Fourier transform of the correlation coefficient of the scatterer.

Equations (26) and (27) give the limitations to the incident light source. The former requires that its spectra along the $x$ and $y$ axes must equal, whereas the latter demands that the real part of the spectral correlation between the two mutually orthogonal components $E_{x}(\mathbf {r}^{\prime },\omega )$ and $E_{y}(\mathbf {r}^{\prime },\omega )$ of the incident electric field must vanish. At first glance, one may be induced to conclude that an incident light source which satisfies these two constraints should be completely unpolarized. However, after some simple calculations for its spectral degree of polarization, we can rule out such an inappropriate impression. The spectral degree of polarization of the incident field can be readily computed as [28]

We see that $\mathcal {P}(\mathbf {r}^{\prime },\omega )$ depends on the squared modulus of $S_{xy}(\omega )$, not just its real part. This fact leads that $\mathcal {P}(\mathbf {r}^{\prime },\omega )$ won’t vanish even if Eq. (26) and Eq. (27) hold, of course, unless $S_{xy}(\omega )$ itself is zero. Therefore, Eqs. (26) and (27) do not impose an explicit constraint on the spectral degree of polarization of the incident field, that is, regardless of whether the incident light source is fully polarized ($\mathcal {P}(\mathbf {r}^{\prime },\omega )=1$), partially polarized ($0<\mathcal {P}(\mathbf {r}^{\prime },\omega )<1$), or completely unpolarized ($\mathcal {P}(\mathbf {r}^{\prime },\omega )=0$), it is able to produce a rotationally symmetric scattered mometum flow in the far zone even if when it is scattered by an anisotropic scatterer. Furthermore, comparing with the constraints for an incident light source to produce an isotropic scattered radiation in the far zone (i.e., the scattered radiation is the same in every polar angle of scattering $\theta$ [32]) in [16], we see that the requirements for producing a rotationally symmetric scattered momentum flow are more rigorous, because Eq. (26) doesn’t allow the situation in which the spectra of incident field along the $x$ and $y$ axes are proportional to each other. This suggests that an incident field capable of producing an isotropic scattered radiation in the far zone, when it is scattered by an anisotropic medium, may not necessarily be capable of producing a rotationally symmetric momentum flow in the far zone.

4. Numerical examples

In the following, some numerical examples will be presented to confirm the possibility of producing a rotationally symmetric scattered momentum flow in the far zone, when a ploychromatic electromagnetic plane wave is scattered by a spatially anisotropic random scatterer.

Figure 2 displays contours of the normalized momentum flow of the light fields in the far zone generated by scattering of partially polarized electromagnetic plane waves on anisotropic Gaussian Schell-model media as a function of the dimensionless scattering polar angle $\theta /\pi$ and scattering azimuthal angle $\phi /\pi$. Figure (2(a)) plots the normalized momentum flow of the scattered field when none of three constraint conditions in Eqs. (25)–(27) holds. It clearly shows that the momentum flow in the far zone generated by scattering of a polychromatic electromagnetic plane wave on a spatially anisotropic random medium has a strong azimuthal angular dependence, which means that the scattered momentum flow in the far zone is indeed rotationally asymmetric. Moreover, we can see from Figs. (2(b)) and (2(c)) that even if the structural parameters of the incident light source or those of the scattering medium meet their individual constraint conditions in Eqs. (25)–(27), the momentum flow of the scattered field in the far zone still lacks rotational symmetry. Only when each of three constraint conditions in Eqs. (25)–(27) holds, will the momentum flow of the scattered field in the far zone be roationally symmetric, as shown in Fig. (2(d)).

 

Fig. 2. Contours of the normalized momentum flow of the far fields generated by scattering of partially polarized plane waves on anisotropic Gaussian Schell-model media as a function of the dimensionless scattering polar angle $\theta /\pi$ and scattering azimuthal angle $\phi /\pi$. $\lambda =632.8 nm$, $k=2\pi /\lambda$, $\sigma _{z}=600\lambda$, $\mu _{z}=590\lambda$, $\text {Im}\bigl [S_{xy}(\omega )\bigr ]=0.5$. The other parameters for calculations are chosen as follows: (a) $\sigma _{x}=400\lambda$, $\sigma _{y}=500\lambda$, $\mu _{x}=380\lambda$, $\mu _{y}=480\lambda$, $S_{xx}(\omega )=1$, $S_{yy}(\omega )=0.8$, $\text {Re}\bigl [S_{xy}(\omega )\bigr ]=0.7$; (b) $\sigma _{x}=300\lambda$, $\sigma _{y}=400\lambda$, $\mu _{x}=280\lambda$, $\mu _{y}=380\lambda$, $S_{xx}(\omega )=S_{yy}(\omega )=1$, $\text {Re}\bigl [S_{xy}(\omega )\bigr ]=0$; (c) $\sigma _{x}=\sigma _{y}=300\lambda$, $\mu _{x}=\mu _{y}=280\lambda$, $S_{xx}(\omega )=1$, $S_{yy}(\omega )=0.8$, $\text {Re}\bigl [S_{xy}(\omega )\bigr ]=0.7$; (d) $\sigma _{x}=\sigma _{y}=300\lambda$, $\mu _{x}=\mu _{y}=280\lambda$, $S_{xx}(\omega )=S_{yy}(\omega )=1$, $\text {Re}\bigl [S_{xy}(\omega )\bigr ]=0$.

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Figure 3 plots contours of the normalized momentum flow of the scattered fields in the far zone generated by Gaussian-correlated quasi-homogeneous anisotropic media as a function of the dimensionless scattering polar angle $\theta /\pi$ and scattering azimuthal angle $\phi /\pi$. Figs. (3(a)) and (3(b)) clearly illustrate that for a quasi-homogeneous anisotropic medium, the realization of the rotationally symmetric scattered momentum flow in the far zone indeed only requires the structural parameters of the medium to satisfy $\mu _{x}=\mu _{y}$ when Eqs. (26) and (27) have been statified in advance, which is consistent with our previous discussion. Furthermore, Fig. (3(b)) also illustrates the possibility of producing a rotationally symmetric distribution of the scattered momentum flow in the far zone when the incident field is completely unpolarized. When the incident light source is completely polarized, the result is plotted in Fig. (3(c)).

 

Fig. 3. Contours of the normalized momentum flow of the far-zone scattered fields generated by Gaussian-correlated quasi-homogeneous anisotropic media as a function of the dimensionless scattering polar angle $\theta /\pi$ and scattering azimuthal angle $\phi /\pi$. $\lambda =632.8 nm$, $k=2\pi /\lambda$, $\sigma _{x}=1200\lambda$, $\sigma _{y}=1500\lambda$, $\sigma _{z}=1600\lambda$, $\mu _{z}=430\lambda$. The other parameters for calculations are chosen as follows: (a) $\mu _{x}=280\lambda$, $\mu _{y}=410\lambda$, $S_{xx}(\omega )=S_{yy}(\omega )=1$, $S_{xy}(\omega )=0$; (b) $\mu _{x}=\mu _{y}=280\lambda$, $S_{xx}(\omega )=S_{yy}(\omega )=1$, $S_{xy}(\omega )=0$; (c) $\mu _{x}=\mu _{y}=345\lambda$, $S_{xx}(\omega )=S_{yy}(\omega )=1$, $\text {Re}\bigl [S_{xy}(\omega )\bigr ]=0$, $\text {Im}\bigl [S_{xy}(\omega )\bigr ]=1$.

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Obviously, the numerical results in Figs. 2 and 3 confirm our theoretical analysis in the previous section.

5. Conclusions

In summary, we have examined the momentum flow of the far field produced by scattering of a polychromatic electromagnetic plane wave on a spatially anisotropic random scatterer. The tensor form of the analytic expression for the angular distribution of the momentum flow of the scattered field in the far zone has been derived, and the far-zone distribution characteristics of the momentum flow of the scattered field have been analyzed. The results have indicated that the momentum flow of plolychromatic electromagnetic light scattered from an anisotropic random medium is short of rotational symmetry. However, we have shown that when the structural parameters of both the scattering medium and the incident light source are selected appropriately, the scattered momentum flow in the far zone can be the same in every azimuthal angle of scattering. We have derived the necessary and sufficient conditions for producing such a rotationally symmetric distribution, and further demonstrated that the scatterers have the same effective widths ($\sigma _{x}$, $\sigma _{y}$) but different effective correlation widths ($\mu _{x}$, $\mu _{y}$), yet all of them are capable of producing rotationally symmetric distributions of the scattered momentum flow in the far zone. The same is true for the complementary situation, i.e., the scattering objects have the same effective correlation widths ($\mu _{x},\mu _{y}$) but different effective widths ($\sigma _{x},\sigma _{y}$), and all of them likewise can produce a rotationally symmetric scattered momentum flow in the far zone. We have also demonstrated that regardless of whether the incoming light source is fully polarized, partially polarized or completely unpolarized, it has the ability to produce a rotationally symmetric scattered momentum flow in the far zone even though when it is scattered by an anisotropic medium. Our results are expected to find useful practical applications in optical micromanipulation [33–35], especially when the optical forces used to manipulate particles need to be rotationally symmetric. In addition, our results may also be of interest in the broader field, such as atmospheric optics, plasma optics, and biooptics.

Finally, it deserves to mention that although the restrictive condition (25) is formally similar to that for a planar anisotropic light source to produce a rotationally symmetric radiant intensity [30], their implications are different. One should bear in mind that we focus on light scattering from a three-dimensional scatterer not light radiation from a planar optical source in free space and that we calculate the momentum flow not just the radiant intensity. Moreover, unlike the results in [30], we have seen that the fact that the scatterers have the same effective widths ($\sigma _{x}$, $\sigma _{y}$) is not enough to determine that they have the same strength distribution of the scattering potential. Likewise, the fact that the media have the same effective correlation widths ($\mu _{x}$, $\mu _{y}$) is also not sufficient to determine that they the same normalized correlation coefficient of the scattering potential. In fact, such a formal similarity mainly originates from the fact that we assume that the direction of the incident electromagnetic plane wave is along that of the $z$ axis. If the incident electromagnetic plane wave doesn’t propagate along the $z$ axis, the necessary and sufficient conditions for producing a rotationally symmetric scattered momentum flow in the far zone will become much more complicated than Eqs. (25)–(27). Although our assumption somewhat limits the application of Eqs. (25)–(27) to this complicated situation, it is enough for our purposes mentioned in the Introduction and our results still work if we consider the spectra and spectral correlations of the incident field and the effective width and the effective correlation width of the scatterer both in a plane oriented at the propagating direction of the incident wave. It also deserves to mention that we currently focus on the far-zone scattered momentum flow, which is a function of a single point. However, when an electromagnetic light wave is scattered by a spatially anisotropic random scatterer, how to achieve rotationally symmetric spectral degree of coherence or correlation intensity fluctuations (both of which are functions of two points) in the far-zone scattered field has not yet been considered in the current research. In addition, how the spatial coherence of the incident field affects the rotational symmetry of the scattered field has not also been revealed. These basic questions are also interesting and could be discussed in future studies.