Condition for invariant spectrum of an electromagnetic wave scattered from an anisotropic random media

Jia Li; Pinghui Wu; Liping Chang

doi:10.1364/OE.23.022123

1. Introduction

Spectral properties of a statistically stationary optical field, which attracted substantial research interests over the past few decades, had shown potential prospects in a variety of scientific areas, e.g. the astronomical science, target recognization and biomedical imaging. Among these investigations, the spectrum of light provided flexible approaches to determine statistical properties of an unknown object [1–4]. The scaling law proposed by Wolf and his collaborators indicated that spectrum of light can remain unchanged as it propagates in free space [5]. Since then, intensive studies had concerned whether it is feasible to remain invariant spectrum of a light wave when it propagates in or scatters from other types of media. As a representative work, the scaling law was further extended to another case, where the propagation of a planar, secondary and quasi-homogeneous (QH) source beam in the far field was considered [6, 7]. Also, a scaling law was obtained for a planar, secondary and stochastic electromagnetic beam scattering upon an anisotropic random media [8, 9].

In addition to above studies, attentions were also paid to spectral properties of light scattered from the spatially random and deterministic media, respectively. It was shown that light scattered from a random media may exhibit spectral shifts toward shorter or longer wavelengths, which can be modulated by changing the scattering angle [10–13]. Furthermore, spectral shifts of light generated by the scattering of plane waves from a QH scatterer, particulate media and spatially deterministic media were investigated in the literature [14–18], respectively. It was reported that scattered field may display either the blue-shifted or red-shifted spectrum, which is induced by correlation properties of the media. Results also indicated that isotropic profiles of spectrum of a far-zone scattered field can be generated provided the second moment of the dielectric susceptibility of the media suffices the scaling law [19]. Such law was further extended to the case where the scattering of an electromagnetic plane wave was concerned [20].

Although the scaling laws for light propagating in or scattering from diverse isotropic media were extensively studied in above literature, to date, however, no literature has addressed the conditions which enable the unchanged spectrum of light as it scatters from an anisotropic media. We aim to derive a scaling law for guaranteeing the invariant spectrum of an electromagnetic plane wave as it scatters upon the media. We assume that the scattering of electromagnetic plane waves from the media is so weak, thus the scattered field can described by using the first-order Born approximation. We particularly explore whether the spectrum of scattered field could be identical to that of incident plane waves, if properties of incident fields and the media suffice certain conditions.

2. Scattering of an electromagnetic plane wave from an anisotropic random media

To begin with, we assume that an electromagnetic plane wave is incident upon an anisotropic random media whose dielectric susceptibility is characterized by a 3 × 3 diagonal matrix [21]:

η (r', ω) = (\begin{matrix} η_{x} (r', ω) & 0 & 0 \\ 0 & η_{y} (r', ω) & 0 \\ 0 & 0 & η_{z} (r', ω) \end{matrix}),

where

r'

denotes a position vector within the incident field. As a special note, we refer to two theoretical models which can be utilized to describe an anisotropic media. The first model stipulated that the correlation function of the scattering potential is spatially separable, that the scatterer occupies different effective radiuses along the Cartesian coordinate axes [15, 17]. Another model demanded that the dielectric susceptibility of the media is represented by a 3 × 3 diagonal matrix [21]. Actually, both definitions were suitable to describe properties of an anisotropic media. In this paper, we employ the second definition to treat the scattering of an electromagnetic plane wave from an anisotropic media. As a result, the elements of the 3D scattering potential matrix of the media can be expressed as [21]:

F_{j} (r', ω) = {(ω / c)}^{2} η_{j} (r', ω), (j = x, y, z),

where ω is the frequency,

η_{j} (r', ω)

is the element of the 3D dielectric susceptibility matrix, as shown by Eq. (1). Typically, the electric components of the incident electromagnetic plane wave are of the following forms:

U_{j}^{(i)} (r', ω) = a_{j} (ω) \exp (i k s_{0} \cdot r'), (j = x, y),

where

a_{j} (ω)

denotes the spectral amplitude of incident fields along the j-axis,

k = ω / c

is the wave number, c is the speed of light propagating in vacuum. As shown in Fig. 1,

s_{0}

represents the unit vector which describes the propagation direction of the incident field. Within the framework of the cylindrical coordinate system (r, θ, φ), we treat the scattering problem based on the system of the scattering plane, which contains the direction of both incident light (

{\overset{⇀}{s}}_{0}

) and scattered waves (

\overset{⇀}{s}

). In this case, profiles of a scattered field are solely dependent of the azimuthal scattering angle θ, i.e. influences of the rotational scattering angle φ on distributions of scattered field are not considered. Such assumption was previously utilized in [22, 23] to treat the weak scattering case. Notably, we emphasize that it is still valid to treat the weak scattering of an electromagnetic plane wave from an anisotropic media, because the media has different influences on the 3D vectorial components of scattered field rather than spatial locations in the 3D Cartesian coordinate system. Actually, such influences are introduced by defining Eq. (1) and (2). As a result, it is reasonable to solely concern the effects of the azimuthal scattering angle θ on far-zone scattered field from an anisotropic media. Accordingly, the space-dependent field components of a 3D scattered field can be simplified to the following forms [21]:

U_{x}^{(s)} (r s, ω) = \int_{D} F_{x} (r', ω) U_{x}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{y}^{(s)} (r s, ω) = \cos^{2} θ \int_{D} F_{y} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{z}^{(s)} (r s, ω) = - \sin θ \cos θ \int_{D} F_{z} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

where the superscript (i) and (s) denote the incident and scattered field components, respectively. In Eqs. (4)-(6), integrations are performed over the scatterer volume D. θ is the azimuthal scattering angle, s represents the unit vector which describes the propagation direction of scattered waves.

G (r s, r', ω)

stands for the outgoing free-space Green function which can be expressed as the far-zone asymptotic form:

G (r s, r', ω) \approx \frac{\exp (i k r)}{r} \exp (- i k s \cdot r') .

According to the unified theory of coherence and polarization of electromagnetic waves, the spectrum of a scattered field can be introduced as [24]:

S^{(s)} (r s, ω) = T r {W_{i j}^{(s)} (r s, r s, ω)}, (i . j = x, y, z),

where

W_{i j}^{(s)} (r_{1} s_{1}, r_{2} s_{2}, ω) = [〈 U_{i}^{(s) *} (r_{1} s_{1}, ω) U_{j}^{(s)} (r_{2} s_{2}, ω) 〉],

is the cross-spectral density matrix of the scattered field. Substituting Eq. (3) and (7) into Eq. (4)-(6) and subsequently into (8),

S^{(s)} (r s, ω)

specified at a position vector rs can be obtained:

S^{(s)} (r s, ω) = \frac{1}{r^{2}} {(\frac{ω}{c})}^{4} {{\tilde{C}}_{x x}^{(η)} (- K, K, ω) S_{x}^{(i)} (ω) + \cos^{4} θ {\tilde{C}}_{y y}^{(η)} (- K, K, ω) S_{y}^{(i)} (ω) + \sin^{2} θ \cos^{2} θ {\tilde{C}}_{z z}^{(η)} (- K, K, ω) S_{y}^{(i)} (ω)},

where

{\tilde{C}}_{i j}^{(η)} (- K, K, ω) = \int_{D} \int_{D} C_{i j}^{(η)} (r_{1}', r_{2}', ω) \exp [- i K \cdot (r_{2}' - r_{1}')] d^{3} r_{1}' d^{3} r_{2}', (i, j = x, y, z),

is the 3D Fourier transform of

C_{i j}^{(η)} (r_{1}', r_{2}', ω)

, which are the elements of the two-point correlation matrix of the dielectric susceptibility:

C_{i j}^{(η)} (r_{1}', r_{2}', ω) = [〈 η_{i}^{*} (r_{1}', ω) η_{j} (r_{2}', ω) 〉] = [C_{i j}^{(η)} (r_{2}' - r_{1}', ω)], (r_{1}', r_{2}') \in V,

where

η_{i} (r', ω)

was introduced in Eq. (1).

K = k (s - s_{0})

is the vector analogous to the momentum transfer vector of the scattering potential.

S_{j}^{(i)} (ω) = 〈 {| a_{j} (ω) |}^{2} 〉

(j = x, y) denotes the spectral component of incident fields along the j-axis. Furthermore, we assume that dimensions of the scatterer are sufficiently large compared with the effective radius of the dielectric susceptibility of the media, the following approximation can be made to the Fourier transform of the elements of the correlation matrix [10, 24]:

{\tilde{C}}_{i j}^{(η)} (- K, K, ω) \approx V {\tilde{C}}_{i j}^{(η)} (K, ω),

where V is the scatterer volume. Based on Eq. (13), the spectrum of scattered field can be further rewritten as the following form:

Fig. 1 Schematic diagram for illustrating the weak scattering theory of a plane wave from an anisotropic random media.

Download Full Size | PDF

S^{(s)} (r s, ω) = \frac{V}{r^{2}} {(\frac{ω}{c})}^{4} {{\tilde{C}}_{x x}^{(η)} (K, ω) S_{x}^{(i)} (ω) + \cos^{4} θ {\tilde{C}}_{y y}^{(η)} (K, ω) S_{y}^{(i)} (ω) + \sin^{2} θ \cos^{2} θ {\tilde{C}}_{z z}^{(η)} (K, ω) S_{y}^{(i)} (ω)} .

Equation (14) establishes the connection between the spectrum of scattered field and diagonal elements of the correlation matrix of the anisotropic media. Compared with Eq. (2.2) of [19] and Eq. (14) of [20], it is noteworthy that each diagonal element of the correlation matrix of the media contributes to spectral distributions of a scattered field. In other words, it was exhibited that the anisotropy of the dielectric susceptibility of the media accounts for changes of the spectrum of a scattered field. In what follows, we aim to obtain the sufficient conditions which have the capacity to remain the invariant spectrum of an electromagnetic plane wave when it scatters upon the media.

3. Sufficient conditions for invariance of spectrum of scattered field

Based on Eq. (14), we further recall the expression for the normalized spectrum of a scattered field [5–9, 19, 20]:

S_{N}^{(s)} (r s, ω) = \frac{S^{(s)} (r s, ω)}{\int_{0}^{\infty} S^{(s)} (r s, ω') d ω'} .

By substituting Eq. (14) into Eq. (15), the normalized spectrum of scattered field results in the following form:

S_{N}^{(s)} (r s, ω) = \frac{ω^{4} S_{x}^{(i)} (ω) {\tilde{C}}_{x x}^{(η)} (K, ω) + \cos^{4} θ ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{y y}^{(η)} (K, ω) + \sin^{2} θ \cos^{2} θ ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{z z}^{(η)} (K, ω)}{\int_{0}^{\infty} ω'^{4} S_{x}^{(i)} (ω') {\tilde{C}}_{x x}^{(η)} (K, ω') d ω' + \cos^{4} θ \int_{0}^{\infty} ω'^{4} S_{y}^{(i)} (ω') {\tilde{C}}_{y y}^{(η)} (K, ω') d ω' + \sin^{2} θ \cos^{2} θ \int_{0}^{\infty} ω'^{4} S_{y}^{(i)} (ω') {\tilde{C}}_{z z}^{(η)} (K, ω') d ω'} .

Generally, the normalized spectrum of scattered field is not identical to that of the incident field. This is because Eq. (16) is complicated and entirely different from the normalized spectrum of the incident electromagnetic plane wave, which is represented as follows:

S_{N}^{(i)} (ω) = \frac{S^{(i)} (ω)}{\int_{0}^{\infty} S^{(i)} (ω') d ω'} = \frac{S_{x}^{(i)} (ω) + S_{y}^{(i)} (ω)}{\int_{0}^{\infty} S_{x}^{(i)} (ω') d ω' + \int_{0}^{\infty} S_{y}^{(i)} (ω') d ω'} .

Comparing Eq. (16) with Eq. (17), we notice that

S_{N}^{(s)} (r s, ω)

is closely dependent of the spectral components of the incident field and the diagonal elements of the correlation matrix of the dielectric susceptibility. Considering Eq. (11), (16) and (17), it is displayed that the spectrum of incident fields generally is not identical to that of scattered waves. Even though, we still devote to explore whether it could be feasible to let Eq. (16) be equivalent to Eq. (17) provided certain conditions are satisfied. Inspired by the scaling laws derived in [19, 20], we particularly consider whether our problem could be solved if the Fourier transforms of elements of the correlation matrix are assumed to have the extended forms of those derived in [19, 20]:

{\tilde{C}}_{j j}^{(η)} (K, ω) = F_{j j} (ω) H_{j j} (K / k), (j = x, y, z),

where

F_{j j} (ω)

and

H_{j j} (K / k)

stand for arbitrary, positive functions which are dependent of the frequency and vectorial positions, respectively. By letting

s = s_{0}

, Eq. (18) results in the following expression:

{\tilde{C}}_{j j}^{(η)} (0, ω) = F_{j j} (ω) H_{j j} (0) .

Based on Eq. (19), Eq. (18) can be further rewritten as:

{\tilde{C}}_{j j}^{(η)} (K, ω) = {\tilde{C}}_{j j}^{(η)} (0, ω) H_{j j}^{- 1} (0) H_{j j} (K / k) .

As the next step, we perform the Fourier transforms to terms on both sides of Eq. (20), it follows that the expressions for the diagonal elements of correlation matrix of the dielectric susceptibility are given by:

C_{j j}^{(η)} (r', ω) = \frac{1}{{(2 π)}^{3}} {\tilde{C}}_{j j}^{(η)} (0, ω) H_{j j}^{- 1} (0) \int H_{j j} (K / k) \exp (i K \cdot r') d^{3} K .

Considering Eq. (21), we change the integral variable from K to

K / k

, the diagonal elements of the correlation matrix result in the following forms:

C_{j j}^{(η)} (r', ω) = k^{3} {\tilde{C}}_{j j}^{(η)} (0, ω) H_{j j}^{- 1} (0) {\tilde{H}}_{j j} (k r'),

where

{\tilde{H}}_{j j} (K) = \int H_{j j} (r') \exp (- i K \cdot r') d^{3} r',

is the 3D Fourier transform of

H_{j j} (r')

. Furthermore, we introduce the degrees of correlations of the dielectric susceptibility:

μ_{j j}^{(η)} (r', ω) = \frac{C_{j j}^{(η)} (r', ω)}{C_{j j}^{(η)} (0, ω)} .

By substituting Eq. (22) into Eq. (24), it is exhibited that the degrees of the correlation matrix yield the following expressions:

μ_{j j}^{(η)} (r', ω) = h_{j j} (k r'), (j = x, y, z),

where

h_{j j} (k r') = H_{j j} (k r') H_{j j}^{- 1} (0) .

Equation (25) is the first primary results in this paper. It indicates that the spectrum of a far-zone scattered field may be identical to that of the incident light provided that the degrees of correlations of the dielectric susceptibility suffice the scaling law. It is noteworthy that Eq. (25) is the generalization of former scaling laws which were shown as Eq. (2.24) of [19] and Eq. (24) of [20]. In other words, we generalize the scaling law obtained from previous literature to an extended one, which additionally stipulates that each element of the correlation matrix of the dielectric susceptibility must suffice the sufficient conditions, i.e. Equation (25) and (26). However, it is worthwhile to note that only the satisfactions of Eq. (25) still cannot guarantee the invariance of spectrum of light as it scatters from an anisotropic media. The reason for explaining the fact is as follows: on the one hand, when we substitute Eq. (18) into Eq. (16), it is found that the resultant spectrum of scattered field still cannot remain the same as that of incident fields (see Eq. (17)). This is because Eq. (18) is the first attempt made to explore whether there is any additional condition which must be sufficed to ensure the spectrum of scattered field be the same as that of incident fields. On the other hand, Eq. (25) and (26) directly result from Eq. (18) by performing strict derivations, so Eq. (25) can be only regarded as a sub-condition of the scaling law. In addition to Eq. (25), we particularly note that other sub-conditions must be derived to let the spectrum of scattered field be identical to that of incident fields. To do this, we further substitute Eq. (20) into Eq. (16) to rewrite the normalized spectrum of scattered field:

\begin{array}{l} S_{N}^{(s)} (r s, ω) = [H_{x x} (s - s_{0}) H_{x x}^{- 1} (0) ω^{4} S_{x}^{(i)} (ω) {\tilde{C}}_{x x}^{(η)} (0, ω) + \cos^{4} θ H_{y y} (s - s_{0}) H_{y y}^{- 1} (0) ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{y y}^{(η)} (0, ω) \\ + \sin^{2} θ \cos^{2} θ H_{z z} (s - s_{0}) H_{z z}^{- 1} (0) ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{z z}^{(η)} (0, ω)] / [H_{x x} (s - s_{0}) H_{x x}^{- 1} (0) \int_{0}^{\infty} ω'^{4} S_{x}^{(i)} (ω') {\tilde{C}}_{x x}^{(η)} (0, ω) d ω' \\ + \cos^{4} θ H_{y y} (s - s_{0}) H_{y y}^{- 1} (0) \int_{0}^{\infty} ω'^{4} S_{y}^{(i)} (ω') {\tilde{C}}_{y y}^{(η)} (0, ω') d ω' + \sin^{2} θ \cos^{2} θ H_{z z} (s - s_{0}) H_{z z}^{- 1} (0) \\ \times \int_{0}^{\infty} ω'^{4} S_{y}^{(i)} (ω') {\tilde{C}}_{z z}^{(η)} (0, ω') d ω'] . \end{array}

Although Eq. (27) is complicated in its form, we can readily obtain the following conditions, which can enable the invariance of spectrum of scattered field:

ω^{4} S_{x}^{(i)} (ω) {\tilde{C}}_{x x}^{(η)} (0, ω) = S^{(i)} (ω),

ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{y y}^{(η)} (0, ω) = S^{(i)} (ω),

ω^{4} S_{y}^{(i)} (ω) {\tilde{C}}_{z z}^{(η)} (0, ω) = S^{(i)} (ω) .

As a special note, we emphasize that Eqs. (28)-(30) and Eq. (25) must be simultaneously satisfied to ensure the unchanged spectrum of light scattered from an anisotropic media. Also, we notice that once these conditions are fulfilled, the spectrum of scattered field is identical to that of the incident field (as shown by Eq. (17)). As a special solution to Eqs. (28)-(30), it is obtained that the spectral components of incident fields must be proportional to each other:

S_{y}^{(i)} (ω) = γ S_{x}^{(i)} (ω),

where γ is a positive constant. Substituted by Eq. (31), Eqs. (28)-(30) can be further rewritten as the following forms:

{\tilde{C}}_{x x}^{(η)} (0, ω) = \frac{γ + 1}{ω^{4}}, {\tilde{C}}_{y y}^{(η)} (0, ω) = {\tilde{C}}_{z z}^{(η)} (0, ω) = \frac{γ + 1}{γ ω^{4}} .

Recalling

{\tilde{C}}_{i i}^{(η)} (0, ω)

is the Fourier transforms of elements of the correlation matrix, it can be denoted by the integral expression:

{\tilde{C}}_{j j}^{(η)} (0, ω) = \int C_{j j}^{(η)} (r', ω) d^{3} r', (i = x, y, z) .

Substituted by Eq. (24), Eq. (33) can be given by an alternative form:

{\tilde{C}}_{j j}^{(η)} (0, ω) = C_{j j}^{(η)} (0, ω) \int μ_{j j}^{(η)} (r', ω) d^{3} r' .

Based on Eq. (25), we change the integral variable from K to

K / k

, then

{\tilde{C}}_{j j}^{(η)} (0, ω)

results in the following expression:

{\tilde{C}}_{j j}^{(η)} (0, ω) = \frac{C_{j j}^{(η)} (0, ω) {\tilde{h}}_{j j} (0)}{k^{3}},

where

{\tilde{h}}_{j j} (0) = \frac{1}{k^{3}} \int h_{j j} (k r') d^{3} r'

is the Fourier transform of

h_{j j} (k r')

, which was introduced by Eq. (26). By substituting Eq. (35) into Eq. (28)-(30), respectively, it follows that the diagonal elements of correlation matrix of the dielectric susceptibility have the following forms:

C_{x x}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} {\tilde{h}}_{x x} (0)}, C_{y y}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} γ {\tilde{h}}_{y y} (0)}, C_{z z}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} γ {\tilde{h}}_{z z} (0)} .

Compared Eq. (37) with Eq. (12), we can readily obtain the condition for the invariance of spectrum of light scattered from an anisotropic media. Such condition requires that the second moments of elements of the dielectric susceptibility are inversely proportional to the frequency:

〈 η_{j}^{*} (r', ω) η_{j} (r', ω) 〉 \propto \frac{1}{ω}, (j = x, y, z) .

Equation (38) is the second primary result in this paper. In addition to the satisfactions of Eq. (25) and (31), the invariance of spectrum of scattered field also requires the second moments of elements of the dielectric susceptibility must suffice certain conditions. It is worthwhile to note that Eqs. (25), (31) and (38) constitute the modified scaling law, which can remain the unchanged spectrum of electromagnetic waves scattered from an anisotropic media. In what follows, discussions will be held by making comparisons between Eq. (38) and Eq. (2.33) of [19], where the scaling law for the scattering of a scalar plane wave from an isotropic media was obtained.

4. Discussions

In the previous section, we showed that the spectrum of light scattered from an anisotropic media can remain unchanged provided that the sufficient conditions, i.e. Equation (25), (31) and (38) are simultaneously satisfied. These conditions constitute the scaling law which enables the spectrum of scattered light be identical to that of incident fields. Remarkably, Eq. (25) can be compared with Eq. (2.20) of [19] and Eq. (24) of [20]. It was demonstrated in [19, 20] that the degree of correlation of an isotropic media must obey the scaling law to enable the spectrum of scattered field be identical to that of incident light. In contrast, we exhibit that the degrees of correlations of an anisotropic media must satisfy the scaling law for the invariance of spectrum of light when it scatters from the media.

As one of the conditions, Eq. (31) is identical to Eq. (26b) of [20]. It is required that the spectral components of the incident wave must be proportional to each other. Even though, we shall emphasize that Eq. (31) is merely one special solution to Eqs. (28)-(30). Actually, there may exist other solutions which also fulfill the scaling law and are different from Eq. (31). We have devoted to this study and anticipated to present results in a future publication.

In addition to the conditions, i.e. Equation (25) and (31), Eq. (38) indicates that the second moments of elements of the dielectric susceptibility must be inversely proportional to the frequency, as we aim to remain the unchanged spectrum of light scattered from an anisotropic media. Also, Eq. (38) can be compared with Eq. (2.33) of [19], which stipulated that the second moment of dielectric susceptibility of an isotropic media must be inversely proportional to the frequency. Altogether, Eqs. (25), (31) and (38) can be regarded as the generalization of all previous scaling laws. Once they are simultaneously satisfied, the spectrum of light scattered from the anisotropic media is identical to that of the incident wave.

The obtained results can find broad applications to a variety of research areas. For example, the scaling law derived for guaranteeing the invariant spectrum of light scattered from an anisotropic media may be well employed in the free space optical communication (FSOC) links. When a polychromatic, electromagnetic wave is utilized in FSOC channels, it can propagate in free space to deliver specific optical signals to receivers. However, when the propagating wave encounters an anisotropic media in intermediate optical paths, the spectrum of light generally changes and differs from the original profile. In such case, the receiver cannot precisely obtain the correct spectral signal of electromagnetic waves. The scaling law derived in this paper provides a useful approach for constructing statistical properties of anisotropic objective in FSOC channels in order to remain original spectral information of electromagnetic waves upon the propagation.

Moreover, numerical calculations can be performed to confirm the validity of the derived scaling law. As typical examples, we particularly consider two types of anisotropic media, of which the second moments of the dielectric susceptibility suffice the following forms, respectively:

C_{x x 1}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} {\tilde{h}}_{x x} (0)}, C_{y y 1}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} γ {\tilde{h}}_{y y} (0)}, C_{z z 1}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} γ {\tilde{h}}_{z z} (0)} .

C_{x x 2}^{(η)} (0, ω) = \frac{γ + 1}{ω c^{3} {\tilde{h}}_{x x} (0)}, C_{y y 2}^{(η)} (0, ω) = \frac{γ + 1}{c^{3} γ {\tilde{h}}_{y y} (0)} ω, C_{z z 2}^{(η)} (0, ω) = \frac{γ + 1}{c^{3} γ {\tilde{h}}_{z z} (0)} ω,

where the subscript “1” and “2” in Eq. (39) and (40) stand for the numbers of two types of media. Evidently, it is shown that the first class of anisotropic media satisfies the scaling law. In contrast, the second sort of media disobeys the derived law. In what follows, we focus on whether such two classes of media could produce the normalized spectrum of scattered field which is identical to that of incident fields, respectively. By substituting Eq. (39) into Eq. (35) and subsequently into Eq. (27), it follows that the normalized spectrum of scattered field induced by the first class of media yields the expression:

S_{N 1}^{(s)} (r s, ω) = \frac{S^{(i)} (ω)}{\int_{0}^{\infty} S^{(i)} (ω) d ω} .

Similarly, the normalized spectrum of scattered field induced by the second sort of media results in the following form:

\begin{array}{l} S_{N 2}^{(s)} (r s, ω) = [H_{x x} (s - s_{0}) / H_{x x} (0) S^{(i)} (ω) + γ \cos^{4} θ H_{y y} (s - s_{0}) / H_{y y} (0) ω^{2} S^{(i)} (ω) \\ + γ \sin^{2} θ \cos^{2} θ H_{z z} (s - s_{0}) / H_{z z} (0) ω^{2} S^{(i)} (ω)] / [H_{x x} (s - s_{0}) / H_{x x} (0) \int_{0}^{\infty} S^{(i)} (ω) d ω \\ + γ \cos^{4} θ H_{y y} (s - s_{0}) / H_{y y} (0) \int_{0}^{\infty} ω^{2} S^{(i)} (ω) d ω + γ \sin^{2} θ \cos^{2} θ H_{z z} (s - s_{0}) / H_{z z} (0) \int_{0}^{\infty} ω^{2} S^{(i)} (ω) d ω] . \end{array}

Comparisons between Eq. (41) and (42) exhibit that the normalized spectrum of scattered field is identical to that of the incident field provided the anisotropic media suffices the scaling law (see Eq. (39)). Conversely, if the anisotropic media disobeys the derived scaling law (see Eq. (40)), the resultant normalized spectrum of scattered field entirely differs from that of the incident field. Based on Eq. (41) and (42), Fig. 2 is presented to show the normalized spectrum of scattered field by employing two types of anisotropic media. For comparisons, the normalized spectrum of incident plane waves is also plotted as the real curve. For the sake of simplicity, we assume that the spectrum of incident electromagnetic plane waves is of the Gaussian profile, of which the spectral bandwidth is 7 × 10¹⁴sec⁻¹. Other calculation parameters are chosen as follows: the central frequency ω₀ = 3.5 × 10¹⁵sec⁻¹, the initial polarization parameter γ = 1, the scattering angle θ = 0. Figure 2 displays that the normalized spectrum of scattered field coincides with that of incident fields, when the second moments of the elements of the dielectric susceptibility suffice the scaling law, i.e. Equation (39). However, when the anisotropic media does not satisfy the scaling law (see Eq. (40)), the resultant normalized spectrum of scattered field displays a blue-shifted spectral profile, as shown by the dash curve in Fig. 2. In such case, the normalized spectrum of scattered field is evidently different from that of the incident polychromatic plane wave, because the blue shifts of spectrum emerge in scattered field.

Fig. 2 Normalized spectrum of scattered field produced by considering two types of anisotropic media, of which the second moments of the elements of the dielectric susceptibility suffice Eq. (39) and (40), respectively.

Download Full Size | PDF

As the final remark, we further lead a discussion connecting our results with the concept of the cross-spectral purity of light. As a well-known result, it is revealed that the normalized spectrum of a far-zone optical field can remain invariant provided a cross-spectrally pure light is incident upon Young’s double pinholes [24–26]. Other reports further extended the concept of cross-spectrally pure fields to the electromagnetic domain [27], described implications of the statistical similarity of an optical field on its cross-spectral purity and cross-spectrally pure fields [28]. As shown by Eqs. (36)-(38) of [28], an electromagnetic, polychromatic plane wave is cross-spectrally pure throughout all space. Therefore, the electromagnetic plane wave concerned in this paper (see Eq. (3)) also is a cross-spectrally pure light. However, by comparing Eq. (17) with Eq. (16) or Eq. (27), we must emphasize that the scattered wave generally is not a cross-spectrally pure light, because the first condition of being a cross-spectrally pure field cannot be satisfied (see Eq. (9) of [29]). Then a question may arise, that whether the scattered field could be cross-spectrally pure, provided the media suffices certain conditions. The first condition of being a cross-spectrally pure light demands that the normalized spectrum of scattered field must be equal at two spatial points. Considering Eq. (27), it follows that the first condition can be satisfied if the scaling law Eq. (37) or (38) holds valid. Even through, the scattered wave still cannot be regarded as a cross-spectrally pure light unless the second condition on the spatial degree of coherence of scattered field is sufficed (see Eq. (10) of [29]). It is noteworthy that such condition can be fulfilled only if the scatterer is spatially isotropic rather than anisotropic, viz.

η_{x} (r', ω) = η_{y} (r', ω) = η_{z} (r', ω) = η (r', ω),

Under the restriction of Eq. (43), our problem simplifies to the case where the scattering of an electromagnetic plane wave from an isotropic random media is concerned. As the extension from the scattering theory of a scalar plane wave to that of an electromagnetic plane wave is straightforward, we can combine Eq. (2), (32) and (43) with Eq. (33) of [29] to obtain the conditions for letting the scattered field be a cross-spectrally pure light, viz.

| {\tilde{C}}_{x x}^{(η)} (- K_{1}, K_{2}, ω) | = | {\tilde{C}}_{y y}^{(η)} (- K_{1}, K_{2}, ω) | = | {\tilde{C}}_{z z}^{(η)} (- K_{1}, K_{2}, ω) | = | {\tilde{C}}^{(η)} (- K_{1}, K_{2}, ω) |,

and

| {\tilde{C}}^{(η)} (- K_{1}, K_{2}, ω) | \propto \frac{C (ω)}{ω^{4}},

where

C (ω)

is a frequency-dependent, non-negative factor which was previously introduced in [29]. Conditions Eq. (44) and (45) demand that the scatterer must be spatially isotropic rather than anisotropic, and the modulus of the Fourier transform of the correlation function of the dielectric susceptibility is independent of spatial locations within scattered field. For a scatterer whose statistical properties satisfy Eq. (38), (44) and (45), the resultant scattered field can be regarded as a cross-spectrally pure light.

5. Conclusion

Within the accuracy of the first-order Born approximation, sufficient conditions are derived to enable the spectrum of light scattered from an anisotropic random media be identical to that of incident plane waves. By assuming that no correlation exists between electric field components of incident waves, and also the scattered field has a symmetrical spectral density distribution in the far-zone region, we show that the scaling law requires restrictions on correlation properties of incident fields and elements of dielectric susceptibility of the media. The obtained results are of importance to study spectral properties of light scattered from an anisotropic media and determine the scattering potential of a spatially random media.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC 61205121, 61304124), the Natural Science Foundation of Zhejiang Province (LY13F010009, LY15F050012), the China Postdoctoral Science Foundation Funded Project (CPSF2012M511386), the Scientific Research Fund of Zhejiang Provincial Education Department (Y201225146) and the Natural Science Foundation of Zhejiang University of Technology (2013XZ003). Authors are especially indebted to Dr. Tao Wang for his helpful discussions and comments.

References and links

1. D. F. V. James, “The Wolf effect and the redshift of quasars,” Pure Appl. Opt. 7(5), 959–970 (1998). [CrossRef]

2. M. Dashtdar and M. T. Tavassoly, “Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces,” J. Opt. Soc. Am. A 26(10), 2134–2138 (2009). [CrossRef] [PubMed]

3. W. Gao, “Spectral changes of the light produced by scattering from tissue,” Opt. Lett. 35(6), 862–864 (2010). [CrossRef] [PubMed]

4. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. 36(21), 4209–4211 (2011). [CrossRef] [PubMed]

5. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). [CrossRef] [PubMed]

6. H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003). [CrossRef]

7. H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004). [CrossRef]

8. J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett. 31(14), 2097–2099 (2006). [CrossRef] [PubMed]

9. J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Opt. Lett. 4, 196–198 (2006).

10. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]

11. J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989). [CrossRef] [PubMed]

12. D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

13. T. Shirai and T. Asakura, “Spectral changes of light induced by scattering from spatially random media under the Rytov approximation,” J. Opt. Soc. Am. A 12(6), 1354–1363 (1995). [CrossRef]

14. Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007). [CrossRef]

15. X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011). [CrossRef] [PubMed]

16. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998). [CrossRef] [PubMed]

17. X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012). [CrossRef]

18. M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012). [CrossRef] [PubMed]

19. E. Wolf, “Far-zone spectral isotropy in weak scattering on spatially random media,” J. Opt. Soc. Am. A 14(10), 2820–2823 (1997). [CrossRef]

20. T. Wang and D. Zhao, “Condition for far-zone spectral isotropy of an electromagnetic light wave on weak scattering,” Opt. Lett. 36(3), 328–330 (2011). [CrossRef] [PubMed]

21. J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012). [CrossRef]

22. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley Press, 2000).

23. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35(14), 2412–2414 (2010). [CrossRef] [PubMed]

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

25. L. Mandel, “Interference and the Alford and Gold effect,” J. Opt. Soc. Am. 52(12), 1335–1340 (1962). [CrossRef]

26. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996). [CrossRef]

27. T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34(24), 3866–3868 (2009). [CrossRef] [PubMed]

28. J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014). [CrossRef]

29. M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014). [CrossRef]

Condition for invariant spectrum of an electromagnetic wave scattered from an anisotropic random media

Abstract

1. Introduction

2. Scattering of an electromagnetic plane wave from an anisotropic random media

3. Sufficient conditions for invariance of spectrum of scattered field

4. Discussions

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Equations (45)

Optics Express