Equivalence theorem of light waves on scattering from media with different types

Hao Wu; Xiaoning Pan; Zhanghang Zhu; Tao Wang; Ke Cheng

doi:10.1364/OE.25.021410

1. Introduction

The weak scattering of light waves not only attracted substantial research interests but also showed potential prospects in a variety of scientific areas such as remote sensing, detection, and medical diagnostics. Over the past three decades, quantities of efforts have been devoted to the investigations of far-zone behaviors of light waves on scattering. For examples, the scattering of light waves from various media has been discussed, and the relations between the properties of the scattered field and the characteristics of the scattering medium have been studied [1–17]; and the far-zone behaviors of light waves on scattering from different types of quasi-homogeneous media have been discussed, and the reciprocity relations of light waves on scattering have been investigated [18–22] (for a review of these researches, please see [23]).

Among all applications of light waves scattering, the inverse scattering problem, i.e. the determination of structural information of an unknown medium from the measurements of its scattered field, has been the focus of attention [24–28]. It should be emphasized that all the discussions on the inverse scattering problem assume that different media may produce far-zone fields with different properties, that is, the characteristics of the scattering medium and the properties of the scattered field are strictly one-to-one correspondence. However, it is shown that when certain conditions between two different media are satisfied, the far-zone normalized spectral density or the far-zone spectral degree of coherence of the scattered field may demonstrate identical distribution, which is called as “Equivalence theorem” [29,30]. This phenomenon is of great importance in the inverse scattering problem because it means that there may be some errors in the determination of structural information of a scatterer. In this manuscript, we will generalize the “equivalence theorem” to a more general case of light waves on scattering from two media with different types, i.e. a continuous medium and a particulate medium. The possibility for these two different types of media to generate scattered field with identical normalized spectral density or to generate scattered field with identical spectral degree of coherence will be discussed, and the condition for identical normalized spectral density and the condition for identical spectral degree of coherence will be presented.

2. Far-zone behaviors of light waves on scattering

Consider a scatterer which occupies a finite domain $D$ is illuminated by a spatially coherent monochromatic plane light wave with a direction specified by a real unit vector $s_{0}$ (see Fig. 1). The property of the incident field at a pair of position vectors ${r^{'}}_{1}$ and ${r^{'}}_{2}$ within the area of the scattering medium should be described by its cross-spectral density function ([31], Sec.6.2), i.e.

W^{(i)} ({r^{'}}_{1}, {r^{'}}_{2}, s_{0}, ω) = S^{(i)} (ω) \exp [i k s_{0} \cdot ({r^{'}}_{2} - {r^{'}}_{1})]

where

ω

is the angular frequency,

S^{(i)} (ω)

is the spectrum of the incident field, and

k = ω / c

is the free-space wave number, with

c

denoting the speed of light in the vacuum. On making use of the accuracy of the first-order Born approximation ([32], Sec.13.1), the cross-spectral density function of the far-zone scattered field, at two points specified by a pair of position vectors

r s_{1}

,

r s_{2}

(

s_{1}

and

s_{2}

are unit vectors), is given by the expression ([31], Sec.6.3)

W^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{S^{(i)} (ω)}{r^{2}} {\tilde{C}}_{F} [- k (s_{1} - s_{0}), k (s_{2} - s_{0}), ω]

where

{\tilde{C}}_{F} (K_{1}, K_{2}, ω) = \iint_{D} C_{F} ({r^{'}}_{1}, {r^{'}}_{2}, ω) \exp [- i (K_{1} \cdot {r^{'}}_{1} + K_{2} \cdot {r^{'}}_{2})] d^{3} {r^{'}}_{1} d^{3} {r^{'}}_{2}

is the six-dimensional spatial Fourier transform of the correlation function of the scattering potential with

K_{1} = - k (s_{1} - s_{0})

,

K_{2} = k (s_{2} - s_{0})

, and

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}, ω)

is the correlation function of the scattering potential, which is defined as ([31], Sec.6.3)

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = 〈 F^{*} ({r^{'}}_{1}, ω) F ({r^{'}}_{2}, ω) 〉

where the asterisk stands for the complex conjugate, the angular brackets denote ensemble average, and

F (r^{'}, ω)

is the scattering potential of the medium.

Fig. 1 Illustration of notations.

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In the following, we will discuss the behaviors of light waves on scattering from two media with different types. Firstly, let us consider that the scatterer is a continuous medium. Among all continuous medium, the quasi-homogeneous medium is frequently mentioned. A medium can be regarded as quasi-homogeneous medium when its correlation function can be factorized into a product that contains a ‘slowly varying’ function $I_{F}$ and a ‘fast varying’ function $μ_{F}$ [18], i.e.

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = I_{F} (\frac{{r^{'}}_{1} + {r^{'}}_{2}}{2}, ω) μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω)

where

I_{F}

and

μ_{F}

are the strength and the normalized correlation coefficient of the scattering potential of the medium, respectively. Upon substituting from Eq. (5) first into Eq. (3), and then into Eq. (2), making use of the variable transforms as

r_{S} = ({r^{'}}_{1} + {r^{'}}_{2}) / 2

,

r_{D} = {r^{'}}_{2} - {r^{'}}_{1}

, we obtain the expression for the cross-spectral density function of the scattered field produced by a quasi-homogeneous medium as

W_{Q}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{S^{(i)} (ω)}{r^{2}} {\tilde{I}}_{F} [(K_{1} + K_{2}), ω] {\tilde{μ}}_{F} [\frac{1}{2} (K_{2} - K_{1}), ω]

where

{\tilde{I}}_{F} (K^{'}, ω) = \int_{D} I_{F} (r_{S}, ω) \exp (- i K^{'} \cdot r_{S}) d^{3} r_{S}

and

{\tilde{μ}}_{F} (K^{″}, ω) = \int_{D} μ_{F} (r_{D}, ω) \exp (- i K^{″} \cdot r_{D}) d^{3} r_{D}

are the three-dimensional spatial Fourier transforms of the strength

I_{F}

and the normalized correlation coefficient

μ_{F}

, respectively.

Now, let us consider the case that the scatterer is a particulate medium. For convenience of following discussion, we assume that all particles in the collection are determinate and identical. In this case, the scattering potential of the whole collection can be expressed as [12]

F (r^{'}, ω) = \sum_{m = 1} f (r^{'} - {r^{'}}_{m}, ω)

where

f (r^{'} - {r^{'}}_{m}, ω)

is the scattering potential of the particle located at a point

{r^{'}}_{m}

. For the simplicity of the following discussion, the scattering potential of the whole collection can be rewritten as follows [17]

F (r^{'}, ω) = f (r^{'}, ω) \otimes \sum_{m = 1} δ (r^{'} - {r^{'}}_{m}, ω)

where

\otimes

denotes the convolution operation, and

δ (\cdot \cdot \cdot)

is the Dirac delta function. Assume that the average over the ensemble of each particle and that over the ensemble of the distribution to be independent of each other. Then, upon substituting from Eq. (9) into Eq. (4), one can obtain the expression of the correlation function of the whole collection as

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = C_{f} ({r^{'}}_{1}, {r^{'}}_{2}, ω) \otimes C_{n} ({r^{'}}_{1}, {r^{'}}_{2}, ω)

where

C_{f} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = f^{*} ({r^{'}}_{1}, ω) f ({r^{'}}_{2}, ω)

is the correlation function of each particle, and

C_{n} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = 〈 \sum_{m = 1} \sum_{n = 1} δ^{*} ({r^{'}}_{1} - {r^{'}}_{n}, ω) δ ({r^{'}}_{2} - {r^{'}}_{m}, ω) 〉

is the distribution function of the particles. If the distribution of particles in the collection is quasi-homogeneous, the distribution function can be expressed as a product of its strength function

S_{n}

and its normalized correlation coefficient

μ_{n}

, i.e.

C_{n} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = S_{n} (\frac{{r^{'}}_{1} + {r^{'}}_{2}}{2}, ω) μ_{n} ({r^{'}}_{2} - {r^{'}}_{1}, ω)

Upon substituting from Eq. (10) together with Eqs. (11) and (13) first into Eq. (3), and then into Eq. (2), averaged over the ensemble of the Fourier transform of the distribution of particles, the cross-spectral density function of the collection of particles can be expressed as

W_{P}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{S^{(i)} (ω)}{r^{2}} {\tilde{f}}^{*} (- K_{1}, ω) \tilde{f} (K_{2}, ω) {\tilde{S}}_{n} [(K_{1} + K_{2}), ω] {\tilde{μ}}_{n} [\frac{1}{2} (K_{2} - K_{1}), ω]

where

\tilde{f} (K, ω) = \int_{D} f (r^{'}, ω) \exp (- i K \cdot r^{'}) d^{3} r^{'}

{\tilde{S}}_{n} (K^{'}, ω) = \int_{D} S_{n} (r_{S}, ω) \exp (- i K^{'} \cdot r_{S}) d^{3} r_{S}

and

{\tilde{μ}}_{n} (K^{″}, ω) = \int_{D} μ_{n} (r_{D}, ω) \exp (- i K^{″} \cdot r_{D}) d^{3} r_{D}

denote the three-dimensional spatial Fourier transforms of

f (r^{'}, ω)

,

S_{n} (r_{S}, ω)

, and

μ_{n} (r_{D}, ω),

respectively. It should be noted that the product of the Fourier transform of the strength and the Fourier transform of the normalized correlation coefficient of the distribution function is also called as the pair-structure factor of the collection [14].

3. Possibility for generating identical far-zone distribution

In this section, the possibility for a quasi-homogeneous medium and a collection of determinate particles with quasi-homogeneous distribution to generate scattered field with identical normalized spectral density or with identical spectral degree of coherence will be discussed.

A. the possibility for identical normalized spectral density

The spectral density of the far-zone scattered field, which can be obtained from the cross-spectral density function of the scattered field, is defined as ([31], Sec.6.2)

S^{(s)} (r s, ω) = W^{(s)} (r s, r s, ω)

For the quasi-homogeneous medium, its spectral density of the scattered field can be obtained by substituting from Eq. (6) into Eq. (16), with a form of

S_{Q}^{(s)} (r s, s_{0}, ω) = \frac{S^{(i)} (ω)}{r^{2}} {\tilde{I}}_{F} (0, ω) {\tilde{μ}}_{F} [k (s - s_{0}), ω]

For the collection of determinate particles with quasi-homogeneous distribution, upon substituting from Eq. (14) into Eq. (16), one can find the spectral density of the scattered field the expression

S_{P}^{(s)} (r s, s_{0}, ω) = \frac{S^{(i)} (ω)}{r^{2}} {\tilde{S}}_{n} (0, ω) {| \tilde{f} [k (s - s_{0}), ω] |}^{2} {\tilde{μ}}_{n} [k (s - s_{0}), ω]

As shown in Eqs. (17) and (18), the scattered spectral density generated by a quasi-homogeneous medium is governed by the Fourier transform of the normalized correlation coefficient of the scattering potential, while the scattered spectral density generated by a collection of particles is governed by the Fourier transform of the scattering potential of each particle and the Fourier transform of the normalized correlation coefficient of the distribution function of particles in the collection. Therefore, one can obtain the density equivalence theorem as: if the Fourier transform of the normalized correlation coefficient of the scattering potential of the quasi-homogeneous medium has the same form as the product of the square of modulus of the Fourier transform of the scattering potential of each particle and the Fourier transform of the normalized correlation coefficient of the distribution function of the particles, the two different media can generate identical normalized spectral density.

B. the possibility for identical spectral degree of coherence

The spectral degree of coherence of the far-zone scattered field can be obtained from the cross-spectral density and the spectral density of the scattered field, with the following definition ([31], Sec.4.2)

μ^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{W^{(s)} (r s_{1}, r s_{2}, s_{0}, ω)}{\sqrt{S^{(s)} (r s_{1}, s_{0}, ω)} \sqrt{S^{(s)} (r s_{2}, s_{0}, ω)}}

Upon substituting from Eqs. (6) and (17) into Eq. (19), one can find the scattered spectral degree of coherence of the quasi-homogeneous medium to be

μ_{Q}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{{\tilde{I}}_{F} [k (s_{2} - s_{1}), ω]}{{\tilde{I}}_{F} (0, ω)} \cdot \frac{{\tilde{μ}}_{F} [k (\frac{s_{1} + s_{2}}{2} - s_{0}), ω]}{\sqrt{{\tilde{μ}}_{F} [k (s_{1} - s_{0}), ω]} \sqrt{{\tilde{μ}}_{F} [k (s_{2} - s_{0}), ω]}}

Because for a quasi-homogeneous medium the normalized correlation coefficient of the scattering potential $μ_{F} (r^{'}, ω)$ is a fast function of its spatial argument, it follows from the well-known reciprocity theorem concerning Fourier-transform pairs that ${\tilde{μ}}_{F} (K, ω)$ should be a slow function of $K$ [18]. In this case

{\tilde{μ}}_{F} [k (s_{1} - s_{0}), ω] \approx {\tilde{μ}}_{F} [k (s_{2} - s_{0}), ω] \approx {\tilde{μ}}_{F} [k (\frac{s_{1} + s_{2}}{2} - s_{0}), ω]

Upon substituting from Eq. (21) into Eq. (20), the spectral degree of coherence of the quasi-homogeneous medium can be simplified as

μ_{Q}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{{\tilde{I}}_{F} [k (s_{2} - s_{1}), ω]}{{\tilde{I}}_{F} (0, ω)}

Similarly, upon substituting from Eqs. (14) and (18) into Eq. (19), and making use of the Fourier-transform approximations as Eq. (21), one can find the scattered spectral degree of coherence of the collection of particles the expression

μ_{P}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \frac{{\tilde{S}}_{n} [k (s_{2} - s_{1}), ω]}{{\tilde{S}}_{n} (0, ω)}

It is shown in Eq. (22) that the spectral degree of coherence of scattered field generated by the quasi-homogeneous medium is governed by the Fourier transform of the strength of the scattering potential of the scatterer, while in Eq. (23) it is shown that the spectral degree of coherence of scattered field generated by the collection of particles is governed by the Fourier transform of the strength of the distribution function of the particles. Analogy to the discussion on the density equivalence theorem, we can conclude the coherence equivalence theorem as follows: a quasi-homogeneous medium and a collection of deterministic particles with quasi-homogeneous distribution can generate scattered field with identical spectral degree of coherence if the Fourier transform of the strength of the scattering potential and the Fourier transform of the strength of the distribution function have the same expression.

4. An example

In this section, an example will be discussed to illustrate the above results, and the condition for identical normalized spectral density and the condition for identical spectral degree of coherence of the scattered field will be investigated. Firstly, let us consider the far-zone spectral density of the scattered field. For a quasi-homogeneous medium, assume that the strength and the normalized correlation coefficient of the scattering potential are given by Gaussian functions [18], i.e.

I_{F} (r_{S}, ω) = A \exp (- r_{S}^{2} / 2 σ_{I}^{2})

and

μ_{F} (r_{D}, ω) = \exp (- r_{D}^{2} / 2 σ_{μ}^{2})

where

A

is a constant,

σ_{I}

and

σ_{μ}

are the effective length and the effective correlation length of the scattering potential of the medium, respectively. Upon substituting from Eq. (24) first into Eq. (7), and then into Eq. (17), after manipulating the Fourier transforms, one can find the scattered spectral density of the quasi-homogeneous medium as

S_{Q}^{(s)} (r s, s_{0}, ω) = \frac{A {(2 π)}^{3} σ_{I}^{3} σ_{μ}^{3} S^{(i)} (ω)}{r^{2}} \exp [- k^{2} {(s - s_{0})}^{2} σ_{μ}^{2} / 2]

For the collection of deterministic particles with quasi-homogeneous distribution, we assume that the scattering potential of each particle, the strength and the normalized correlation coefficient of the distribution function are all Gaussian-centered [17], i.e.

f (r^{'}, ω) = B \exp (- {r^{'}}^{2} / 2 σ^{2})

S_{n} (r_{S}, ω) = C \exp (- r_{S}^{2} / 2 σ_{n s}^{2})

and

μ_{n} (r_{D}, ω) = \exp (- r_{D}^{2} / 2 σ_{n η}^{2})

where

B

and

C

are both constant,

σ

is the effective width of the scattering potential of each particle, and

σ_{n s}

,

σ_{n η}

are the effective length and the effective correlation length of the distribution function, respectively. Upon substituting from Eq. (26) first into Eq. (15), and then into Eq. (18), one can obtain the scattered spectral density of the collection of particles, with the following form

S_{P}^{(s)} (r s, s_{0}, ω) = \frac{B^{2} C {(2 π)}^{6} σ^{6} σ_{n s}^{3} σ_{n η}^{3} S^{(i)} (ω)}{r^{2}} \exp [- k^{2} {(s - s_{0})}^{2} (2 σ^{2} + σ_{n η}^{2}) / 2]

It follows at once from Eqs. (25) and (27) that, the condition for the two different media to produce identical normalized spectral density can be expressed as

σ_{μ}^{2} = 2 σ^{2} + σ_{n η}^{2}

The normalized correlation functions for a quasi-homogeneous medium and that for a collection of particles whose structural characteristic parameters are governed by Eq. (28) are plotted in Figs. 2(a) and 2(b), respectively. As shown in Fig. 2, the distributions of the correlation functions of the two media are different even if the condition of Eq. (28) is satisfied.

Fig. 2 Normalized correlation functions for the two different media whose structural characteristic parameters are governed by Eq. (28). The parameters for calculations are chosen as follows: $(a) σ_{I} = 20 λ,$ $σ_{μ} = 4 λ;$ $(b) σ = λ,$ $σ_{n s} = 25 λ,$ $σ_{n η} = \sqrt{14} λ .$

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In Fig. 3, the normalized spectral density of the scattered field is plotted. Figure 3(a) is the normalized spectral density of light waves on scattering from the Gaussian-centered quasi-homogeneous medium described by Fig. 2(a), while Fig. 3(b) is the normalized spectral density of light waves on scattering from the Gaussian-centered determinate particles with Gaussian-centered quasi-homogeneous distribution described by Fig. 2(b). Comparing with Figs. 3(a) and 3(b), it is shown that these two different media indeed produce scattered field with identical normalized spectral density when the condition of Eq. (28) is satisfied.

Fig. 3 Normalized spectral densities of the far-zone scattered field of two different media. The parameters for calculations in (a) and (b) are the same as those in Figs. 2(a) and 2(b), respectively.

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Now, let us consider the spectral degree of coherence of the scattered field. For the quasi-homogeneous medium, its spectral degree of coherence of the scattered field can be obtained by substituting from Eq. (24)a) first into Eq. (7)a), and then into Eq. (22), with a form of

μ_{Q}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \exp [- k^{2} {(s_{2} - s_{1})}^{2} σ_{I}^{2} / 2]

For the collection of deterministic particles with random distribution, upon substituting from Eq. (26)b) first into Eq. (15)b), and then into Eq. (23), we obtain for the spectral degree of coherence of the scattered field the expression

μ_{P}^{(s)} (r s_{1}, r s_{2}, s_{0}, ω) = \exp [- k^{2} {(s_{2} - s_{1})}^{2} σ_{n s}^{2} / 2]

Combining Eq. (29) with Eq. (30), the condition for identical spectral degree of coherence generated by the two different media can be expressed as follows

σ_{I}^{2} = σ_{n s}^{2}

The normalized correlation functions for a quasi-homogeneous medium and that for a collection of particles whose structural characteristic parameters are governed by Eq. (31) are plotted in Figs. 4(a) and 4(b), respectively. In Fig. 5, the spectral degrees of coherence of the far-zone scattered field are plotted. Figure 5(a) is the spectral degree of coherence produced by a medium that described by Fig. 4(a), while Fig. 5(b) is the spectral degree of coherence produced by a medium that described by Fig. 4(b). It is shown from Figs. 4 and 5 that the Gaussian-centered quasi-homogeneous medium and the collection of particles with Gaussian-centered quasi-homogeneous distribution, which are different from each other, can generate scattered field with identical spectral degree of coherence when the condition of Eq. (31) is satisfied.

Fig. 4 Normalized correlation functions of the two different media whose structural characteristic parameters are governed by Eq. (31). The parameters for calculations are chosen as follows: $(a) σ_{I} = 25 λ,$ $σ_{μ} = 5 λ;$ $(b) σ = λ,$ $σ_{n s} = 25 λ,$ $σ_{n η} = 3 λ .$

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Fig. 5 Spectral degrees of coherence of the far-zone scattered field of the two different media. The parameters for calculations in (a) and (b) are the same as those in Figs. 4(a) and 4(b), respectively.

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In this paper, the condition for a quasi-homogeneous medium and a quasi-homogeneous distributed particles collection to generate scattered field with identical normalized spectral density and the condition for them to generate scattered field with identical spectral degree of coherence are discussed. If the two conditions between these two media are satisfied simultaneously, the scattered field may demonstrate identical normalized spectral density and identical spectral degree of coherence simultaneously. To experiment verification of our results, one should first design different media that satisfy the conditions. A potential method may be inspired by the research made by Korotkova [33]. It is shown that a prescribed scattering medium can be designed by changing the correlation functions, and a practical realization of this medium can be accomplished with some modern manufacturing techniques such as liquid crystal light modulators or 3D printing [33–35].

5. Conclusion

In summary, the possibility for identical far-zone behaviors of light waves scattered from continuous media and from particulate media has been investigated. It is shown that when certain conditions between these two different media are satisfied, the far-zone normalized spectral density or the far-zone spectral degree of coherence will demonstrate an identical distribution. An example of light waves scattered from a Gaussian-centered quasi-homogeneous medium and from a collection of particles with Gaussian-centered quasi-homogeneous distribution has been discussed. These results may have potential applications in the inverse scattering problem.

Funding

National Natural Science Foundation of China (NSFC) (11404231, 61475105).

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Equivalence theorem of light waves on scattering from media with different types

Abstract

1. Introduction

2. Far-zone behaviors of light waves on scattering

3. Possibility for generating identical far-zone distribution

A. the possibility for identical normalized spectral density

B. the possibility for identical spectral degree of coherence

4. An example

5. Conclusion

Funding

References and links

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Optics Express