In the light of the perspective of statistical similarity, we examine the maintenance of the second-order coherence of a light wave on weak scattering from a random medium. Some new and nontrivial results relating to properties of the scattered field which remains the second-order coherence of the incident field are presented. By assuming that the scattered field remains the second-order coherence, we can show that all of higher order correlation functions of Fourier component of the scattering potential can reduce to the like-factorization forms with a series of constant coefficients. These coefficients furnish an efficient and direct way to describe the higher order coherence property of the scattered field. We also show that the combination of the maintenance of second- and fourth-order coherence implies the scattered field coherence to all orders. Finally, the structure feature of the random medium is also discussed when the coherence of the incident field is retained up to 2nth order, in particular, in the case of the second-order coherence. Our theory is an important contribution for understanding of spatially fully coherent scattered fields, and also gives a general and new method to discuss the variation of the coherence of the scattered field.
1. Introduction
Weak scattering theory is not only a topic of considerable importance but also has potential applications in many fields, such as remote sensing, climate research, medical diagnosis, and so on. During the past three decades, the research works are mainly concerned with the discussion of the statistical properties of the scattered field for various types of incident waves and scatterers [1–18] (for a review of this research, please see [19]). Recently, some great breakthrough has been made on this theory. For example, a lot of novel methods designing a scattering medium which can produce the expected scattered intensity patterns have been proposed [20–26]. The higher order correlation of the scattered field and its application to determine the structure feature of the scattering medium have been discussed intensively [27–32]. Equivalence theorem for light waves on weak scattering has also been proposed [33,34].
On the other hand, in recent studies, the maintenance of the coherence of a light wave on weak scattering has stimulated considerable interests. In 2010, Wang et al discussed the condition for the invariance of the spectral degree of coherence of a fully coherent light wave on weak scattering [35]. It is shown that if the modulus of the six-dimensional Fourier transform of the correlation function of scattering potential can be factorized, the spectral degree of coherence of the scalar plane wave on weak scattering will remain invariant. Soon after, this condition was generalized to the scattering of electromagnetic plane waves by Li et al [36]. It is shown that if the scattered field retains the second-order coherence of the incident field, other than the limitation for the scattering medium should be met, more restrictions regarding scattering angle and polarization of the incident plane wave should also be required. It is clear from these remarks that the notion of the fully coherent scattered field is an important one, and evidently should be studied more carefully than has been done up to now. In this paper, we propose a new perspective, i.e., based on the fluctuating Fourier component of the scattering potential rather than its correlation function, to examine the maintenance of the second-order coherence of a light wave on weak scattering from a random medium. We obtain many new results different from the earlier works, and reveal the genuine physical implication of the maintenance of the second-order coherence. By assuming that the scattered field remains the second-order coherence, we generalize our analysis to higher order coherence, and show that all of higher order correlation functions of Fourier component of the scattering potential can reduce to the like-factorization forms with a series of constant coefficients. These coefficients give an efficient and direct way to describe the higher order coherence property of the scattered field. We also show that the combination of the maintenance of second- and fourth-order coherence implies the scattered field coherence to all orders. Finally, the structure feature of the random medium will also be discussed when the coherence of the incident field is retained up to 2nth order, in particular, in the case of the second-order coherence.
2. Maintenance of the second-order coherence of a light wave on weak scattering from a random medium: from the perspective of statistical similarity
We begin by assuming that a monochromatic scalar plane wave, propagating in a direction specified by a real unit vector $\mathbf {s}_{0}$, is incident on a medium occupying a finite three-dimensional domain (see Fig. 1). The incident field within the scatterer at a point $\mathbf {r}^{\prime }$ can be represented as [37]
(1)$$U^{(i)}(\mathbf{r}^{\prime},\mathbf{s}_{0};\omega)=a(\omega)\exp\left( ik\mathbf{s}_{0}\cdot\mathbf{r}^{\prime}\right) ,$$
where $a(\omega )$, in general, is a complex quantity, $\omega$ being the frequency of the incident light and will be dropped in the following for brevity, and $k=\omega /c$, $c$ being the speed of light in vacuum.Let $F\left ( \mathbf {r}^{\prime }\right )$ be the scattering potential of the medium, which is related to the refractive index of the medium as [37,38]
(2)$$F\left( \mathbf{r}^{\prime}\right) =\frac{k^{2}}{4\pi}\left[ n^{2}\left( \mathbf{r}^{\prime}\right) -1\right] ,$$
where $n\left ( \mathbf {r}^{\prime }\right )$ is the refractive index of the medium. For a random medium whose scattering potential $F\left ( \mathbf {r}^{\prime }\right )$ varies randomly with position $\mathbf {r} ^{\prime }$ within the scatterer, its physical properties can be described by the second moment of the scattering potential $F\left ( \mathbf {r}^{\prime }\right ) ,$ which is given by formula [37] (3)$$C_{F}^{(1,1)}\left( \mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime}\right) =\left\langle F^{{\ast}}\left( \mathbf{r}_{1}^{\prime}\right) F\left( \mathbf{r}_{2}^{\prime}\right) \right\rangle _{M},$$
where the angular brackets with subscript M denote the average taken over the ensemble of sample media. Assume that the scatterer is so weak that the scattering can be treated within the accuracy of the first-order Born approximation [37,38]. The cross-spectral density function of the scattered field in the far zone at two points, specified by position vectors $r\mathbf {s}_{1}$ and $r\mathbf {s}_{2}\left ( \left \vert \mathbf {s} _{1}\right \vert ^{2}=\left \vert \mathbf {s}_{2}\right \vert ^{2}=1\right )$, has the following the expression [10,37] (4)$$W^{(s)}\left( r\mathbf{s}_{1},\;r\mathbf{s}_{2},\mathbf{s}_{0}\right) =\frac{\left\vert a\right\vert ^{2}}{r^{2}}\widetilde{C}_{F}^{(1,1)}\left[{-}k(\mathbf{s}_{1}-\mathbf{s}_{0}),\;k(\mathbf{s}_{2}-\mathbf{s}_{0})\right] ,$$
where (5)$$\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) = {\displaystyle\int_{D}} {\displaystyle\int_{D}} \left\langle F^{{\ast}}\left( \mathbf{r}_{1}^{\prime}\right) F\left( \mathbf{r}_{2}^{\prime}\right) \right\rangle _{M}\exp\left[{-}i\left( \mathbf{K}_{2}\cdot\mathbf{r}_{2}^{\prime}+\mathbf{K}_{1}\cdot\mathbf{r} _{1}^{\prime}\right) \right] d^{3}r_{1}^{\prime}d^{3}r_{2}^{\prime}$$
is the six-dimensional Fourier transform of the correlation function of the scattering potential of the medium, and (6)$$\mathbf{K}_{1}={-}k(\mathbf{s}_{1}-\mathbf{s}_{0}),\quad \mathbf{K}_{2}=k(\mathbf{s}_{2}-\mathbf{s}_{0})\textrm{.}$$
The vectors $\mathbf {K}_{1}$ and $\mathbf {K}_{2}$ are analogous to the momentum transfer vector of quantum mechanical theory of potential scattering, with $\left \vert \mathbf {K}_{1}\right \vert \leq {\textstyle 2k}$ and $\left \vert \mathbf {K}_{2}\right \vert \leq {\textstyle 2k}$. On interchanging the order of integration and ensemble average in Eq. (5), we can reformulate the six-dimensional Fourier transform of the correlation function of the scattering potential of the medium as [7] (7)$$\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{2}\right) \right\rangle _{M},$$
where $\widetilde {F}\left ( \mathbf {K}\right ) = {\textstyle \int _{D}} F\left ( \mathbf {r}^{\prime }\right ) \exp \left [ -i\mathbf {K\cdot r}^{\prime }\right ] d^{3}r^{\prime }$ is the Fourier component of the scattering potential of the medium [38].Let $r\mathbf {s}_{1}=r\mathbf {s}_{2}=r\mathbf {s}$ in Eq. (4), the cross-spectral density function can reduce to spectral density function, and combine Eqs. (6) and (7), then the spectral density function of the scattered field can be expressed as
(8)$$S^{(s)}\left( r\mathbf{s,\;s}_{0}\right) =\frac{\left\vert a\right\vert ^{2} }{r^{2}}\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}\right) \widetilde{F}\left( \mathbf{K}\right) \right\rangle _{M}.$$
The spectral degree of coherence of the far-zone scattered field can also be obtained from its cross-spectral density function by the following definition [37,39] (9)$$\mu^{(s)}\left( r\mathbf{s}_{1},\;r\mathbf{s}_{2},\mathbf{s}_{0}\right) =\frac{W^{(s)}\left( r\mathbf{s}_{1},\;r\mathbf{s}_{2},\mathbf{s}_{0}\right) }{\sqrt{S^{(s)}\left( r\mathbf{s}_{1}\;\mathbf{,s}_{0}\right) }\sqrt {S^{(s)}\left( r\mathbf{s}_{2}\;\mathbf{,s}_{0}\right) }}.$$
On substituting from Eqs. (4), (7) and (8) into Eq. (9), one can find the spectral degree of coherence of the scattered field (10)$$\mu^{(s)}\left( r\mathbf{s}_{1},\;r\mathbf{s}_{2},\mathbf{s}_{0}\right) =\mu_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) ,$$
where (11)$$\mu_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\frac {\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{2}\right) \right\rangle _{M}}{\sqrt {\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{1}\right) \right\rangle _{M}}\sqrt {\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{2}\right) \widetilde{F}\left( \mathbf{K}_{2}\right) \right\rangle _{M}}}$$
is the normalized correlation coefficient of $\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) .$ One can readily show that $0\leq$ $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) \right \vert \leq 1.$ It is evident, from Eq. (10), that when $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K} _{1},\mathbf {K}_{2}\right ) \right \vert =1$, the scattered field at two scattering directions $\left ( \mathbf {s}_{1},\mathbf {s}_{2}\right )$ is spectrally fully coherent. When $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) \right \vert =0,$ the scattered field at that pair of scattering directions is spectrally fully incoherent. In the intermediate case $0<\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K} _{1},\mathbf {K}_{2}\right ) \right \vert <1$, the scattered field at that pair of scattering directions is spectrally partially coherent.We now examine the maintenance of the coherence of a light wave on weak scattering from the perspective of statistical similarity [40,41]. We will fisrt propose the following theorem
Theorem: A coherent plane wave on weak scattering from a random medium, if the scattered field remains the second-order coherence of the incident field at a pair of scattering directions $\left ( \mathbf {s} _{1},\mathbf {s}_{2}\right )$, the associated Fourier components of the scattering potential $\widetilde {F}\left ( \mathbf {K}\right )$ labelled by that pair of scattering directions will satisfy the following relation
(12)$$\widetilde{F}\left( \mathbf{K}_{2}\right) =\Delta\left( \mathbf{K} _{1},\mathbf{K}_{2}\right) \widetilde{F}\left( \mathbf{K}_{1}\right) ,$$
where $\Delta \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ is a deterministic function of its arguments.Proof. To prove that $\widetilde {F}\left ( \mathbf {K}_{2}\right ) =\Delta \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) \widetilde {F}\left ( \mathbf {K}_{1}\right )$ is a necessary condition that the scattered field at a pair of scattering directions remains the second-order coherence of the incident field, we start with the famous Cauchy–Schwarz inequality, applied to Fourier component of the scattering potential $\widetilde {F}\left ( \mathbf {K}\right )$, which reads
(13)$$\left\vert \left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{2}\right) \right\rangle _{M}\right\vert ^{2}\leq\left\langle \left\vert \widetilde{F}\left( \mathbf{K}_{1}\right) \right\vert ^{2}\right\rangle _{M}\left\langle \left\vert \widetilde{F} (\mathbf{K}_{2})\right\vert ^{2}\right\rangle _{M}.$$
It is well-known that both sides of the inequality above are equal to each other, if and only if $\widetilde {F}\left ( \mathbf {K}_{2}\right )$ and $\widetilde {F}\left ( \mathbf {K}_{1}\right )$ are proportional to each other. From Eq. (11), $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K}_{1} ,\mathbf {K}_{2}\right ) \right \vert =1$ indicates (14)$$\left\vert \left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{2}\right) \right\rangle _{m}\right\vert ^{2}=\left\langle \left\vert \widetilde{F}\left( \mathbf{K}_{1}\right) \right\vert ^{2}\right\rangle _{m}\left\langle \left\vert \widetilde{F}\left( \mathbf{K}_{2}\right) \right\vert ^{2}\right\rangle _{M},$$
which shows that $\widetilde {F}\left ( \mathbf {K}_{1}\right )$ and $\widetilde {F}\left ( \mathbf {K}_{2}\right )$ are associated by Eq. (12). This completes the proof that $\widetilde {F}\left ( \mathbf {K}_{2}\right ) =\Delta \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) \widetilde {F}\left ( \mathbf {K}_{1}\right )$ is a necessary condition that the scattered field at a pair of scattering directions retains the second-order coherence of the incident field.Next we prove the sufficiency. On substituting from Eq. (12) into Eq. (11), it is obvious that $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K}_{1} ,\mathbf {K}_{2}\right ) \right \vert =1$ establishes. Thus, we complete the proof of the theorem.
Now let us determine the deterministic function $\Delta \left ( \mathbf {K} _{1},\mathbf {K}_{2}\right )$ in Eq. (12). On substituting Eqs. (12) into (7), after some simple calculations, we can readily find that the deterministic function $\Delta \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ can be expressed as
(15)$$\Delta\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\frac{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) }{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{1}\right) }.$$
From Eqs. (7) and (11), one can find that $\widetilde {C}_{F}^{(1,1)}\left (\mathbf {K}_{1},\mathbf {K}_{2}\right ) =\sqrt {\widetilde {C}_{F}^{(1,1)}\left (\mathbf {K}_{1},\mathbf {K}_{1}\right ) }\sqrt {\widetilde {C}_{F}^{(1,1)}\left (\mathbf {K}_{2},\mathbf {K}_{2}\right ) }\mu _{F}^{(1,1)}$ $\left ( \mathbf {K} _{1},\mathbf {K}_{2}\right )$, thus we can express $\Delta \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ as (16)$$\Delta\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\sqrt{\frac {\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{2},\mathbf{K}_{2}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{1}\right) }} \mu_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) .$$
Recall that $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K} _{2}\right ) \right \vert =1,$ we arrive at (17)$$\Delta\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\sqrt{\frac {\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{2},\mathbf{K}_{2}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{1}\right) } }\exp\left[ i\psi\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) \right] ,$$
where $\psi \left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ is the phase of the normalized correlation coefficient $\mu _{F}^{(1,1)}\left ( \mathbf {K} _{1},\mathbf {K}_{2}\right )$, which can be determined from interference experiments [10,37,39].From the relation (12), it can be seen that the scattered field remains the second-order coherence of the incident field at a pair of scattering directions, which means the fluctuations of Fourier components of the scattering potential at that pair of scattering directions have a deterministic phase and amplitude relationship. Now we use the relation (12) to derive an important result that if the scattered field is full coherence at any pair of scattering directions in the whole scattering space.
Assume that $\mathbf {K}_{a}$ is a momentum transfer vector labelled by an arbitrary scattering direction $\mathbf {s}_{a}$ in the whole scattering space, of course, which is here spectrally fully coherent. If we consider a pair of momentum transfer vectors $\mathbf {K}_{1}$ and $\mathbf {K}_{2}$ labelled by any other two scattering directions $\mathbf {s}_{1}$ and $\mathbf {s}_{2}$, respectively, in this scattering space. The relations of the Fourier component of the scattering potential $\widetilde {F}\left ( \mathbf {K}_{1}\right )$ and $\widetilde {F}\left ( \mathbf {K}_{2}\right )$ to $\widetilde {F}\left ( \mathbf {K}_{a}\right )$ can be expressed by the following forms, respectively
(18)$$ \widetilde{F}\left( \mathbf{K}_{1}\right) =\Delta\left( \mathbf{K} _{a},\mathbf{K}_{1}\right) \widetilde{F}\left( \mathbf{K}_{a}\right) ,$$
(19)$$ \widetilde{F}\left( \mathbf{K}_{2}\right) =\Delta\left( \mathbf{K} _{a},\mathbf{K}_{2}\right) \widetilde{F}\left( \mathbf{K}_{a}\right) . $$
On using Eq. (15), Eqs. (18) and (19) can be expressed as (20)$$ \widetilde{F}\left( \mathbf{K}_{1}\right) =\frac{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{1}\right) }{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\widetilde{F} \left( \mathbf{K}_{a}\right) ,$$
(21)$$ \widetilde{F}\left( \mathbf{K}_{2}\right) =\frac{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{2}\right) }{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\widetilde{F} \left( \mathbf{K}_{a}\right) . $$
On substituting from Eqs. (20) and (21) into Eq. (7), and using the Hermitian property of correlation function, i.e., $[\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{a},\mathbf {K}_{1}\right ) ]^{\ast } =\widetilde {C}_{F} ^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{a}\right )$, we can obtain (22)$$\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\frac{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{a}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\frac{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{2}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{a}\right) \widetilde{F}\left( \mathbf{K}_{a}\right) \right\rangle _{M}.$$
It is clear, from Eq. (22), that the Fourier transform of the correlation function of the scattering potential $\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ meets the following functional equation (23)$$\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\frac{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{a}\right) \widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{2}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }.$$
Let us now define the complex function $\widetilde {\Re }(\mathbf {K})$ as (24)$$\widetilde{\Re}(\mathbf{K})=\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K} _{a},\mathbf{K}\right) /\sqrt{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K} _{a},\mathbf{K}_{a}\right) }.$$
With this definition, we can formulate our results as follows: There exists a complex function such that $\widetilde {\Re }(\mathbf {K})$ for all $\mathbf {K}_{1}$ and $\mathbf {K}_{2}$ we have (25)$$\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{1},\mathbf{K}_{2}\right) =\widetilde{\Re}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{\Re }(\mathbf{K}_{2}).$$
It seems that, from Eq. (24), the factorized form of $\widetilde {C} _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ depends on the choice of $\mathbf {K}_{a}$. However, we can see that if Eq. (25) holds for all $\mathbf {K}_{1}$ and $\mathbf {K}_{2}$, this dependence will become trivial. Assume that $\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K} _{2}\right )$ has the second factorized form in which $\widetilde {\Re }\left ( \mathbf {K}\right )$ is replaced by $\widetilde {\Re }^{^{\prime }}\left ( \mathbf {K}\right ) .$Thus we have relations (26)$$\widetilde{\Re}^{{\ast}^{\prime}}\left( \mathbf{K}_{1}\right) \widetilde{\Re }^{^{\prime}}\left( \mathbf{K}_{2}\right) =\widetilde{\Re}^{{\ast}}\left( \mathbf{K}_{1}\right) \widetilde{\Re}(\mathbf{K}_{2})$$
or (27)$$\frac{\widetilde{\Re}^{{\ast}^{\prime}}\left( \mathbf{K}_{1}\right) }{\widetilde{\Re}^{{\ast}}\left( \mathbf{K}_{1}\right) }=\frac{\widetilde{\Re }(\mathbf{K}_{2})}{\widetilde{\Re}^{^{\prime}}\left( \mathbf{K}_{2}\right) }.$$
Equation (27) indicates that there must exist a constant $\alpha$ such that $\widetilde {\Re }^{^{\prime }}\left ( \mathbf {K}\right ) =\alpha \widetilde {\Re }\left ( \mathbf {K}\right ) .$ From Eq. (26), it follows that $\left \vert \alpha \right \vert ^{2}=1.$ The change of $\mathbf {K}_{a}$ only corresponds to multiplying $\widetilde {\Re }\left ( \mathbf {K}\right )$ by a phase factor. Apparently, this phase factor disappears in the calculation of $\widetilde {C} _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) .$ Therefore, the factorized form of $\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{1} ,\mathbf {K}_{2}\right )$ is independent on the choice of $\mathbf {K}_{a}$.3. Higher order correlation property of the scattered field which remains the second-order coherence of the incident field
Although the relation (12) or Eq. (25) is only imposed on the maintenance of second-order coherence, in fact, its consequence also includes a set of identities which must be obeyed by the higher order correlation function $\widetilde {C}_{F}^{(n,\;n)} (\mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K}_{n};\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n})$. These identities permit us to simplify the $\widetilde {C}_{F}^{(n,\;n)}(\mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K} _{n};\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n})$ for arbitrary arguments as a like-factorization form with a constant coefficient. We now show these nontrivial identities. First we give the higher order correlation function of Fourier component of the scattering potential
(28)$$\widetilde{C}_{F}^{(n,\;n)}(\mathbf{K}_{1},\cdot{\cdot}\cdot,\mathbf{K} _{n};\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n})=\left\langle \widetilde{F}^{{\ast}}\left( \mathbf{K}_{1}\right) \cdot{\cdot}\cdot \widetilde{F}^{{\ast}}\left( \mathbf{K}_{n}\right) \widetilde{F}\left( \mathbf{K}_{n+1}\right) \cdot{\cdot}\cdot\widetilde{F}\left( \mathbf{K} _{2n}\right) \right\rangle _{M}.$$
We start by still assuming that $\mathbf {K}_{a}$ is a momentum transfer vector labelled by an arbitrary scattering direction $\mathbf {s}_{a}$. Since the scattered field remains the second-order coherence of the incident field in the whole scattering space, the relation of any other Fourier component of scattering potential $\widetilde {F}\left ( \mathbf {K}_{i}\right )$ to $\widetilde {F}\left ( \mathbf {K}_{a}\right )$ can be written as (29a)$$ \widetilde{F}\left( \mathbf{K}_{i}\right) =\frac{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{i}\right) }{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\widetilde{F} \left( \mathbf{K}_{a}\right) ,$$
(29b)$$ \widetilde{F}^{{\ast}}\left( \mathbf{K}_{i}\right) =\frac{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{i},\mathbf{K}_{a}\right) }{\widetilde{C} _{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\widetilde{F} ^{{\ast}}\left( \mathbf{K}_{a}\right) . $$
We now apply each of the identities Eqs. (29a) and (29b) $n$ times to Eq. (28), then we can give the higher order correlation function of Fourier component of the scattering potential as (30)$$\widetilde{C}_{F}^{(n,\;n)}(\mathbf{K}_{1},\cdot{\cdot}\cdot,\mathbf{K} _{n};\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n} )=\overset{n}{\underset{i=1}{ {\textstyle\prod} }}\frac{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{i},\mathbf{K}_{a}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) }\frac{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K} _{i+n}\right) }{\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a} ,\mathbf{K}_{a}\right) }\left\langle \left[ \widetilde{F}^{{\ast}}\left( \mathbf{K}_{a}\right) \right] ^{n}\left[ \widetilde{F}\left( \mathbf{K}_{a}\right) \right] ^{n}\right\rangle _{M}.$$
By recalling the definition of $\widetilde {\Re }(\mathbf {K})$ in Eq. (24), we can finally write Eq. (30) in the following form (31)$$\widetilde{C}_{F}^{(n,\;n)}(\mathbf{K}_{1},\cdot{\cdot}\cdot,\mathbf{K} _{n};\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n})=\mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a};\mathbf{K} _{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\overset{n}{\underset{i=1}{ {\textstyle\prod} }}\widetilde{\Re}^{{\ast}}\left( \mathbf{K}_{i}\right) \widetilde{\Re }(\mathbf{K}_{i+n}),$$
where (32)$$\mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K} _{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})=\frac{\widetilde{C} _{F}^{(n,\;n)}\left( \mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a} ,\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}\right) }{\left[ \widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) \right] ^{n}}.$$
From the definition of $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a} ,\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a})$, it is shown that the quantities $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ can be viewed as constants because they are independent of $\mathbf {K}_{1}\cdot \cdot \cdot \mathbf {K}_{n}$. It is easy to show that $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ don’t depend on the choice of $\mathbf {K}_{a}$ either. To see this we need only note that $\widetilde {C}_{F}^{(n,\;n)}(\mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K} _{n};\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n})$ are independent of $\mathbf {K}_{a}$ and recall that in the last section we showed that the products of the $\widetilde {\Re }(\mathbf {K})$ such as $\widetilde {C} _{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right )$ in Eq. (25) are independent of $\mathbf {K}_{a}$. In effect, $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ are constants determined only by the physical property of the medium. We can also see that, from Eq. (32), $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K} _{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ are real and non-negative.We have shown that a coherent plane wave on scattering from a random medium, if the scattered field retains the second-order coherence of the incident field, the higher order correlation functions of Fourier component of the scattering potential have necessarily the like-factorization form (31). It is clear, for the scattered field remaining the second-order coherence of the incident field, that $\mu _{F}^{\left ( 1,1\right ) }(\mathbf {K}_{a} ,\mathbf {K}_{a})=1,$ but $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a} ,\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a})=1$ for $n\neq 1$, which depend on the physical property of the scattering medium. For a deterministic medium, for example, one can readily find $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})=1$ for all values of $n,$ where $\widetilde {C}_{F}^{(n,\;n)}\left ( \mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K}_{n},\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n}\right ) =$ $\widetilde {F}^{\ast }\left ( \mathbf {K}_{1}\right ) \cdot \cdot \cdot \widetilde {F}^{\ast }\left ( \mathbf {K}_{n}\right ) \widetilde {F}\left ( \mathbf {K}_{n+1}\right ) \cdot \cdot \cdot \widetilde {F}\left ( \mathbf {K} _{2n}\right )$ are in accordance with the definition of ${2n}$th-order coherence in [42]. Hence, $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a} ,\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a})=1$ for all values of ${n}$ can be taken as the criterion that the scattered field remains the complete coherence of the incident field up to $2n$th order. We can see that the quantities $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K} _{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ play a significant role in the description of the higher order coherence of the scattered field which remains the second-order coherence of the incident field. We next give detailed comments on the quantities $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$.
A genuine and realizable correlation function is restricted by the constraint of nonnegative definiteness, which can be fulfilled if the correlation function can be expressed as a superposition integral of the form $\widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{1},\mathbf {K}_{2}\right ) = {\textstyle \int }$ $p\left ( \mathbf {v}\right ) \widetilde {\Re }^{\ast }\left ( \mathbf {K} _{1},\mathbf {v}\right ) \widetilde {\Re }(\mathbf {K}_{2},\mathbf {v})d^{3}v$, where $p\left ( \mathbf {v}\right )$ is an arbitrary nonnegative weight function, and $\int p\left ( \mathbf {v}\right ) d^{3}v=1.$ This condition can be generalized to the ${2n}$th-order correlation function of Fourier component of the scattering potential, which can be expressed as
(33)$$\widetilde{C}_{F}^{\left( n,\;n\right) }(\mathbf{K}_{1},\cdot{\cdot} \cdot,\mathbf{K}_{n};\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n})=\int p\left( \mathbf{v}\right) \underset{i=1}{\overset{n}{{\textstyle\prod}} }\widetilde{\Re}^{{\ast}}\left( \mathbf{K}_{i},\mathbf{v}\right) \widetilde{\Re}(\mathbf{K}_{i+n},\mathbf{v})d^{3}v.$$
Here, we refer to the generalization of Tchebycheff inequality in [43], and further generalize it from two dimensional version to three dimensional one, viz., (34)$${\textstyle\int} p\left( \mathbf{v}\right) A\left( \mathbf{v}\right) B(\mathbf{v} )d^{3}v\geq {\textstyle\int} p\left( \mathbf{v}\right) A\left( \mathbf{v}\right) d^{3}v {\textstyle\int} p\left( \mathbf{v}^{\prime}\right) B(\mathbf{v}^{\prime})d^{3}v^{\prime}$$
If we set $A\left ( \mathbf {v}\right ) =\left \vert \widetilde {\Re } (\mathbf {K}_{a},\mathbf {v})\right \vert ^{2m},$ $B\left ( \mathbf {v}\right ) =\left \vert \widetilde {\Re }(\mathbf {K}_{a},\mathbf {v})\right \vert ^{2(n-m)},$ where ${n}\geq {m}\geq 0$, one can find the following inequality (35)$$\begin{aligned} & \int p\left( \mathbf{v}\right) \left\vert \widetilde{\Re}(\mathbf{K} _{a},\mathbf{v})\right\vert ^{2m}\left\vert \widetilde{\Re}(\mathbf{K} _{a},\mathbf{v})\right\vert ^{2(n-m)}d^{3}v\\ & \geq\int p\left( \mathbf{v}\right) \left\vert \widetilde{\Re} (\mathbf{K}_{a},\mathbf{v})\right\vert ^{2m}d^{3}v\int p\left( \mathbf{v} ^{^{\prime}}\right) \left\vert \widetilde{\Re}(\mathbf{K}_{a},\mathbf{v} ^{^{\prime}})\right\vert ^{2(n-m)}d^{3}v^{^{\prime}}, \end{aligned}$$
After some simple calculations, we can rewrite inequality (35) as (36)$$\begin{aligned} & \int p\left( \mathbf{v}\right) \left[ \widetilde{\Re}^{{\ast}} (\mathbf{K}_{a},\mathbf{v})\widetilde{\Re}(\mathbf{K}_{a},\mathbf{v})\right] ^{n}d^{3}v\\ & \geq\int p\left( \mathbf{v}\right) \left[ \widetilde{\Re}^{{\ast} }(\mathbf{K}_{a},\mathbf{v})\widetilde{\Re}(\mathbf{K}_{a},\mathbf{v})\right] ^{m}d^{3}v\int p\left( \mathbf{v}^{^{\prime}}\right) \left[ \widetilde{\Re }^{{\ast}}(\mathbf{K}_{a},\mathbf{v}^{^{\prime}})\widetilde{\Re}(\mathbf{K} _{a},\mathbf{v}^{^{\prime}})\right] ^{n-m}d^{3}v^{^{\prime}}, \end{aligned}$$
On using Eq. (33), one can readily find that the inequality (36) represents (37)$$\begin{aligned} & \widetilde{C}_{F}^{(n,\;n)}\left( \mathbf{K}_{a},\cdot{\cdot}\cdot \mathbf{K}_{a},\mathbf{K}_{a},\cdot{\cdot}\cdot\mathbf{K}_{a}\right)\\ & \geq\widetilde{C}_{F}^{(n-m,\;n-m)}\left( \mathbf{K}_{a},\cdot{\cdot} \cdot\mathbf{K}_{a},\mathbf{K}_{a},\cdot{\cdot}\cdot\mathbf{K}_{a}\right) \widetilde{C}_{F}^{(m,\;m)}\left( \mathbf{K}_{a},\cdot{\cdot}\cdot\mathbf{K} _{a},\mathbf{K}_{a},\cdot{\cdot}\cdot\mathbf{K}_{a}\right) . \end{aligned}$$
If we multiply the factor $\left [ \widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{a},\mathbf {K}_{a}\right ) \right ] ^{-n}$ on both sides of the inequality (37), then set $m=1$ and recall the assumption of $\mu _{F} ^{(1,1)}(\mathbf {K}_{a},\mathbf {K}_{a})=1$ , one can find that (38)$$\mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K} _{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\geq\mu_{F}^{\left( n-1,\;n-1\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a};\mathbf{K} _{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}).$$
We can see that $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ form a monotonically increasing sequence.Another set of inequalities can be derived from Cauchy–Schwarz inequality, whose integral form can be expressed as
(39)$$\left\vert {\textstyle\int} p\left( \mathbf{v}\right) A^{{\ast}}\left( \mathbf{v}\right) B(\mathbf{v} )d^{3}v\right\vert ^{2}\leq {\textstyle\int} p\left( \mathbf{v}\right) \left\vert A\left( \mathbf{v}\right) \right\vert ^{2}d^{3}v {\textstyle\int} p\left( \mathbf{v}\right) \left\vert B\left( \mathbf{v}^{^{\prime}}\right) \right\vert ^{2}d^{3}v^{^{\prime}}$$
If we set $A\left ( \mathbf {v}\right ) =\frac {\left [ \widetilde {\Re }(\mathbf {K}_{a},\mathbf {v})\right ] ^{n-m}}{\left [ \widetilde {C}_{F}\left ( \mathbf {K}_{a},\mathbf {K}_{a}\right ) \right ] ^{n}}$ and $B(\mathbf {v} )=\left [ \widetilde {\Re }^{\ast }(\mathbf {K}_{a},\mathbf {v})\right ] ^{m}\left [ \widetilde {\Re }(\mathbf {K}_{a},\mathbf {v})\right ] ^{n}$ and combine the definition of $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a} ,\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a}),$ i.e., Eq. (32), we can arrive at (40)$$\begin{aligned} \left[ \mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\right] ^{2} & \leq\mu_{F}^{\left( n-m,\;n-m\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\\ & \times\mu_{F}^{\left( n+m,\;n+m\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}).\end{aligned}$$
According to this inequality, we have (41)$$\begin{aligned} 2\ln\mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot ,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}) & \leq\ln \mu_{F}^{\left( n-m,\;n-m\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot ,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\\ & +\ln\mu_{F}^{\left( n+m,\;n+m\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}).\end{aligned}$$
This inequality indicates that $\ln \mu _{F}^{\left ( n,\;n\right ) } (\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ is convex function of $n$. The convexity property allows us to set $n\geq q\geq 0,$ and $m\geq 0,$ we have [44] (42)$$\begin{aligned} \left( q+m\right) \ln\mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a} ,\cdot{\cdot}\cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}) & \leq q\ln\mu_{F}^{\left( n+m,\;n+m\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\\ & +m\ln\mu_{F}^{\left( n-q,\;n-q\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\end{aligned}$$
Now we set $q=n-1$ and $h=n+m$ in inequality (42), respectively, and recall that $\mu _{F}^{(1,1)}(\mathbf {K}_{a},\mathbf {K}_{a})=1,$ we can find the inequality (43)$$\left[ \mu_{F}^{\left( n,\;n\right) }(\mathbf{K}_{a},\cdot{\cdot} \cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\right] ^{\frac{h-1}{n-1}}\leq\mu_{F}^{\left( h,\;h\right) }(\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a};\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a})\quad\textrm{for}\;h\geq n$$
This inequality gives lower limits to the rapidity with which the $\mu _{F}^{\left ( h,\;h\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ increase. Moreover, from this inequality, it is shown that if scattered fields cannot retain the fourth-order coherence of the incident field, i.e., $\mu _{F}^{\left ( 2,2\right ) }(\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a})>1,$ the $\mu _{F}^{\left ( h,\;h\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ increase without bound as $h\rightarrow \infty .$Finally we will show an intriguing property of the scattered field remaining the second-order coherence of the incident field is that the combination of the maintenance of second- and fourth-order coherence, i.e., $\mu _{F} ^{(1,1)}(\mathbf {K}_{a},\mathbf {K}_{a})=$ $\mu _{F}^{\left ( 2,2\right ) }(\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a})=1$, implies the scattered field coherence to all orders. To show this property we note that as a result of the maintenance of the second-order coherence the correlation functions $\widetilde {C}_{F}^{(n,\;n)}(\mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K}_{n};\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n})$ meet the like-factorization forms Eq. (31) with a series of constants $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K} _{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$ which characterize the scattered field. We therefore only need to examine the quantities $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})$. From Eq. (32), $\mu _{F}^{\left ( 2,2\right ) }(\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K}_{a})=1$ implies $\widetilde {C}_{F}^{(2,2)}\left ( \mathbf {K}_{a},\mathbf {K}_{a},\mathbf {K} _{a},\mathbf {K}_{a}\right ) =\left [ \widetilde {C}_{F}^{(1,1)}\left ( \mathbf {K}_{a},\mathbf {K}_{a}\right ) \right ] ^{2}.$ Thus, on substituting from Eq. (33) into the left-hand of this identity, one can obtain
(44)$$\int p\left( \mathbf{v}\right) \left\vert \widetilde{\Re}(\mathbf{K} _{a},\mathbf{v})\right\vert ^{4}d^{3}v=\left[ \widetilde{C}_{F} ^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) \right] ^{2}.$$
Using $\int p\left ( \mathbf {v}\right ) \widetilde {\Re }^{\ast }(\mathbf {K} _{a},\mathbf {v})\widetilde {\Re }(\mathbf {K}_{a},\mathbf {v})d^{3}v=\widetilde {C} _{F}^{(1,1)}\left ( \mathbf {K}_{a},\mathbf {K}_{a}\right ) ,$ and $\int p\left ( \mathbf {v}\right ) d^{3}v=1$, we can rewrite Eq. (44) as the following form (45)$$\int p\left( \mathbf{v}\right) \left[ \left\vert \widetilde{\Re} (\mathbf{K}_{a},\mathbf{v})\right\vert ^{2}-\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) \right] ^{2}d^{3}v=0.$$
One can readily find that Eq. (45) implies the following nontrivial result (46)$$\left\vert \widetilde{\Re}(\mathbf{K}_{a},\mathbf{v})\right\vert ^{2}=\widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) .$$
We now, from Eq. (33), calculate the correlation function $\widetilde {C} _{F}^{\left ( n,\;n\right ) }\left ( \mathbf {K}_{a},\cdot \cdot \cdot \mathbf {K}_{a},\mathbf {K}_{a},\cdot \cdot \cdot \mathbf {K}_{a}\right ) ,$ making use of Eq. (46) and $\int p\left ( \mathbf {v}\right ) d^{3}v=1$ in whole integral calculations, one can arrive at (47)$$\widetilde{C}_{F}^{(n,\;n)}\left( \mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K} _{a},\mathbf{K}_{a},\cdot{\cdot}\cdot,\mathbf{K}_{a}\right) =\left[ \widetilde{C}_{F}^{(1,1)}\left( \mathbf{K}_{a},\mathbf{K}_{a}\right) \right] ^{n}.$$
By recalling the definition of $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K} _{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a}),$ Eq. (32), one can find that Eq. (47) gives rise to $\mu _{F}^{\left ( n,\;n\right ) }(\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K} _{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})=1,$ for all values of ${n}$ higher than 2. Thus, we demonstrate that the combination of the maintenance of second- and fourth-order coherence implies scattered field coherence to all orders.4. Structure feature of the random medium when the coherence of the incident field is retained up to 2nth order
Based on the previous discussions, it is shown that when a coherent plane wave on weak scattering from a deterministic medium, the scattered field can remain not only the second-order coherence but also all orders coherence of the incident field. As for a random medium, there exists such a expectation that if second- and fourth-order coherence of the incident field are preserved, the scattered field will remain coherence to all orders. It is natural to, in such a case, ask what is the structure feature of the random medium. This is an important question, because answering this question clearly will not only help us to understand the fully coherent scattered field more deeply, but also contribute to the inverse scattering problem. In the following, we will solve this question. First let us discuss $\mu _{F}^{\left ( n,\;n\right ) } (\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a};\mathbf {K}_{a},\cdot \cdot \cdot ,\mathbf {K}_{a})=1,$ in Eq. (32) for all values of ${n}$ in detail. In this case, Eq. (31) becomes
(48)$$\widetilde{C}_{F}^{(n,\;n)}\left( \mathbf{K}_{1},\cdot{\cdot}\cdot,\mathbf{K} _{n},\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n}\right) =\widetilde{\Re }^{{\ast}}\left( \mathbf{K}_{1}\right) \cdot{\cdot}\cdot\widetilde{\Re}^{{\ast} }\left( \mathbf{K}_{n}\right) \widetilde{\Re}\left( \mathbf{K} _{n+1}\right) \cdot{\cdot}\cdot\widetilde{\Re}\left( \mathbf{K}_{2n}\right) .$$
In addition, the $2n$th-order moment of the scattering potential of the random medium can be defined as (49)$$C_{F}^{(n,\;n)}\left( \mathbf{r}_{1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{n}^{\prime},\mathbf{r}_{n+1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{2n}^{\prime}\right) =\left\langle F^{{\ast}}\left( \mathbf{r}_{1}^{\prime }\right) \cdot{\cdot}\cdot F^{{\ast}}\left( \mathbf{r}_{n}^{\prime}\right) F\left( \mathbf{r}_{n+1}^{\prime}\right) \cdot{\cdot}\cdot F\left( \mathbf{r}_{2n}^{\prime}\right) \right\rangle _{M}$$
From Eqs. (3) and (5), we note that Eqs. (48) and (49) are Fourier transforms of each other, i.e., the inverse Fourier transform of $\widetilde {C} _{F}^{(n,\;n)}\left ( \mathbf {K}_{1},\cdot \cdot \cdot ,\mathbf {K}_{n} ,\mathbf {K}_{n+1},\cdot \cdot \cdot ,\mathbf {K}_{2n}\right )$ is equal to $C_{F}^{(n,\;n)}\left ( \mathbf {r}_{1}^{\prime },\cdot \cdot \cdot ,\mathbf {r} _{n}^{\prime },\mathbf {r}_{n+1}^{\prime },\cdot \cdot \cdot ,\mathbf {r} _{2n}^{\prime }\right ) ,$ which can be expressed as (50)$$\begin{aligned} & C_{F}^{(n,\;n)}\left( \mathbf{r}_{1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{n}^{\prime},\mathbf{r}_{n+1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{2n}^{\prime}\right)\\ & =\overset{2n}{\overbrace{ {\textstyle\int_{V\left( \mathbf{K}_{1}\right) }} \cdot{\cdot}\cdot {\textstyle\int_{V\left( \mathbf{K}_{2n}\right) }} }}\widetilde{C}_{F}^{(n,\;n)}\left( \mathbf{K}_{1},\cdot{\cdot}\cdot ,\mathbf{K}_{n},\mathbf{K}_{n+1},\cdot{\cdot}\cdot,\mathbf{K}_{2n}\right)\\ & \times\exp\left[ i\left( \mathbf{K}_{1}+\mathbf{K}_{2}+{\cdot}{\cdot} \cdot{+}\mathbf{K}_{2n}\right) \right] d^{3}K_{1}\cdot{\cdot}\cdot d^{3}K_{2n}, \end{aligned}$$
where $V\left ( \mathbf {K}_{i}\right )$ denotes the region of integration of $\mathbf {K}_{i}$ $(i=1,2,\cdot \cdot \cdot 2n)$ [10]$\mathbf {.}$ On substituting Eq. (48) into Eq. (50), and performing this inverse Fourier transform, we can obtain an important result (51)$$C_{F}^{(n,\;n)}\left( \mathbf{r}_{1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{n}^{\prime},\mathbf{r}_{n+1}^{\prime},\cdot{\cdot}\cdot,\mathbf{r} _{2n}^{\prime}\right) =\Re^{{\ast}}\left( \mathbf{r}_{1}^{\prime}\right) \cdot{\cdot}\cdot\Re^{{\ast}}\left( \mathbf{r}_{n}^{\prime}\right) \Re\left( \mathbf{r}_{n+1}^{\prime}\right) \cdot{\cdot}\cdot\Re\left( \mathbf{r} _{2n}^{\prime}\right) .$$
This deceptively simple result is quite profound. It tells us that when a coherent plane wave on scattering from a random medium, if the $2n$th-order coherence of the incident field is remained, the $2n$th-order moment of the scattering potential $C_{F}^{(n,\;n)}\left ( \mathbf {r}_{1}^{\prime },\cdot \cdot \cdot ,\mathbf {r}_{n}^{\prime },\mathbf {r}_{n+1}^{\prime },\cdot \cdot \cdot ,\mathbf {r}_{2n}^{\prime }\right )$ can be factorized into the product of $2n$ equivalent scattering potential $\Re \left ( \mathbf {r}^{\prime }\right ) .$In particular, for $n=1$, we have
(52)$$C_{F}^{(1,1)}\left( \mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime}\right) =\Re^{{\ast}}\left( \mathbf{r}_{1}^{\prime}\right) \Re\left( \mathbf{r} _{2}^{\prime}\right) .$$
The normalized correlation coefficient of the scattering potential of the scatterer is defined as [37] (53)$$\mu_{F}^{(1,1)}\left( \mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime}\right) =\frac{C_{F}^{(1,1)}\left( \mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime }\right) }{\sqrt{C_{F}^{(1,1)}\left( \mathbf{r}_{1}^{\prime},\mathbf{r} _{1}^{\prime}\right) }\sqrt{C_{F}^{(1,1)}\left( \mathbf{r}_{2}^{\prime },\mathbf{r}_{2}^{\prime}\right) }}.$$
On substituting from Eq. (52) into Eq. (53), it is clear that $\left \vert \mu _{F}^{(1,1)}\left ( \mathbf {r}_{1}^{\prime },\mathbf {r}_{2}^{\prime }\right ) \right \vert =1.$ From Cauchy–Schwarz inequality $\left \vert \left \langle F^{\ast }\left ( \mathbf {r}_{1}^{\prime }\right ) F\left ( \mathbf {r} _{2}^{\prime }\right ) \right \rangle _{M}\right \vert ^{2}\leq \left \langle \left \vert F\left ( \mathbf {r}_{1}^{\prime }\right ) \right \vert ^{2}\right \rangle _{M}\left \langle \left \vert F\left ( \mathbf {r}_{2}^{\prime }\right ) \right \vert ^{2}\right \rangle _{M}$, we know that both sides of the inequality above are equal to each other, if and only if $F\left ( \mathbf {r}_{1}^{\prime }\right )$ and $F\left ( \mathbf {r}_{2}^{\prime }\right )$ are proportional to each other. Hence $\left \vert \mu _{F} ^{(1,1)}\left ( \mathbf {r}_{1}^{\prime },\mathbf {r}_{2}^{\prime }\right ) \right \vert =1\rightarrow \left \langle F^{\ast }\left ( \mathbf {r}_{1}^{\prime }\right ) F\left ( \mathbf {r}_{2}^{\prime }\right ) \right \rangle _{M}=$ $\left \langle \left \vert F\left ( \mathbf {r}_{1}^{\prime }\right ) \right \vert ^{2}\right \rangle _{M}\left \langle \left \vert F\left ( \mathbf {r}_{2}^{\prime }\right ) \right \vert ^{2}\right \rangle _{M}$ implies the following expression can be established (54)$$F\left( \mathbf{r}_{2}^{\prime}\right) =\kappa\left( \mathbf{r}_{1} ^{\prime},\mathbf{r}_{2}^{\prime}\right) F\left( \mathbf{r}_{1}^{\prime }\right) ,$$
where $\kappa \left ( \mathbf {r}_{1}^{\prime },\mathbf {r}_{2}^{\prime }\right )$ is a deterministic function of its arguments, which can be determined in the same way as the determination of $\Delta \left ( \mathbf {K}_{1},\mathbf {K} _{2}\right )$. Equation (54) reveals that the hidden physical implication of the maintenance of the second-order coherence, i.e., the fluctuations of the scattering potential $F\left ( \mathbf {r}^{\prime }\right )$ at arbitrary two points $\left ( \mathbf {r}_{1}^{\prime },\mathbf {r}_{2}^{\prime }\right )$ within the scatterer are either identical or are proportional to each other. Hence, we can have the following statementA coherent plane wave on weak scattering from a random medium, if the scattered field remains the second-order coherence of the incident field, the scattering potential $F\left ( \mathbf {r}^{\prime }\right )$ at arbitrary two points $\left ( \mathbf {r}_{1}^{\prime },\mathbf {r}_{2}^{\prime }\right )$ within the scatterer will have the relation (54), and vice versa.
5. Conclusion
In summary, based on the perspective of statistical similarity, we have considered the maintenance of the second-order coherence of a light wave on weak scattering from a random medium. It is shown that the scattered field remains the second-order coherence of the incident field at a pair of scattering directions, which means the fluctuations of Fourier components of the scattering potential at that pair of scattering directions have a deterministic phase and amplitude relationship. By assuming that the scattered field remains the second-order coherence, we further discuss its higher order correlation property, and find that all of higher order correlation functions of Fourier component of the scattering potential can reduce to the like-factorization forms with a series of constants. These coefficients furnish an efficient and direct way to describe the higher order coherence property of the scattered field. We have also demonstrated that the combination of the maintenance of second- and fourth-order coherence implies the scattered field coherence to all orders. Finally, the structure feature of the random medium has also been revealed when the coherence of the incident field is preserved up to 2nth order, in particular, in the case of the second-order coherence. Our theory is an important contribution for understanding of spatially fully coherent scattered fields, and develops a generalisation of the results obtained in Refs [40,41]. Especially, the sections 3 and 4 are genuine important generalisation of the theory in Refs. [40,41]. At the same time, our theory also gives a general and new method to discuss the variation of the coherence of the scattered field. Work of generalizing our theory to the scattering of an electromagnetic plane wave is under way, with results to be reported subsequently.
Funding
National Natural Science Foundation of China (11474253, 11874321).
Acknowledgments
We are very grateful to Professor J. H. Eberly for some helpful suggestions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produce by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]
2. W. H. Cater and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988). [CrossRef]
3. D. F. V. James, M. F. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359(1), 67–71 (1990). [CrossRef]
4. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998). [CrossRef]
5. G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light-4,” Opt. Commun. 168(1-4), 39–45 (1999). [CrossRef]
6. 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]
7. M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014). [CrossRef]
8. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75(5), 056609 (2007). [CrossRef]
9. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008). [CrossRef]
10. 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]
11. 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]
12. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35(14), 2412–2414 (2010). [CrossRef]
13. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010). [CrossRef]
14. Y. Ding and D. Zhao, “Far-zone behaviors of light waves on scattering from a particulate medium with different types of particles,” J. Opt. Soc. Am. B 34(11), 2376–2380 (2017). [CrossRef]
15. C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42-43), 2697–2702 (2012). [CrossRef]
16. X. Wang, Z. Liu, and K. Huang, “Weak scattering of a random electromagnetic source of circular frames upon a deterministic medium,” J. Opt. Soc. Am. B 34(8), 1755–1764 (2017). [CrossRef]
17. X. Wang, Z. Liu, and K. Huang, “Scattering of partially coherent radially polarized beams upon a deterministic medium,” J. Quant. Spectrosc. Radiat. Transfer 217, 105–110 (2018). [CrossRef]
18. X. Wang, Z. Liu, and K. Huang, “Multiple spectral switches generated by the scattering of a novel electromagnetic random source of circular frame upon a semisoft boundary medium,” Laser Phys. Lett. 16(6), 066005 (2019). [CrossRef]
19. D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012). [CrossRef]
20. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015). [CrossRef]
21. O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015). [CrossRef]
22. 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]
23. Y. Ding and D. Zhao, “Random medium model for producing optical coherence lattice,” Opt. Express 25(21), 25222–25233 (2017). [CrossRef]
24. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). [CrossRef]
25. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016). [CrossRef]
26. J. Li and O. Korotkova, “Random medium model for cusping of plane waves,” Opt. Lett. 42(17), 3251–3254 (2017). [CrossRef]
27. Y. Ding and D. Zhao, “Correlation between intensity fluctuations of polychromatic electromagnetic light waves on weak scattering,” Phys. Rev. A 97(2), 023837 (2018). [CrossRef]
28. J. Li and Y. Shi, “Application of third-order correlation between intensity fluctuations to determination of scattering potential of quasi-homogeneous medium,” Opt. Express 25(19), 22191–22205 (2017). [CrossRef]
29. J. Li, F. Chen, and L. Chang, “Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium,” Opt. Express 24(21), 24274–24286 (2016). [CrossRef]
30. H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011). [CrossRef]
31. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). [CrossRef]
32. D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013). [CrossRef]
33. T. Wang and D. Zhao, “Equivalence theorem for the spectral density of light waves on weak scattering,” Opt. Lett. 39(13), 3837–3840 (2014). [CrossRef]
34. H. Wu and T. Wang, “Equivalence theorem of light waves on scattering from media with different types,” Opt. Express 25(18), 21410–21418 (2017). [CrossRef]
35. T. Wang and D. Zhao, “Condition for the invariance of the spectral degree of coherence of a completely coherent light wave on weak scattering,” Opt. Lett. 35(6), 847–849 (2010). [CrossRef]
36. J. Li, J. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun. 284(3), 724–728 (2011). [CrossRef]
37. E. Wolf, “Introduction to the Theory of Coherence and Polarization of Light,” (Cambridge University, 2007).
38. M. Born and E. Wolf, “Principles of Optics,” (Cambridge University, 1999), 7th ed.
39. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
40. S. A. Ponomarenko, H. Roychowdhury, and E. Wolf, “Physical significance of complete spatial coherence of optical fields,” Phys. Lett. A 345(1-3), 10–12 (2005). [CrossRef]
41. M. Lahiri and E. Wolf, “Implications of complete coherence in the space-frequency domain,” Opt. Lett. 36(13), 2423–2425 (2011). [CrossRef]
42. R. J Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130(6), 2529–2539 (1963). [CrossRef]
43. U. M. Titulaer and R. J. Glauber, “Correlation functions for coherent fields,” Phys. Rev. 140(3B), B676–B682 (1965). [CrossRef]
44. G. H. Hardy, J. K. Littlewood, and G. Polya, “Inequalities” (Cambridge University, 1952), 2nd ed., pp. 43, 168.
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