Weak ghost scattering with stochastic electromagnetic beams

1. Introduction

Light scattering can be used to study a lot of phenomena related to the interaction of light waves with materials, and has been treated in Ref [1–6] in great detail. Many other aspects of the application of light scattering also have been actively investigated, such as in particle characterization [7], material analysis [8], atmospheric optics [9], nanometer optics [10], condensed matter physics [11], physical chemistry [12], biomedicine [13], and astrophysics [14]. Further development of novel light scattering techniques is important since it can provide new diagnostic methods in different research fields.

Recently, based on the idea of ghost imaging, we have proposed a kind of ghost scattering scheme to study light scattering with either scalar incoherent light fields or biphoton quantum states [15,16]. Ghost imaging is a kind of nonlocal and indirect imaging technique. One can use the spatial correlation properties of fluctuating light fields (either thermal or entangled) to retrieve the information of the object via spatial intensity correlation measurements between two spatially correlated light beams (the reference beam and the test beam) [17–28]. Ghost imaging has attracted noteworthy interest and stimulated lots of studies, such as methods to improve imaging quality [29–38], and imaging through different circumstance [39–49].

In our ghost scattering scheme, we replace the objects in the GI scheme with scatterers and place the detectors far from the scatterers, then we perform nonlocal correlation measurement of the far-zone intensity fluctuations in the test and reference scattering paths. We can obtain the scattering information of the test scatterer in the reference arm with the help of a fixed point like detector in the test arm, just like the case in GI. Since the nature of light fields is vector electromagnetic fields, it is interesting to study the influence of the polarization properties of the incident light fields on the behavior of ghost scattering. In the past decades, the unified theory of coherence and polarization of random electromagnetic fields has been established [50], and the propagation and scattering properties of stochastic electromagnetic beams have been widely investigated [51–59]. With the help of the weak scattering theory of stochastic electromagnetic beams [1], the aim of this paper is to generalize the scalar ghost scattering theory with classically incoherent light fields to the case of stochastic electromagnetic beams.

The paper is organized as follows. In section 2, under the first-order Born approximation of stochastic electromagnetic beams scattering, we present the ghost scattering model and derive the expressions for the correlation of intensity fluctuations of the far-zone scattered fields, showing how the scattering information of the test scatterer can be obtained from the measurement in the reference arm. Then in section 3, we provide numerical examples of ghost scattering with the so called stochastic electromagnetic Gaussian Schell-model (EMGS) beams and discuss the influences of polarization properties and other parameters on the ghost scattering results. Finally, conclusions are given in section 4.

2. Model and theory

The electromagnetic ghost scattering model is shown in Fig. 1, which is very similar as the ghost scattering model with scalar incoherent fields [15] except the source is replaced by a stochastic electromagnetic beam. A stochastic electromagnetic beam is split by a beam splitter into two beams which are propagating in two different paths called as test path and reference path. In the test path, an unknown test scatterer with the refractive index distribution $n(\vec {r}_a)$ is illumined and the scattered field is detected by a point-like detector $D_t$ located at the far-zone position $\vec {r}_1$. In the reference path, we place a known object with the refractive index distribution $n(\vec {r}_b)$ as a reference scatterer, and measure the scattered fields from this scatterer via a point-like detector $D_r$ fixed at the far-zone position $\vec {r}_2$. Experimentally, $D_t$ can be scanned over the space, and $D_r$ also can be changed. The scattering intensity signals recorded in $D_t$ and $D_r$ are correlated by a correlator to obtain the correlation of intensity fluctuations (CIF). We will derive the formula of CIF under the weak scattering condition and show that the scattering information of the unknown test scatterer can be retrieved from it.

 

Fig. 1. Geometry of the electromagnetic ghost scattering model. An incident stochastic electromagnetic beam is split into two beams by the beam splitter (BS). An unknown test scatterer is illuminates by one of the beams and the far-zone scattered fields are detected by the test detector $D_t$ located at $\vec {r}_1$. The other beam is scattered by a reference scatterer and considered as a reference beam, with the far-zone scattered fields detected by the reference detector $D_r$ located at $\vec {r}_2$. The scattered signals measured by the test and reference detector $D_t$ and $D_r$ are further processed by the correlator to realize ghost scattering.

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From the vector electromagnetic scattering theory [1], for a quasi-monochromatic vector incident field (centered at wavelength $\lambda$) $\vec {E}^i(\vec {r}_0)=\begin {bmatrix} \vec {E}^i_x & \vec {E}^i_y & \vec {E}^i_z \\ \end {bmatrix}^T$ and the scatterer refractive index distribution $n(\vec {r}_0)$ within the region $V$, under the first-order Born approximation, the far-zone scattered field $\vec {E}^s(\vec {r})$ has the form,

Now, considering the ghost scattering scheme, we are interested with the correlation matrix of intensity fluctuation ($\overleftrightarrow {\mathbf {G}}$) between the scattered field from the reference scatterer and the test scatterer. The components of $\overleftrightarrow {\mathbf {G}}$ take the form

Experimentally often the total intensity is measured, $I_t(\vec {r})=I_x(\vec {r})+I_y(\vec {r})+I_z(\vec {r})$, then the corresponding correlation of intensity fluctuation (CIF) is given by

The electromagnetic wave is a kind of transverse waves, so that in the spherical coordinator the far-zone scattered fields only have two components $E^s_{\theta }$ and $E^s_{\phi }$. Suppose the incident beam propagates along z direction and has no z component ($E_z^i=0$), under the weak scattering condition, the far-zone scattered field has the form [1]

From Eq. (13), we can see that the scattering properties of the scatterers are contained in the 4 different $\widetilde {\mathbf {F}}_{\alpha \beta }$. Due to the statistical correlation of the stochastic electromagnetic fields, these $\widetilde {\mathbf {F}}_{\alpha \beta }$ may be different each other. Including the angle dependent factors $\overleftrightarrow {S^{(2)}}$, the CIF in the electromagnetic ghost scattering may be very complicated and the ghost scattering results may be modified.

Finally, we briefly discuss the differences between our work and the theory of ghost imaging with electromagnetic stochastic beams in Ref. [60]. Since we are concerned with light scattering in the far-zone, the free-space outgoing Green’s function is approximated by the simple form of $\frac {e^{jkr}}{r}e^{-jk\hat {s}\cdot \vec {r}_0}$ in Eq. (1) (see Eqs. (2.10) and (5.2) in Ref. [1]). It is this $\hat {s}\cdot \vec {r}_0$ term leading to the Fourier transform like formulas in Eqs. (11,12). On the other hand, for ghost imaging, the Green’s function in the paraxial region can be simplified as the Fresnel diffraction, leading to the response functions (Eqs. (8,9)) in Ref. [60]. Thus our results on ghost scattering are different with the case of ghost imaging.

3. Numerical examples

Now, we use numerical examples to demonstrate the electromagnetic ghost scattering properties. To be compared with the case of scalar ghost scattering, we choose a 8-rectangle scatterer shown in Fig. 2(a) as the test scatter, which has been used in Ref. [15]. The reference scatterer is assumed to be a uniformly transmissive plane, i.e., $F_r\propto 1$. The wavelength used in calculations is set to be $\lambda =632$ nm. The distances from the source to the test scatterer and reference scatterer are the same value 100mm. The parameters of the incident electromagnetic Gauss-Schell model beams are varied to see how these factors can affect the behavior of electromagnetic ghost scattering. In following figures, the length unit is mm, and the frequency unit is $k=2\pi /\lambda$. The incident stochastic field along the Z direction is supposed to be described by the electromagnetic Gauss-Schell model (EMGS) beam [50], with the cross-spectral density matrix in the source plane ($z_1=z_2=0$) given by

 

Fig. 2. (a)The two dimensional test scatterer $F_t(x,y)$ used in numerical simulations, length unit in mm. (b) The corresponding spatial spectrums of (a) shown as the ideally scattering distribution. (c) The calculated DSP with the parameters $A_x=1$, $A_y=0.5$, $B_{xy}=0.3$, $\sigma =20$, $\delta _{xx}=5\times 10^{-4}$, $\delta _{xy}=7.5\times 10^{-4}$, $\delta _{yy}=6\times 10^{-4}$. (d) The calculated GSP with the same parameters in (c). (e) The calculated GSP with the parameters used in (c) except $\delta _{xy}=17.5\times 10^{-4}$, $\delta _{yy}=26\times 10^{-4}$, corresponding to the case of very different coherence lengths. (f) The normalized $s_x$ axial distributions in (c,d,e), corresponding to the blue solid, red dashed, and black dotted lines respectively.

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Based on these equations, we can numerically calculate the results of electromagnetic ghost scattering and discuss the effects of different parameters such as the coherence lengths $\delta _{\alpha \beta }$, beam widths $\sigma$, polarization amplitudes $A_{x,y}$, and degree of coherence $B_{xy}$. From Eqs. (5,12,13), the CIF is the function of both $\hat {s}_{\perp 1}$ in the test path and $\hat {s}_{\perp 2}$ in the reference path. As in Ref. [15], we define two kinds of ghost scattering outputs, the ghost scattering pattern (GSP) and the direct scattering pattern (DSP). The GSP is $I_G(s_{rx},s_{ry})=\mathbf {C}(r_1\hat {s}_{\perp 1}=(0,0),r_2\hat {s}_{\perp 2}=r_2(s_{rx},s_{ry}))$ corresponding to the correlation between the scattered field in the test path detected only at the forward direction ($\hat {s}_{\perp 1}=(0,0)$) and the arbitrary scattered fields scanned in the reference path $(s_{rx},s_{ry})$, so that the output seems to be emerged from the reference path in which contains no test scatterers to be studied. Similarly, the DSP is $I_D(s_{tx},s_{ty})=\mathbf {C}(r_1\hat {s}_{\perp 1}=r_1(s_{tx},s_{ty}),r_2\hat {s}_{\perp 2}=(0,0))$, and the output can be intuitively regarded as the results in the direct scattering configuration contains the test scatterer.

Firstly, in Fig. 2(a,b), we plot the test scatterer and its spectrum. Then in Fig. 2(c) we show that the DSP is quite similar as the ideal spectrum under the used parameters, while the GSP in Fig. 2(d) is degraded with the outer tiny structures suppressed. Here, the coherent lengths are very similar ($\delta _{xx}=5\times 10^{-4}$ mm, $\delta _{xy}=7.5\times 10^{-4}$ mm, $\delta _{yy}=6\times 10^{-4}$ mm), so that the changes of $\widetilde {\mathbf {F}}_{\alpha \beta }$ in Eq. (13) may be not very obvious, then the main structures are similar as the scalar cases in Ref. [15]. However, when $\delta _{\alpha \beta }$ are very different from each other, $\widetilde {\mathbf {F}}_{\alpha \beta }$ may have noticeable changes, then the GSP obtained from the combinations in Eq. (5) should be degraded. As shown in Fig. 2(e), with $\delta _{xx}=5\times 10^{-4}$ mm, $\delta _{xy}=17.5\times 10^{-4}$ mm, $\delta _{yy}=26\times 10^{-4}$ mm, the GSP deviates from the scalar case significantly. In Fig. 2(f), we plot the $s_x$ axial distributions in Fig. 2(c,d,e) to see their differences more clearly. The DSP is more accurate compared with the ideal spectrum because the output is measured in the direct test path. The GSP with similar $\delta _{\alpha \beta }$ is degraded to some degree, while the very different $\delta _{\alpha \beta }$ lead to significant distortion of the ghost scattering results.

Now we know the coherent lengths $\delta _{\alpha \beta }$ can affect the electromagnetic ghost scattering. In Fig. 3, we choose two sets of coherent length parameters (very similar and very different $\delta _{\alpha \beta }$), then change other parameters to see the dependence of GSPs on the amplitude $A_{x,y}$, the degree of coherence parameter $B_{xy}$, and the beam width parameter $\sigma$. For simplicity, we set $A_x=1$.

 

Fig. 3. Normalized GSP $I_G(s_{x},0)$ to see the effects of parameters $A_y$, $B_{xy}$ and $\sigma$ on electromagnetic ghost scattering, with $A_x=1$. The coherence lengths in the left figures (a,c,e) are $\delta _{xx}=5\times 10^{-4}$, $\delta _{xy}=7.5\times 10^{-4}$, $\delta _{yy}=6\times 10^{-4}$, and in the right figures (b,d,f) are $\delta _{xx}=5\times 10^{-4}$, $\delta _{xy}=17.5\times 10^{-4}$, $\delta _{yy}=26\times 10^{-4}$. (a,b) $s_x$ axial distributions with $A_y=0$, 0.25, 0.5, 1, corresponding to the black dotted, red dashed, blue dash-dot, and magenta solid curves, $B_{xy}=0.3$, $\sigma =20$. (c,d) $s_x$ axial distributions with $B_{xy}=0.1$, 0.3, 0.6, 0.9, corresponding to the black dotted, red dashed, blue dash-dot, and magenta solid curves, $A_{y}=0.5$, $\sigma =20$. (a,b) $s_x$ axial distributions with $\sigma =5$, 10, 20, 50, corresponding to the black dotted, red dashed, blue dash-dot, and magenta solid curves, $A_y=0.5$, $B_{xy}=0.3$.

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In Fig. 3(a,b), $A_{y}$ is varied from $A_y/A_x=0$ to $A_y/A_x=1$. In the case of similar $\delta _{\alpha \beta }$, the GSPs have almost the same distributions (Fig. 3(a)) because $\widetilde {\mathbf {F}}_{\alpha \beta }$ have similar distributions so that the combinations from Eq. (5) only lead to very tiny and almost unobservable differences in this figure. But for very different $\delta _{\alpha \beta }$, $A_{y}=0$ leads to the best behavior of the scattering distribution since only the most accurate component $\widetilde {\mathbf {F}}_{xx}$ affects the GSP and the other two worse components are not included in the CIF. When $A_y$ increases, the influences of $\widetilde {\mathbf {F}}_{xy}$ and $\widetilde {\mathbf {F}}_{yy}$ will be more and more important, so that the quality of GSPs will be degraded (Fig. 3(b)).

The findings in Fig. 3(c,d) can be understood with the similar discussions. Now, the changes of $B_{xy}$ only changes the influences of $\widetilde {\mathbf {F}}_{xy}$ in Eq. (5) in Eq. (5), and the contributions from $\widetilde {\mathbf {F}}_{xx}$ and $\widetilde {\mathbf {F}}_{yy}$ are fixed, so that the variations in Fig. 3(d) is relatively small. Further we find the larger $B_{xy}$ corresponding to the better GSP, which can be interpreted from the fact that $\delta _{xy}$ is smaller than $\delta _{yy}$, making $\widetilde {\mathbf {F}}_{xy}$ better than $\widetilde {\mathbf {F}}_{yy}$, so that the increase of $B_{xy}$ will increase the contribution of the relatively accurate component $\widetilde {\mathbf {F}}_{xy}$ in the GSPs, as shown in Fig. 3(d).

Finally, in Fig. 3(e,f), we compare the results with different beam widths. Here, for similar $\delta _{\alpha \beta }$ in Fig. 3(e), the larger $\sigma$ leads to the better scattering distribution, similar as the results in the scalar ghost scattering. On the other hand, in Fig. 3(f), the curves with different $\sigma$ are very similar and highly degraded. This can be understood from the observation that these very different $\delta _{\alpha \beta }$ already lead to the significant suppression of the scattering distributions so that the further influences from the change of $\sigma$ are not important.

4. Conclusion

In conclusion, we have derived general expressions of ghost scattering with stochastic electromagnetic fields under the first-order Born approximation. With the help of these formulas, using the stochastic electromagnetic Gaussian Schell-model beam as the incident field, we have numerically obtained the far-zone correlation of scattering intensity fluctuations and demonstrated the electromagnetic ghost scattering properties. We have discussed how the polarization properties and other parameters of the incident EMGS beams can modify the ghost scattering results compared with the early investigated scalar cases. If the coherence lengths of different components in the cross-spectral density matrix of the EMGS beam are very different, the ghost scattering patterns may be significantly degraded. We can retrieve the scalar ghost scattering theory under the conditions of linear polarization and paraxial approximation.