Measuring complex correlation matrix of partially coherent vector light via a generalized Hanbury Brown&#x2013;Twiss experiment

Zhen Dong; Zhen Dong; Zhaofeng Huang; Yahong Chen; Yahong Chen; Fei Wang; Fei Wang; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.398185

1. Introduction

Partially coherent vector light is a vectorial extension of the scalar random light and it presents rich polarization and coherence features during its propagation, interference, and light-matter interaction [1–6]. The electromagnetic coherence theory [7,8] has been well developed to describe the statistical properties of such random vectorial light. The complex correlation matrix, among the theory, characterizes in detail the correlations existing in the components of the fluctuating light field at a pair of points and has found uses in a wide array of applications, e.g., crystallography [9,10], incoherent source reconstruction [11], and optical imaging [12–14]. Distinguished from the field correlation in the scalar random light, the correlation matrix involves not only the correlations among a single field component at two different points, but also the correlations between two orthogonal field components at a pair of points. The former correlations can be measured by evaluating the visibility and the displacement of the intensity interference fringes formed by a traditional Young’s two-pinhole experiment [7], whereas the later correlations, demonstrated only recently, can be determined from the Stokes-parameter interference fringes formed by a modified Young’s experiment [15,16]. Although the approaches based on Young’s two-pinhole interferometry have several limitations, e.g., low light efficiency and time-consuming, other methods have been developed to overcome a part of these limitations [17–19].

At the same time, for the partially coherent vector light fields obeying Gaussian statistics [20], their (second-order) field correlations can be recovered from the measurement of the field higher-order correlations (e.g., the intensity correlation) via a classic Hanbury Brown–Twiss (HBT) experiment [21]. However, during the measurement, only the modulus of the field correlations can be extracted. The phase information is missing [22]. Recently, Singh and coauthors proposed that by introducing a reference random light into the classic HBT experiment and by applying the Fourier fringe analysis technique, the phase information of the coherence function of a random light field can be obtained [23,24]. Lately, they extended the method to measure the vectorial complex correlation functions of the random vector light [25,26] and applied it to the polarization imaging applications [27,28]. Yet lately, we proposed a generalized HBT experiment by introducing a pair of fully coherent reference fields with controllable phase difference to robustly and effectively measure both the amplitude and the phase information of the complex correlation among a scalar random light field [29]. In this work, we extend such generalized HBT experiment into the vectorial case and apply it to measure the complex correlation matrix of the partially coherent vector light field. To this end, we introduce a pair of coherent reference vector fields within the framework of a classic HBT type experiment and measure the intensity-intensity cross-correlation of the composite fields that are superposed by the reference vectors and the field realization of the partially coherent vector light. We test our general method by measuring the complex correlation matrix for a special correlated partially coherent vector beam. Moreover, we demonstrate that the advanced protocol of measuring the entire information of complex correlation matrix can be used in the reconstruction of the polarization state hidden behind a thin-layer diffuser (e.g., a ground-glass disk).

2. Theory

Let us consider a (stationary) partially coherent vector light beam that propagates closely along the $z$ direction. The (second-order) statistical properties of such partially coherent vector light beam, in the space-frequency domain, at (angular) frequency $\omega$ and a pair of spatial positions $\mathbf {r}_1 = (x_1,y_1)$ and $\mathbf {r}_2 = (x_2,y_2)$, are involved in a $2\times 2$ spectral coherence matrix [7,8]

(1)$$\mathbf{W}(\mathbf{r}_1,\mathbf{r}_2 , \omega) = \langle \mathbf{E}^\ast(\mathbf{r}_1, \omega) \mathbf{E}^{\mathrm{T}} (\mathbf{r}_2, \omega) \rangle,$$

where the asterisk, superscript T, and angle brackets denote complex conjugate, matrix transpose, and ensemble average, respectively, and $\mathbf {E} (\mathbf {r}, \omega ) = [E_x(\mathbf {r}, \omega ), E_y(\mathbf {r}, \omega ) ]^{\mathrm {T}}$ is a field realization with $E_x(\mathbf {r}, \omega )$ and $E_y(\mathbf {r}, \omega )$ being the components of the realization vector, along two mutually orthogonal $x$ and $y$ directions perpendicular to the $z$ axis. From now on we do not, for brevity, explicitly show the frequency dependence of spectral quantities.

The spatial correlations among the partially coherent vector light field can be evaluated by the (normalized) correlation matrix with each element being written as

(2)$$\mu_{\alpha \beta}(\mathbf{r}_1,\mathbf{r}_2) = \frac{W_{\alpha \beta}(\mathbf{r}_1,\mathbf{r}_2)}{\sqrt{S_\alpha(\mathbf{r}_1) S_\beta(\mathbf{r}_2) }},$$

where $\alpha , \beta \in \{ x,y \}$, $W_{\alpha \beta }(\mathbf {r}_1,\mathbf {r}_2) = \langle E_\alpha ^\ast (\mathbf {r}_1) E_\beta (\mathbf {r}_2) \rangle$ is an element in the coherence matrix $\mathbf {W}(\mathbf {r}_1,\mathbf {r}_2)$, and $S_x(\mathbf {r}) = \langle I_x (\mathbf {r})\rangle$ and $S_y(\mathbf {r}) = \langle I_y (\mathbf {r})\rangle$ are the spectral densities of the $x$ and $y$ components of the field, while $I_x (\mathbf {r}) = |E_x(\mathbf {r})|^2$ and $I_y (\mathbf {r}) = |E_y(\mathbf {r})|^2$ are the random intensities of the $x$ and $y$ components of the field realization. Based on the Siegert relation [20], for a partially coherent vector light obeying Gaussian statistics, the intensity-intensity correlation of the field components is closely linked with the correlation matrix element $\mu _{\alpha \beta }(\mathbf {r}_1,\mathbf {r}_2)$, i.e.,

(3)$$\frac{\langle I_\alpha(\mathbf{r}_1) I_\beta(\mathbf{r}_2) \rangle}{S_\alpha(\mathbf{r}_1)S_\beta(\mathbf{r}_2) } = 1 + |\mu_{\alpha \beta}(\mathbf{r}_1,\mathbf{r}_2)|^2,$$

where the angle brackets denotes ensemble average, as before. It is found from the above relation that the conventional HBT experiment gives us only the modulus of the correlation matrix. While the phase information is lacked. Here, we note that a traditional HBT experiment, such as the total intensity-intensity correlation of the field, $\langle I (\mathbf {r}_1) I (\mathbf {r}_2) \rangle$ with $I (\mathbf {r}) = I _x(\mathbf {r}) + I _y(\mathbf {r})$, allows us to get the information of the electromagnetic degree of coherence, which is defined in [30]. However, such electromagnetic degree of coherence is a real quantity and still contains no phase information among the field correlations.

To obtain both the modulus and phase information of the correlation matrix, we now consider a generalized HBT-type experiment, in which we firstly introduce two coherent reference field vectors, $\mathbf {E}^{\textrm {(R1)}}(\mathbf {r})$ and $\mathbf {E}^{\textrm {(R2)}}(\mathbf {r})$, where the superscripts ‘R1’ and ‘R2’ denote the first and the second reference fields, respectively. We then combine the field realization of the partially coherent vector light with the reference field vectors. The field realizations of the two composite light therefore can be written as

(4)$$\mathbf{E}^{\textrm{(C1)}}(\mathbf{r}) = \mathbf{E}^{\textrm{(R1)}}(\mathbf{r}) + \mathbf{E}(\mathbf{r}),$$

(5)$$\mathbf{E}^{\textrm{(C2)}}(\mathbf{r}) = \mathbf{E}^{\textrm{(R2)}}(\mathbf{r}) + \mathbf{E}(\mathbf{r}).$$

Different from a conventional HBT experiment, we now measure the intensity-intensity cross-correlation of the two composite fields at a pair of points, i.e.,

(6)$$G^{\textrm{(C)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) = \langle I_{\alpha}^{\textrm{(C1)}} (\mathbf{r}_1) I_{\beta}^{\textrm{(C2)}} (\mathbf{r}_2) \rangle,$$

where $I_{\alpha }^{\textrm {(C1)}} (\mathbf {r})$ denotes the random intensity of the $\alpha$ component of the first composite field and $I_{\beta }^{\textrm {(C2)}} (\mathbf {r})$ denotes the random intensity of the $\beta$ component of the second composite field. Taking Eqs. (4) and (5) into Eq. (6) and applying the Gaussian moment theorem [7,20], we obtain (after the tedious but straightforward derivations) that

(7)$$\begin{aligned} G^{\textrm{(C)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) = &\langle I_\alpha^{\textrm{(U1)}}(\mathbf{r}_1) \rangle \langle I_\beta^{\textrm{(U2)}}(\mathbf{r}_2) \rangle + |W_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) |^2 \\ & + 2 \sqrt{I_\alpha^{\textrm{(R1)}}(\mathbf{r}_1) I_\beta^{\textrm{(R2)}}(\mathbf{r}_2) } \mathrm{Re} [\mathrm{e}^{\mathrm{i} \Delta \phi_{\alpha \beta}} W_{\alpha \beta}(\mathbf{r}_1,\mathbf{r}_2)], \end{aligned}$$

where $I_\alpha ^{\textrm {(U1)}}(\mathbf {r}) = I_\alpha ^{\textrm {(R1)}}(\mathbf {r}) + I_\alpha (\mathbf {r})$, $I_\beta ^{\textrm {(U2)}}(\mathbf {r}) = I_\beta ^{\textrm {(R2)}}(\mathbf {r}) + I_\beta (\mathbf {r})$, $I_\alpha ^{\textrm {(R1)}}(\mathbf {r})$ and $I_\beta ^{\textrm {(R2)}}(\mathbf {r})$ are the intensities of the first and the second reference fields along the $\alpha$ and the $\beta$ directions, respectively, $\mathrm {Re}$ denotes the real part, and $\Delta \phi _{\alpha \beta } = \mathrm {Arg}[E_\alpha ^{\textrm {(R1)}}(\mathbf {r}_1)] -\mathrm {Arg}[E_\beta ^{\textrm {(R2)}}(\mathbf {r}_2)]$ denotes the phase delay between the two reference fields’ components. From the term $\mathrm {Re} [\mathrm {e}^{\mathrm {i} \Delta \phi _{\alpha \beta }} W_{\alpha \beta }(\mathbf {r}_1,\mathbf {r}_2)]$ in Eq. (7), we find that by adjusting the phase delay $\Delta \phi _{\alpha \beta }$ to be 0 or $\pi /2$, the real or imaginary part of the complex coherence matrix $\mathbf {W}(\mathbf {r}_1, \mathbf {r}_2)$ of the partially coherent vector light can be obtained. Therefore, both the modulus and phase information of the correlation matrix are involved in $G^{\textrm {(C)}}_{\alpha \beta } (\mathbf {r}_1, \mathbf {r}_2)$.

However, we find in Eq. (7) that apart from the information of the complex coherence matrix, $G^{\textrm {(C)}}_{\alpha \beta } (\mathbf {r}_1, \mathbf {r}_2)$ contains additional terms that can be regarded as the unnecessary background in the complex correlation recovery. Such background terms can be removed by doing the following intensity-intensity cross-correlation,

(8)$$G^{\textrm{(U)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) = \langle I_\alpha^{\textrm{(U1)}}(\mathbf{r}_1) I_\beta^{\textrm{(U2)}}(\mathbf{r}_2) \rangle,$$

where $I_\alpha ^{\textrm {(U1)}}(\mathbf {r})$ and $I_\beta ^{\textrm {(U2)}}(\mathbf {r})$, defined as before, can be regarded as the intensities of the components of the fields that are superposed incoherently by the partially coherent vector light and the reference fields. Again, by using the Gaussian moment theorem, we obtain

(9)$$G^{\textrm{(U)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) = \langle I_\alpha^{\textrm{(U1)}}(\mathbf{r}_1) \rangle \langle I_\beta^{\textrm{(U2)}}(\mathbf{r}_2) \rangle + |W_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) |^2.$$

It follows from Eqs. (7) and (9) that the difference between $G^{\textrm {(C)}}_{\alpha \beta } (\mathbf {r}_1, \mathbf {r}_2)$ and $G^{\textrm {(U)}}_{\alpha \beta } (\mathbf {r}_1, \mathbf {r}_2)$ can be expressed as

(10)$$\begin{aligned} \Delta G_{\alpha \beta} &(\mathbf{r}_1, \mathbf{r}_2, \Delta \phi_{\alpha \beta}) = G^{\textrm{(C)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) - G^{\textrm{(U)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2), \\ & = 2 \sqrt{I_\alpha^{\textrm{(R1)}}(\mathbf{r}_1) I_\beta^{\textrm{(R2)}}(\mathbf{r}_2) } \mathrm{Re} [\mathrm{e}^{\mathrm{i} \Delta \phi_{\alpha \beta}} W_{\alpha \beta}(\mathbf{r}_1,\mathbf{r}_2)], \end{aligned}$$

where we write the difference $\Delta G_{\alpha \beta } (\mathbf {r}_1, \mathbf {r}_2, \Delta \phi _{\alpha \beta })$ as a function of the phase delay $\Delta \phi _{\alpha \beta }$. Based on the definition of the correlation matrix in Eq. (2) and by modulating the phase delay, we obtain from Eq. (10) that the real and imaginary parts of the correlation matrix can be expressed as

(11)$$\mu^{\prime}_{\alpha \beta}(\mathbf{r}_1, \mathbf{r}_2) = \frac{\Delta G_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2, \Delta \phi_{\alpha \beta} = 0 )}{2\sqrt{I_\alpha^{\textrm{(R1)}}(\mathbf{r}_1) I_\beta^{\textrm{(R2)}}(\mathbf{r}_2) S_\alpha(\mathbf{r}_1) S_\beta(\mathbf{r}_2) }},$$

(12)$$\mu^{\prime\prime}_{\alpha \beta}(\mathbf{r}_1, \mathbf{r}_2) = \frac{\Delta G_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2, \Delta \phi_{\alpha \beta} = \frac{\pi}{2} )}{2\sqrt{I_\alpha^{\textrm{(R1)}}(\mathbf{r}_1) I_\beta^{\textrm{(R2)}}(\mathbf{r}_2) S_\alpha(\mathbf{r}_1) S_\beta(\mathbf{r}_2) }},$$

where the prime and double prime denote the real and imaginary parts, respectively.

Up to now, we have showed that the complex correlation matrix of the partially coherent vector light obeying Gaussian statistics can be recovered from the generalized HBT experiment by introducing a pair of coherent reference vector fields with a controllable phase delay. Next, we study the experimental implementation of our general theory. The random field discussed in this work is statistically stationary. Thus, the ensemble average in Eqs. (6) and (8) can be replaced with the time average [8]. Therefore, by measuring a large set of intensities, i.e., $I_{\alpha }^{\textrm {(C1)}} (\mathbf {r})$, $I_{\beta }^{\textrm {(C2)}} (\mathbf {r})$, $I_{\alpha }^{\textrm {(R1)}} (\mathbf {r})$, $I_{\beta }^{\textrm {(R2)}} (\mathbf {r})$, $I_{\alpha } (\mathbf {r})$, and $I_{\beta } (\mathbf {r})$, respectively, within the time period that is longer that the characteristic coherence time of the light field [31], with the aid of a fast response rate charge-coupled device (CCD) camera, the complex correlation matrix can be recovered effectively. In a certain circumstance, i.e., the partially coherent vector light is created by a Fresnel or Fourier transformation optical system, the ensemble average in Eqs. (6) and (8) can be equivalent to a spatial average of a speckle field at a single time [32]. Therefore, the complex correlation matrix now can be recovered in near real-time.

3. Experiment

In this section, we test our general theory by measuring the complex correlation matrix of a recently introduced partially coherent vector light beam, namely specially correlated radially polarized (SCRP) vector beam [22]. Due to the unconventional correlations embedded in the field, such vector beam exhibits extraordinary propagation properties in free space, such as the beam is unpolarized at source but becomes progressively more polarized on propagation; in far field the beam can display a very pure radial polarization state when the initial spatial coherence of the source is very low. The experimental setup for generating SCRP vector beam and measuring its complex correlation matrix is shown in Fig. 1, which will be described in detail later. The coherence matrix elements of such a SCRP vector beam are expressed as [22,33]

(13)$$\begin{aligned} W_{\alpha \alpha} (\mathbf{r}_1, \mathbf{r}_2) = & ~S_0 \exp\left( -\frac{\mathbf{r}_1^2 + \mathbf{r}_2^2}{4 \sigma_0^2} \right)\left[ 1-\frac{(\alpha_2-\alpha_1)^2}{\delta_0^2} \right] \\ & \times \exp \left[ -\frac{(\mathbf{r}_1-\mathbf{r}_2)^2}{2\delta_0^2}\right] \exp \left[ - \mathrm{i} (\mathbf{r}_1-\mathbf{r}_2) \cdot \mathbf{v}_0 \right], \end{aligned}$$

(14)$$\begin{aligned} W_{xy}(\mathbf{r}_1, \mathbf{r}_2) = & - S_0 \exp\left( -\frac{\mathbf{r}_1^2 + \mathbf{r}_2^2}{4 \sigma_0^2} \right) \frac{(x_2-x_1)(y_2-y_1)}{\delta_0^2} \\ & \times \exp \left[ -\frac{(\mathbf{r}_1-\mathbf{r}_2)^2}{2\delta_0^2}\right] \exp \left[ - \mathrm{i} (\mathbf{r}_1-\mathbf{r}_2) \cdot \mathbf{v}_0 \right],\\ \end{aligned}$$

(15)$$W_{yx}(\mathbf{r}_1, \mathbf{r}_2) = ~ W_{xy}^\ast(\mathbf{r}_2, \mathbf{r}_1),$$

where $S_0$ is a positive constant, $\sigma _0$ and $\delta _0$ denote the beam width and the transverse coherence width, respectively, and $\mathbf {v}_0$ is a constant vector. Here we notice that a linear phase factor, i.e., $\exp \left [ - \mathrm {i} (\mathbf {r}_1-\mathbf {r}_2) \cdot \mathbf {v}_0 \right ]$, is added in the coherence matrix to make sure that the correlation matrix is complex. Taking Eqs. (13)–(15) into Eq. (2) and by using the relation $S_\alpha (\mathbf {r}) = W_{\alpha \alpha } (\mathbf {r}, \mathbf {r})$, we can obtain the complex correlation matrix for the SCRP vector beam. We show in the top panels of Figs. 2–4 that the calculated spatial distribution of the real, imaginary, and square modulus of the complex correlations $\mu _{xx}(\Delta \mathbf {r})$, $\mu _{yy}(\Delta \mathbf {r})$, and $\mu _{xy}(\Delta \mathbf {r})$, respectively, where $\Delta \mathbf {r} = \mathbf {r}_1 - \mathbf {r}_2$ since the beam is a Schell-model type [5]. As $\mu _{yx}(\Delta \mathbf {r})$ is a complex conjugate of $\mu _{xy}(\Delta \mathbf {r})$, it is not displayed here. In the calculation, we let $\delta _0 = 0.08$ mm and $\mathbf {v}_0 = (15,20)$ $\mathrm {mm}^{-1}$. We find that both the real and imaginary parts of the field correlations show the intricate spatial distributions. The spatial patterns for $\mu _{xx}(\Delta \mathbf {r})$ and $\mu _{yy}(\Delta \mathbf {r})$ are ‘orthogonal’ to each other, i.e., the correlation distributions for $\mu _{yy}(\Delta \mathbf {r})$ can be obtained by rotating clockwise of these for $\mu _{xx}(\Delta \mathbf {r})$. Whereas, the spatial distributions for $\mu _{xy}(\Delta \mathbf {r})$ in general display the four-petal patterns.

Fig. 1. The schematic of the experimental setup for generating a SCRP vector beam and measuring the complex correlation matrix. BS, beam splitter; NDF, neutral-density filter; BE, beam expander; RPC, radial polarization converter; L1, L2, L3, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; M, reflecting mirror; LP, linear polarizer; QWP, quarter-wave plant; PBS, polarization beam splitter; CCD1, CCD2, charge-coupled devices.

Download Full Size | PDF

Fig. 2. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{xx}(\Delta \mathbf {r})$ of a SCRP vector beam.

Download Full Size | PDF

Fig. 3. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{yy}(\Delta \mathbf {r})$ of a SCRP vector beam.

Download Full Size | PDF

Fig. 4. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{xy}(\Delta \mathbf {r})$ of a SCRP vector beam.

Download Full Size | PDF

Figure 1 shows the experimental setup for the preparation of the SCRP vector beam and the measurement of its complex correlation matrix. A vertically polarized ($y$-polarized) He-Ne laser of wavelength $\lambda = 632$ nm is firstly split by a beam splitter (BS) into two beams that go respectively into the top and bottom arms shown in Fig. 1. The top arm is intended to create the SCRP vector beam, while the bottom arm is used for preparing the coherent reference vector fields. In the top arm, after modulated by a neutral-density filter (NDF) and a beam expander (BE), the reflected light beam from the BS is impinged onto a radial polarization converter (RPC) for generating a coherent radially polarized beam. We note here the generated radially polarized beam is off-axis with the spatial displacement being $\boldsymbol {\rho }_0$. The off-axis radially polarized beam passes through a thin lens L1 of focal length $f_1 = 150$ mm and then illuminates a rotating ground-glass disk (RGGD), producing a spatially incoherent off-axis radially polarized beam with its coherence matrix has the form $\mathbf {W}(\boldsymbol {\rho }_1-\boldsymbol {\rho }_0, \boldsymbol {\rho }_2-\boldsymbol {\rho }_0)$ with

(16)$$\mathbf{W}(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2) = \frac{\delta(\boldsymbol{\rho}_1 - \boldsymbol{\rho}_2)}{\omega_0^2} \exp\left( - \frac{\boldsymbol{\rho}_1^2+\boldsymbol{\rho}_2^2}{\omega_0^2}\right) { \left( \begin{array}{cc} \rho_{1x}\rho_{2x} & \rho_{1x}\rho_{2y} \\ \rho_{1y}\rho_{2x} & \rho_{1y}\rho_{2y} \end{array} \right) }.$$

Above, $\boldsymbol {\rho }_1 = (\rho _{1x}, \rho _{1y})$ and $\boldsymbol {\rho }_2 = (\rho _{2x}, \rho _{2y})$ are two arbitrary position vectors in the transverse plane immediately after the RGGD, $\delta (\boldsymbol {\rho }_1 - \boldsymbol {\rho }_2)$ is the Dirac delta function, and $\omega _0$ is the beam width of the radially polarized beam on the RGGD, which is controlled by the distance between L1 and the RGGD. In our experiment, $\omega _0$ is about $0.6$ mm and the spatial displacement $\boldsymbol {\rho }_0 = (0.38, 0.5)$ mm. We stress that the beam width $\omega _0$ on the RGGD in our experiment is larger than the inhomogeneity scale of the RGGD, otherwise the transmitted beam from the RGGD cannot be regarded as an incoherent beam [20].

After the incoherent off-axis radially polarized beam have passed through a thin lens L2 of the focal distance $f_2=250$ mm and a Gaussian amplitude filter (GAF) of the transmission function $T(\mathbf {r}) = \exp (-\mathbf {r}^2/4\sigma _0^2)$, the coherence matrix of the output beam can be expressed by [22]

(17)$$\mathbf{W}(\mathbf{r}_1, \mathbf{r}_2) = \iint \mathbf{W}(\boldsymbol{\rho}_1-\boldsymbol{\rho}_0, \boldsymbol{\rho}_2-\boldsymbol{\rho}_0) H^\ast(\mathbf{r}_1, \boldsymbol{\rho}_1) H (\mathbf{r}_2, \boldsymbol{\rho}_2) \mathrm{d}^2\boldsymbol{\rho}_1\mathrm{d}^2\boldsymbol{\rho}_2,$$

where $H (\mathbf {r}, \boldsymbol {\rho }) = \frac {-\mathrm {i} T(\mathbf {r})}{\lambda f_2} \exp [\frac {\mathrm {i}\pi }{\lambda f_2}(\boldsymbol {\rho }^2-2\mathbf {r}\cdot \boldsymbol {\rho })]$ is the response function of the optical system between the incoherent source and the output field. It follows from Eqs. (16) and (17) that the coherence matrix of the output beam has the same form of that for a SCRP vector beam in Eqs. (13)–(15) with $\delta _0 = \lambda f_2/(\pi \omega _0)$ and $\mathbf {v}_0 = 2\pi \boldsymbol {\rho }_0 / (\lambda f_2)$. Therefore, the SCRP vector beam is generated immediately after the GAF. At the same time, we record the random intensity distributions $I_x(\mathbf {r})$ and $I_y(\mathbf {r})$ of the $x$ and $y$ components of the SCRP vector beam by the CCD. The generated SCRP vector beam satisfies the Gaussian statistics since the beam is generated by an incoherent source from a RGGD [20]. Further, in our experiment, since the SCRP vector beam is generated by a Fourier transformation optical system, the ensemble average in the following data processing can be replaced with the spatial average of a single realization intensity distribution.

To measure the complex correlation matrix of the generated SCRP vector beam, we introduce the reference fields in the bottom arm of Fig. 1. The $y$-polarized light beam is transmitted through a NDF, a linear polarizer (LP), a quarter-wave plant (QWP), and a BE. The transmission angle of the LP is set to be $45^\circ$ with respect to the $x$ axis and the fast axis of the QWP is set to be parallel or perpendicular to the polarization direction of the beam transmitted from the LP. Therefore, in the two cases, the transmitted reference beams are both of $45^\circ$ polarized but are endowed with a $\pi /2$ phase delay. We now record the intensity distributions of the $x$ and $y$ components of both the reference beams, i.e., $I_x^{\textrm {(R1)}}(\mathbf {r})$, $I_y^{\textrm {(R1)}}(\mathbf {r})$, $I_x^{\textrm {(R2)}}(\mathbf {r})$, and $I_y^{\textrm {(R2)}}(\mathbf {r})$. Next we combined the SCRP and the reference vector beams by a BS. The $x$ and $y$ components of the composite fields are then separated by a polarization beam splitter (PBS) and imaged onto CCD1 and CCD2, respectively, by a 2$f$ imaging system formed by a thin lens L3 of focal distance $f_3 = 150$ mm. The distances from the second BS to L3 and from L3 to CCD1 and CCD2 are all equal to $2f_3$. We record, respectively, the intensity distributions $I_x^{\textrm {(C1)}}(\mathbf {r})$, $I_y^{\textrm {(C1)}}(\mathbf {r})$, $I_x^{\textrm {(C2)}}(\mathbf {r})$, and $I_y^{\textrm {(C2)}}(\mathbf {r})$ of the $x$ and $y$ components of the composite fields. We note here that the rotating speed of the ground-glass disk is slow enough so that the CCD can capture the instantaneous intensity speckles.

The intensity-intensity correlations in Eqs. (6) and (8) therefore can be obtained by

(18)$$G^{\textrm{(C)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2, \Delta \phi_{\alpha \beta} = 0) = \langle I_{\alpha}^{\textrm{(C1)}} (\mathbf{r}_1) I_{\beta}^{\textrm{(C1)}} (\mathbf{r}_2) \rangle_{\textrm{s}},$$

(19)$$ G^{\textrm{(C)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2, \Delta \phi_{\alpha \beta} = \frac{\pi}{2}) = \langle I_{\alpha}^{\textrm{(C1)}} (\mathbf{r}_1) I_{\beta}^{\textrm{(C2)}} (\mathbf{r}_2) \rangle_{\textrm{s}},$$

(20)$$ G^{\textrm{(U)}}_{\alpha \beta} (\mathbf{r}_1, \mathbf{r}_2) = \langle I_\alpha^{\textrm{(U1)}}(\mathbf{r}_1) I_\beta^{\textrm{(U2)}}(\mathbf{r}_2)\rangle_{\textrm{s}},$$

where $\langle \cdots \rangle _{\textrm {s}}$ denotes the spatial average, $I_\alpha ^{\textrm {(U1)}}(\mathbf {r}) = I_\alpha ^{\textrm {(R1)}}(\mathbf {r}) + I_\alpha (\mathbf {r})$, and $I_\beta ^{\textrm {(U2)}}(\mathbf {r}) = I_\beta ^{\textrm {(R2)}}(\mathbf {r}) + I_\beta (\mathbf {r})$, as before. Taking the results obtained from Eqs. (18)–(20) into Eqs. (10)–(12), we finally recover the complex correlation matrix of the SCRP vector beam. The experimental results of the measured spatial distribution of the real, imaginary, and square modulus of the correlation matrix elements $\mu _{xx}(\Delta \mathbf {r})$, $\mu _{yy}(\Delta \mathbf {r})$, and $\mu _{xy}(\Delta \mathbf {r})$ are shown in the bottom panels of Figs. 2–4, respectively. On comparing the experimental results with the calculated ones shown in the top panels of Figs. 2–4, we find the measured distribution of the complex correlation matrix has a very good overlap with the theoretical prediction. We note that the noise especially in the distribution of $\mu _{xy}(\Delta \mathbf {r})$ comes from the imperfection of the polarization state of the fully coherent radially polarized beam from the RPC.

4. Polarization recovery

In this section, we show the measured complex correlation matrix can be used to reconstruct the polarization state hidden behind the RGGD. The main principle is shown in Eq. (17), in which the coherence matrix $\mathbf {W}(\boldsymbol {\rho }_1-\boldsymbol {\rho }_0, \boldsymbol {\rho }_2-\boldsymbol {\rho }_0)$ for the incoherent vector beam can be written as

(21)$$\mathbf{W}(\boldsymbol{\rho}_1-\boldsymbol{\rho}_0, \boldsymbol{\rho}_2-\boldsymbol{\rho}_0) = \delta(\boldsymbol{\rho}_1 - \boldsymbol{\rho}_2) \boldsymbol{\Phi}(\boldsymbol{\rho}_1-\boldsymbol{\rho}_0, \boldsymbol{\rho}_1-\boldsymbol{\rho}_0),$$

where $\boldsymbol {\Phi }$ is the polarization matrix of the field hidden behind the RGGD. Taking Eq. (21) and $H (\mathbf {r}, \boldsymbol {\rho })$ into Eq. (17), we obtain after a straightforward integration that

(22)$$\mathbf{W}(\mathbf{r}_1, \mathbf{r}_2) = \frac{T(\mathbf{r}_1)T(\mathbf{r}_2)}{\lambda^2 f_2^2} \int \boldsymbol{\Phi}(\boldsymbol{\rho}-\boldsymbol{\rho}_0, \boldsymbol{\rho}-\boldsymbol{\rho}_0) \exp\left[ \frac{\mathrm{i}2 \pi}{\lambda f_2} (\mathbf{r}_1 - \mathbf{r}_2) \cdot \boldsymbol{\rho}\right] \mathrm{d}^2\boldsymbol{\rho}.$$

We find in Eq. (22) that the polarization matrix of the field hidden behind the RGGD and the complex coherence matrix of the beam generated after the RGGD form a Fourier-transformation pair. Thus, once the complex coherence matrix is measured, we can reconstruct the polarization matrix of the field hidden behind the RGGD. The polarization state of the field can be extracted from the polarization matrix through the four Stokes parameters $\mathcal {S}_j = \mathrm {tr}(\boldsymbol {\Phi } \boldsymbol {\sigma }_j)$ with $j \in (0,\ldots ,3)$, where $\mathrm {tr}$ denotes the matrix trace, $\boldsymbol {\sigma }_0$ is the $2 \times 2$ unit matrix, and $\boldsymbol {\sigma }_1$, $\boldsymbol {\sigma }_2$, $\boldsymbol {\sigma }_3$ are the three Pauli matrices [8]. The quantity $\mathcal {S}_0$ denotes the spectral density of the field, while other three quantities $\mathcal {S}_1$, $\mathcal {S}_2$, and $\mathcal {S}_3$ denote respectively the differences between the $x$- and $y$-polarized, $+\pi /4$- and $-\pi /4$-linearly polarized, and right- and left-handed circularly polarized spectral densities. In Figs. 5(a)–5(d) we display the four Stokes parameters recovered from the measured field complex correlations shown in the bottom panels of Figs. 2–4. In Fig. 5(e) we show the spatial distribution of the reconstructed polarization state hidden behind the RGGD. We find the recovered polarization state of the field hidden behind the RGGD is indeed an off-axis radial polarization and it is consistent with our predication.

Fig. 5. Recovery of the polarization state of the field hidden behind a RGGD through the complex correlations measurement. (a)–(d), the recovered Stokes parameters $\mathcal {S}_0$, $\mathcal {S}_1$, $\mathcal {S}_2$, and $\mathcal {S}_3$; (e), the spatial distribution of the reconstructed polarization state.

Download Full Size | PDF

5. Conclusions

In summary, we have demonstrated both theoretically and experimentally that the spatial distribution of the complex correlation matrix of a partially coherent vector light beam obeying Gaussian statistics can be recovered through the intensity-intensity cross-correlations of the composite fields synthesized by the random light realization and a pair of reference vectors with the controllable phase delay. We have further demonstrated experimentally that our complex correlations measurement can be applied to reconstruct the polarization properties of the light field hidden behind the ground-glass disk. Different from Young-type interferometers and diffraction-based methods, e.g., in [15–18], our intensity-intensity correlations measurement is free of the diffraction elements, e.g., the pinholes or obstacles, therefore has high light efficiency. As the random fluctuations of most natural sources obey Gaussian statistics, our protocol may be used for measuring the complex correlation information embedded in other light sources, e.g., the light shined by the glowing stars.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11874046, 11904247, 11974218, 91750201); Innovation Group of Jinan (2018GXRC010); Natural Science Research of Jiangsu Higher Education Institutions of China (19KJB140017); China Postdoctoral Science Foundation (2019M661915); Natural Science Foundation of Shandong Province (ZR2019QA004); Priority Academic Program Development of Jiangsu Higher Education Institutions; Qinglan Project of Jiangsu Province of China.

Disclosures

The authors declare no conflicts of interest.

References

1. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

2. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

3. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37(20), 4179–4181 (2012). [CrossRef]

4. G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019). [CrossRef]

5. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017). [CrossRef]

6. Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020). [CrossRef]

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

8. A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33(12), 2431–2442 (2016). [CrossRef]

9. E. Wolf, “Solution of the phase problem in the theory of structure determination of crystals from X-ray diffraction experiments,” Phys. Rev. Lett. 103(7), 075501 (2009). [CrossRef]

10. O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020). [CrossRef]

11. R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018). [CrossRef]

12. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008). [CrossRef]

13. X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015). [CrossRef]

14. Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). [CrossRef]

15. H. Partanen, B. J. Hoenders, A. T. Friberg, and T. Setälä, “Young’s interference experiment with electromagnetic narrowband light,” J. Opt. Soc. Am. A 35(8), 1379–1384 (2018). [CrossRef]

16. H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019). [CrossRef]

17. K. A. Sharma, G. Costello, E. Vélez-Juárez, T. G. Brown, and M. A. Alonso, “Measuring vector field correlations using diffraction,” Opt. Express 26(7), 8301–8313 (2018). [CrossRef]

18. K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020). [CrossRef]

19. X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019). [CrossRef]

20. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

21. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19(16), 15188–15195 (2011). [CrossRef]

22. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]

23. R. K. Singh, R. V. Vinu, and A. M. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014). [CrossRef]

24. R. K. Singh, A. M. Sharma, and P. Senthilkumaran, “Vortex array embedded in a partially coherent beam,” Opt. Lett. 40(12), 2751–2754 (2015). [CrossRef]

25. R. V. Vinu and R. K. Singh, “Experimental determination of generalized Stokes parameters,” Opt. Lett. 40(7), 1227–1230 (2015). [CrossRef]

26. R. V. Vinu and R. K. Singh, “Synthesis of statistical properties of a randomly fluctuating polarized field,” Appl. Opt. 54(21), 6491–6497 (2015). [CrossRef]

27. D. Singh and R. K. Singh, “Lensless Stokes holography with the Hanbury Brown–Twiss approach,” Opt. Express 26(8), 10801–10812 (2018). [CrossRef]

28. D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018). [CrossRef]

29. Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020). [CrossRef]

30. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef]

31. H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020). [CrossRef]

32. M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014). [CrossRef]

33. H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019). [CrossRef]

Measuring complex correlation matrix of partially coherent vector light via a generalized Hanbury Brown–Twiss experiment

Abstract

1. Introduction

2. Theory

3. Experiment

4. Polarization recovery

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (22)

Optics Express