April 15, 2026
Here we derive the relation between the space-time domain COAM matrix \(\overleftrightarrow{\Gamma}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)\) and its space-frequency domain counterpart. By using Eqs. (7) and (8), the following relations are found between the MCF and COAM matrix elements: \[\begin{align} \Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)=&\sum_{l=-\infty}^\infty\sum_{m=-\infty}^\infty \Gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)e^{i(m\phi_2-l\phi_1)}, \tag{1}\\ \Gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=&\frac{1}{(2\pi)^2}\iint_0^{2\pi}\Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)e^{-i(m\phi_2-l\phi_1)}d\phi_1d\phi_2.\tag{2} \end{align}\] In the space-frequency domain, the spatial coherence between the fields at \(\mathbf{r}_1\) and \(\mathbf{r}_2\) and at frequency \(\omega\) is characterized by the cross-spectral density function (CSDF) \(W(\mathbf{r}_1,\mathbf{r}_2,\omega)\), defined by the Wiener–Khintchine relations [28] \[\begin{align} W(\mathbf{r}_1,\mathbf{r}_2,\omega)=&\frac{1}{2\pi}\int_{-\infty}^\infty \Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)e^{i\omega\tau}d\tau, \tag{3} \\ \Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)=&\int_{0}^\infty W(\mathbf{r}_1,\mathbf{r}_2,\omega) e^{-i\omega\tau}d\omega. \tag{4} \end{align}\] Substituting Eq. (4 ) to Eq. (2 ) leads to \[\begin{align} \label{A5} \Gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{1}{(2\pi)^2}\iint_0^{2\pi} \left[\int_0^\infty W(\mathbf{r}_1,\mathbf{r}_2,\omega)e^{-i\omega\tau}d\omega \right] e^{-i(m \phi_2-l \phi_1)}d\phi_1d\phi_2. \end{align}\tag{5}\] Similarly to the MCF in Eq. (1 ), the CSDF can be expressed as [31] \[\begin{align} \label{W-coam} W(\mathbf{r}_1,\mathbf{r}_2,\omega)=\sum_{l=-\infty}^\infty\sum_{m=-\infty}^\infty W_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\omega)e^{i(m\phi_2-l\phi_1)}, \end{align}\tag{6}\] where \(W_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\omega)\) are the elements of the space-frequency domain COAM matrix. Combining Eqs. (6 ) and (5 ), changing the order of integrations and summations, and employing the integral \[\begin{align} \frac{1}{2\pi}\int_0^{2\pi}e^{i(\alpha-\beta)x}dx=\delta_{\alpha\beta}, \end{align}\] we find that \[\begin{align} \label{GammalmWlm} \Gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\int_0^\infty W_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\omega)e^{-i\omega\tau}d\omega. \end{align}\tag{7}\] The space-time and space-frequency domain COAM matrices thus satisfy a Wiener–Khintchine type relation in the polar Fourier space.
In the following we derive the intensity at detector \(\mathrm{D}_1\) in the setup depicted in Fig. 1(b). A similar result can be naturally established for the intensity at detector \(\mathrm{D}_2\) by following the approach presented here. Assume that a quasi-monochromatic, paraxial field \(U(\pmb{\rho},t)\), \(\pmb{\rho}=[\rho,\phi]\) at a plane \(z=0\) enters a combination of a spiral phase plate of order \(-l\), a thin lens with focal length \(f\), and a ring aperture of radius \(\rho_0\) and width \(\Delta\) placed directly behind the lens. The detector is at the distance \(f\) from the system. The instantaneous intensity at the focal plane of the lens is obtained as (see Sect. 7.1 of [30]) \[\begin{align} I(\pmb{\rho},t)=\frac{1}{(\lambda f)^2}\iiiint T^\ast(\pmb{\rho}_1^\prime)T(\pmb{\rho}_2^\prime)J(\pmb{\rho}_1^\prime,\pmb{\rho}_2^\prime,t)\exp\left[-\frac{ik}{f}\pmb{\rho}\cdot(\pmb{\rho}_2^\prime-\pmb{\rho}_1^\prime)\right]d^2\pmb{\rho}_1^\prime d^2\pmb{\rho}_2^\prime, \end{align}\] where \(\lambda\) is the central wavelength, \(k=2\pi/\lambda\), \(T(\pmb{\rho})\) is the combined transmission function of the ring aperture and spiral phase plate, and \(J(\pmb{\rho}_1,\pmb{\rho}_2,t)=U^\ast(\pmb{\rho}_1,t)U(\pmb{\rho}_2,t)\) is the instantaneous mutual intensity of the beam at the input plane of the system. We further assume that the slit width \(\Delta\) is narrow enough such that the transmission function can be approximated as \(T(\pmb{\rho})=e^{-il\phi}\delta(\rho-\rho_0)\Delta\), where \(\delta(x)\) is the Dirac delta. Then, evaluating the form above at the focal point \(\rho=0\), we find that \[\begin{align} I(0,t)=\frac{4\pi^2\Delta^2\rho_0^2}{\lambda^2 f^2}\iint_0^{2\pi}J(\rho_0,\phi_1^\prime,\rho_0,\phi_2^\prime,t)e^{-il(\phi_2^\prime-\phi_1^\prime)}d\phi_1^\prime d\phi_2^\prime=\frac{4\pi^2\Delta^2\rho_0^2}{\lambda^2 f^2}I_l(\rho_0,t), \end{align}\] where the non-averaged version of Eq. (2 ) has been used to obtain the second equality. Thus, we see that measuring the instantaneous intensity at the focal spot of the lens gives the intensity \(I_l(\rho_0,t)\) of the mode \(U_l(\rho_0,t)\).
In this section, we show that the limits \(\bar{\gamma}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=1\) and \(\bar{\gamma}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=0\) correspond to all pairs of modes \(U_l(\pmb{\xi}_1,t)\), \(U_m(\pmb{\xi}_2,t+\tau)\), being completely correlated and uncorrelated, respectively. We begin by introducing the correlation coefficients [31] \[\begin{align} \gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{\langle U_l^\ast(\mathbf{r}_1,t)U_m(\mathbf{r}_2,t+\tau)\rangle}{\sqrt{\langle |U_l(\pmb{\xi}_1,t)|^2\rangle \langle |U_m(\pmb{\xi}_2,t)|^2\rangle}}, \end{align}\] which are limited as \(0\leq |\gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|\leq 1\), with the lower and upper bounds corresponding to \(U_l(\pmb{\xi}_1,t)\) and \(U_m(\pmb{\xi}_2,t+\tau)\) being completely uncorrelated and fully correlated, respectively. We may now write the (squared) OAMDC in the form \[\begin{align} &\bar{\gamma}^2(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{\sum\limits_{l,m\in\mathcal{L}}|\gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2 \langle |U_l(\pmb{\xi}_1,t)|^2\rangle \langle |U_m(\pmb{\xi}_2,t)|^2\rangle}{\sum\limits_{l,m\in\mathcal{L}} \langle |U_l(\pmb{\xi}_1,t)|^2\rangle \langle |U_m(\pmb{\xi}_2,t)|^2\rangle}. \end{align}\] From this expression it is clear that \(\bar{\gamma}^2(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=1\) if and only if \(|\gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|=1\) for all \((l,m)\), i.e., when all the components \(U_l(\pmb{\xi}_1,t)\) and \(U_m(\pmb{\xi}_2,t+\tau)\) are fully correlated. In addition, \[\begin{align} &\mathrm{tr}[\overleftrightarrow{\Gamma}^\dagger(\pmb{\xi}_1,\pmb{\xi}_2,\tau)\overleftrightarrow{\Gamma}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)]=\sum_{l,m\in\mathcal{L}}|\langle U_l^\ast(\pmb{\xi}_1,t)U_m(\pmb{\xi}_2,t+\tau)\rangle|^2, \end{align}\] which attains the value of zero if and only if \(\gamma_{lm}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=0\) for all \((l,m)\). Consequently, \(\bar{\gamma}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=0\) corresponds to all the modes \(U_l(\pmb{\xi}_1,t)\) and \(U_m(\pmb{\xi}_2,t+\tau)\) being completely uncorrelated.
In this section we show the derivation of Eq. (23). From Eqs. (18) and (22) we see that \[\begin{align} \bar{g}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{1}{L}|\bar{\gamma}_0(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2+\frac{1}{2}\sum_{n=1}^{L^2-1} |\bar{\gamma}_n(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2, \end{align}\] which, after taking the common factor and using Eq. (21), can be written as \[\begin{align} \label{S15} \bar{g}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{1}{L}|\bar{\gamma}_0(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2\left[1+\frac{L}{2}\frac{\sum_{n=1}^{L^2-1} |\bar{\mathcal{S}}_n(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2}{|\bar{\mathcal{S}}_0(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2}\right]. \end{align}\tag{8}\] Next, we define the quantity \(\mathcal{Q}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)\) as in Eq. (24): \[\begin{align} \label{S16} \mathcal{Q}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\sqrt{\frac{L}{2(L-1)}\frac{\sum_{n=1}^{L^2-1} |\bar{\mathcal{S}}_n(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2}{|\bar{\mathcal{S}}_0(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2}}. \end{align}\tag{9}\] Substituting Eq. (9 ) in Eq. (8 ) immediately leads to \[\begin{align} \bar{g}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\frac{1}{L}|\bar{\gamma}_0(\pmb{\xi}_1,\pmb{\xi}_2,\tau)|^2\left[1+(L-1)\mathcal{Q}^2(\pmb{\xi}_1,\pmb{\xi}_2,\tau)\right], \end{align}\] which completes the derivation.
Here we show that \(Q(\pmb{\xi})\) can be interpreted as the spectral distance of the OM from a uniform distribution. We begin by noting that the normalized eigenvalues \(\sigma(\pmb{\xi})\) satisfy \[\label{muncond} \sum\limits_{n=1}^{L}\sigma_n(\pmb{\xi})=1.\tag{10}\] Consequently, \(0\leq \sigma_l(\pmb{\xi}) \leq 1\), defining a probability distribution, i.e., vector \(\vec{\sigma}(\pmb{\xi})=[\sigma_l(\pmb{\xi})]\) lies on the simplex. We then consider the spectral Euclidean distance (variance) \(\| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2\), where \(\vec{I}=\vec{1}/L\) and \(\vec{1}\) is an \(L\)D vector of ones, normalized to its maximum value: \[\label{M1} \frac{\| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2}{\text{max}\| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2}.\tag{11}\]
Since \(\text{max} \| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2\) is achieved in the corners of the simplex, i.e., if \(\sigma_1(\pmb{\xi})=1\) and \(\sigma_l(\pmb{\xi})=0\), \(l\geq 2\), we find that \[\begin{align} \text{max}\| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2^2& = \left( \frac{L-1}{L}\right)^2-\frac{L-1}{L^2}=\frac{L-1}{L}. \end{align}\] Then, after opening the brackets and using Eq. (10 ), the numerator of Eq. (11 ) becomes \[\| \vec{\sigma}(\pmb{\xi})-\vec{I} \|_2^2=\sum\limits_{l=1}^{L}\left[ \sigma_l(\pmb{\xi})-\frac{1}{L}\right]^2=\sum\limits_{l=1}^{L}\sigma_l^2(\pmb{\xi})-\frac{1}{L}.\] Thus, as is evident from Eqs. (28) and 10 , the ratio in Eq. 11 coincides with \(Q(\pmb{\xi})\): \[\label{M2} Q^2(\pmb{\xi})=\frac{L}{L-1}\left[ \sum\limits_{l=1}^{L}\sigma_l^2(\pmb{\xi})-\frac{1}{L}\right].\tag{12}\]
Here we establish the COAM matrix displayed in Eq. (29). The MCF for the incoherent superposition of LG and \(I_l\)-Bessel correlated modes is given by \[\begin{align} \Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)=\Gamma_\mathrm{LG}(\mathbf{r}_1,\mathbf{r}_2,\tau)+\Gamma_\mathrm{B}(\mathbf{r}_1,\mathbf{r}_2,\tau). \end{align}\] Above, \[\begin{align} \label{GammaLG} \Gamma_\mathrm{LG}(\mathbf{r}_1,\mathbf{r}_2,\tau)=U_\mathrm{LG}^\ast(\mathbf{r}_1)U_\mathrm{LG}(\mathbf{r}_2)e^{-i\omega\tau} \end{align}\tag{13}\] is the portion of the MCF associated with the fully coherent LG modes, with \(U_\mathrm{LG}(\mathbf{r})=\sum_{l\in\mathcal{L}}U_l(\pmb{\xi})e^{il\phi}\) and [18] \[\begin{align} U_l(\pmb{\xi})=& \frac{w_0}{w(z)} \sqrt{\frac{2}{\pi|l|!}} \left[\frac{\rho\sqrt{2}}{w(z)}\right]^{|l|} \exp\left[- \frac{\rho^2}{w^2(z)} \right] \exp\left[i(|l|+1)\Phi(z) - i\frac{\omega_0}{c} \frac{\rho^2}{2R(z)}\right]. \label{Ul} \end{align}\tag{14}\] Here \(w(z) = w_0\left[ 1+(z/z_{R})^2 \right]^{1/2}\), \(w_0\) is the beam waist, \(z_{R}=\pi w_0/\lambda_0\) is the Rayleigh range, \(\lambda_0\) and \(\omega_0\) are the central wavelength and frequency, respectively, \(c\) denotes the speed of light, \(R(z)=z[1+(z_{R}/z)^2]\) is the radius of curvature, and \(\Phi(z)=arctan(z/z_{R})\). Employing Eq. (2 ), the COAM matrix corresponding to \(\Gamma_\mathrm{LG}(\mathbf{r}_1,\mathbf{r}_2,\tau)\) is found to be \[\begin{align} \overleftrightarrow{\Gamma}_\mathrm{LG}(\pmb{\xi}_1,\pmb{\xi}_2,\tau)=\vec{U}^\dagger(\pmb{\xi}_1)\vec{U}(\pmb{\xi}_2)e^{-i\omega\tau}, \end{align}\] where \(\vec{U}(\pmb{\xi})=[U_l(\pmb{\xi})]\). In addition, \[\begin{align} \Gamma_\mathrm{B}(\mathbf{r}_1,\mathbf{r}_2,\tau)=\int_0^\infty W_\mathrm{B}(\mathbf{r}_1,\mathbf{r}_2,\omega)e^{-i\omega\tau}d\omega \end{align}\] represents the MCF contribution from the \(I_l\)-Bessel correlated part of the beam, with the related CSDF taken to be \[\begin{align} W_\mathrm{B}(\mathbf{r}_1,\mathbf{r}_2,\omega) =\sum_{l\in\mathcal{L}}W_l(\pmb{\xi}_1,\pmb{\xi}_2,\omega)e^{il(\phi_2-\phi_1)}, \end{align}\] where [49] \[\begin{align} W_l(\pmb{\xi}_1,\pmb{\xi}_2,\omega)=&W_f(\omega)\frac{\zeta^{-l/2}}{1-\zeta}\frac{w_0^2}{w(z_1)w(z_2)} \exp\left(i\left\{\frac{\omega}{c}\left(z_1-z_2\right)-(l+1)[\Phi(z_1)-\Phi(z_2)]\right\}\right) \nonumber \\&\times \exp\left\{i\frac{\omega}{c}\left[\frac{\rho_1^2}{2R(z_1)}-\frac{\rho_2^2}{2R(z_2)}\right]\right\} \exp\left\{-\frac{1+\zeta}{1-\zeta}\left[\frac{\rho_1^2}{w^2(z_1)}+\frac{\rho_2^2}{w^2(z_2)} \right]\right\}\nonumber\\ &\times I_l\left[\frac{4\zeta^{1/2}}{1-\zeta}\frac{\rho_1\rho_2}{w(z_1)w(z_2)} \right]. \end{align}\] Here \(W_f(\omega)\) is a spectral weight factor, \(I_l(x)\) is the modified Bessel function of order \(l\), \(\zeta\), \(0<\zeta <1\), is a coherence parameter associated with the modes, with larger \(\zeta\) corresponding to more incoherent light, and the other parameters are same as above. In particular, we may choose the spectral factor to be a Gaussian function \[\begin{align} \label{Wf-Gauss} W_f(\omega)= \frac{1}{\delta_\omega\sqrt{2\pi}}\exp\left[-\frac{(\omega-\omega_0)^2}{2\delta_\omega^2}\right], \end{align}\tag{15}\] where \(\delta_\omega\) is spectral bandwidth. The components \(W_l(\pmb{\xi}_1,\pmb{\xi}_2,\omega)\) represent the diagonal COAM matrix elements in the space-frequency domain, with the off-diagonals taken to be zero. We then obtain by evaluating Eq. (7 ) the diagonal elements of the space-time domain COAM matrix at a plane of constant \(z\): \[\begin{align} \label{GammaB} \Gamma_{ll}^{(\mathrm{B})}(\rho_1,\rho_2,z,\tau)=&\frac{\zeta^{-l/2}}{1-\zeta}\left[\frac{w_0}{w(z)}\right]^2 \exp\left\{i\left[\frac{\omega_0}{c}\frac{\rho_1^2-\rho_2^2}{2R(z)}-\omega_0\tau\right]\right\} \exp\left\{-\frac{\delta_\omega^2}{2}\left[\frac{1}{c}\frac{\rho_1^2-\rho_2^2}{2R(z)}-\tau\right]^2\right\} \nonumber\\&\times \exp\left\{-\frac{1+\zeta}{1-\zeta}\left[\frac{\rho_1^2+\rho_2^2}{w^2(z)}\right]\right\} I_l\left[\frac{4\zeta^{1/2}}{1-\zeta}\frac{\rho_1\rho_2}{w^2(z)} \right]. \end{align}\tag{16}\] Combining Eqs. (13 ) and (16 ) and setting \(\tau=0\), we obtain Eq. (29): \[\begin{align} \overleftrightarrow{\Gamma}(\rho_1,\rho_2,z)=\vec{U}^\dagger(\rho_1,z)\vec{U}(\rho_2,z)+\overleftrightarrow{R}(\rho_1,\rho_2,z), \end{align}\] where \(\overleftrightarrow{R}(\rho_1,\rho_2,z)=\mathrm{diag}[R_l(\rho_1,\rho_2,z)]\) and \[\begin{align} R_l(\rho_1,\rho_2,z)=&\frac{\zeta^{-l/2}}{1-\zeta}\left[\frac{w_0}{w(z)}\right]^2 \exp\left[i\frac{\omega_0}{c}\frac{\rho_1^2-\rho_2^2}{2R(z)}\right] \exp\left[-\frac{\delta_\omega^2}{8c}\frac{(\rho_1^2-\rho_2^2)^2}{R^2(z)}\right] \nonumber\\&\times \exp\left\{-\frac{1+\zeta}{1-\zeta}\left[\frac{\rho_1^2+\rho_2^2}{w^2(z)}\right]\right\} I_l\left[\frac{4\zeta^{1/2}}{1-\zeta}\frac{\rho_1\rho_2}{w^2(z)} \right]. \end{align}\]