## 2 Target-in-the-Loop Atmospheric Sensing Concept  
The theoretical foundation of the TILAS concept is based on the optical reciprocity principle [10] and the associated conservation law for the so-called interference metric \(J_{\rm int}\) coupling the complex amplitudes of the counter-propagating optical waves [11; 12; 13]. As was recently shown [8; 9; 14], the square modulus of the interference metric can be measured using the TILAS system depicted in Figure 1a. This system is composed of a laser, fiber-optics circulator, fiber-coupled transceiver, and fiber-coupled photodetector. The circulator separates the outgoing and incoming laser beams by sending the outgoing, linearly-polarized Gaussian beam from the laser to the transceiver, while the entering wave is directed to the photodetector. The power signal measured at the TILAS photodetector is given by the expression [8]  
\[P_{\rm TILAS}(t)\ =\ \alpha\Big{|}J_{\rm int}(t)\Big{|}^{2}\ =\ \alpha\Big{|}\!\!\int A( \mathbf{r},z\ =\ 0)\psi(\mathbf{r},z\ =\ 0,t)\,d^{2}\mathbf{r}\Big{|}^{2}. \tag{1}\]  
where \(\alpha\) is a coefficient dependent on the transmitted power and the photodetector's sensitivity, \(A(\mathbf{r},z\ =\ 0)\) is the stationary complex amplitude of the outgoing wave transmitted along the optical \(z\)-axis, \(\psi(\mathbf{r},z\ =\ 0,t)\) is the amplitude of the incoming wave backscattered off a target located at the plane \(z=L\) back to the receiver (transceiver) plane \(z=0\), \(\mathbf{r}=\{x,y\}\) is the spatial vector in the plane orthogonal to the propagation direction, \(t\) is the time, and the integration is performed over the entire transverse plane.  
In this paper, we consider the stationary TILAS transceiver and the target with a glint, moving perpendicularly to the sensing path direction with velocity \(\mathbf{v}_{T}\) (Figure 1b). For this type of target, the following quantities are computed based on the TILAS received power signal measurements:  
1. Autocorrelation function of the signal, \[B(\tau)\ = \frac{\left\langle P_{\text{TILAS}}(t)P_{\text{TILAS}}(t+\tau) \right\rangle-\left\langle P_{\text{TILAS}}(t)\right\rangle^{2}}{\left\langle p _{\text{TILAS}}^{2}(t)\right\rangle-\left\langle P_{\text{TILAS}}(t)\right\rangle ^{2}},\] (2) where \(\tau\) is the time delay between the power signal values measured at different moments of time \(t\) and angle brackets denote the averaging over the integration time interval \(T.\) Here we assume that \(T\gg\tau_{\text{at}},\) where \(\tau_{\text{at}}\) is the characteristic correlation time of the atmospheric turbulence pattern change inside the transceiver aperture [8].
2. Interference metric variance, also referred to here as the TILAS received power signal variance, given by the expression [8] \[\sigma_{J}^{2}\ =\ \left\langle P_{\text{TILAS}}(t)\right\rangle_{T}\! \left\langle\sqrt{P_{\text{TILAS}}(t)}\right\rangle_{T}^{2}-1.\] (3)  
In this paper, we focus on the atmospheric turbulence model that can be represented by a single spatially localized turbulence layer, referred to as here as the single layer model. The homogeneously distributed turbulence model is also briefly considered. We show that parameters \(B(\tau)\) and \(\sigma_{J}^{2}\) given by the equations above can be used for estimation of both the strength and position of the turbulence layer along the optical propagation path, and, potentially, can lead to the turbulence profiling. We also demonstrate that consecutive analysis of the correlation properties of the TILAS received power signal \(P_{\text{TILAS}}\) (\(t\)) results in the estimation of the atmospheric turbulence outer scale \(L_{0}\) of the layer.  
Numerical simulation approach for the analysis is explained in the following section.