Complex Reflections Ratio Analysis

• Complex Reflections Ratio is a metric that quantifies the relationship between reflected signals by preserving amplitude and phase information.
• It is applied across stratified media, scattering experiments, and VLC systems to provide insights for interferometric imaging and wireless performance analysis.
• Its mathematically defined expressions and experimental setups enable precise estimation of material properties and system-level metrics.

The complex reflections ratio is a central analytical and experimental metric that quantifies the relationship between reflected signals—either fields or intensities—across disparate physical configurations or in the presence of multiple propagation paths, including coherent and incoherent contributions. It is foundational in problems ranging from stratified optical media, wave transport in random materials, to interference and coverage analyses in wireless and visible light communication systems. Its mathematical definition, physical significance, and practical measurement vary by field, but its core function remains the comparative quantification of reflected components in complex settings.

1. Formal Definitions Across Domains

The "complex reflections ratio" has established, context-specific mathematical forms:

• Stratified Media / Thin Films: For incident electromagnetic radiation on a multilayer system, the complex reflection coefficient $r_{\rm total}$ encapsulates amplitude and phase. The reflections ratio $R_{\rm ratio}$ compares two configurations (a) and (b) with identical ambient medium and wavelength as

$$R_{\rm ratio} = \frac{r_{\rm total}^{(a)}}{r_{\rm total}^{(b)}}$$

with $r_{\rm total}$ determined by characteristic-matrix or multiple-beam recursive formalisms (Nahmad-Rohen et al., 2019).

• Disordered and Scattering Media: In time- or frequency-resolved experiments using the reflection matrix $K$, the complex weight of single or recurrent scattering is estimated via

$\hat{\rho} = \frac{\|K_f\|_F^2}{\|K\|_F^2}$

where $K_f$ isolates confocal or memory-effect contributions. In a focused basis,

$\hat{\rho}_f = \frac{\|R_f\|^2}{\|R\|^2}$

robustly quantifies confocal (single+recurrent) reflections (Brütt et al., 2022).

• Visible Light Communications (VLC): The reflections–ratio at user location $y$ is defined as

$\Xi(y) = \frac{P_{\rm ref}(y)}{P_{\rm dir}(y)}$

where $P_{\rm ref}(y)$ aggregates all reflected power (from $k$-th order image LEDs) and $P_{\rm dir}(y)$ is the direct LOS component (Gupta et al., 2018).

2. Theoretical Frameworks and Computational Approaches

Stratified Layer Optics

The total complex reflection coefficient $r_{\rm total}$ is computed via the characteristic matrix approach, in which the stack of $N$ parallel layers is represented by the product $M = M_1 M_2 \cdots M_N$, with each $M_j$ introducing phase advance and admittance scaling:

$$M_j = \begin{pmatrix} \cos\delta_j & i \sin\delta_j / q_j \ i q_j \sin\delta_j & \cos\delta_j \end{pmatrix}$$

The full stack yields

$$r_{\rm total} = \frac{q_0 (M_{11} + M_{12} q_{N+1}) - (M_{21} + M_{22} q_{N+1})}{q_0 (M_{11} + M_{12} q_{N+1}) + (M_{21} + M_{22} q_{N+1})}$$

The reflections ratio between two stacks directly probes modification of the reflected field by internal changes (e.g., adding a thin layer), with phase and amplitude information preserved, which is essential for interferometric reflectometry (Nahmad-Rohen et al., 2019).

Multiple Scattering and the Reflection Matrix

The reflection matrix $K$ decomposes into orders of scattering via the Born expansion:

$K(\omega) = \sum_{n=1}^\infty K^{(n)}(\omega)$

with $K^{(1)}$ (single scattering) maintaining unique anti-diagonal phase coherence (memory effect), and higher orders $K^{(n)}$, $n\ge2$, encoding complex multiple- and recurrent-scattering paths. Projection of $K$ onto the anti-diagonal subspace yields $\hat{\rho}$, but recurrent scattering biases it to a "confocal scattering ratio". Focusing-based projections $\hat{\rho}_f$ avoid far-field and memory-effect assumptions and are robust against near-field or strongly aberrant regimes (Brütt et al., 2022).

Indoor VLC Network Model

Reflection contributions in VLC systems are incorporated via a superposition of line-of-sight (direct) and higher-order, attenuated, image-based paths:

$$P_{\rm ref}(y) = \sum_{k=1}^K \sum_{x \in \Phi_k} P_{\rm tx} \alpha^2 \eta^k (\|x-y\|^2 + h^2)^{-\beta}$$

with $\Phi_k$ the point process of $k$th-order image sources, and attenuation $\eta^k$. The resulting reflections ratio $\Xi(y)$ is a critical parameter controlling SINR, achievable rate, and spatially-varying interference fields (Gupta et al., 2018).

3. Physical Interpretation and Key Properties

Amplitude and Phase Sensitivity

Unlike intensity-based ratios, the complex reflections ratio retains phase information, which is essential for resolving structural attributes (e.g., thickness, refractive index) in layered materials (Nahmad-Rohen et al., 2019). In interferometric schemes, both $|r|$ and $\arg\,r$ are measurable, enabling retrieval of subsurface properties not accessible via intensity alone.

Confocal and Recurrent Contributions

In strongly scattering media, the estimator $\hat{\rho}$ captures not only true single scattering but also "recurrent" events sharing the same entry and exit resolution cell, which exhibit identical memory-effect signatures. Thus, the complex reflections ratio in this setting becomes a confocal indicator rather than a pure single-scattering weight. Its decay with depth approximately scales as $-\frac{z}{\ell_s}$, enabling estimation of the scattering mean free path $\ell_s$ (Brütt et al., 2022).

Spatial Inhomogeneity

In VLC environments, $\Xi(y)$ quantifies the locally varying impact of wall and higher-order reflections on user performance. Edge and corner locations exhibit heightened reflections ratios, and thus suffer degraded SINR and lower rate coverage, especially as wall reflectivity $\eta$ or image order $K$ increases. The reflections ratio therefore directly ties geometric room features and user position to system-level metrics (Gupta et al., 2018).

4. Applications in Measurement and Imaging

Interferometric Reflectometry

The ratio of complex reflection coefficients between a layered and bare interface,

$$R_{\rm ratio} = \frac{r_{\text{sample}}}{r_{\text{reference}}}$$

extracts nanoscale thickness or composition, as in characterizing lipid bilayers. Measured modulations in $|r|$ (few $10^{-3}$–$10^{-2}$) and phase shifts (tens of degrees) are resolvable by modern interferometric techniques at visible wavelengths (Nahmad-Rohen et al., 2019).

Matrix Imaging and Ultrasonic Probes

In ultrasonic imaging, $\hat{\rho}$ or $\hat{\rho}_f$ determine the usable fraction of backscattered signals for depth-resolved reconstruction. By monitoring their decline with depth, the transition from single- to multiple-scattering dominated regimes is mapped, guiding imaging protocol selection and frequency bandwidth choices (Brütt et al., 2022).

Network Performance Estimation

In indoor VLC, $\Xi(y)$ integrates directly into semi-closed form SINR and rate coverage probabilities, impacting system planning and resource allocation for reliable coverage (Gupta et al., 2018).

5. Limitations, Biases, and Control Strategies

Recurrent Scattering Bias

Conventional estimators relying on memory-effect anti-diagonal projections are systematically biased upward by recurrent scattering. Focused-basis estimators, which project onto explicit single-scatter responses at a chosen depth, provide more accurate measures of the confocal scattering ratio, robust to near-field and strong-scattering effects (Brütt et al., 2022).

Geometric and Boundary Sensitivities

The complex reflections ratio is tightly controlled by geometric parameters. In stratified media, layer sequence, thickness, and index set the phase accrual and interference regime. In VLC systems, wall reflectivity, room shape, and user location modulate the ratio. In random media, array configuration, resolution cell size, and depth together dictate measurement sensitivity and interpretation (Nahmad-Rohen et al., 2019, Brütt et al., 2022, Gupta et al., 2018).

6. Representative Analytical Expressions

Context	Reflection Ratio Expression	Principal Variables
Multilayer optics	$$R_{\rm ratio} = \frac{r_{\rm total}^{(a)}}{r_{\rm total}^{(b)}}$$	$n_j$, $d_j$, $\lambda$, $r_{j,j+1}$
Disordered media imaging	$\hat{\rho} = \frac{\|K_f\|_F^2}{\|K\|_F^2}$	$K$, projection basis, scattering order
Indoor VLC networks	$\Xi(y) = \frac{P_{\rm ref}(y)}{P_{\rm dir}(y)}$	$\eta$, $K$, $y$, room geometry

In all contexts, the complex reflections ratio serves as a powerful, model-dependent parameter for comparing reflected signals, diagnosing channel or material properties, and guiding both measurement and system design strategies.