Probability Distribution of Transmittance

• Probability Distribution of Transmittance (PDT) is a statistical model that quantifies random fluctuations in optical transmittance caused by beam wandering, spot distortions, and absorption variations.
• It is derived using analytical methods, simulations, and empirical models that integrate spatial, temporal, and modal correlations to predict system performance.
• The insights from PDT facilitate optimized adaptive protocols in quantum communications and aid in the diagnostic evaluation of thin films and layered media.

The probability distribution of transmittance (PDT) describes the statistical distribution of optical transmittance in systems where the direct propagation of waves or particles is subject to random spatial, temporal, or structural fluctuations. PDT constitutes a central concept in wave and quantum transport, atmospheric optics, thin-film physics, and random media, where it encodes the complete statistical characterization of fluctuating losses, beam distortions, and multi-path effects. In quantum communication through turbulent atmosphere, the PDT determines how nonclassical features (e.g. squeezing, entanglement) and channel capacities are degraded by random transmission. In thin films and layered media, PDT provides diagnostics of homogeneity and effective light-blocking capacity. Across these domains, the mathematical structure, limiting forms, and model selection for PDT depend intricately on the underlying physical mechanisms of scattering, absorption, or random wave propagation.

1. Fundamental Definitions and General Framework

Let η∈0,1 denote the random fraction of incident optical power transmitted through a given region—typically a receiver aperture, slab, or interface. The PDT, P(η), is formally defined such that P(η)dη is the probability that a single measurement of transmittance yields a value in [η,η+dη]. In quantum optics, η parameterizes the single-mode input–output relation:

$$\hat{a}_\text{out} = \sqrt{\eta}\,\hat{a}_\text{in} + \sqrt{1-\eta}\,\hat{c}$$

and in phase space,

$$P_\text{out}(\alpha) = \int_0^1 P(\eta)\,\frac{1}{\eta}\,P_\text{in}\left(\frac{\alpha}{\sqrt{\eta}}\right) d\eta$$

Knowledge of P(η) enables predictive calculations for all statistical and security-critical observables in quantum communication and allows optimization of post-selection and adaptive protocols under fluctuating-loss conditions (Vasylyev et al., 2018, Semenov et al., 7 Sep 2025, 1804.00172).

2. Physical Origins and Decomposition of Fluctuations

The statistical nature of PDT arises from physical fluctuations that can be decomposed into distinct mechanisms, especially in atmospheric channels:

• Beam wandering: Random transverse displacement of the beam centroid, typically modeled as a bivariate Gaussian process with variance σ_bw² determined by fourth-order spatial field correlations (1804.00172, Vasylyev et al., 2018).
• Beam-shape (spot) distortions: Variations in beam width, ellipticity, and higher-order speckle, parametrized via conditional distributions given the beam centroid position (1804.00172).
• Absorption/thickness variation: In films, statistical variations in thickness or agent distribution map into fluctuations of local transmittance via Beer–Lambert law or its analogues (Amano, 2020).
• Coherent multi-mode or multi-path interference: In multi-port systems or layered media, the input coherence and eigenvalue spectrum of the transmission operator define the distribution of output transmittance via B-spline and large-deviation theories (Wang et al., 6 Nov 2025, Pradas et al., 2017, Park et al., 2022).

The law of total probability is employed to factor the overall PDT:

$P(\eta) = \int d^2r_0\,\rho(r_0)\,P(\eta \mid r_0)$

where ρ(r_0) captures the statistics of centroid displacement and P(η|r_0) encodes conditional spot or shape-induced fluctuations (1804.00172, Vasylyev et al., 2018, Semenov et al., 7 Sep 2025).

3. Analytical and Empirical Models for PDT

Several established forms for PDT arise in limiting regimes or as empirical fits to simulation and experiment:

PDT Models and Key Applicability Regimes

Model	Formulation	Main Applicability
Log-negative Weibull	Beam-wandering regime; analytical form for Gaussian beams	Weak distortion, wandering-dominated regime (1804.00172, Vasylyev et al., 2018, Klen et al., 2023)
Truncated log-normal	Fluctuating spot/broadening; two parameters matched to ⟨η⟩,⟨η²⟩	Strong turbulence, small-aperture, spot-dominated (1804.00172, Semenov et al., 7 Sep 2025, Klen et al., 2023)
Elliptic-beam	Full beam-shape fluctuations; sampling centroid & spot axes	Intermediate regime, moderate apertures (Klen et al., 2023, Semenov et al., 7 Sep 2025, Dutta et al., 2023)
Circular-beam	Gaussian spot with log-normal radius, centroid wandering	Broad aperture range, computational efficiency (Pechonkin et al., 17 Jul 2025, Semenov et al., 7 Sep 2025)
Beta distribution	Empirical two-moment fit, flexible support on [0,1]	Wide regime, accurate for many PDE-realistic P(η) (Klen et al., 2023, Semenov et al., 7 Sep 2025)
Law of total probability	Hierarchical model, integrating over beam wander and conditional distortions	Universally valid, numerical evaluation (1804.00172, Semenov et al., 7 Sep 2025, Vasylyev et al., 2018)

For thin films and coatings, Beer–Lambert law with fluctuating thickness t and PDF p(t) yields

$$P_T(T) = p(-\frac{1}{\alpha}\ln T)\cdot \frac{1}{\alpha T}$$

which, for Gaussian or uniform thickness, results in log-normal or $1/T$ distributions, respectively (Amano, 2020).

In random layered 1D media, P(ln T) exhibits a universal (log-normal) central segment controlled by s = L/ℓ and a nonuniversal low-transmittance tail set by the stack parameters (Park et al., 2022).

In coherent random multi-mode systems, P(T) is given exactly by a fundamental B-spline of degree N−1 with knots at the transmission eigenvalues; in the large N limit, this converges to a Gaussian distribution (Wang et al., 6 Nov 2025).

4. Derivation and Parameter Identification

PDT model parameterization relies on a hierarchy of statistical moments and physical correlation functions:

• First and Second moments: ⟨η⟩, ⟨η²⟩ extracted from second- and fourth-order intensity correlations, often via analytic weak-turbulence approximations or direct simulation (1804.00172, Semenov et al., 7 Sep 2025, Klen et al., 2023).
• Centroid variance (σ_bw²): Computed from spatial cross-correlation integrals or Monte Carlo sampling based on physical link geometry (e.g., up-link/down-link specifics in satellite QKD) (Vasylyev et al., 2018, Dutta et al., 2023).
• Conditional spot-width distribution: Log-normal or Gaussian, marginalized over in the circular- and elliptic-beam models (Pechonkin et al., 17 Jul 2025, Klen et al., 2023).
• Empirical fits: Beta-distribution parameters $a,b$ are set by moment-matching, yielding high fidelity to simulation data in many regimes (Klen et al., 2023).

For coherent transport through arbitrary N-port systems, eigenvalues of the transmittance matrix fully determine the B-spline shape and all moments of the PDT (Wang et al., 6 Nov 2025).

5. Numerical Simulation and Validation

The primary numerical tool is the phase-screen/split-step method, simulating paraxial wave propagation through a sequence of random phase screens with spectral statistics set by the Kolmogorov or von Kármán spectrum. At each realization, the optical field at the receiver is evaluated, integrated over the aperture to yield η, and the ensemble of η-values is histogrammed to approximate P(η) (Klen et al., 2023, Semenov et al., 7 Sep 2025).

Validation of analytical models employs the Kolmogorov–Smirnov metric D_M to quantify the discrepancy between model and simulation. Empirical Beta models often yield D_M < 0.05 over a wide range of parameters. Elliptic- and circular-beam models are more accurate near the long-term beam radius W_LT; truncated log-normal and Beta models are preferred for small-aperture, strong-distortion, or large-wander regimes (Klen et al., 2023, Pechonkin et al., 17 Jul 2025, Semenov et al., 7 Sep 2025).

6. Applications in Quantum Communication, Thin Films, and Random Media

• Quantum communication: PDT is the core input for modeling nonclassical state transmission, postselection strategies, and security analysis in QKD schemes. Channel key rates are integrated over P(η); high-η tails preserve nonclassicality and raise postselected protocol success rates (1804.00172, Semenov et al., 7 Sep 2025, Dutta et al., 2023). Adaptive optics and beam tracking modify the effective PDT, often reducing the wandering variance and concentrating probability at higher η (1804.00172, Vasylyev et al., 2018).
• Thin films, coatings: PDT enables rigorous comparison of light-blocking in the presence of geometric and compositional heterogeneity. Broader PDTs indicate less predictable and potentially compromised performance in sunscreen films or optical coatings (Amano, 2020).
• Layered/random media: PDTs in one-dimensional layered systems are used to distinguish universal single-parameter scaling classes (log-normal central regime for P(ln T)) from nonuniversal behavior at the low-transmittance edge, with direct consequences for localization theory, random lasing, and beyond (Park et al., 2022, Pradas et al., 2017).

7. Future Directions and Open Problems

Current developments focus on:

• Analytic models for strong turbulence: Extending moment-based analytical PDT predictions beyond σ_R² ≳ 1; closed-form solutions remain challenging (Semenov et al., 7 Sep 2025, Pechonkin et al., 17 Jul 2025).
• Multi-time and temporally correlated PDTs: Bridging between ensemble- and time-averaged PDTs to account for realistic detector integration and non-Markovian turbulence (Semenov et al., 7 Sep 2025).
• Spatiotemporal, multimodal, and polarization-resolved PDTs: Incorporating joint statistics across spatial, temporal, and internal degrees of freedom (Semenov et al., 7 Sep 2025).
• Physical interpretation of heavy-tailed and multimodal PDTs: Identifying operator-theoretic and phase-space origins of complex PDT morphologies in deeply random or strongly fluctuating media (Pradas et al., 2017, Park et al., 2022, Wang et al., 6 Nov 2025).
• Improved diagnostics and engineering: Leveraging PDT modeling to optimize link geometries, adaptive control, coating composition, or protocol design.

PDT modeling remains a crucial enabling tool for both understanding light-matter interaction under randomness and for the engineering of robust photonic and quantum-communication architectures (1804.00172, Pechonkin et al., 17 Jul 2025, Semenov et al., 7 Sep 2025, Amano, 2020, Wang et al., 6 Nov 2025, Park et al., 2022).