Overview of Transmittance-Based Metrics

• Transmittance-based metric is a quantitative measure that uses the ratio of transmitted to incident light to assess material performance and device efficiency.
• It employs analytical methods such as the Beer–Lambert law, transfer matrix formalism, and Fabry–Pérot models for precise optical characterization.
• Applications include optimizing metamaterials, photovoltaic devices, thin films, and astronomical calibrations, enabling rapid diagnostics and enhanced design.

A transmittance-based metric is a quantitative measure or framework that uses the optical transmittance—i.e., the fraction of incident electromagnetic energy transmitted through a material or structure—to characterize, optimize, and compare the performance of diverse physical systems, ranging from metamaterials and photovoltaic devices to thin films, ceramics, and advanced photonic structures. The metric may be a direct measurement (e.g., $T(\lambda)$ at specific wavelengths), an integrated band-averaged value, or an analytic expression coupling transmittance to underlying structural or physical parameters. Transmittance-based metrics play a central role in experimental characterization, theoretical modeling, material optimization, and the engineering of device functionalities.

1. Definition and Fundamental Concepts

Transmittance ($T$) is defined as the ratio of transmitted to incident intensity: $$T(\lambda) = \frac{I_{\text{transmitted}}(\lambda)}{I_{\text{incident}}(\lambda)}$$ The precise nature of $T$—whether it refers to spectral, angle-resolved, hemispherical, or polarization-resolved transmittance—depends on the context and measurement geometry.

A transmittance-based metric, as used in recent literature, refers to any figure, function, or parameter derived from $T$ (or its derivatives, e.g., the derivative with respect to a structural or environmental parameter) to index performance, material quality, device efficiency loss, functional response, or phase transitions. It may incorporate physical modeling, statistical properties, or material-specific weighting schemes.

2. Theoretical Frameworks and Mathematical Modeling

In complex media, the relationship of transmittance to device structure or microstructure is central. Analytical models typically take the form:

• Beer–Lambert Law: $T = \exp(-aL)$, where $a$ is the absorbance and $L$ the path length, forming the basis for absorbance estimation in quantum and classical regimes (Allen et al., 2019).
• Transfer Matrix Formalism (TMM): $T$ is computed via products of interface and propagation matrices (e.g., $T = |\det(M)/M_{22}|^2$ for multilayers), enabling full spectrum and angle-resolved predictions in layered optical systems, photonic crystals, and ENZ metamaterials (Briere et al., 2016, Carrera-Escobedo et al., 2016, Torres-Guzmán et al., 29 Feb 2024).
• Fabry–Pérot/Cavity Models: Transmittance maxima arise from resonance conditions, as in $$T = 1 / \left[1 + \left(4|r|^2/(1 - |r|^2)^2\right) \sin^2(\Delta\phi/2)\right]$$, with phase and amplitude terms reflecting cavity geometry (Briere et al., 2016).
• Stochastic/Statistical Models: Where film thickness is heterogeneous, the overall $T$ can be computed by averaging over the relevant probability density function, e.g.,

$$T = \int_{t_{\min}}^{t_{\max}} e^{-K \ln 10 \cdot t} p(t) dt$$

with $p(t)$ the PDF for thickness (Amano, 2020).

In composite and polycrystalline systems (e.g., transparent ceramics), multi-mechanism models—such as combined Rayleigh (for pores) and Rayleigh–Gans–Debye (for grains) scattering—are combined additively by exponentials in $$T = (1 - R) \exp[-(y_{\text{pore}} + y_{\text{grain}} + \alpha) t]$$ (Xiong et al., 30 Apr 2025).

3. Measurement Methodologies and Experimental Designs

Thin Films and Coatings:

Direct measurement of $T(\lambda)$ uses spectrophotometers—often with integrating spheres to acquire hemispherical transmittance data that incorporate both direct and scattered light components (Micheli et al., 2019, Jansson et al., 15 Nov 2024).

Photonics, 2D Materials:

Micro-spectroscopy platforms provide high spatial resolution (down to $\sim 1\,\mu$m), enabling $T(\lambda)$ mapping of exfoliated or CVD-grown flakes to extract excitonic resonances and count atomic layers by comparing sample and substrate spectra ($T_{\text{flake}}/T_{\text{substrate}}$) (Frisenda et al., 2016).

Multilayer and Metamaterial Structures:

TMM-based setups or simulation pipelines are used to compute and fit $T$ over spectra and incidence angles, evaluating polarization-resolved $T_{s,p}$, and quantifying high-transmittance "band" or "bubble" formation (Carrera-Escobedo et al., 2016).

Ultradilute Gases:

In TDLAS, $T$ is affected by both classical (optics, pathlength, pressure) and quantum (wavepacket spreading, detector size) effects, with statistical significance in $T$-versus-aperture experiments (Ratajczak, 2020).

Ceramics and Microstructure:

Real in-line $T$ is acquired with tight angular selectivity (e.g., $<0.5^\circ$) to resolve forward transmission and separate scattering losses, further linked to grain and pore statistics (Xiong et al., 30 Apr 2025).

Astronomy:

Absolute photometric calibration is achieved by fitting the entire system transmission $T(\lambda)$ using a combination of observed stellar counts and synthetic photometry generated from calibrator spectra (e.g., Gaia XP), with piecewise and polynomial corrections applied for field non-uniformities (Garrappa et al., 17 Dec 2024).

4. Metric Formulations and Device Figures-of-Merit

Solar Cells (ST-SCs):

Transmittance-based figures-of-merit ($\mathrm{FoM}$) integrate optical and photovoltaic parameters. For BIPV: $$\mathrm{FoM}_{\text{BIPV}} = \frac{n_{\mathrm{Avg}}}{\mathrm{SQL}} \cdot \frac{n_{\min}}{n_{\max}} \cdot \frac{\sqrt{T_{\mathrm{Avg}} T_{550}}}{T^*}$$ where $n$ are PCEs, $T_{\mathrm{Avg}}$ is averaged (visible or broad) transmittance, $T_{550}$ is the 550 nm transmittance, and $T^*$ is an estimated transmittance ceiling based on the absorption limit set by the active layer's bandgap (Kumar et al., 2022).

PV Soiling Losses:

Soiling-induced transmittance degradation is tracked via the hemispherical $T(\lambda)$ of soiled versus clean coupons, with the soiling ratio $r_s$ calculated as $r_s = I_{\text{sc,soil}} / I_{\text{sc,ref}}$, where the currents are integrals over $T(\lambda)$ weighted by spectral response. Strong correlations (R$^2 > 0.98$) exist between single-wavelength or band-averaged $T$ and electrical yield loss, enabling rapid, non-electrical soiling diagnostics (Micheli et al., 2019).

Wavefront Shaping (ENZ Metamaterials):

Transmittance efficiency is defined as the ratio of Poynting vector flow across an output facet to total emitted power. Optimization (e.g., via TMM) can raise this from $2 \times 10^{-5}$ in a single ENZ layer to 15\% with a resonant multilayer air-gap structure (Briere et al., 2016).

Astronomical Calibration:

Transmittance-fitting frameworks reconstruct $T(\lambda)$ per image by minimizing residuals between observed and synthetic stellar photometry, leveraging a high-density calibrator sample (e.g., Gaia) to achieve millimagnitude-level absolute zero-point accuracy (Garrappa et al., 17 Dec 2024).

5. Optimization, Sensitivity, and Trade-offs

Methods for optimizing transmittance-based metrics vary with system:

• Resonant optimization (ENZ/fabry-Perot): Numerical and analytical methods (e.g., adaptation layers, phase-matched air gaps, facet engineering) can yield orders-of-magnitude transmission improvements (Briere et al., 2016).
• Polarization/orientation tuning: In low-symmetry or topologically tunable materials (e.g., strained black phosphorus), light polarization and strain modulate $T$ radically—from near-zero to unity (Nualpijit et al., 2017).
• Microstructure control: In TPCs, reducing pore size (keeping $d \ll \lambda$), minimizing birefringence contrast ($\Delta n$), and controlling grain dimensions can suppress scattering and maximize $T$ (Xiong et al., 30 Apr 2025).
• Environmental adaptation: Real-time beam expansion in atmospheric channels stabilizes $T$ fluctuations at the expense of absolute transmission, requiring joint optimization to maximize quantum key rates or entanglement sharing (Usenko et al., 2020).
• Physical constraints: Path length in absorption spectroscopy ($L$) should be tailored to maximize Fisher information, and the possible quantum advantage in estimation is ultimately capped by the optimization of system parameters (e.g., $L_{\text{opt}}$ for sample absorbance) (Allen et al., 2019).

6. Interpretation and Application Across Domains

Transmittance-based metrics enable:

• Layer counting and thickness identification in 2D materials via excitonic feature tracking in $T(\lambda)$ (Frisenda et al., 2016).
• Detection and discrimination of topological phase transitions in 2D Dirac systems, with extremal transmittance and simultaneous inflection/sign reversal in the Faraday angle marking TPTs (Calixto et al., 2023).
• Comparative evaluation of coated/engineered surfaces: E.g., ITO$+$AR films for ion trap optics are benchmarked by UV transmittance ($T_{370\,\mathrm{nm}} \approx 80\%$), resistance, and scattering (Jansson et al., 15 Nov 2024).
• Accurate model inversion: Inverse transmittance modeling reconstructs underlying film thickness PDFs, or refines microstructural parameters (e.g., grain size, pore density) from experimental $T(\lambda)$ (Amano, 2020, Xiong et al., 30 Apr 2025).
• Photometric precision in astronomy, where image-by-image fit of $T(\lambda)$ with hundreds of calibrator spectra enables high-accuracy cross-telescope and temporal flux calibration (Garrappa et al., 17 Dec 2024).
• Material and appearance capture: SVBSDF estimation, from single-scan images, recovers spatially-varying $T$ maps for photo-realistic rendering, enabling material digitization for virtual manufacturing and design (Rodriguez-Pardo et al., 20 Feb 2025).
• Rapid soiling diagnostics and energy yield forecasting for PV installations (Micheli et al., 2019).

7. Limitations and Future Directions

Key challenges include:

• Ambiguity in parameter extraction: Inverse recovery of microstructural or compositional parameters from $T$ is often ill-posed and may require multi-angular or multi-modal datasets (e.g., axial and non-axial transmittance in chiral thin films) (Sherwin et al., 2019).
• Angular and geometric effects: Real-world devices experience angle-of-incidence sensitivity, non-uniform soiling, and non-ideal optical configurations that can complicate direct transfer of laboratory metrics to field environments (Micheli et al., 2019, Ratajczak, 2020).
• Spectral and environmental variability: Model fidelity depends on wavelength-dependent indices, environmental conditions (humidity, temperature), and quantum-classical crossovers (relevant for low-pressure gases or quantum-enhanced sensing) (Allen et al., 2019, Ratajczak, 2020).
• Physical constraints in optimization: In quantum measurements, inherent trade-offs restrict achievable gains; for photonic structures, fabrication tolerances and material imperfections constrain maximally achievable $T$ (Briere et al., 2016, Allen et al., 2019).

Future work is directed at:

• Integrating hybrid models that blend statistical, transfer-matrix, quantum, and computational (neural) approaches.
• Multi-modal and multi-spectral calibration, particularly where cross-validation with synthetic standards or high-resolution reference data is available.
• Application-driven metric refinement, e.g., to guide PV cleaning schedules, to optimize transparent ceramics, or to ensure photometric consistency in increasingly large astronomical surveys.

Table: Representative Transmittance-Based Metrics in Key Domains

Application	Metric/Formula	Reference
PV Soiling Loss	$r_s = I_{\text{sc,soil}} / I_{\text{sc,ref}}$; AST	(Micheli et al., 2019)
ST Solar Cell FoM (BIPV)	See Section 4 above	(Kumar et al., 2022)
ENZ Wavefront Shaping	Efficiency: $P_{\text{out}}/P_{\text{emitted}}$	(Briere et al., 2016)
Quantum Absorbance Estimation	Classical/Quantum Fisher; $T = \exp(-aL)$	(Allen et al., 2019)
Scattering in TPCs	$$T = (1 - R) \exp[-(y_{\text{pore}} + y_{\text{grain}} + \alpha)t]$$	(Xiong et al., 30 Apr 2025)
Astronomy Photometric Calibration	$T(\lambda)$ fit to Gaia XP synthetic photometry	(Garrappa et al., 17 Dec 2024)
2D Material Layer Counting	Excitonic resonance dips in $T(\lambda)$	(Frisenda et al., 2016)
Anisotropic 2D Materials	$T_\mu = |1 + (\pi\alpha/2)f_{\mu\mu}|^{-2}$	(Nualpijit et al., 2017)

Summary

Transmittance-based metrics are fundamental for the quantitative assessment, optimization, and comparison of the optical properties and energy performance of a wide range of advanced materials and device architectures. Their formulation ranges from simple ratios and averages to highly structured analytical expressions and sophisticated machine learning models. Their application spans experimental diagnostics, inverse parameter inference, optimization of photonic and optoelectronic components, and the design of device-specific figures-of-merit. The continued refinement of transmittance-based metrics, informed by the interplay of measurement capability, physical modeling, and computational advances, remains vital for innovations in optical engineering and materials science.