TMMax: Thin-Film & ML Simulation

• TMMax is a polysemous term defining different domain-specific methods in photonics, video analysis, and graphical models dependent on local context.
• In photonics, TMMax is a high-performance, fully vectorized JAX library for simulating multilayer thin films via the transfer matrix method, achieving speedups up to 700×.
• In video and graphical applications, TMMax refers to parameter-free temporal max pooling and truncated max-of-convex models that optimize efficiency without complex computations.

TMMax is not a single universally fixed acronym across the literature represented here. It appears explicitly as the name of a JAX-based Python library for modeling multilayer optical thin-film structures with the transfer matrix method, and it also appears as shorthand or analogy in several other research contexts, including TemporalMaxer’s temporal max-pooling block for temporal action localization, truncated max-of-convex models in high-order Markov random fields, and several “TMMax-type” constructions in stochastic-process and multiscale laser-matter settings (Danis et al., 15 Jul 2025, Tang et al., 2023, Pansari et al., 2015, Hees et al., 2017, Benhayoun et al., 28 Jan 2026). In the most explicit naming usage, TMMax denotes a high-performance, fully vectorized, differentiable thin-film simulation library implemented in JAX (Danis et al., 15 Jul 2025).

1. Nomenclature and scope

The designation “TMMax” is domain-dependent rather than standardized. One explicit use is the library name “TMMax: High-performance modeling of multilayer thin-film structures using transfer matrix method with JAX” (Danis et al., 15 Jul 2025). In another usage, the TemporalMaxer paper states that the authors never explicitly define an acronym “TMMax”; in that context, it is best understood as shorthand for the TemporalMaxer temporal max-pooling block, namely a parameter-free temporal max pooling used as the only temporal context modeling operation in the backbone (Tang et al., 2023). In computer vision, “Truncated Max-of-Convex Models” are abbreviated TMCM and are described as “sometimes called TMMax” in the supplied exposition (Pansari et al., 2015). Other supplied materials use “TMMax-type” only analogically, not as the formal title of the method (Hees et al., 2017, Benhayoun et al., 28 Jan 2026).

Usage	Domain	Meaning
TMMax	Photonics / computational optics	JAX-based transfer-matrix thin-film library
TMMax	Video understanding	TemporalMaxer temporal max-pooling block
TMCM, sometimes called TMMax	Computer vision / graphical models	Truncated Max-of-Convex Models
“TMMax-type”	Stochastic processes	CTRM/OCTRM maxima framework analogy
“TMMax-type”	Ultrafast laser-matter simulation	Maxwell–TTM–MD coupling analogy

This polysemy creates a recurrent source of confusion. A common misconception is that TMMax names a single method family spanning these domains. The supplied literature does not support that reading. Instead, it supports a narrower conclusion: the string “TMMax” is reused across otherwise unrelated technical contexts, and interpretation must be taken from local disciplinary usage.

2. TMMax as a thin-film simulation library

In the thin-film literature, TMMax is a high-performance Python library for simulating multilayer optical thin films with the transfer matrix method, implemented on top of JAX (Danis et al., 15 Jul 2025). Its stated purpose is to model structures such as distributed Bragg reflectors, anti-reflection coatings, spectral filters, and decorative coatings while addressing three limitations of traditional TMM implementations: scalar treatment of wavelength and angle, lack of automatic differentiation, and limited infrastructure for material data and analysis.

TMMax uses the standard Abeles form of the transfer matrix method for planar, homogeneous, isotropic layers. For a stack of $N$ media, including incident and exit semi-infinite media, the forward and backward field amplitudes satisfy

$$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$

with system matrix

$$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$

Each layer matrix is decomposed into interface and propagation terms,

$$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$

where the phase is

$$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$

From the total matrix, TMMax follows the conventional Abeles approach to recover complex reflection and transmission amplitudes $r$ and $t$, and then computes

$$R = |r|^2, \qquad T = \Re\left(\frac{n_{N-1}\cos\theta_{N-1}}{n_0 \cos\theta_0}\right) |t|^2,$$

with absorbance $A = 1 - R - T$ in the stated setting (Danis et al., 15 Jul 2025). The implementation supports both TE and TM polarization and uses complex refractive indices $n_i(\lambda)$, so absorption is handled directly through $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$0.

The library description also emphasizes infrastructure beyond the core TMM. TMMax includes a curated material database of approximately 30 widely used thin-film materials, stored as ready-to-use .npy files containing wavelength, $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$1, and $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$2, together with analysis tools for reflectance, transmittance, absorbance, color prediction, and sensitivity analysis (Danis et al., 15 Jul 2025).

3. Vectorization, JAX execution model, and differentiability

A central design feature of TMMax is full vectorization over wavelength and angle of incidence (Danis et al., 15 Jul 2025). Traditional NumPy-based TMM codes are described as scalar in wavelength and angle, so dense $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$3-$$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$4 sweeps require nested Python loops. TMMax instead treats wavelengths and angles as vectorized axes and expresses all intermediate quantities—internal angles, phase delays, Fresnel coefficients, and batched $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$5 transfer matrices—as array operations.

The implementation relies on three JAX mechanisms. First, jax.jit just-in-time compiles the TMM computation into an XLA kernel. Second, jax.lax.scan replaces explicit Python loops over layers in the cumulative matrix product. Third, broadcasting over batch dimensions yields tensorized computations across wavelength, angle, and layer axes. The paper describes this organization as especially well matched to TMM because the method is fundamentally built from matrix multiplications and linear transforms (Danis et al., 15 Jul 2025).

The code base is modular. The supplied description assigns internal-angle computation to angle.py, wavevector and phase handling to wavevector.py, Fresnel/interface terms to fresnel.py and reflect_transmit.py, chained matrix multiplication to cascaded_matmul.py, material loading and interpolation to data.py, and visualization helpers to plot.py (Danis et al., 15 Jul 2025). The library is written in functional style rather than with internal classes, although users may wrap functions in their own object-oriented interfaces.

Automatic differentiation is a second major consequence of the JAX implementation. Because the full TMM pipeline is differentiable, TMMax permits gradients such as

$$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$6

and more generally $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$7 for a user-defined loss over layer thicknesses or other design parameters (Danis et al., 15 Jul 2025). The supplied example uses jax.grad on a loss that compares a computed reflectance spectrum against a target. This suggests direct compatibility with gradient-based inverse design and with JAX-native optimizers such as Optax.

4. Benchmarks, application domains, and operational scope

The TMMax paper reports benchmarking against Steven Byrnes’ NumPy-based tmm library under identical stack, wavelength, and angle inputs (Danis et al., 15 Jul 2025). In one benchmark, 20 multilayer structures with layer counts from 2 to 400 were generated, with layer materials randomly chosen from 7 materials and thicknesses randomly drawn between 100 nm and 500 nm; the wavelength array had 20 points from 500 to 1000 nm and the angle array had 20 points from $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$8 to $$\begin{bmatrix} E_0^{+} \ E_0^{-} \end{bmatrix} = \mathbf{M} \begin{bmatrix} E_{N-1}^{+} \ E_{N-1}^{-} \end{bmatrix},$$9. On a single Intel Core i9 core, runtime for the NumPy baseline grew steeply with layer count, whereas TMMax runtime grew slowly and was nearly constant at approximately 1.0–1.2 s for 2, 22, and 42 layers. The reported speedup ranged from approximately $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$0 at 2 layers to approximately $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$1 at 400 layers (Danis et al., 15 Jul 2025).

A second benchmark varied wavelength and angle array sizes from $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$2 to $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$3, for fixed 8-layer and 80-layer stacks (Danis et al., 15 Jul 2025). For the 8-layer case, the NumPy baseline reached about 138 s at $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$4, while TMMax remained below 3 s across the entire grid; at the smallest $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$5 grid, however, the baseline was faster, at about 0.1 s versus about 0.6 s for TMMax, a difference attributed to JAX compilation and dispatch overhead. For the 80-layer stack, the baseline exceeded 760 s for large grids, whereas TMMax remained below 8 s.

These benchmarks define the library’s intended operating regime. TMMax is presented as especially advantageous for large-scale thin-film simulation: many layers, dense wavelength-angle grids, or optimization-heavy workflows (Danis et al., 15 Jul 2025). The cited application classes include distributed Bragg reflectors, anti-reflection coatings, bandpass and bandstop filters, decorative coatings, color prediction, and fabrication-tolerance analysis. The analysis module can quantify how percentage deviations in layer thickness affect transmittance, reflectance, and resultant color, and it can convert reflectance spectra to perceived color using the Python ColorPy library (Danis et al., 15 Jul 2025).

The same source also states clear scope limitations. TMMax is restricted to planar, laterally infinite, homogeneous layers; it does not treat gratings, metasurfaces, or nonplanar geometries. Material models are isotropic and scalar in $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$6 and $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$7; anisotropy and birefringence are not mentioned in the current version. Standard $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$8 transfer-matrix multiplication may become ill-conditioned for extremely thick or highly absorbing stacks, and the paper does not describe S-matrix or logarithmic stabilization schemes (Danis et al., 15 Jul 2025).

5. TMMax in machine learning and graphical models

In temporal action localization, the string “TMMax” is used informally for TemporalMaxer’s temporal max-pooling block rather than as an author-defined acronym (Tang et al., 2023). TemporalMaxer is a temporal action localization backbone built on the claim that extracted clip-level features from pre-trained 3D CNNs are already sufficiently informative that heavy long-term temporal context modeling is not required. In this setting, TMMax denotes a parameter-free one-dimensional temporal max pooling with kernel size $$\mathbf{M} = \mathbf{I}_0 \cdot \prod_{i=1}^{N-2} \mathbf{M}_i.$$9 and stride $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$0, used as the sole temporal context modeling operation in a multi-scale temporal pyramid (Tang et al., 2023). The block is defined by local channelwise maximization over a temporal neighborhood and has no learnable weights, no FFN, no attention, and no normalization inside the temporal context module itself.

This usage is significant because it is tied to an explicit empirical argument against the necessity of Transformer-style long-range self-attention for TAL when strong 3D CNN features are already available (Tang et al., 2023). On THUMOS14, the paper reports Avg mAP $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$1 for TemporalMaxer versus $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$2 for ActionFormer over IoU $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$3–$$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$4, alongside GMACs $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$5 versus $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$6, parameters $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$7M versus $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$8M, and backbone time $$\mathbf{M}_i = \mathbf{I}_i \mathbf{P}_i = \begin{bmatrix} \alpha_{i,i+1} & \gamma_{i,i+1} \ \gamma_{i,i+1} & \alpha_{i,i+1} \end{bmatrix} \begin{bmatrix} e^{-j\delta_i} & 0 \ 0 & e^{j\delta_i} \end{bmatrix},$$9 ms versus $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$0 ms in the cited ablation. The same source reports improvements over ActionFormer on EPIC-Kitchens 100, MUSES, and MultiTHUMOS (Tang et al., 2023). In that literature, therefore, TMMax signifies a specific nonparametric pooling operator embedded in a TAL architecture, not the optical thin-film package.

A separate use appears in computer vision under the model family “Truncated Max-of-Convex Models” (TMCM), described in the supplied exposition as “sometimes called TMMax” (Pansari et al., 2015). TMCM generalizes pairwise truncated convex models to high-order cliques. Its energy combines arbitrary unary potentials with clique potentials defined as the weighted sum of the $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$1 largest truncated convex distances over disjoint label pairs in a clique: $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$2 For clique $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$3, after sorting labels, the clique potential is

$$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$4

The model reduces to standard pairwise TCM when clique size is $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$5 and $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$6, and it recovers robust $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$7-type behavior for $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$8 in the supplied account (Pansari et al., 2015). Inference uses a range expansion algorithm in which each move is solved by an s-t min-cut. The stated multiplicative approximation bounds are $$\delta_i = \frac{2\pi}{\lambda} n_i d_i \cos\theta_i.$$9 for truncated max-of-linear with $r$0, $r$1 for the linear case with general $r$2, and $r$3 for truncated max-of-quadratic with $r$4, where $r$5 is the largest clique size (Pansari et al., 2015).

These machine-learning uses share only the lexical element “TMMax.” One concerns parameter-free temporal max pooling in sequence modeling; the other concerns high-order energy design in MRF/CRF optimization. Neither is methodologically related to the JAX thin-film library.

6. Analogical and extended “TMMax-type” usages

The supplied materials also use “TMMax-type” analogically in two additional areas. In stochastic-process theory, “Coupled Continuous Time Random Maxima” develops CTRM and OCTRM processes for maxima observed at random heavy-tailed times (Hees et al., 2017). The exposition explicitly labels this a “TMMax-type” framework, but the paper’s own terminology is CTRM/OCTRM rather than TMMax. The central objects are

$r$6

with scaling limits

$r$7

Here $r$8 is an extremal process and $r$9 is the inverse of a $t$0-stable subordinator (Hees et al., 2017). In that interpretation, “TMMax-type” refers to continuous-time maxima under random waiting times and possible coupling between waiting times and marks.

In ultrafast laser-matter modeling, the Maxwell–Two-Temperature Model–Molecular Dynamics framework is described in the supplied exposition as a “TMMax-type” approach because it couples Maxwell’s equations with TTM and MD in a closed feedback loop (Benhayoun et al., 28 Jan 2026). The formal name in the paper is M-TTM-MD, not TMMax. The framework solves Maxwell’s equations via FDTD, couples the absorbed power density

$t$1

into a TTM electron energy balance, and transfers electron-phonon energy to atoms through an MD coupling term in Newton’s equations (Benhayoun et al., 28 Jan 2026). In that literature, the analogy emphasizes “Maxwell + TTM + structural dynamics,” not a standardized acronym.

These analogical uses reinforce the broader terminological point. “TMMax” can function as a local shorthand for “tail maximum,” “temporal max,” or “Maxwell-plus-TTM,” but those are interpretive overlays rather than a single cross-disciplinary formalism.

7. Conceptual synthesis

Across the supplied literature, TMMax is best understood as a polysemous technical label whose meaning is fixed by local context rather than by any universal definition. In photonics, it is the explicit name of a high-performance JAX library for fully vectorized, differentiable transfer-matrix simulation of multilayer thin films, with material-database support, analysis tooling, and reported $t$2–$t$3 speedups over a baseline NumPy implementation in large-scale regimes (Danis et al., 15 Jul 2025). In temporal action localization, it denotes TemporalMaxer’s parameter-free temporal max-pooling block, used to replace heavier temporal context modeling and reported to improve both efficiency and mAP over transformer-based alternatives on several TAL benchmarks (Tang et al., 2023). In high-order graphical modeling, it designates, or is at least associated with, truncated max-of-convex clique energies optimized by range expansion and s-t min-cut (Pansari et al., 2015). In other supplied texts it appears only as an analogy for random-maxima or Maxwell–TTM couplings (Hees et al., 2017, Benhayoun et al., 28 Jan 2026).

A plausible implication is that the persistence of the string “TMMax” across such distinct areas reflects a recurring technical motif—maximization under structure, whether over temporal neighborhoods, label spreads, tail probabilities, or field-coupled dynamics—rather than a shared methodological lineage. For precise usage, however, the only reliable rule is contextual: in contemporary arXiv usage represented here, “TMMax” names different objects in different fields, and any encyclopedia treatment must preserve that distinction.