Unified Temporal Coupled-Mode Framework

• Unified Temporal Coupled‐Mode Framework is a comprehensive formulation extending traditional TCMT to arbitrary modes, ports, and resonator couplings.
• It integrates phenomenological and rigorous Maxwellian/QNM-based theories to capture resonant dynamics, nonlocal scattering, and energy conservation constraints.
• The framework enables advanced photonic applications such as metasurfaces, dynamic modulation, and thermal emitters while maintaining a clear mode–channel structure.

A unified temporal coupled-mode framework denotes the family of formulations that retain the standard resonant input–output architecture of temporal coupled-mode theory (TCMT)—modal amplitudes, external channels, direct/background scattering, and coupling operators—while extending it beyond the original weakly coupled, few-mode, high-$Q$ setting. In the recent literature, this unification proceeds along several axes: arbitrary numbers of modes and ports, arbitrarily coupled resonators, nonlocal metasurfaces, dispersive and spatiotemporally modulated media, and ab initio Maxwellian formulations based on quasinormal modes (QNMs). The resulting picture is not a single equation but an organized hierarchy of models ranging from phenomenological TCMT to exact Maxwell-derived evolution laws (Wang, 2024, Christopoulos et al., 2023, Wu et al., 2024, Zhang et al., 2020).

1. Canonical mode–channel structure

At its core, TCMT reduces a distributed electromagnetic problem to ordinary differential equations for a small set of modal amplitudes. In the basic single-mode normalization, the mode amplitude is chosen so that

$|a|^2 = W_{\mathrm{res}},$

while the incident port amplitude satisfies

$|s^+|^2 = P_{\mathrm{in}}.$

The corresponding cavity evolution and output relations are

$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$

For multiple ports and modes, the same structure becomes

$$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$

with full scattering matrix

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$

This decomposition identifies two coherent pathways: a background/direct pathway through $C$, and a resonant pathway through $D$ and $K$ (Christopoulos et al., 2023, Wang, 2024).

This canonical structure is broad enough to encompass standing-wave cavities, ring resonators, metasurfaces, and free-space scatterers. In the tutorial treatment of nonlinear resonant photonics, the same backbone is used as the starting point for self- and cross-phase modulation, saturable absorption, frequency generation, gain, carrier effects, and thermal dynamics, with the resonator physics encoded as perturbative shifts of resonance frequency and loss rate (Christopoulos et al., 2023).

The canonical few-mode picture also supports coupled-resonator supermodes. For instance, for two interacting modes the amplitudes obey

$$\frac{d\tilde a_1}{dt}=-j(\omega-\omega_1)\tilde a_1+\mu_{c,12}\tilde a_2,\qquad \frac{d\tilde a_2}{dt}=-j(\omega-\omega_2)\tilde a_2+\mu_{c,21}\tilde a_1,$$

with the energy-conservation constraint

$|a|^2 = W_{\mathrm{res}},$0

This already foreshadows the general framework: a linear resonant subsystem, explicit ports, and constraints imposed by conservation laws (Christopoulos et al., 2023).

2. Conservation laws, reciprocity, and symmetry constraints

The unification of TCMT is inseparable from the constraints imposed by energy conservation and time-reversal symmetry. In the arbitrary-$|a|^2 = W_{\mathrm{res}},$1 formulation with $|a|^2 = W_{\mathrm{res}},$2 localized modes and $|a|^2 = W_{\mathrm{res}},$3 ports,

$|a|^2 = W_{\mathrm{res}},$4

energy conservation gives

$|a|^2 = W_{\mathrm{res}},$5

With time-reversal symmetry, the additional reciprocity constraints are

$|a|^2 = W_{\mathrm{res}},$6

and hence

$|a|^2 = W_{\mathrm{res}},$7

In this formulation, reciprocity acts mode-by-mode. If the coupling vector of a mode vanishes, the mode is a hidden mode; for any non-hidden mode, both coupling magnitudes and phases are restricted by the background matrix $|a|^2 = W_{\mathrm{res}},$8 (Wang, 2024).

A particularly strong result concerns the coupling-phase geometry. Writing $|a|^2 = W_{\mathrm{res}},$9, the reciprocity condition becomes a singularity condition on

$|s^+|^2 = P_{\mathrm{in}}.$0

so non-hidden modes require $|s^+|^2 = P_{\mathrm{in}}.$1. In double-port systems, this yields tight closed-form bounds on both coupling phases and strengths. The same analysis gives a geometric interpretation in terms of generalized reflections: time-reversal symmetry forces $|s^+|^2 = P_{\mathrm{in}}.$2 projected generalized reflections to cancel out (Wang, 2024).

Recent high-$|s^+|^2 = P_{\mathrm{in}}.$3 metasurface theory generalizes these constraints further by allowing asymmetry between coupling and decoupling coefficients while preserving basis covariance. In that invariant form,

$|s^+|^2 = P_{\mathrm{in}}.$4

together with

$|s^+|^2 = P_{\mathrm{in}}.$5

This extension is essential for asymmetric dielectric metasurfaces, unidirectionally guided resonances, and the dual unidirectional coupled resonant states introduced in that context (Maksimov et al., 1 May 2025).

3. From phenomenology to Maxwellian and QNM-based formulations

Classical TCMT is widely successful, but several recent works emphasize that it is fundamentally phenomenological. In the standard resonator form,

$|s^+|^2 = P_{\mathrm{in}}.$6

the quantity $|s^+|^2 = P_{\mathrm{in}}.$7 is interpreted as stored energy, $|s^+|^2 = P_{\mathrm{in}}.$8 is a complex resonance frequency, $|s^+|^2 = P_{\mathrm{in}}.$9 is the input-channel vector, and $$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$0 is the coupling vector. The critique is not that this equation is useless, but that it is not an exact Maxwellian law (Wu et al., 2024).

The most direct Maxwell-based replacement is the exact Maxwell evolution (EME) equation derived from QNM theory. Starting from source-free Maxwell equations for QNMs with outgoing-wave conditions and expanding the scattered field as

$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$1

one obtains, for nondispersive materials,

$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$2

Here the compact notation means

$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$3

Two physical differences from classical CMT are emphasized. First, the excitation is nonlocal: it is a spatial overlap over the resonator volume, not a field sampled at a single coupling point. Second, the forcing can be decomposed into a CMT-like part plus a derivative-related part; in the authors’ words, the EME excitation is “not directly proportional to the driving field, but to its temporal derivative.” The same work also stresses that the QNM amplitude does not admit the usual stored-energy interpretation because QNMs are non-Hermitian and diverge outside the resonator (Wu et al., 2024).

A broader QNM-based generalization is Quasinormal Coupled Mode Theory (QCMT). QCMT keeps the CMT architecture—modes, channels, background scattering, and scattering matrices—but derives it ab initio from Maxwell’s equations. Its couplings are frequency dependent, it contains an additional Born-like direct-scattering term with no standard TCMT counterpart, and it does not assume isolated high-$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$4 resonances or weak coupling. In time domain it becomes a convolutional theory, and ordinary TCMT is recovered only when the couplings and direct term can be approximated as delta-like kernels (Zhang et al., 2020).

Black-box Coupled-Mode Theory (BBCMT) pushes the same agenda into open, lossy, dispersive resonators. Its central object is a resonator Hamiltonian matrix $$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$5, obtained from QNM fields and rigorous normalization via Poynting’s theorem and conjugated reciprocity. BBCMT also incorporates non-resonant scattering and absorption through a background perturbation decomposition,

$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$6

and allows subcomponents to be treated as black boxes specified only by their input–output transfer characteristics (Tali et al., 2020).

4. Main generalizations of the framework

The unifying role of TCMT is most visible in the diversity of extensions built on the same mode–channel template.

Extension axis	Representative object	Representative papers
Arbitrarily coupled thermal emitters	$$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$7-resonator TCMT with complex coupling matrix $$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$8	(Huang et al., 2022)
Nonlocal metasurfaces	Spatially resolved amplitude $$\frac{da}{dt}=j\omega_0 a-\gamma a+\mu_e s^+, \qquad s^- = c\, s^+ + d_e\, a.$$9 and kernel $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$0	(Overvig et al., 2023)
Dynamic modulation in dispersive media	Spatiotemporal phase matching with $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$1 and $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$2	(Dana et al., 2014)
Pulses under arbitrary perturbations	Exact time-domain coupled equations for $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$3	(Sivan et al., 2016)
Temporal photonic interfaces	Floquet-sheet TCMT with photon-number correction $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$4	(Wang et al., 22 Feb 2026)

For thermal emission, the generalized $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$5-resonator theory writes

$$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$6

where each resonator has its own resonance frequency, intrinsic loss, external radiative loss, and pairwise complex couplings $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$7. The emitted spectral power density is then expressed in closed form through the same matrices. This formulation was introduced as a unified TCMT for thermal emission from multiple arbitrarily coupled resonators, including different resonance frequencies, different losses, and both real and imaginary coupling components (Huang et al., 2022).

For diffractive nonlocal metasurfaces, spatio-temporal coupled mode theory (STCMT) promotes the lumped amplitude $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$8 to a field $$\dot{\mathbf a} = (j\Omega - \Gamma)\mathbf a + K^T \mathbf s_+, \qquad \mathbf s_- = C\mathbf s_+ + D\mathbf a,$$9. Under the nonlocal metasurface specialization,

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$0

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$1

This leads to a Green-function description and a nonlocal scattering kernel $S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$2. The characteristic nonlocality length

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$3

sets the in-plane propagation distance of the resonance before radiative leakage, and the formalism reduces to ordinary TCMT when the device is effectively local or spatially invariant (Overvig et al., 2023).

Dynamic modulation in dispersive media introduces a second layer of unification: spatial and temporal phase matching appear on equal footing. In the spatiotemporally modulated formalism,

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$4

with

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$5

This framework covers purely spatial coupling, purely temporal coupling, and fully spatiotemporal quasi-phase matching in a single notation (Dana et al., 2014).

A related exact pulse formalism derives coupled-mode equations for broadband pulses in dispersive media with arbitrary spatio-temporal perturbations. Its key point is that the correct pulsed ansatz must preserve the full frequency dependence of both $S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$6 and the mode profiles, otherwise spurious coupling appears even in the unperturbed system. The resulting equations were validated numerically against FDTD and used to describe pulse broadening, shortening, and more complex shaping (Sivan et al., 2016).

Time-varying photonic interfaces provide another specialized extension. For a single Floquet-sheet resonator governed by time-modulated conductivities, the resonator obeys

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$7

Because the process is frequency conversion, the coefficients are fixed by photon-number conservation rather than ordinary energy conservation, yielding

$S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$8

This produces a distinctive frequency-ratio correction absent from static TCMT and establishes a direct bridge to the Floquet transfer-matrix method (Wang et al., 22 Feb 2026).

5. Approximation hierarchy, dark states, and known failure regimes

A unified framework also requires an explicit account of when ordinary TCMT is valid and when it fails. In the Maxwellian reinterpretation of resonator dynamics, classical CMT is recovered only under two approximations: the incident wavepacket must have a slowly varying envelope with carrier frequency close to resonance, and the overlap-based driving must be approximated locally at a coupling center $S = C + D[j(\omega-\Omega)+\Gamma]^{-1}K^T.$9. Under those assumptions the overlap forcing reduces to a CMT-like coupling constant. For pure harmonic excitation, the same work notes that the resulting coupling formula remains accurate regardless of resonator size, which helps explain the empirical success of TCMT for monochromatic driving even in large racetrack resonators (Wu et al., 2024).

The same study also delimits the failure regimes. Deviations become important for short pulses, off-resonance pulses, and resonators comparable to or larger than the wavelength. In the reported calculations, a subwavelength cavity with a gentle envelope is described well by both CMT and EME; a steeper pulse can still be fitted successfully by refitting the coupling coefficient; but for a cavity of order one wavelength even rescaling cannot reproduce the EME response, including a double-peak structure. The conclusion is explicit: CMT can fail for “short and off-resonance pulses” and for “resonators of sizes comparable to or greater than the wavelength,” although fitted coefficients often preserve excellent agreement in many practical settings (Wu et al., 2024).

A different limitation appears in quasi-dark states. In a three-microring system composed of one outer ring coupled to a bus waveguide and two identical embedded rings, standard TCMT predicts that the central supermode is a strict dark state because it has zero amplitude in the outer ring and the bus couples only through that ring. Experiment, transfer-matrix calculations, and FDTD instead show a high-$C$0 quasi-dark state with weak but finite excitation. The identified cause is the lumped-element nature of standard TCMT: it omits non-resonant circulating power in the outer ring. The proposed correction adds indirect couplings,

$C$1

together with off-diagonal cross-decay terms in $C$2, thereby restoring agreement in both steady-state and transient dynamics (Souza et al., 2016).

Standard phenomenological CMT also breaks down for classes of scatterers where weak coupling, high $C$3, and frequency-independent couplings are untenable. QCMT identifies these classes explicitly: lossy or non-Hermitian materials, plasmonics, metasurfaces and metagratings with many strongly coupled resonances, random media, high-radiative-loss resonators, and even relatively simple multi-resonant objects such as a Mie sphere. In such cases, modal overlap, strong radiation leakage, dispersion, and direct scattering cannot be ignored without losing quantitative and sometimes qualitative accuracy (Zhang et al., 2020).

6. Applications, scope, and emerging directions

Within photonics, unified TCMT frameworks are used for thermal emitters, nonlocal metasurfaces, wavefront shaping, thermal focusing, resonant filters, optical modulation, integrated photonic circuits, plasmonic resonators, metasurfaces, metagratings, photovoltaic absorbers, transparent displays, and random or disordered media (Huang et al., 2022, Overvig et al., 2023, Souza et al., 2016, Zhang et al., 2020). In several of these cases, the value of the framework is not only computational economy relative to full-wave simulation, but also the explicit separation of resonant and nonresonant pathways, which gives a compact interpretation of Fano interference, subradiance, superradiance, symmetry-protected bound states in the continuum, and directional resonant coupling (Maksimov et al., 1 May 2025).

The framework is also explicitly presented as extensible beyond electromagnetism. The Maxwellian EME work states that the regularized QNM formalism provides a unified mathematical framework anticipated to be applicable to all electromagnetic resonator geometries, and that the same theoretical approach can be extended to other wave physics. Likewise, the exact pulse theory under arbitrary spatio-temporal perturbations is described as valid across the electromagnetic spectrum and directly adaptable to other wave systems (Wu et al., 2024, Sivan et al., 2016).

A notable current direction is dynamic gain and self-time-modulated resonant systems. In exceptional-point lasers, the gain medium can support an oscillating population inversion above threshold, which couples different frequencies and therefore cannot be captured by conventional TCMT based on static saturable gain. The QNM-based perturbative analysis developed for this regime interprets dynamic inversion as repeated resonant excitation of passive modes, producing a compact model for self-modulated lasing dynamics, frequency-comb generation, time-varying scattering, and non-reciprocal transmission (He et al., 2024).

Taken together, these developments suggest a precise synthesis. Ordinary TCMT remains the compact phenomenological core; exact Maxwellian and QNM-based theories provide rigorous reference formulations; and spatial, temporal, thermal, Floquet, and nonlinear extensions enlarge the admissible class of systems without discarding the underlying mode–channel language. In that sense, the unified temporal coupled-mode framework is best understood as an approximation hierarchy with a common architecture rather than as a single closed formalism (Wu et al., 2024, Zhang et al., 2020, Christopoulos et al., 2023).