Light Propagation Coupling

• Light propagation coupling is a phenomenon where optical channels and modes interact via overlap integrals, effective Hamiltonians, and coupling operators.
• It encompasses diverse mechanisms such as evanescent tunneling, dissipative exchanges, and nonlinear mixing, which are analyzed using transmission matrices and coupled-mode theories.
• Applications include impedance matching in nanocouplers, robust multimode fiber control, and engineered routing in photonic lattices for efficient optical signal transfer.

Light propagation coupling denotes a class of optical transport phenomena in which propagation is governed by explicit coupling between channels, modes, resonators, interfaces, or internal material degrees of freedom. In the literature, this encompasses impedance-matched transfer between conventional and topological waveguides, transmission-matrix control in strongly coupled multimode fibers, dissipative exchange through common reservoirs, nonlinear and nonreciprocal coupling in waveguides, atom-light and magnon-photon interactions in dispersive media, and hopping in photonic lattices and cavity superlattices. Across these settings, propagation is characterized not only by local refractive index and geometry, but by coupling operators, overlap integrals, effective Hamiltonians, or effective metrics that determine transmission, group delay, bandwidth, directionality, and robustness (Yoshimi et al., 2023, Xiong et al., 2016, Mukherjee et al., 2017).

1. Formal descriptions of coupled propagation

Several mathematical descriptions recur across the field.

Formalism	Representative relation	Role
Transmission-matrix / time-delay	$E_{\rm out},m}(\omega)=\sum_n T_{mn}(\omega)E_{\rm in},n}(\omega)$, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$	Mode mixing, eigenchannels, principal modes
Coupled-mode / tight-binding	$i\,dA_n/dz+\sum_m C_{nm}A_m=0$, $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$	Waveguide arrays, state transfer, lattice transport
Dissipative coupling	$$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$	Diffusive-yet-coherent transport and steady-state engineering
Effective-metric optics	$g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$	Birefringence, deflection, anisotropic propagation

In multimode fibers, the transmission matrix $T(\omega)$ provides a complete linear mapping between guided input and output channels, while the Wigner–Smith operator $Q$ yields delay eigenstates, the principal modes (Xiong et al., 2016, Cao et al., 2023). In waveguide arrays and photonic lattices, nearest-neighbor couplings $g_j$ or $C_{nm}$ determine diffraction, Rabi-like transfer, and perfect state transfer (Rodríguez-Lara, 2013, Rojas et al., 2014). In dissipative photonics, the coupling is not Hamiltonian hopping but Lindblad exchange through a common reservoir, producing a discrete diffusion equation for complex amplitudes rather than probabilities (Mukherjee et al., 2017). In nonlinear electrodynamics and rotating media, propagation follows null geodesics of an effective optical metric rather than those of the background geometry (Guzman-Herrera et al., 2023, Mieling, 2019).

A common implication is that “coupling” is not a single mechanism. Depending on the platform, it may mean modal overlap, evanescent tunneling, engineered loss, nonlinear wave mixing, spin-orbit locking, or matter-assisted frequency-dependent hybridization.

2. Guided-wave interfaces and nanocoupling

A central problem is coupling between strongly mismatched modes. In valley photonic crystals, efficient transfer from a conventional Si-wire waveguide into a topological slow-light edge state was realized by inserting a short “filled-hole” taper at the bearded interface between two VPhCs of opposite valley Chern number (Yoshimi et al., 2023). The platform uses a $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$0-thick Si slab, a honeycomb lattice of equilateral triangles with period $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$1, and a wire-waveguide width $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$2. The optimum coupler length is $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$3 ($Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$4). In the moderate slow-light window $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$5–30, corresponding to wavelengths $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$6–$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$7, numerical simulation gave an average insertion loss of $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$8, while experiment gave $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$9. At $i\,dA_n/dz+\sum_m C_{nm}A_m=0$0, direct butt coupling with $i\,dA_n/dz+\sum_m C_{nm}A_m=0$1 gave $i\,dA_n/dz+\sum_m C_{nm}A_m=0$2 ($i\,dA_n/dz+\sum_m C_{nm}A_m=0$3), whereas $i\,dA_n/dz+\sum_m C_{nm}A_m=0$4 raised $i\,dA_n/dz+\sum_m C_{nm}A_m=0$5 ($i\,dA_n/dz+\sum_m C_{nm}A_m=0$6). The underlying mechanism is mode matching: the filled-hole region creates an intermediate Bloch mode that overlaps both the wire mode and the slow-light edge mode, but excessive taper length increases out-of-plane leakage because the in-gap Bloch mode lies above the light line (Yoshimi et al., 2023).

A distinct solution is the bent metal-clad “L-coupler,” designed for fiber-to-waveguide and 3D chip-to-chip coupling (Lu et al., 2016). Its key element is a $i\,dA_n/dz+\sum_m C_{nm}A_m=0$7 bent, metal-clad dielectric channel with a horn input port. In 3D FDTD, the TE coupling efficiency peaks at $i\,dA_n/dz+\sum_m C_{nm}A_m=0$8 at $i\,dA_n/dz+\sum_m C_{nm}A_m=0$9, remains at or above $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$0 over $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$1–$$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$2, and yields an insertion loss of about $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$3 with return loss about $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$4. The coupler is strongly polarization-dependent: TM efficiency is $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$5–$$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$6. The footprint is less than $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$7, and back-to-back 3D interconnects reach $$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$8–$$\hat H=\sum_j \omega_j\hat n_j+\sum_j g_j(\hat a_j^\dagger\hat a_{j+1}+\hat a_{j+1}^\dagger\hat a_j)$$9, or about $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$0 when Si tapers are inserted into metal-clad glass guides (Lu et al., 2016).

At a broader level, nanocoupling strategies fall into four principal categories: adiabatic tapered waveguides, direct evanescent or directional couplers, lens-based couplers, and scatterer-based couplers such as antennas and gratings (Andryieuski et al., 2012). The comparison is fundamentally a trade-off between efficiency, footprint, bandwidth, and fabrication complexity. The review reports, for example, dielectric tapers with $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$1 at $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$2, directional couplers around $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$3, resonant-stub end-fire coupling around $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$4 but with $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$5, and vertical grating couplers with $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$6–$$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$7 efficiency over $$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$8–$$d\rho/dt=\sum_{j=1}^N \gamma_j D[a_j-a_{j+1}]\,\rho$$9 (Andryieuski et al., 2012).

Polarization can itself be a coupling variable. In elliptical femtosecond-laser-written waveguides, the asymmetry of the spatial transverse profiles of linearly polarized modes produces polarization-dependent coupling coefficients $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$0 and $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$1, with an effective coupling $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$2 (Rojas et al., 2014). This was linked to a compact polarizing beam splitter. For semi-axes $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$3, $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$4, and spacing $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$5, the reported coefficients are $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$6 and $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$7, giving $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$8 and a splitting length of about $g_{\rm eff}^{\mu\nu}k_\mu k_\nu=0$9 (Rojas et al., 2014).

3. Multimode fibers, strong mode coupling, and principal modes

In multimode fibers, coupling is often represented as a high-dimensional linear map between modal bases. Perturbations such as bending, stress, temperature, and index inhomogeneity generate off-diagonal mode-coupling coefficients $T(\omega)$0, and the full transmission matrix $T(\omega)$1 then relates input and output modal amplitudes (Cao et al., 2023). This framework generalizes naturally to temporal, spectral, and polarization coupling.

A particularly important construct is the Wigner–Smith time-delay matrix. In the absence of back-reflection, it takes the form

$T(\omega)$2

and its eigenvectors are the principal modes (Xiong et al., 2016). For a principal mode at $T(\omega)$3, the output pattern is frequency-independent to first order under small detuning, yielding a plateau in the spectral autocorrelation $T(\omega)$4 with $T(\omega)$5, in contrast to the generic $T(\omega)$6 behavior of arbitrary inputs (Xiong et al., 2016).

The experimental demonstration used a $T(\omega)$7 step-index fiber with $T(\omega)$8 core diameter, $T(\omega)$9, and about $Q$0 supported modes, with strong coupling introduced by stress clamps (Xiong et al., 2016). The complex transmission matrix was measured interferometrically with a tunable CW laser and a phase-only SLM. Launching the measured principal-mode eigenvectors produced output speckle patterns that remained fixed over a finite bandwidth. For short pulses, random spatial inputs suffered severe modal dispersion, whereas principal-mode excitation gave identical temporal traces in all output channels up to a constant factor, so that the full output reproduced the input pulse shape shifted by the corresponding delay time $Q$1, with negligible broadening (Xiong et al., 2016).

The broader multimode-fiber control literature extends this approach to imaging, spectroscopy, endoscopy, optical trapping, microfabrication, and optical computing (Cao et al., 2023). Transmission matrices may be measured by off-axis holography, phase stepping, co-propagating references, or phase retrieval; inversion can be performed by least-squares regularization or singular-value decomposition. Reported performance metrics include focusing enhancement $Q$2, power ratio $Q$3, and fidelity $Q$4. The same review reports multimode fibers used as spectrometers with $Q$5 at $Q$6 length and $Q$7 at $Q$8 (Cao et al., 2023).

These results establish a key distinction: strong mode coupling does not preclude control. When the full operator $Q$9 is known, random-looking propagation can be recast into eigenchannels with predictable spatio-temporal behavior.

4. Dissipative, nonlinear, and nonreciprocal coupling

Not all light propagation coupling is unitary. In dissipatively coupled waveguide networks, the fundamental interaction is an engineered coupling to a common reservoir rather than coherent hopping (Mukherjee et al., 2017). The Lindblad operators are differences of neighboring modes, $g_j$0, and for initial coherent-product states the complex amplitudes obey

$g_j$1

which is formally identical to a discrete diffusion equation, but for complex amplitudes (Mukherjee et al., 2017). In a homogeneous chain, the only steady mode is the uniform superposition, and any initial $g_j$2 relaxes to $g_j$3, with $g_j$4. The same framework supports channel-selective routing in a “quantum distributor”; perfect routing occurs when the control amplitudes satisfy $g_j$5 (Mukherjee et al., 2017). Experimentally, this was implemented in femtosecond-laser-written borosilicate-glass structures with auxiliary reservoir arrays and sample length $g_j$6 (Mukherjee et al., 2017).

A different non-Hermitian mechanism appears in actively coupled nonlinear waveguides (Alexeeva et al., 2013). Here two identical lossy Kerr waveguides are embedded in an active medium that amplifies the in-phase component of the overlapping evanescent fields. In the symmetric and antisymmetric basis, the in-phase mode $g_j$7 has net growth rate $g_j$8, whereas the out-of-phase mode $g_j$9 remains damped with rate $C_{nm}$0. The amplification threshold is $C_{nm}$1. Above threshold, Kerr mixing creates a feedback loop between the amplified symmetric mode and the damped antisymmetric mode, producing stable stationary or oscillatory regimes. The origin loses stability in a pitchfork bifurcation at $C_{nm}$2, and each symmetry-broken fixed point undergoes a Hopf bifurcation at $C_{nm}$3 for $C_{nm}$4. The device can act as a comparator or integrate-and-fire oscillator, and the reported switching sensitivity extends to power differences of order $C_{nm}$5 or less (Alexeeva et al., 2013).

Nonreciprocal coupling can also be induced mechanically. A subwavelength spinning dielectric cylinder near a slab waveguide yields different transmissions for opposite propagation directions because the cylinder’s chiral modes couple unidirectionally to the guided wave via transverse spin-orbit interaction (Yang et al., 2022). In the temporal coupled-mode description, the CW and CCW resonances split by the Sagnac effect, and the forward and backward coupling coefficients differ. Full-wave simulations show that higher-order chiral modes and larger spinning speed generally give stronger nonreciprocity, and that the coupling gap has a non-monotonic optimum: very small gaps spoil unidirectionality through symmetry breaking, while large gaps suppress evanescent coupling. The reported optimum is $C_{nm}$6–$C_{nm}$7, and for $C_{nm}$8 the maximum isolation ratio reaches about $C_{nm}$9; at $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$00, the isolation near the CCW resonance is about $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$01 (Yang et al., 2022).

A recurrent misconception is that “dissipative” and “nonreciprocal” necessarily imply incoherent transport. The dissipatively coupled networks explicitly preserve off-diagonal coherence and support decoherence-free subspaces, while the spinning-cylinder system relies on coherent resonant interference and spin-momentum locking rather than stochastic scattering (Mukherjee et al., 2017, Yang et al., 2022).

5. Coupling in dispersive and hybrid matter systems

In dispersive atomic media, the coupling fields themselves sculpt the propagation law. A four-level double $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$02-type system in $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$03, driven by a weak probe and three strong coupling fields, produces a susceptibility $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$04 whose spatial structure depends on the interference of plane-wave and Laguerre–Gaussian couplings (Sabegh et al., 2020). For an LG probe, the group velocity is defined by $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$05. When one of the strong couplings is also an LG mode, the medium acquires a petal-shaped gain and dispersion pattern through terms such as $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$06 and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$07, which distort the probe’s helical phase front. The reported result is that the local group-delay per unit length can become negative even though the global dispersion is normal, so that the probe LG field can exceed the speed of light in free space inside the medium (Sabegh et al., 2020). The same analysis shows that $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$08 on the optical axis, at the waist, and at the Rayleigh range (Sabegh et al., 2020).

A related, but structurally richer, setting is the five-level combined tripod–$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$09 coupling scheme (Hamedi et al., 2017). The four control fields define interference parameters $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$10 and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$11, and a global dark state exists when both $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$12 and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$13. In that regime the system exhibits EIT, slow light, and a steep linear dispersion. If $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$14, the system reduces to an $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$15-type four-level absorber; if $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$16, it reduces to an ordinary $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$17-scheme. The coupled Maxwell–Bloch equations yield a nonlinear Schrödinger equation for the probe envelope, supporting stable slow-light optical solitons. In the cesium-vapor example reported in the paper, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$18, so the relevant solutions are dark solitons with ultraslow group velocity $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$19 (Hamedi et al., 2017).

Optomechanical coupling provides another route to group-delay control. In a double-ended optomechanical cavity driven by a strong coupling laser and a weak probe in an EIT configuration, the transmitted probe amplitude is $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$20, and the group delay is $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$21 (Tarhan et al., 2012). For the parameter set based on Thompson et al., with $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$22, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$23, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$24, and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$25, the reported values at $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$26 are $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$27 in transmission and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$28 in reflection (Tarhan et al., 2012).

Hybridization with magnetic excitations leads to dispersive optomagnonic coupling. In a Faraday-active dispersive medium, the magnon–photon interaction Hamiltonian can be derived for both degenerate and non-degenerate optical modes, with coupling constants proportional to the zero-point magnon fluctuation $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$29, the overlap integral $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$30, and frequency derivatives of $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$31 and the Faraday coefficient $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$32 (Bittencourt et al., 2021). In a Lorentz dispersion model, the degenerate coupling $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$33 peaks sharply near the epsilon-near-zero frequency $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$34, and for YIG-like parameters with $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$35 the reported estimate is $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$36, כלומר single-magnon strong coupling in a micron-scale volume. The same theory shows that non-degenerate Voigt-mode coupling vanishes at frequencies satisfying

$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$37

which the authors attribute to polarization selection rules controlled by dispersion (Bittencourt et al., 2021).

6. Lattices, superlattices, and engineered transport networks

In one-dimensional photonic lattices, coupling is frequently cast as a tight-binding Hamiltonian with site-dependent propagation constants $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$38 and nearest-neighbor couplings $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$39 (Rodríguez-Lara, 2013). Because the coupling matrix is a Jacobi matrix, the propagator can be written in closed form through orthogonal polynomials. Several special lattices then become analytically tractable. In the uniform lattice, the eigenvalues are $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$40. In the perfect-transfer lattice, the engineered couplings

$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$41

guarantee $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$42, i.e. perfect transfer from site $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$43 to site $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$44 at $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$45, provided the input state has no vacuum component (Rodríguez-Lara, 2013). The same formalism was used to compare the propagation of single-photon, coherent, path-entangled, and two-mode squeezed-vacuum states (Rodríguez-Lara, 2013).

Three-dimensional cavity superlattices introduce a distinct regime. In the inverse-woodpile 3D photonic band-gap crystal, five coupled-cavity bands arise from quadrupole-like resonances, and for the three converged central bands the dispersion bandwidth is largest in the $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$46-diagonal directions (Hack et al., 2018). Tight-binding analysis yields seven independent couplings, including $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$47, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$48, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$49, and diagonal terms such as $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$50. The dominant nonzero couplings occur along $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$51, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$52, and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$53, while $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$54, $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$55, and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$56 couplings are one to two orders of magnitude smaller. Because positive and negative hoppings coexist, large couplings can coexist with vanishing bandwidths. This is why the authors distinguish “Cartesian light” from ordinary Bloch-wave transport, band-gap tunneling, one-dimensional CROW propagation, and diffusive edge transport (Hack et al., 2018).

At a much smaller scale, arrays of seven $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$57 clusters provide a self-consistent quantum-classical model of coupled propagation and exciton transfer (Lisinetskaya et al., 2018). Each cluster obeys a time-dependent Schrödinger equation, while the total electric field is propagated by coupling the external driving field to the retarded dipole fields radiated by all clusters. Ab initio LR-TDDFT supplies the on-site energies and dipole matrix elements, and a genetic algorithm optimizes the spectral phase of a femtosecond pulse to steer energy through a T-shaped structure. The reported result is selective switching of light localization in a structure of about $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$58, with switching on a $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$59 timescale (Lisinetskaya et al., 2018).

Open quantum spin chains extend the notion of coupled propagation to strongly interacting driven-dissipative media. Using quantum Langevin equations for a Heisenberg-like chain with nearest-neighbor $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$60 and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$61 couplings, the reported steady-state transmission is ballistic at $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$62 and again at $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$63, but acquires an apparent system-size dependence at intermediate interactions $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$64 under high incident power (Manasi et al., 2017). This establishes that nonlinear many-body interactions can change the transport law itself, even when the optical input remains monochromatic.

7. Geometric and relativistic coupling

Coupling can also be induced by motion, rotation, and nonlinear effective geometry. In cylindrical step-index fibers viewed in a slowly rotating frame, Earth’s rotation generates weak coupling between a mode $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$65 and its $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$66 sidebands (Mieling, 2019). The perturbative calculation yields a corrected dispersion relation with

$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$67

in agreement with geometrical optics (Mieling, 2019). The induced sideband amplitudes scale as $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$68. For typical parameters, the estimates are $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$69 for Earth’s spin and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$70 for Earth’s orbital motion, while $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$71 is of order $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$72 at the equator for Earth’s spin and $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$73 for Earth’s orbit (Mieling, 2019). The same study concludes that the sidebands are likely too weak to observe in ordinary fibers, even though the phase shifts are measurable interferometrically (Mieling, 2019).

In ModMax nonlinear electrodynamics coupled to gravity, light propagation near a charged black hole is determined by two optical metrics rather than one background metric (Guzman-Herrera et al., 2023). The static, spherically symmetric solution has

$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$74

and the eikonal condition is $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$75 for two polarization branches (Guzman-Herrera et al., 2023). One branch coincides with the background metric, while the other is deformed, producing vacuum birefringence. Purely radial rays show no birefringence, but angular propagation does. The deflection angles satisfy $Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$76, and the two shadow radii obey

$Q(\omega)=-i\,T^{-1}(\omega)\,dT(\omega)/d\omega$77

This identifies coupling between nonlinear electrodynamics and gravity as a propagation-coupling problem in the effective-metric sense, not a waveguide or cavity problem (Guzman-Herrera et al., 2023).

Taken together, these examples show that light propagation coupling ranges from nanometer-scale impedance matching to effective-geometric birefringence. The unifying feature is the explicit role of couplings—between modes, channels, reservoirs, resonances, matter excitations, or optical metrics—in setting how light is transmitted, delayed, redirected, localized, or split across complex photonic and optically active systems.