Tight-Binding-Like Photonics Overview

• Tight-binding-like photonics is a framework that reduces continuum electromagnetic problems to discrete lattice models where localized modes act as orbitals and couplings as hopping terms.
• It integrates a variety of platforms—from coupled resonators and waveguide arrays to photonic crystals and synthetic dimensions—to emulate lattice Hamiltonians familiar from condensed matter physics.
• This approach facilitates detailed analysis of spectra, topological edge states, and transport properties while addressing complications like non-orthogonality and extended coupling corrections.

Searching arXiv for recent and foundational papers on tight-binding-like photonics. Tight-binding-like photonics denotes a class of photonic theories and platforms in which electromagnetic propagation is reduced to an effective lattice Hamiltonian or discrete recurrence, with localized photonic modes playing the role of orbitals, resonant frequencies or propagation constants acting as onsite terms, and evanescent, resonant, or modulation-induced transfer acting as hopping. Across coupled resonator optical waveguides, waveguide arrays, microwave resonator lattices, synthetic dimensions, photonic crystals, and Maxwell-derived interface models, the central move is the same: replace a continuum field problem by a finite- or countable-dimensional matrix problem whose spectrum, eigenmodes, and topology can be analyzed with the methods of band theory, transfer matrices, and tight-binding transport (Li et al., 6 Aug 2025). In some settings this reduction is phenomenological, while in others it is exact or asymptotically exact, as in interface-amplitude mappings of 1D photonic crystals or void–channel networks formed by narrow gaps between perfect conductors (Henriques et al., 2020, Palmer et al., 2021).

1. Discrete photonic Hamiltonians and the tight-binding analogy

The basic tight-binding analogy is explicit in several photonic settings. In programmable coupled-cavity arrays, each optical cavity is identified with a lattice site, its resonance frequency with the onsite potential $\mu_n$, and the evanescent coupling between neighboring cavities with the hopping rate $J_n$, yielding the finite 1D Hamiltonian

$$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$

for a nearest-neighbor chain (Saxena et al., 2022). In coupled resonator optical waveguides used as scattering structures, single-photon amplitudes on cavity sites satisfy a discrete Schrödinger equation,

$(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$

which for nearest-neighbor coupling reduces to a cosine-band lattice problem with plane-wave lead states and standard reflection and transmission amplitudes (Jiang et al., 2017).

This same logic appears in waveguide arrays, where the paraxial Helmholtz equation becomes a Schrödinger-like evolution equation in the propagation coordinate $z$, and after projection onto localized guide modes yields

$$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$

for site amplitudes $\psi_m$ (Li et al., 6 Aug 2025). In synthetic frequency lattices, modulation of a ring resonator at the free spectral range couples adjacent resonant frequency bins and gives

$$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$

which is exactly a 1D tight-binding model with a tunable Peierls phase $\phi$ (Li et al., 6 Aug 2025).

A recurring refinement is that photonic reductions are often generalized rather than strictly electronic-style. In exciton-polariton micropillar lattices, localized $s$ and $J_n$0 photonic orbitals are non-orthogonal, so the spectrum is obtained from

$J_n$1

rather than an ordinary Hermitian eigenproblem (Mangussi et al., 2019). In waveguide photonics, the same issue arises because isolated guided modes are not exactly orthogonal at finite separation, and the correct reduced problem is

$J_n$2

with overlap matrix $J_n$3 (Tschernig et al., 29 May 2026). These cases show that “tight-binding-like” does not always mean a naive nearest-neighbor matrix with orthonormal orbitals; it may instead require overlap matrices, energy-dependent couplings, or orthogonalization procedures.

2. Maxwell-to-lattice reductions

One major branch of tight-binding-like photonics starts directly from Maxwell’s equations and derives a discrete lattice representation. A particularly clean example is a 1D photonic crystal made of alternating dielectric slabs $J_n$4 and $J_n$5. For this system, the magnetic-field amplitudes at the two interfaces within each unit cell,

$J_n$6

obey the exact difference equations

$J_n$7

with

$J_n$8

(Henriques et al., 2020). Here the “sites” are interface amplitudes, and the resulting two-sublattice recurrence has SSH form near a band edge. Expanding near the gap closing gives a Dirac-like Hamiltonian with mass proportional to $J_n$9, so the metal/photonic-crystal interface supports one Tamm state per band gap in the topologically non-trivial regime and none in the trivial regime (Henriques et al., 2020).

An asymptotically exact continuum-to-discrete mapping appears in void–channel photonics. In 2D TE polarization, the out-of-plane magnetic field $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$0 in the voids between closely spaced perfect conductors satisfies

$$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$1

with Neumann boundary conditions. In the low-frequency, narrow-channel limit, each void acts as a lumped node and each narrow channel as the only significant coupling path, leading to a mass–spring graph with effective parameters

$$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$2

(Palmer et al., 2021). Because only directly connected voids interact at leading order, the asymptotic graph is genuinely short-ranged, which is precisely the regime required for chiral-symmetric models such as SSH and square-root semimetals (Palmer et al., 2021).

A more indirect Maxwell-to-lattice route is used in 3D photonic crystals. There, the vectorial transversality constraint

$$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$3

obstructs ordinary Wannierization for the lowest transverse bands. The resulting transversality-enforced tight-binding construction uses topological quantum chemistry to express the band symmetry data as a difference of elementary band representations, introduces auxiliary longitudinal modes, and fits a scalar effective Hamiltonian in $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$4 whose positive-energy sector reproduces the physical bands (Morales-Pérez et al., 2023). This does not yield literal photonic Wannier orbitals, but it does produce a compact pseudo-orbital basis that matches dispersion, symmetry content, and topology over the Brillouin zone (Morales-Pérez et al., 2023).

3. Canonical physical platforms

Coupled-cavity and resonator arrays are among the most direct realizations. A silicon photonic array of 8 strongly coupled racetrack resonators on silicon-on-insulator implements a finite nearest-neighbor chain with $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$5, and thermo-optic island heaters provide site-resolved tuning of onsite terms $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$6 (Saxena et al., 2022). Because the measured device is open and lossy, the relevant operator is the non-Hermitian effective Hamiltonian

$$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$7

whose complex eigenvalues are directly reconstructed by reflection-based Hamiltonian tomography (Saxena et al., 2022). The platform reports a $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$8 reduction in nearest-neighbor thermal crosstalk compared to traditional heaters, and a calibrated programming model achieves greater than $$H = \sum_n \mu_n a_n^\dagger a_n + \sum_n J_n \left(a_{n+1}^\dagger a_n + a_n^\dagger a_{n+1}\right)$$9 accuracy, or an error of only $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$0 of the mean hopping rate (Saxena et al., 2022).

Single-photon transport in coupled resonator optical waveguides provides a complementary scattering formulation. Parallel-connected resonator waveguides between left and right leads reduce to transfer-matrix channels whose amplitudes are recombined coherently at splitter and combiner junctions (Jiang et al., 2017). In the presence of embedded two-level atoms, the local Jaynes–Cummings interaction generates an energy-dependent onsite correction proportional to $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$1, so the structure becomes a photonic analog of a tight-binding impurity network (Jiang et al., 2017). The paper identifies a perfect-reflection regime “determined by the number of coupled resonator waveguides,” and shows numerically that changing the atom transition frequency can shift the window of perfect reflection to cover almost all incoming photon energy (Jiang et al., 2017).

SNAP microresonator chains realize a different kind of 1D lattice. For a chosen axial mode order $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$2, the localized axial mode in the $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$3-th bottle resonator serves as the site orbital $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$4, and adjacent resonators couple with a hopping extracted from the symmetric/antisymmetric splitting,

$(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$5

The same physical array then hosts several effective SSH lattices, one per axial order $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$6, because higher-order axial modes are more extended and therefore more strongly coupled (Fried et al., 8 Jan 2026). This platform combines $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$7, propagation loss $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$8 dB/cm, and sub-angstrom fabrication precision, and experimentally realizes both a 5-site SSH2 chain and a 21-site SSH2–SSH4 heterojunction (Fried et al., 8 Jan 2026).

Microwave dielectric-resonator arrays realize lattice Hamiltonians at a macroscopic scale. In a graphene-like structure built from high-index dielectric disks between metallic plates, each disk contributes a single localized TE resonance, neighboring disks couple through exponentially decaying fields proportional to $(\epsilon_j-E)\psi(j)=\sum_n J_{j,n}\psi(j+n),$9, and a finite honeycomb lattice reproduces the Dirac point, linearly vanishing density of states near it, and zigzag/corner states (Kuhl et al., 2010). The measured dimer splitting yields $z$0, $z$1, and $z$2, while next-nearest-neighbor coupling is about $z$3 of nearest-neighbor coupling (Kuhl et al., 2010).

4. Topological models and their photonic realization

The SSH model is the most ubiquitous tight-binding template in photonics. In Sagnac-loop-reflector Fabry–Perot lattices, alternating two SLR types with different cross-coupling coefficients $z$4 and $z$5 produces an SSH chain in the weak-coupling limit. The exact wave-optics description is given by a unit-cell transfer matrix, while the reduced coupled-mode equations take the form

$z$6

equivalent to the Hamiltonian

$z$7

(Kim et al., 14 May 2026). In a 20-site lattice, the topological phase supports an isolated midgap resonance at $z$8 with edge-localized power profiles, and disorder in the SLR couplers acts primarily as symmetry-preserving hopping disorder (Kim et al., 14 May 2026).

The same SSH logic appears in topological Tamm states. For a 1D photonic crystal terminated by a metal, the exact interface-amplitude recurrence reduces near the relevant gap to an SSH-like Bloch Hamiltonian with off-diagonal structure $z$9, winding number $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$0, and a Dirac-like mass term $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$1 (Henriques et al., 2020). The non-trivial regime for the $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$2-terminated interface is $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$3, where one Tamm state appears in each band gap; for $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$4, the system is topologically trivial and the gaps host no Tamm states (Henriques et al., 2020).

Reduced-symmetry topological photonics extends this SSH/Kane–Mele program beyond high-symmetry honeycomb lattices. A Babinet-dual metallic surface-wave structure on a rhombic lattice realizes a spin-type photonic topological insulator whose pseudospin fields are

$$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$5

Its reduced-symmetry tight-binding model is an anisotropic Kane–Mele Hamiltonian with unequal nearest-neighbor hoppings $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$6 and $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$7 and intrinsic second-neighbor spin-orbit-like coupling $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$8 (Davis et al., 26 Jul 2025). Although the reduced symmetry removes the symmetry-enforced Dirac cone at $$i\frac{\partial\psi_m}{\partial z} + \sum_{<m,n>} J_{m,n} \psi_n=0$$9, the Wilson loop still winds nontrivially and the tight-binding model has $\psi_m$0, with robust edge states verified in full-wave simulations (Davis et al., 26 Jul 2025).

Higher-order and multiband topological physics also fit naturally into the tight-binding-like program. The SNAP heterojunction between SSH2 and SSH4 lattices required generalized topological polarization methods based on sublattice phases and sublattice displacement rather than only the usual $\psi_m$1 winding number (Fried et al., 8 Jan 2026). This suggests that multiband photonic heterostructures can demand bulk markers adapted to local, rather than global, lattice symmetry. A plausible implication is that photonic lattice simulators with multiple simultaneous mode families, such as SNAP, are especially suited to studying topological junctions beyond the minimal two-band setting (Fried et al., 8 Jan 2026).

5. Beyond the nearest-neighbor approximation

A central technical issue in tight-binding-like photonics is the validity of the reduced basis. In conventional coupled-mode theory for waveguide arrays, isolated single-waveguide modes are assumed orthonormal, so one writes

$\psi_m$2

The 2026 Löwdin-orthogonalization analysis shows that this is mathematically inconsistent whenever neighboring guided modes overlap, because the correct projected equation is

$\psi_m$3

with nontrivial overlap matrix $\psi_m$4 (Tschernig et al., 29 May 2026). The remedy is to build an orthonormal localized basis

$\psi_m$5

which yields the Löwdin Hamiltonian

$\psi_m$6

(Tschernig et al., 29 May 2026). This corrected model captures renormalized onsite terms, enhanced long-range couplings, and negative next-nearest-neighbor hoppings that carry an effective phase $\psi_m$7, and it keeps propagation fidelity above $\psi_m$8 in the showcased dimer, 5-site chain, and $\psi_m$9 array, whereas standard tight binding can fall to about $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$0 in larger systems (Tschernig et al., 29 May 2026).

Non-orthogonality is equally important in polariton lattices. A multi-orbital model based on $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$1 and $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$2 non-orthogonal photonic orbitals reproduces the observed bulk bands, edge states, and polarization-resolved dispersions of polariton graphene ribbons (Mangussi et al., 2019). It identifies two distinct physical corrections to the simplest orthogonal nearest-neighbor model: $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$3-$$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$4 inter-orbital coupling reshapes the central bands, while non-orthogonality produces the asymmetry of the outer $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$5 bands without destroying flat edge states (Mangussi et al., 2019). The overlap integrals are not negligible; for a representative geometry the paper reports $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$6, $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$7, $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$8, and $$i\frac{d a_m}{d t} = g e^{i\phi} a_{m-1} + g e^{-i\phi} a_{m+1},$$9 (Mangussi et al., 2019).

The reliability problem extends to optical response more broadly. A study of electronic tight-binding models for monolayer $\phi$0 shows that band-structure agreement is necessary but not sufficient for accurate dielectric and shift-current responses, because those observables depend on eigenvectors, interband velocity matrix elements, and Berry-connection-related quantities, not only on energies (Ghosh et al., 2024). This suggests a direct caution for photonic tight-binding models: matching dispersion alone may be insufficient for predicting excitation strengths, polarization response, nonlinear conversion, or geometric observables. A plausible implication is that photonic reduced models should be validated against the property of interest, especially when field profiles and mode overlaps, rather than frequencies alone, control the experiment (Ghosh et al., 2024).

6. Scope, significance, and emerging directions

Taken together, these works show that tight-binding-like photonics is not a single method but a family of reductions linking continuum electromagnetism to lattice physics with varying levels of rigor. At one end are exact and asymptotically exact mappings, such as Maxwell-to-interface SSH chains and void–channel mass–spring networks (Henriques et al., 2020, Palmer et al., 2021). In the middle are highly accurate projected models, such as programmable coupled-cavity arrays with full Hamiltonian tomography or SNAP axial-mode lattices with extracted hopping parameters (Saxena et al., 2022, Fried et al., 8 Jan 2026). At the more approximate end are phenomenological orbital models for polariton lattices and reduced-symmetry topological insulators, which remain useful precisely because they organize spectra, edge states, and polarization textures in a compact matrix form (Mangussi et al., 2019, Davis et al., 26 Jul 2025).

A major recent trend is the move from qualitative analogy to quantitatively faithful photonic tight binding. The 2025 review explicitly frames this as a transition from solving only Maxwell equations numerically to constructing matrix Hamiltonians that capture spectra and wavefunctions across a wide range of photonic platforms (Li et al., 6 Aug 2025). The 2026 Löwdin work sharpens that program by showing how overlap corrections can repair the breakdown of conventional coupled-mode theory in dense or large arrays (Tschernig et al., 29 May 2026). The 2023 transversality-enforced model addresses an analogous obstruction in 3D vectorial photonics by replacing unavailable Wannier functions with symmetry-guided pseudo-orbitals and auxiliary longitudinal modes (Morales-Pérez et al., 2023).

Another direction is the expansion from literal spatial lattices to synthetic and hybrid ones. Synthetic frequency dimensions already realize exact hopping Hamiltonians with tunable gauge phases (Li et al., 6 Aug 2025). SNAP devices host several effective SSH lattices simultaneously, one per axial order, and use generalized polarization methods for multiband heterojunctions (Fried et al., 8 Jan 2026). In continuum photonic crystals, strained Dirac cones can produce Landau-level physics and synthetic gauge fields even without a microscopic tight-binding starting point; the low-energy theory is still Dirac- and lattice-like, but it emerges from a multiscale reduction of the full wave equation rather than from discrete resonator sites (Guglielmon et al., 2020). This suggests that “tight-binding-like” can include both literal photonic lattices and continuum systems whose relevant asymptotic envelope theory has the same algebraic structure.

The resulting significance is twofold. Methodologically, tight-binding-like models provide a low-dimensional language for scattering, spectroscopy, topology, and inverse design. Physically, they make photonic mirrors, filters, edge channels, corner states, Dirac points, Tamm states, Landau levels, and non-Hermitian transport intelligible in terms of orbitals, hoppings, symmetry breaking, and bulk–boundary correspondence. The common lesson is that the usefulness of the tight-binding analogy depends on how carefully the photonic basis is chosen and how honestly its limitations—non-orthogonality, long-range coupling, vectorial constraints, or property-specific matrix elements—are handled (Jiang et al., 2017, Saxena et al., 2022, Tschernig et al., 29 May 2026, Li et al., 6 Aug 2025).