1D Photonic Quasicrystals

• One-dimensional photonic quasicrystals are deterministic, aperiodic dielectric structures with long-range order and unique spectral properties.
• They exhibit fractal spectra with pseudo-gaps and critically localized modes, enabling slow-light propagation and multi-wavelength filtering.
• Topological invariants in these systems facilitate robust edge states, offering promising avenues for advanced optical device engineering.

A one-dimensional photonic quasicrystal (1D PQC) is a deterministic, nonperiodic dielectric structure characterized by a lack of translational symmetry but the presence of long-range order, leading to distinct and highly nontrivial photonic spectral and localization properties. These systems are constructed according to mathematical rules—such as Fibonacci, Octonacci, or Dodecanacci sequences, or incommensurate harmonic superpositions—where the spatial modulation of the refractive index is neither periodic nor random. The consequence is a hierarchy of photonic pseudo-band-gaps, critically localized modes, and in certain architectures, topologically nontrivial edge states with quantized invariants. 1D PQCs are central both as model systems for understanding quasiperiodic physics and as platforms for novel photonic devices exhibiting fractal spectra, slow-light propagation, and robust high-Q modes.

1. Mathematical Constructions and Sequence Types

Canonical 1D PQCs are generated by deterministic aperiodic sequences, each with distinct inflation or substitution rules. The most thoroughly studied example is the Fibonacci quasicrystal, generated by inflation $A \to AB$, $B \to A$. The $n$-th generation word is $S_n = S_{n-1}S_{n-2}$, with the number of layers following the Fibonacci series; the A:B ratio converges to the golden mean $\varphi = (1+\sqrt{5})/2$ as $n\to\infty$ (Ghulinyan, 2015). Other deterministic sequences include the Octonacci (rule $S_n = S_{n-1} S_{n-2} S_{n-1}$, A:B inflation limit $1:\sqrt{2}$) (Brandão et al., 2015), and Dodecanacci (more complex inflation rules yielding higher-order Pisot ratios) (Nayak et al., 2020).

Alternatively, a broad and general class arises by superposing two periodic lattices of incommensurate periods $T_1$, $T_2$ with irrational ratio $\beta = T_1/T_2$ to define a continuous modulation, e.g.,

$$\varepsilon^{-1}(x) = \mu(x) = 1 + \tfrac12[\cos x + \cos(\beta x)]$$

where $0<\beta<1$ sets the degree of quasiperiodicity (Quan et al., 10 Jan 2026). The choice of $\beta$ governs fractal spectral properties and spatial localization of modes.

All deterministic PQC sequences are ultimately defined by recursive inflation/substitution or cut-and-project algorithms, resulting in structures with dense Bragg spectra and self-similar scaling.

2. Electromagnetic Formalism and Spectral Analysis

The electromagnetic properties of 1D PQCs are treated via the transfer-matrix method due to the lack of translation invariance. For each dielectric layer $j$ (refractive index $n_j$, thickness $d_j$), the transfer matrix $M_j$ relates the tangential fields across the interface. For a stack of $t$ layers, the total transfer matrix $M_\text{tot} = \prod_{j=1}^t M_j$ yields the complex reflection and transmission amplitudes. For normal incidence,

$$T(\omega) = \left|\frac{1}{M_\text{tot}^{(11)}(\omega)}\right|^2$$

$R = |M_\text{tot}^{(21)}/M_\text{tot}^{(11)}|^2$ (Brandão et al., 2015, Ghulinyan, 2015, Bellingeri et al., 2017).

For quasiperiodic modulations given by incommensurate harmonics, Bloch's theorem is inapplicable. Instead, the generalized spectral (projection) method embeds the problem in a higher-dimensional periodic superspace, mapping the 1D nonperiodic dielectric profile to a projection of a higher-dimensional lattice. The Maxwell eigenproblem is then cast as an infinite-dimensional matrix, truncated and diagonalized numerically to yield the spectrum (Quan et al., 10 Jan 2026).

3. Fractal Spectra, Pseudo-Gaps, and Localization

One-dimensional PQCs exhibit singular continuous (fractal/Cantor-set) spectra, with a hierarchy of pseudo-gaps whose widths scale as a negative power of the approximant generation or follow Pisot scaling. For Fibonacci and Octonacci QCs, the transmission spectrum shows multiple mini-gaps and a self-similar structure around a central frequency, with each inflation generation contracting the gap pattern in frequency by $1/\varphi$ (Fibonacci) or $1/(1+\sqrt{2})$ (Octonacci) (Brandão et al., 2015, Ghulinyan, 2015). Dodecanacci PQCs show even more complex gap hierarchies (Nayak et al., 2020).

Superposed incommensurate harmonics produce a characteristic "butterfly"-shaped spectral plot as a function of irrational ratio $\beta$, with abundant self-similar band gaps (Quan et al., 10 Jan 2026). In all these cases, the spectrum is densely gapped: the density of states forms a zero-measure set interleaved with fractal mini-gaps. Critically localized modes appear at the edges of these gaps, with localization quantified by the inverse participation ratio (IPR) and Lyapunov exponent $$\gamma(\omega) = \lim_{j\to\infty} \frac{1}{j} \ln\|M^{(j)}(\omega)\|$$ (Ghulinyan, 2015).

The location and size of the primary spectral gap in incommensurate-harmonic PQCs is governed by linear laws in the irrational parameter: the structure factor $Q(\beta)$ shows piecewise-linear behavior $Q=1-\beta$ for $0<\beta<\beta_c$, $Q=\beta$ for $\beta_c<\beta<1$ with a critical point $\beta_c\approx0.424$ (Quan et al., 10 Jan 2026).

4. Topological Invariants and Boundary States

A subset of 1D PQCs possess nontrivial topological properties. In the resonator-lattice realization, onsite energies are quasiperiodically modulated as $$\omega_x = \omega_{c0} + \lambda\cos(2\pi b x + \phi)$$ with $b$ irrational (Zhang, 2015). The tight-binding Hamiltonian

$$H_0 = \sum_{x=1}^{L} \omega_x|\psi_x\rangle\langle\psi_x| + \sum_{x=1}^{L-1} t(|\psi_x\rangle\langle\psi_{x+1}| + \text{h.c.})$$

when equipped with twisted boundary conditions (flux $\theta$, phase $\phi$), enables the computation of the first Chern number $C_1$ over the $(\phi,\theta)$ torus:

$$C_1 = \frac{1}{2\pi i} \int_0^{2\pi} d\phi\, d\theta\, \operatorname{Tr}\{P[\partial_\phi P, \partial_\theta P]\}$$

with projector $P$ onto below-gap states. The nontrivial $C_1$ implies the presence of exponentially localized edge states at open boundaries, whose frequency traverses the mid-gap as $\phi$ is varied (“bulk–boundary correspondence”) (Zhang, 2015, Verbin et al., 2014).

Topological pumping has been implemented in photonic waveguide arrays, where a family interpolating smoothly between the Fibonacci and Harper models through a continuous deformation parameter $\beta$ preserves the Chern number of the central gap, and the adiabatic variation of phase (the "phason") $\phi$ yields quantized photon pumping (Verbin et al., 2014).

5. Bandgap Engineering, Localization Phenomena, and Reentrant Transitions

PQCs provide a rich platform for multi-gap engineering and field localization. Under resonance Bragg conditions, deterministic aperiodic sequences of quantum wells (e.g., Fibonacci) yield multiple photonic reflection bands, distinct from the single band of periodic systems. Doping (e.g., with carriers giving rise to Mahan singularity) induces asymmetric line shapes and broad low-frequency stop-bands, with superradiant or photonic-crystal scaling depending on the number of wells (Voronov et al., 2010).

Mode localization can be controlled via the generation/order of the sequence, material parameters, and incommensuration ratio. In addition to critical localization characteristic of quasicrystals, specialized models with dimerized tight-binding lattices and long-range hopping demonstrate mobility edges and a reentrant localization–delocalization transition: upon increasing quasiperiodic modulation, certain modes localize, then re-delocalize at higher modulation, corresponding to a non-monotonic IPR (Vaidya et al., 2022). This has been both theoretically predicted and experimentally observed in Si/SiO$_2$ multilayer PQCs.

6. Experimental Implementations and Device Applications

1D PQCs are generally fabricated via high-precision thin-film deposition (RF-sputtering, electron-beam evaporation, sol–gel), with A/B layers drawn from dielectric, semiconducting, superconducting, or metamaterial materials. Femtosecond-written photonic waveguide arrays enable implementation of complex coupling sequences and nontrivial topological phases (Verbin et al., 2014). Superconductor–metamaterial Dodecanacci stacks require low-temperature control and mm/nm-scale layer engineering (Nayak et al., 2020).

Experimentally, PQCs exhibit broadband reflective behavior, multifrequency filtering (due to mini-gaps), high-Q narrowband transmission (localized states), and tunable pseudo-gap positioning via the irrational parameter $\beta$ or inflation order (Quan et al., 10 Jan 2026, Brandão et al., 2015, Nayak et al., 2020). Topological boundary states can be detected by near-field scanning or pump–probe resonance as localized spectral peaks, robust to moderate disorder (Zhang, 2015).

Device applications include multi-wavelength optical filters, omnidirectional reflectors (notably in Octonacci for TE polarization), slow-light channels near pseudo-gap edges, defect-assisted nanocavities, and reconfigurable photonic sensors. In superconductor/metamaterial stacks, the operating temperature and layer thickness tune both gap width and filter quality factor. Integration of intentional defects or refractive-index clustering permits fine spectral tailoring or controlled insertion of defect modes (Bellingeri et al., 2017, Brandão et al., 2015, Nayak et al., 2020).

7. Comparative and Universal Aspects of 1D PQCs

PQCs are distinguished from both periodic photonic crystals (which feature a single band gap and Bloch modes) and disordered multilayers (which present Anderson localization and no long-range order). The singular continuous spectrum, self-similar scaling, and deterministic location of critically localized states are hallmarks unique to PQCs. The role of the irrational parameter $\beta$ in controlling the global spectral structure is universal for incommensurate-harmonic PQCs, and the general behaviors—including gap hierarchy, localization scaling, and fractal spectrum—are largely independent of the precise irrational chosen, except near the periodic (weak-quasiperiodic) limits (Quan et al., 10 Jan 2026).

Topologically nontrivial PQCs extend the correspondence between non-periodic band theory and higher-dimensional topological invariants (Chern numbers), supporting boundary modes inaccessible in periodic or random systems. The ability to interpolate adiabatically between different topological classes under continuous deformation permits robust device behavior across large parameter spaces (Zhang, 2015, Verbin et al., 2014).

A plausible implication is that further advances in 1D PQC engineering and theory will continue to bridge the gap between mathematical quasicrystal models and practical photonic platforms, both for exploring fundamental aspects of spectral theory and for realizing next-generation multi-functional photonic devices.