One-Dimensional Photonic Quasicrystals (1D PQC)

• One-dimensional photonic quasicrystals are aperiodic multilayer systems defined by deterministic inflation rules (e.g., Fibonacci, Dodecanacci) that produce fractal transmission spectra.
• They exhibit unique optical properties including dense clusters of transmission resonances, pseudo-gaps, and critically localized modes that transition between extended and localized states.
• Transfer-matrix and topological methods reveal robust photonic transport, inspiring applications in ultra-narrow spectral filtering, cavity formation, and quantized light pumping mechanisms.

A one-dimensional photonic quasicrystal (1D PQC) is a deterministic aperiodic multilayer structure that controls the propagation of electromagnetic waves through a quasiperiodic modulation of dielectric properties. Distinct from periodic photonic crystals and totally random (disordered) systems, 1D PQCs such as those based on the Fibonacci or Dodecanacci sequences exhibit nontrivial spectral features, critical localization of modes, and—through connections to higher-dimensional topological models—support robust topological invariants with physical manifestations in photonic transport and localization phenomena (Ghulinyan, 2015, Bellingeri et al., 2017, Verbin et al., 2014, Jagannathan, 14 Jan 2026).

1. Fundamentals and Construction Principles

A 1D PQC is formed by stacking alternating layers of materials (A, B)—each with fixed or frequency-dependent refractive indices—according to a deterministic inflation rule. Unlike periodic photonic crystals (PCs) with strict translational symmetry, PQCs lack any finite repeat unit but display long-range order. Canonical examples include:

• Fibonacci sequence: Constructed via substitution rules (A→AB, B→A) or word concatenations $S_n = S_{n-1}S_{n-2}$, producing a structure where the ratio of adjacent block counts converges to the golden mean $\tau = (1+\sqrt{5})/2$. The resulting spatial order ensures self-similarity and a dense hierarchical set of spectral gaps (Ghulinyan, 2015, Bellingeri et al., 2017).
• Dodecanacci sequence: Defined by the inflation $D_x = \{A D_{x-2} D_{x-1}\}^2 D_{x-1}$, with a rapid growth in layer count and more complex hierarchical structure, giving rise to a richer spectrum with increasing generations (Nayak et al., 2020).

Mathematically, the general dielectric profile of a 1D PQC can be realized via:

• Binary substitution sequences (e.g., Fibonacci, Thue–Morse).
• Fourier superpositions with incommensurate frequencies, such as $$\epsilon^{-1}(x) = \frac{1}{2}[\cos x + \cos(\beta x)] + 1$$ for irrational $\beta$ (Quan et al., 10 Jan 2026).
• Tight-binding Hamiltonians with quasiperiodic modulation of hopping or onsite terms, as in the off-diagonal Harper and Fibonacci models: $t_n = t_0[1+\lambda d_n]$, with $d_n$ taking forms from smooth cosine to binary step for interpolation (Verbin et al., 2014).

2. Optical and Spectral Properties

A 1D PQC exhibits a fundamentally different optical response from periodic or random stacks:

• Spectral characteristics: The spectrum comprises dense clusters of transmission resonances and a self-similar hierarchy of pseudo-gaps rather than a finite set of broad photonic band gaps. In the strongly quasiperiodic regime, the band structure forms a fractal (“butterfly-shaped”) plot as a function of the irrational parameter $\beta$, with the principal gap indices depending linearly on $\beta$ in distinct regimes: $Q=1-\beta$ for $\beta<\beta_c$ and $Q=\beta$ for $\beta>\beta_c$, where $\beta_c \approx 0.424$ (Quan et al., 10 Jan 2026).
• Mode localization: Modes near gap edges are critically localized, typically characterized via participation ratio metrics. Unlike exponential Anderson localization found in random stacks, quasiperiodic localization obeys power laws; the envelope of the mode falls off slower than exponential and is associated with multifractal eigenstates (Ghulinyan, 2015, Vaidya et al., 2022).
• Delocalization transitions: The interplay of dimerization, long-range coupling, and quasiperiodic modulation may induce reentrant transitions—where increasing the modulation drives a transition from extended to localized states, followed by relapse back to extended states and finally full localization. These reentrant phenomena are observed both numerically and experimentally in silicon/silica multilayer PQCs (Vaidya et al., 2022).

The evolution of bandgaps and localization with system parameters is summarized in the table below:

PQC Parameter	Effect on Gaps/Localization	Reference
Higher inflation (Fibonacci, Dodecanacci)	More and wider pseudo-gaps, higher self-similarity	(Nayak et al., 2020)
Increasing refractive index contrast	Wider/deeper pseudo-gaps	(Bellingeri et al., 2017)
Layer thickness shift ($d_A$, $d_B$)	Blue-shift of spectral features, narrower or broader resonances	(Nayak et al., 2020)
Modulation strength (cosine amplitude $A$ or $\alpha$)	Non-monotonic extended-localized-extended transitions (mobility edges)	(Vaidya et al., 2022)
Irrational $\beta$ value	Principal gap index linear in $\beta$ in strong regime, mode localization at gap edges	(Quan et al., 10 Jan 2026)

3. Transfer-Matrix and Spectral Analysis

Propagation of light in a 1D PQC is modeled by the transfer-matrix formalism. For a stack of $N$ layers with refractive indices $n_j$ and thicknesses $d_j$, the characteristic matrix is:

$$M_j = \begin{pmatrix} \cos\varphi_j & -i\,\sin\varphi_j / p_j \ -i\,p_j\,\sin\varphi_j & \cos\varphi_j \end{pmatrix},$$

with $\varphi_j = \frac{\omega}{c} n_j d_j$, and $p_j = n_j$ (normal incidence). The total transfer matrix $M$ is the ordered product. The transmission and reflection coefficients are derived from $M$ and the boundary indices. Pseudo-gaps are identified as frequencies where $|{\rm Tr} M|>2$ (Bellingeri et al., 2017, Ghulinyan, 2015, Nayak et al., 2020).

For deterministic aperiodic sequences, such as the Fibonacci chain, a two-wave approximation incorporating the dominant Fourier component is often used for analytical estimates of gap positions and widths, accounting for the lower structure factor and reduced width compared to periodic stacks (Voronov et al., 2010).

The generalized spectral method (projection method) embeds the 1D quasiperiodic structure into a higher-dimensional (e.g., 2D) periodic superspace, mapping the eigenproblem into a matrix representation that allows analysis of spectral features, gap indices, and participation ratios (Quan et al., 10 Jan 2026).

4. Topological Classification and Pumping

The topological classification of 1D PQCs is rooted in dimensional extension—associating synthetic parameters (often phase shifts or “phasons”) that correspond to a second momentum coordinate, thus mapping the system to a higher-dimensional (2D) ancestor Hamiltonian. The seminal connection is to the 2D Quantum Hall problem:

• Mapping: The 1D Fibonacci chain is the dimensional reduction of a 2D “Fibonacci–Hall” model, where the geometric flux in the cut-and-project method plays the role of the 2D magnetic flux (Jagannathan, 14 Jan 2026).
• Chern numbers: Gaps of the 1D PQC inherit Chern numbers from the 2D ancestor; these are calculated from the Berry curvature over the synthetic Brillouin zone. Gap-labeling theorems provide the precise mapping: each gap’s integrated density of states relates linearly to the irrational frequency and carries a topological invariant (Jagannathan, 14 Jan 2026, Verbin et al., 2014).
• Thouless pumping: By adiabatically varying the synthetic phase parameter (e.g., introducing a running phason or “flux”) along the photonic propagation axis in a waveguide array, robust boundary modes can be pumped across the sample—manifested as quantized, topologically protected transport of light. Experimental realizations sweep the deformation parameter $\beta$ between Harper and Fibonacci limits to allow practical pumping despite the sharply localized edge states in the pure Fibonacci limit (Verbin et al., 2014).

5. Experimental Realizations

Laboratory implementations of 1D PQCs span a range of material platforms and fabrication strategies:

• Dielectric multilayers: PECVD, e-beam evaporation, sputtering, and pulsed-laser deposition are used to stack alternating high and low-index materials (e.g., Si/SiO₂, TiO₂/SiO₂), with precise thickness control monitored by quartz-crystal microbalance. Quasiperiodic patterns may be implemented by direct control of deposition time or subsequent etch cycles (Bellingeri et al., 2017, Vaidya et al., 2022).
• Photonic waveguide arrays: Femtosecond-laser direct writing inscribes single-mode waveguides with tunable spacing to realize site-to-site couplings as prescribed by the desired quasiperiodic modulation. Synthetic parameter control (e.g., phason sweeping, deformation from Harper-to-Fibonacci) enables observation of quantized light pumping and boundary state localization (Verbin et al., 2014).
• Superconductor–metamaterial multilayers: Structures combining negative-index metamaterial and superconducting layers arranged in, e.g., Dodecanacci order, are analyzed in the microwave regime. Temperature dependence of the superconducting penetration depth allows spectral tuning of the photonic bands (Nayak et al., 2020).
• Quantum-well structures: Doped multiple quantum wells arranged in quasiperiodic order exhibit both the Mahan singularity in absorption and a characteristic double-peaked reflection spectrum, with self-similar sidebands and substantial spectral asymmetry due to doping (Voronov et al., 2010).

Measurement techniques include transmission/reflection spectroscopy, time-domain pulse propagation, and direct imaging of intensity distributions at sample output.

6. Spectral Engineering, Tuning, and Applications

Systematic control and engineering of spectral features in 1D PQCs are achievable via manipulation of sequence type, inflation generation, refractive index contrast, defect introduction, and cluster statistics of high-index layers (Bellingeri et al., 2017, Ghulinyan, 2015):

• Tailoring gaps and resonances: Inflation order dictates the density and width of pseudo-gaps; higher order ($N$) yields finer spectral structure. Defect layers or controlled clustering can localize states or produce narrow pass-bands for filters.
• Thermal/structural tunability: Structures incorporating superconducting or index-tunable layers enable dynamic spectral shifts (e.g., temperature-induced red-shift and narrowing of gaps) (Nayak et al., 2020).
• Reentrant transitions: Varying the amplitude of the quasiperiodic modulation can induce controllable transitions between extended, localized, and reentrant extended states, useful for dynamic state confinement or high-$Q$ nanocavity engineering (Vaidya et al., 2022).
• Photonic devices: PQCs enable applications in ultra-narrow spectral filtering, high-$Q$ cavity formation, distributed feedback, slow-light delay lines, multi-line lasers, and sensors exploiting the fractal “fingerprint” of transmission spectra (Ghulinyan, 2015, Bellingeri et al., 2017).

7. Theoretical and Physical Significance

1D PQCs reveal the interface between order, aperiodic order, and randomness in wave physics. They realize deterministic structures with multifractal spectra, host critical states neither fully localized nor fully extended, and embody topological phases via dimensional lifting and adiabatic continuity to established models such as the quantum Hall effect (Jagannathan, 14 Jan 2026). The unique confluence of spectral properties, tunable localization, and topological transport in 1D PQCs continues to inspire both experimental exploration and theoretical developments in photonics, condensed matter, and beyond.