Fibonacci Waveguides

• Fibonacci waveguides are one-dimensional photonic structures whose material or geometric parameters follow Fibonacci substitution rules, resulting in fractal spectral gaps and multifractal modes.
• They are analyzed using transfer-matrix and tight-binding formalisms that elucidate quasiperiodic wave propagation and energy localization in various experimental realizations.
• Experimental implementations demonstrate controlled defect engineering, topological pumping, and tailored transport properties, highlighting the versatility of Fibonacci ordering in photonic design.

Searching arXiv for the cited Fibonacci-waveguide and related quasicrystal papers to ground the article in fresh source metadata. arXiv search: "Fibonacci waveguides photonic quasicrystals experimental observation multifractality Fibonacci chains topological pumping" Fibonacci waveguides are wave-propagation media whose constitutive segments, couplings, or spacings are arranged according to a Fibonacci substitution tiling rather than a periodic lattice. In the strict one-dimensional setting, a “Fibonacci waveguide” can be defined as any one-dimensional wave-propagation medium whose cross-sectional or material parameters are arranged along the axis according to a generalised Fibonacci substitution tiling (Davies et al., 2023). In photonics, this notion encompasses dielectric multilayers, coupled optical-waveguide arrays, microwave resonator chains, and related quasiperiodic media in which the elementary building blocks $A$ and $B$ are concatenated by the classical substitution $A\to AB,\;B\to A$ or by generalised Fibonacci–Lucas rules. Across these realizations, Fibonacci order produces a fractal or Cantor-like spectral organization, pseudo-gaps or super band gaps, critical or multifractal modes, and transport properties that differ sharply from both periodic and disordered systems (Ghulinyan, 2015).

1. Fibonacci ordering and geometric construction

The classical Fibonacci word is generated from two symbols $A$ and $B$ by the substitution rule

$S:\;A\to AB,\;B\to A,$

with successive approximants such as $A$, $AB$, $ABA$, $ABAAB$, and so forth (2207.13755). An equivalent recursion widely used in photonic multilayers is

$B$0

with associated lengths $B$1, $B$2, $B$3 (Ghulinyan, 2015). In the quasiperiodic limit, the ratio of symbol counts approaches the golden-mean-derived constant $B$4 (2207.13755).

In one-dimensional photonic quasicrystals, the letters $B$5 and $B$6 are mapped to dielectric layers of refractive indices $B$7 and thicknesses $B$8. A standard quarter-wave construction imposes

$B$9

so that a high-order approximant such as $A\to AB,\;B\to A$0 contains 377 layers ordered by the Fibonacci word (Ghulinyan, 2015). In coupled-waveguide implementations, the same symbolic sequence usually modulates nearest-neighbour bond lengths or couplings rather than layer composition. In finite photonic chains, distinct inter-waveguide separations $A\to AB,\;B\to A$1 and $A\to AB,\;B\to A$2 realize couplings $A\to AB,\;B\to A$3 or $A\to AB,\;B\to A$4 through an approximately exponential law $A\to AB,\;B\to A$5 (Ghosh et al., 13 May 2026).

Generalised constructions replace the classical rule by

$A\to AB,\;B\to A$6

yielding generalised Fibonacci numbers

$A\to AB,\;B\to A$7

with precious-mean and metal-mean subclasses as special cases (Davies et al., 2023). This broader definition accommodates structured rods, mass–spring systems, multi-supported beams, and other one-dimensional waveguiding media described by $A\to AB,\;B\to A$8 transfer matrices.

Two-dimensional Fibonacci lattices are constructed by combining independent one-dimensional Fibonacci words along the transverse $A\to AB,\;B\to A$9- and $A$0-axes. In optically induced arrays, adjacent waveguides are separated by two distances $A$1 and $A$2, where $A$3, and the coordinates are generated by cumulative sums of the corresponding letters (Boguslawski et al., 2015). This produces deterministic aperiodicity with a pure-point Fourier spectrum, while still hampering transverse transport relative to periodic lattices.

2. Governing models and analytical formalisms

Two complementary descriptions dominate the analysis of Fibonacci waveguides: transfer-matrix formalisms for continuous or layered media, and tight-binding Hamiltonians for coupled resonators or waveguides.

For layered photonic quasicrystals at normal incidence, a single dielectric layer $A$4 is represented by the transfer matrix

$A$5

and the total matrix of a Fibonacci stack is the ordered product $A$6 (Ghulinyan, 2015). Because the Fibonacci word obeys concatenation, the matrices satisfy

$A$7

The corresponding traces $A$8 obey the trace map

$A$9

which is central to the spectral theory of Fibonacci multilayers (Ghulinyan, 2015).

An analogous transfer-matrix approach appears in continuous quasicrystalline waveguides governed by the one-dimensional Helmholtz equation

$B$0

For a Fibonacci medium with piecewise-constant wave speed $B$1, the transfer matrices over the unit $B$2- and $B$3-segments are

$B$4

and the traces $B$5 again satisfy

$B$6

(Davies et al., 2022).

In coupled-resonator and waveguide chains, the standard model is a nearest-neighbour tight-binding Hamiltonian

$B$7

with $B$8 chosen by the Fibonacci word (2207.13755). In optical-waveguide realizations under the paraxial approximation, Maxwell’s equations reduce to a Schrödinger-type evolution in propagation distance $B$9,

$S:\;A\to AB,\;B\to A,$0

which maps directly to tight-binding dynamics after mode expansion (Ghosh et al., 13 May 2026). This mapping is also used in optically induced two-dimensional Fibonacci lattices, where the weak probe field satisfies a paraxial Schrödinger-type equation in a refractive-index landscape written by nondiffracting Bessel beams (Boguslawski et al., 2015).

A specialized construction relevant to the analysis of finite Fibonacci chains is conumbering, in which each site is assigned a “conumber” by projecting the cut-and-project construction onto the perpendicular direction. Conumbering reorders sites by local environment, placing atomic sites near the centre of the perpendicular axis and molecular sites at the edges, thereby exposing the hierarchical local density of states (LDOS) structure (2207.13755).

3. Spectra, gaps, and self-similarity

The most distinctive spectral feature of Fibonacci waveguides is the replacement of ordinary Bloch bands by hierarchically fragmented pass and stop regions. In periodic systems the Bloch condition gives

$S:\;A\to AB,\;B\to A,$1

with gaps where $S:\;A\to AB,\;B\to A,$2 (Ghulinyan, 2015). For Fibonacci approximants one may still define an effective Bloch wave number $S:\;A\to AB,\;B\to A,$3 through

$S:\;A\to AB,\;B\to A,$4

and forbidden pseudo-gaps are those intervals where $S:\;A\to AB,\;B\to A,$5 in the limit $S:\;A\to AB,\;B\to A,$6 (Ghulinyan, 2015).

In the generalized setting, a super band gap is a frequency $S:\;A\to AB,\;B\to A,$7 that remains in a gap for every periodic approximant beyond some index. Equivalently, if there exists $S:\;A\to AB,\;B\to A,$8 such that

$S:\;A\to AB,\;B\to A,$9

then $A$0 lies in the super band gap $A$1 (Davies et al., 2023). A sufficient condition is

$A$2

For the classical golden-mean case, the growth conditions

$A$3

force monotone growth and therefore membership in a super band gap (Davies et al., 2023). This formalizes the long-used observation that periodic approximants faithfully reproduce the main persistent gaps of the underlying quasicrystal.

In layered photonic Fibonacci structures, the transmission spectrum $A$4 exhibits a cascade of deepening pseudo-gaps whose widths follow power-law scaling $A$5, and within each gap the number of mini-resonances grows as $A$6, reproducing on successive scales (Ghulinyan, 2015). The density of states likewise splits into narrow spikes around critical modes and vanishing plateaux in the gaps (Ghulinyan, 2015). In finite Fibonacci-type photonic chains, the spectrum versus phason angle shows the characteristic fractal, Cantor gaps of a Fibonacci quasicrystal, with defect states appearing as non-dispersing lines inside a gap (Ghosh et al., 13 May 2026).

Microwave realizations of the Fibonacci hopping model directly confirm the hierarchical spectrum in a finite platform. The conumber-reordered LDOS reveals self-similar and recursive embeddings of atomic and molecular bands, and the measured LDOS is well described by renormalization formulas in the strong- and weak-coupling-dominant limits (2207.13755). This establishes that the fractal organization is not only an asymptotic property of infinite models but also experimentally accessible in finite quasiperiodic devices.

4. Critical states, multifractality, and transport

Fibonacci waveguides are associated with eigenstates that are typically neither Bloch-extended nor exponentially localized. In the microwave-resonator realization of the Fibonacci tight-binding chain, multifractality is quantified through the mass exponents

$A$7

with generalized fractal dimensions

$A$8

and a singularity spectrum obtained by the Legendre transform

$A$9

(2207.13755). Experimentally, box-counting on the measured $AB$0 LDOS and a Chhabra–Jensen analysis of the site intensities yield $AB$1 and $AB$2 in good agreement with perturbative theory for $AB$3, with deviations at large $AB$4 attributed mainly to finite size and discrepancies at $AB$5 to low-intensity noise floor (2207.13755).

A related characterization appears in Fibonacci waveguide quantum electrodynamics, where the photonic spectrum of an aperiodic resonator array is singular continuous and the eigenstates are critical. There the inverse participation ratio scales as

$AB$6

signalling a state that is neither fully extended nor localized (Bönsel et al., 8 Jul 2025). The singularity spectrum $AB$7 is continuous over a finite interval of $AB$8, unlike the trivial cases associated with uniform bands or pure-point localization (Bönsel et al., 8 Jul 2025).

Transport reflects this intermediate spectral character. In optically induced two-dimensional Fibonacci lattices, transverse light transport is significantly hampered relative to a periodic square lattice, even though the Fibonacci structure remains highly ordered and has a pure-point Fourier spectrum (Boguslawski et al., 2015). Averaged over 36 input sites, the effective beam width grows more slowly in the Fibonacci array than in the periodic array, reaching $AB$9 at $ABA$0, compared with $ABA$1 in the periodic case (Boguslawski et al., 2015). The stated mechanism is the breaking of translational invariance, which fragments minibands and generates pseudo-band-gaps and localized eigenmodes (Boguslawski et al., 2015).

Coupled-chain Fibonacci systems sharpen this point by exhibiting several transport classes in a single architecture. Depending on the coupling scheme, two coupled Fibonacci chains may display a richer hierarchical spectrum, a coexistence of Bloch and critical eigenstates, or a large number of degenerate compact localized states (Moustaj et al., 2022). In waveguide language, extended modes lead to ballistic spreading $ABA$2, critical modes to anomalous diffusion $ABA$3, $ABA$4, and compact states to diffraction-free confinement (Moustaj et al., 2022). This demonstrates that quasiperiodic order can be used not only to suppress transport globally but also to engineer coexistence between distinct transport channels.

5. Experimental realizations and measurement strategies

Fibonacci waveguides have been implemented in several experimentally distinct platforms, each emphasizing a different aspect of quasiperiodic wave physics.

In microwave experiments on Fibonacci hopping chains, the physical sites are high-index cylindrical dielectric rods made of TiZrNbZnO with refractive index $ABA$5, radius $ABA$6, and height $ABA$7, sandwiched between metal plates to enforce evanescent coupling (2207.13755). Each isolated resonator supports a TE$ABA$8-like mode near $ABA$9 with linewidth $ABAAB$0, and the couplings $ABAAB$1 are set by two centre-to-centre distances. Reflection $ABAAB$2 is measured with a loop antenna at each site and inverted by harmonic inversion to extract the resonance frequencies, linewidths, and mode intensities $ABAAB$3, from which the LDOS is assembled (2207.13755).

In femtosecond-laser-written photonic chains designed for end-to-end pumping, the Fibonacci sequence is encoded in bond lengths $ABAAB$4 and $ABAAB$5 in an array of $ABAAB$6 waveguides of length $ABAAB$7 (Ghosh et al., 13 May 2026). The devices are fabricated in glass using a 1030 nm laser with 350 fs pulse duration and 1 MHz repetition rate; the resulting waveguides are single-mode at 780 nm, with mode-field diameter $ABAAB$8, $ABAAB$9, and loss $B$00 (Ghosh et al., 13 May 2026). The input and output are coupled and imaged at 780 nm, and robustness is tested by replacing selected bonds with an intermediate spacing $B$01 (Ghosh et al., 13 May 2026).

Optically induced two-dimensional Fibonacci lattices use a different methodology. A Ce:SBN photorefractive crystal is written sequentially by a set of zero-order nondiffracting Bessel beams centred on the Fibonacci lattice sites, producing an effective intensity

$B$02

and an induced index change

$B$03

under an external bias field $B$04 (Boguslawski et al., 2015). Probe-beam localization is then quantified through the inverse participation ratio and effective width, with numerical propagation computed by a split-step Fourier method.

Continuous quasiperiodic waveguides based on reflection symmetry are analysed through finite-element or transfer-matrix calculations rather than direct fabrication data in the cited work, but the design pathway is explicit: one constructs a Fibonacci half-space, reflects it about $B$05, and solves for gap frequencies whose stable eigenvectors satisfy Dirichlet or Neumann conditions at the symmetry plane (Davies et al., 2022). This platform emphasizes the production of localized edge modes from a bulk quasicrystal rather than bulk transport alone.

A broader implication is that “Fibonacci waveguide” is not tied to a single fabrication technology. The same substitutional order has been realized or proposed in dielectric stacks, microwave metamaterial chains, femtosecond-written integrated photonics, photorefractive lattices, and related metamaterial or circuit platforms (Ghulinyan, 2015).

6. Waveguiding functions, defect engineering, and extensions

Fibonacci order supports several distinct waveguiding mechanisms. In conventional one-dimensional photonic quasicrystals, every occurrence of the substring “AA” in a Fibonacci stack acts as a half-wave cavity embedded in unbalanced quasiperiodic mirrors, producing minibands of critically localized modes with nonuniform spectral splitting and non-exponential temporal decay (Ghulinyan, 2015). The same structures support high-$B$06 nanocavities, slow-light states, multi-wavelength narrow-line lasing, optical filtering with fractal passbands, and refractometric sensing through pseudo-gap shifts (Ghulinyan, 2015).

A different mechanism arises from imposed reflection symmetry. When a Fibonacci medium on $B$07 is mirrored to produce $B$08, frequencies inside bulk spectral gaps can support decaying edge modes satisfying either $B$09 or $B$10, depending on parity (Davies et al., 2022). Their decay rate is controlled by the smaller eigenvalue of the hyperbolic transfer matrix, yielding an exponential envelope

$B$11

and numerical examples with contrast $B$12 and $B$13 show edge modes near $B$14 and $B$15 (Davies et al., 2022). Robustness comparisons indicate that Fibonacci guides may outperform simple periodic guides in some settings, although the advantage becomes less clear when decay rates are matched (Davies et al., 2022). This directly addresses a common misconception that quasiperiodicity is automatically more robust than periodicity in every comparable design.

Topological pumping in Fibonacci-type photonic chains uses yet another mechanism. In a finite $B$16 quasiperiodic chain, a defect-type state appears inside a spectral gap and can be transferred from one end to the other by a small structural change. In the photonic implementation, only two waveguides are bent so that selected bond lengths interpolate smoothly with propagation distance $B$17, and measured output intensities confirm near-unity transfer efficiency from the leftmost to the rightmost guide (Ghosh et al., 13 May 2026). The pumped state is described as a topologically protected defect or boundary mode whose robustness against global phason shifts and local bond defects stems from the topological gap-labelling theorem of Sturmian sequences (Ghosh et al., 13 May 2026).

Quantum-optical extensions show that the same aperiodic environment can mediate structured emitter–emitter interactions. In Fibonacci waveguide QED, giant emitters coupled to a Fibonacci resonator chain form vacancy-like dressed states only for specific coupling configurations determined by local Fibonacci subwords, and the resulting effective atomic Hamiltonian inherits the Fibonacci structure (Bönsel et al., 8 Jul 2025). In an SSH-like aperiodic variant generated by a $B$18-Lucas rule, off-resonantly coupled emitters interact through bound states with aperiodically modulated profiles, and the effective coupling matrix has a singular-continuous density of states and multifractal properties (Bönsel et al., 8 Jul 2025). This suggests that the quasiperiodic geometry can be transferred from the photonic bath to the induced many-body interactions.

The concept also extends beyond optics in the narrow sense. Generalised Fibonacci tilings have been used to analyse persistent gaps in rods, beams, and mass–spring chains (Davies et al., 2023), while quasiperiodic cylinder arrays based on Fibonacci geometries have been studied for water-wave transmission and guiding, where multiple narrow gaps and resonance-based wave steering emerge from the same substitutional order (Smerdon et al., 2024). A plausible implication is that “Fibonacci waveguide” denotes a structural principle rather than a material class: the same deterministic aperiodicity can organize wave transport whenever the underlying physics admits transfer matrices or tight-binding-type couplings.

7. Conceptual significance and open directions

Several broad conclusions recur across the literature. First, Fibonacci waveguides occupy an intermediate regime between periodic order and randomness. Their spectra are described as fractal, Cantor-like, or singular continuous; their eigenmodes are critical or multifractal rather than conventionally extended or exponentially localized; and their transport can be sub-ballistic, highly site-dependent, or defect-mediated (2207.13755). These properties are already visible in finite approximants and can be analysed with transfer-matrix trace maps, renormalization recursions, inverse-participation and singularity spectra, or direct LDOS measurements (Ghulinyan, 2015).

Second, periodic approximants are not merely numerical conveniences. The theory of super band gaps shows that sufficiently large periodic repetitions of finite Fibonacci words reproduce the persistent gaps of the infinite quasiperiodic system exactly in the relevant sense (Davies et al., 2023). This provides a rigorous foundation for an experimental practice that is common across photonic quasicrystals, namely the fabrication of finite or periodically repeated approximants rather than mathematically infinite structures.

Third, Fibonacci waveguides do not realize a single localization paradigm. Depending on platform and coupling scheme, they can support critical bulk states, compact localized states, reflection-induced edge states, defect minibands, or topological pumping states (Moustaj et al., 2022). Common shorthand descriptions such as “quasiperiodic localization” therefore require qualification: some devices exploit hierarchical pseudo-gaps, some exploit symmetry-induced defect confinement, and some exploit finite-size gap modes that are adiabatically transported.

Finally, the experimental literature points toward controlled departures from the ideal Fibonacci structure. The microwave-resonator platform is stated to extend naturally to controlled phason or bond disorder and to higher-dimensional codimension-1 tilings such as the Rauzy tiling (2207.13755), while integrated photonic chains already demonstrate robustness to local bond deformations (Ghosh et al., 13 May 2026). This suggests that future work will continue to treat Fibonacci waveguides not simply as static quasiperiodic samples, but as designable platforms for critical wave phenomena, robust state transfer, structured light confinement, and engineered effective Hamiltonians across classical and quantum regimes.