Self-Similar Photonic Lattices

• Self-similar photonic lattices are photonic structures generated by recursive rules such as inflation or fractal iteration, yielding scale-invariant architectures.
• They differentiate between geometric self-similarity, which defines the lattice’s recursive structure, and dynamical self-imaging, where light reconstructs itself via spectral commensurability.
• Applications include controlling light localization, studying anomalous diffusion in fractal baths, and designing quasicrystalline photonic patterns based on quadratic quasilattices.

Searching arXiv for the cited works to ground the article in the current literature. Self-similar photonic lattices are photonic structures in which the geometry, spacing law, or coupling graph is generated by inflation, recursive substitution, or fractal iteration, so that the lattice reproduces itself under a change of scale. In the literature considered here, that strict structural meaning is distinct from two neighboring usages: dynamical self-reproduction, where a field reconstructs itself during propagation in an otherwise regular lattice, and self-induced lattice formation, where the optical field writes a transient photonic structure into a medium and is subsequently localized by it. The distinction is substantive: quasilattice inflation rules and fractal graphs address geometric self-similarity directly, whereas Talbot revivals, Bloch-Zener self-imaging, and plasma-lattice self-action concern recurrence or self-organization without recursive lattice geometry (Boyle et al., 2016, Bönsel et al., 22 May 2026).

1. Terminology and scope

In the cited work, “self-similar photonic lattices” appears in more than one technical sense. The most restrictive meaning is geometric: the lattice itself is generated recursively or by an inflation rule, as in quadratic one-dimensional quasilattices or fractal photonic graphs. A second meaning is dynamical: the field profile reproduces itself after propagation because of commensurate modal phases, even though the underlying lattice is not self-similar. A third, broader neighboring notion is self-induced or spatially variant lattice formation, where the structure is generated optically or prescribed as a position-dependent interference pattern, but does not repeat across scales.

Usage in the literature	Defining feature	Representative works
Structural self-similarity	Inflation, recursion, fractal generation	(Boyle et al., 2016, Bönsel et al., 22 May 2026)
Dynamical self-imaging	Recurrent field reconstruction	(Ramezani et al., 2012, Bender et al., 2015)
Self-induced or spatially variant lattices	Self-organization or position-dependent periodicity	(Kupfer et al., 2013, Kumar et al., 2016)

This taxonomy is not merely semantic. Structural self-similarity determines the graph on which transport, localization, and spectral measures are defined. By contrast, dynamical self-imaging is governed by phase commensurability in the propagation constants, and self-induced lattices are controlled by nonlinear medium response and optical interference. The cited literature repeatedly emphasizes that these should not be conflated.

2. Geometric self-similarity: quasilattices and fractal photonic graphs

A rigorous one-dimensional generator of self-similar lattices is provided by quadratic quasilattices with two spacings, $L>S>0$, arranged by the closed-form “floor form”

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$

where $0<\kappa<1$ is irrational (Boyle et al., 2016). Because

$x_{n+1}-x_n \in \{S,L\},$

the structure is binary but aperiodic. The same work gives two equivalent geometric constructions: dualization of a one-dimensional bi-grid and a cut-and-project construction from a two-dimensional lattice using projection of the midpoints of intersected parallelograms. It further distinguishes three equivalence notions—lattice equivalent, self-similar, and self-same—and derives explicit transformation laws under inflation. Self-similarity is encoded by a unimodular integer matrix

$$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$

with expanding eigenvalue

$$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$

and the work tabulates ten special self-similar one-dimensional quasilattices relevant for Ammann patterns and Penrose-like tilings (Boyle et al., 2016).

That paper is mathematical rather than photonic, but it explicitly identifies these quasilattices as building blocks for two-, three-, and higher-dimensional quasicrystalline tessellations. This makes them directly relevant as exact spacing laws for deterministic aperiodic photonic chains, line families in quasicrystalline photonic patterns, or other binary-spacing photonic architectures. The important point is that self-similarity there is intrinsic to the lattice rule itself, not to the propagation dynamics through a separate uniform structure.

A distinct but directly photonic realization of structural self-similarity appears in fractal baths formed by self-similar graphs of coupled resonators or waveguides, including Sierpiński gaskets, the Sierpiński pyramid, the Vicsek graph, and Sierpiński carpets (Bönsel et al., 22 May 2026). These are generated recursively by iteration of a finite substitution rule, and the work distinguishes nested finitely ramified fractals from infinitely ramified carpets. In that setting, the relevant notion of distance is the chemical distance $d(\mathbf r,\mathbf r')$, and the photonic Hamiltonian is treated as a graph operator. Here, self-similarity is again geometric and multiscale, but the physical consequences are analyzed through Green’s functions and anomalous diffusion rather than through inflation rules alone.

3. Localization and impurity physics in fractal photonic lattices

The most explicit physical theory for self-similar photonic lattices in the strict sense is a waveguide-QED treatment of atom-photon bound states in fractal baths (Bönsel et al., 22 May 2026). The starting point is a two-level emitter coupled to a structured bosonic bath. In a translationally invariant periodic lattice near a quadratic band edge, one would expect

$\xi\sim \Delta^{-1/2}.$

The fractal case lacks translational invariance, lacks a good quasimomentum, and generally lacks a band-edge effective-mass expansion. The cited work replaces Bloch reasoning with a real-space Green’s-function formulation written as a Laplace transform of the heat kernel: $$G(\mathbf r)=-\int_0^\infty d\tau\, e^{-\Delta\tau}K(\mathbf r,\tau).$$

Using sub-Gaussian heat-kernel bounds, the paper shows that the far-field localization length is controlled by the walk dimension $d_w$ of anomalous diffusion,

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$0

rather than by band curvature (Bönsel et al., 22 May 2026). This scaling is one of the clearest signatures that self-similar photonic lattices define a distinct localization regime. The far-field photonic component obeys an exponential envelope in chemical distance with an algebraic prefactor,

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$1

where $$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$2 is the spectral dimension. In the near field, the Green’s-function difference scales algebraically. For nested finitely ramified fractals the exponent is

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$3

with $$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$4 the fractal dimension, and the resulting near-field behavior agrees with resistance/first-passage scaling. Sierpiński carpets deviate from this simple law and numerically appear closer to the marginal two-dimensional logarithmic trend (Bönsel et al., 22 May 2026).

A central technical point is that the universal heat-kernel framework applies to Laplacian-type operators, not generically to a bare adjacency Hamiltonian. The work therefore “renders the bath Hamiltonian Laplacian-like” by compensating local connectivity inhomogeneity with on-site terms,

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$5

On regular Euclidean lattices, the distinction between $$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$6 and $$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$7 is an irrelevant constant shift; on fractals with strongly varying site degree, it is not. This identifies an explicit photonic design principle: if one wants impurity physics controlled by fractal diffusion exponents, degree-dependent on-site compensation is essential (Bönsel et al., 22 May 2026).

4. Dynamical self-imaging in non-self-similar lattices

A separate literature studies self-imaging in photonic lattices whose geometry is not self-similar. In a $$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$8-symmetric dimerized waveguide chain with gain-loss dimers, the governing dispersion is

$$x_n=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor,$$9

with threshold

$0<\kappa<1$0

(Ramezani et al., 2012). Talbot revivals require a periodic input

$0<\kappa<1$1

which discretizes the Bloch momenta to

$0<\kappa<1$2

and exact revival occurs only when the corresponding eigenvalues are rationally commensurate. At the spontaneous $0<\kappa<1$3-symmetry-breaking point, exact Talbot revivals occur only for $0<\kappa<1$4; $0<\kappa<1$5 is excluded because it excites the defective $0<\kappa<1$6 mode. In the exact phase, $0<\kappa<1$7 can revive for suitable $0<\kappa<1$8, and the Talbot length becomes $0<\kappa<1$9-dependent (Ramezani et al., 2012). The recurring pattern is thus a self-imaged field, not a self-similar lattice.

A related but distinct mechanism appears in locally $x_{n+1}-x_n \in \{S,L\},$0-symmetric dimer lattices with a transverse index gradient $x_{n+1}-x_n \in \{S,L\},$1 (Bender et al., 2015). There the key propagation scales are the Bloch period

$x_{n+1}-x_n \in \{S,L\},$2

and the inter-ladder phase period

$x_{n+1}-x_n \in \{S,L\},$3

Stable Bloch-Zener oscillations occur in the exact $x_{n+1}-x_n \in \{S,L\},$4-symmetric phase, and self-imaging arises when these scales are commensurate,

$x_{n+1}-x_n \in \{S,L\},$5

The recurrent evolution consists of wavepacket splitting, Zener tunneling at

$x_{n+1}-x_n \in \{S,L\},$6

and giant recombinations, which can be enhanced by a localized defect (Bender et al., 2015). This is a paradigmatic example of dynamical self-reproduction engineered spectrally rather than geometrically.

The conceptual boundary is explicit in both works. Neither introduces a fractal hierarchy, inflation rule, or scale-invariant lattice geometry. Their relevance to self-similar photonic lattices is therefore indirect: they show how self-reconstruction can emerge from spectral arithmetic even in structurally simple arrays.

5. Self-induced and spatially variant photonic lattices

Self-induced plasma photonic lattices provide another neighboring concept. In underdense plasma driven by two pairs of counterpropagating femtosecond Gaussian laser pulses, the interference pattern of the pulses creates a ponderomotive-optical lattice, and the cycle-averaged relativistic ponderomotive force

$x_{n+1}-x_n \in \{S,L\},$7

redistributes electrons into a periodic plasma structure (Kupfer et al., 2013). The same pulses both generate the plasma-index modulation and are later localized by it, so the lattice is “self-induced.” For a uniform initial electron density $x_{n+1}-x_n \in \{S,L\},$8, formation begins around $x_{n+1}-x_n \in \{S,L\},$9, the initial lattice spacing is $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$0, and near peak interaction half the peaks disperse, leaving an effective lattice constant of $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$1. The peak plasma density reaches about $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$2, slightly below critical density (Kupfer et al., 2013).

The optical consequence is transient light localization by the plasma-index lattice. At $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$3, the field is confined near the interference center with a $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$4 intensity enhancement relative to the surrounding region; at $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$5, the enhancement reaches $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$6. The work also reports defect-like mode displacement under an imposed density bump or dip, and preferred-direction emission under a linear density gradient of slope $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$7 over $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$8, while the center still experiences more than $$\tau=\begin{pmatrix}a&b\ c&d\end{pmatrix},$$9 intensity enhancement (Kupfer et al., 2013). Yet the same paper explicitly states that there is no recursive hierarchy of scales, no fractal law, and no renormalization-like scaling. The result is better classified as pattern formation in a nonlinear plasma-optical system than as self-similar lattice generation.

Spatially variant lattices fabricated by interference lithography provide a second neighboring case (Kumar et al., 2016). There the local lattice orientation $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$0 and period $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$1 are prescribed as continuous functions of position, and the local reciprocal vectors are

$$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$2

Because $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$3 varies spatially, the phase cannot be written as $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$4; instead one must solve an integrated-gradient problem,

$$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$5

using finite differences and a least-squares solve (Kumar et al., 2016). The resulting phase-only kinoform is displayed on a phase-only SLM in a $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$6 Fourier filtering system. The demonstrated example is a square lattice whose orientation changes from $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$7 to $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$8 as per the azimuth angle, with the stated possibility of guiding a Gaussian beam through a $$\lambda_{\parallel}=\frac12\left[a+d+\sqrt{(a+d)^2-4(ad-bc)}\right],$$9 bend by photonic crystal self-collimation. This is digitally reconfigurable and completely scalable, but it is not fractal or scale-invariant (Kumar et al., 2016).

Together, these works delimit the concept sharply. Self-induced plasma lattices and spatially variant lattices are programmable, emergent, or reconfigurable; they are not self-similar in the strict recursive sense.

6. Scalable structured families, isospectral reshaping, and conceptual boundaries

Two additional lines of work broaden the design space around self-similar photonic lattices without directly entering it. One realizes all-band-flat photonic lattices of arbitrary size by engineering a finite honeycomb lattice with site-dependent couplings that mimic a bosonic Fock-state lattice (Yang et al., 2023). The sites are indexed by nonnegative integers constrained by

$d(\mathbf r,\mathbf r')$0

with $d(\mathbf r,\mathbf r')$1 on the A sublattice and $d(\mathbf r,\mathbf r')$2 on the B sublattice, so the total number of resonators is

$d(\mathbf r,\mathbf r')$3

The couplings obey a square-root law,

$d(\mathbf r,\mathbf r')$4

and analogously in the other two bond directions, leading to the Hamiltonian

$d(\mathbf r,\mathbf r')$5

Its spectrum collapses into flat levels

$d(\mathbf r,\mathbf r')$6

with degeneracy $d(\mathbf r,\mathbf r')$7 (Yang et al., 2023). The lattice is scalable and rule-based, but the paper explicitly does not claim fractal geometry, exact scale invariance, or recursive self-similarity.

Another neighboring development uses discrete supersymmetry to reshape one-dimensional $d(\mathbf r,\mathbf r')$8 lattices into compact two-dimensional isospectral partner lattices (Wolterink et al., 2022). A conventional $d(\mathbf r,\mathbf r')$9 chain has couplings

$\xi\sim \Delta^{-1/2}.$0

and an equidistant eigenvalue ladder of spacing $\xi\sim \Delta^{-1/2}.$1. By repeated SUSY transformations and inverse reattachment in an orthogonal direction, the resulting two-dimensional partner has Hamiltonian described by a Kronecker sum

$\xi\sim \Delta^{-1/2}.$2

or more generally

$\xi\sim \Delta^{-1/2}.$3

These arrays preserve the spectrum of the original one-dimensional lattice and therefore preserve perfect imaging and perfect state transfer at

$\xi\sim \Delta^{-1/2}.$4

while exhibiting different transient dynamics (Wolterink et al., 2022). This is iterative isospectral reshaping, not self-similarity.

Taken together, these neighboring literatures suggest a precise conceptual boundary. Strict self-similar photonic lattices are those in which scale recursion is built into the geometry or graph, as in fractal photonic baths or inflation-generated quasilattices. Dynamical self-imaging, self-induced cavity formation, spatial grading, Fock-state coupling laws, and SUSY isospectrality are technically adjacent because they also produce recurrent, programmable, or multiscale-looking behavior, but they do not by themselves define self-similar lattices. A plausible implication is that future work may combine these currently separate design logics—recursive geometry, anomalous-diffusion control, spectral commensurability, and coupling-law engineering—within a single photonic platform, but the cited literature treats them as distinct categories rather than as a unified theory (Bönsel et al., 22 May 2026, Boyle et al., 2016).