Isospectral and square-root Cholesky photonic lattices

Cholesky factorization provides photonic lattices that are the isosectral partners or the square root of other arrays of coupled waveguides. The procedure is similar to that used in supersymmetric quantum mechanics. However, Cholesky decomposition requires initial positive-definite mode-coupling matrices and the resulting supersymmetry is always broken. That is, the isospectral partner has the same range as that of the initial mode-coupling matrix. It is possible to force a decomposition where the range of the partner is reduced but the characteristic supersymmetric intertwining is lost. As an example, we construct a Cholesky isospectral partner and the square root of a waveguide necklace with cyclic symmetry. We use experimental parameters from the telecommunication $C$ band to construct a finite-element model of these Cholesky photonic lattices to good agreement with our analytic prediction.

Figure 1

Finite-element modeling of normal modes (a), (b) of a two-waveguide necklace and (c), (d) its isospectral partner. The propagation constant is (a)–(c) ${\beta }_a=5.87648\times {10}^6\mathrm{rad}/\mathrm{m}$ for asymmetric and (b)–(d) ${\beta }_s=5.87731\times {10}^6\mathrm{rad}/\mathrm{m}$ for symmetric modes; see text for more details.

Figure 2

Sketch of (a) the square-root Cholesky lattice associated to a four-element necklace (note the negative coupling), and (b) its nine-waveguide realization; see text for more details.

Figure 3

(a) Propagation constant ${\beta }_j$ obtained from the analytic Cholesky square-root array with negative coupling strength (triangles), its analytic nine-waveguide realization (circles), and its finite-element model simulation using comsol (diamonds) and our in-house Mathematica routines (squares). (b) Fidelity overlap between analytic and numerical normal modes, ${\mathcal{F}}_{j,k}=|{\mathbf{a}}_j^{*}·{\mathbf{n}}_k|$.

Figure 4

Propagation constant ${\alpha }_j$ for the effective matrix $\mathbf{M}(\epsilon ,g)$, its partner $\mathbf{P}$, and its Cholesky square-root array ${\mathbf{H}}_X$ for (a) broken SUSY, $\epsilon >0$, and (b) forcing a pseudo-zero-energy mode, $\epsilon =0$.