Explicit Construction of Floquet-Bloch States from Arbitrary Solution Bases of the Hill Equation

Abstract

For the Hill equation describing one-dimensional periodic systems, a constructive formulation is developed for generating Floquet-Bloch states directly from arbitrary pairs of linearly independent solutions. One-dimensional photonic crystals are used as a concrete illustration. Explicit closed-form formulas map an arbitrary fundamental system to the corresponding Floquet-Bloch basis via the monodromy matrix, including the generic Jordan band-edge case, without reliance on canonically normalized solutions. The construction can be expressed directly in terms of the transfer matrix, making the residual representation freedom transparent and providing an implementation-ready framework for analytical and numerical studies of periodic systems.

Figures (4)

• Figure 1: Schematic representation of a binary photonic crystal, deposited on a substrate with refractive index $n_s$. The light of vacuum wavenumber $k$ impinges on the crystal parallel to its axis $z$.
• Figure 2: Transmittance and $\cos(\mu d)$ for infrared light incident normally from air ($n_i = 1$) on a binary photonic crystal of $N = 6$ periods. Each period consists of Ge ($n_1 = 4.0$, $d_1 = 0.55~\mu m$) and ZnS ($n_2 = 2.2$, $d_2 = 1.00~\mu m$) layers, deposited on a glass substrate with refractive index $n_s = 1.5$. The Bloch wavenumber $\mu$ is obtained from Eq. (52). Allowed bands ($\lvert \cos(\mu d) \rvert < 1$) and bandgaps ($\lvert \cos(\mu d) \rvert > 1$) are clearly observed in the transmission spectrum. The second bandgap is suppressed due to condition $n_1d_1 = n_2d_2$.
• Figure 3: Absolute values of the Floquet-Bloch waves over the first three periods of the binary photonic crystal, with parameters as in Fig. 2, for a representative wavenumber $k = 0.53~\mu m^{-1}$ in an allowed band. The states $F_{1,2}^{(\mathrm{n})}(z)$, shown by black lines, are constructed using Choice A (identity fundamental matrix), and the states $F_{1,2}^{(\mathrm{pw})}(z)$, shown by thick gray lines, are constructed using Choice B (traveling-wave fundamental matrix). The two constructions differ only by constant complex factors $\alpha_{1,2}$, in full agreement with the invariance analysis of Section \ref{['sec:change']}. The states $F_{1}^{(\rm pw)}(z)$ and $F_{2}^{(\rm pw)}(z)$ form a complex-conjugate pair in allowed bands; as a result, their absolute values coincide. Vertical dotted lines mark unit-cell boundaries at $z = d, 2d, 3d$, with $d = d_1 + d_2$.
• Figure 4: Absolute values of the Floquet-Bloch waves over the first three periods of the binary photonic crystal, with parameters as in Fig. 2, for a representative wavenumber $k = 0.83~\mu m^{-1}$ in a bandgap. In this regime the Bloch wavenumber $\mu$ is complex, and one Floquet-Bloch wave is evanescent while the other grows exponentially. As expected, the profiles obtained using Choices A and B differ only by constant complex factors $\alpha_{1,2}$, in accordance with the invariance analysis of Section \ref{['sec:change']}.