Electromagnetic modes of a disordered photonic crystal

We present a systematic numerical approach to compute the eigenmodes and the related eigenfrequencies of a disordered photonic crystal, characterized by small fluctuations of the otherwise periodic dielectric profile. The field eigenmodes are expanded on the basis of Bloch modes of the corresponding periodic structure, and the resulting eigenvalue problem is diagonalized on a truncated basis including a finite number of Bloch bands. The Bloch-mode expansion is very effective for modeling modes of very extended disordered structures in a given frequency range, as only spectrally close bands must be included in the basis set. The convergence can be easily verified by repeating the diagonalization for an increased band set. As illustrations, we apply the method to the W1 line-defect waveguide and to the L3 coupled-cavity waveguide, both based on a photonic crystal slab with a triangular lattice of circular holes. We compute and characterize the eigenfrequencies, spatial field profiles, and radiation loss rates of the localized modes induced by disorder. For the W1 waveguide, we demonstrate that radiation losses, at the bottom of the spatially even guided band, are determined by a small hybridization with Bloch modes of the spatially odd band, induced by disorder in spite of their frequency separation. The Bloch-mode expansion method has a very broad range of applications: It can be also used to accurately compute the modes of structures with systematic modulations of the periodic dielectric constant, as in several designs of high-$Q$ cavities.

Figure 1

(a) Sketch of the W1 line-defect waveguide. The shaded region denotes the supercell used for computing the bands of the regular structure. (b) Sketch of the disorder model used in this work. The dashed and full circles indicate holes of the regular and disordered structure respectively. (c) Band structure of the W1 waveguide with $d=0.5a$, $\epsilon =12$, $r=0.3$, and $b=10$. Band 1 (red) is the spatially even guided band. Bands labeled 1 to 4 are those that were included in the Bloch-mode expansion of the disordered structure.

Figure 2

Spectral function (see text) of the disordered W1 waveguide. (a) $\sigma =0.002a$, plotted on a $\log {}_{10}$ color scale. Bands 1 and 2 of the regular structure are also plotted (solid lines). The dotted line is the boundary of the light cone. (b)–(e) Detail (on a linear color scale) of the spectral function at the lower edge of band 1 for $\sigma =0.001a$, $\sigma =0.002a$, $\sigma =0.004a$, $\sigma =0.008a$, respectively.

Figure 3

(a) Density of states in the vicinity of the edge of band 1. Blue, $\sigma =0.002a$, averaged over 500 disorder realizations; red, the corresponding quantity for the regular structure, showing the characteristic Van Hove singularity at the band edge. (b) The spectral function and the band dispersion are plotted as a reference.

Figure 4

(a)–(c) Electric field spatial profile of three selected eigenmodes computed for $\sigma =0.002a$, at $\omega a/2\pi c=0.272\hspace{0.5em}67$, $0.272\hspace{0.5em}83$, and $0.275\hspace{0.5em}18$, respectively [indicated by horizontal lines in Fig. 2c]. The main panels display the field intensity computed at the center of each elementary cell, while insets show the full field profile for selected portions of the W1 waveguide.

Figure 5

(a) Radiation loss rates of disordered eigenmodes, computed for the four values of $\sigma$ considered in this work. Modes for which the intrinsic radiation loss mechanism is dominant are those showing a loss rate independent of $\sigma$. (b) Band dispersion (solid lines) and light cone (dashed line) are plotted for reference.

Figure 6

(a)–(c) Eigenfrequencies of four selected eigenmodes computed as a function of the number of bands included in the Bloch-mode expansion for $\sigma =0.002a$. (d) Computed radiation loss rates of the same modes, as a function of the number of included bands. (e) Coefficients of the Bloch-mode expansion $|U_{\beta }(k,n)|$ computed for the fundamental mode $\beta =1$ and $n=1,2,3,4$.

Figure 7

(a) Circles, IPN values of the lowest eigenmodes computed for one disorder realization with $\sigma =0.002a$; blue line, IPN averaged over 500 disorder realizations; red line, average attenuation length of the transmission along the guide. (Inset) Detail in the vicinity of the edge of band 1 (denoted by a vertical line). (b) The spectral function and the band dispersion are plotted as a reference.

Figure 8

(a) Sketch of the L3 coupled-cavity waveguide. The shaded region denotes the supercell used for computing the bands of the regular structure. (b) Spectral function computed for $\sigma =0.002a$, showed as a linear scale color plot. The dispersion of the guided band of the regular structure is plotted as a white line. (c) Radiation loss rates of disordered eigenmodes, computed for the four values of $\sigma$ considered in this work.

Figure 9

(a)–(c) Electric field spatial profile of three selected eigenmodes computed for $\sigma =0.002a$. The three corresponding values of $\omega a/2\pi c$ are denoted by horizontal lines in Fig. 8b. The main panels display the field intensity computed at the center of each elementary cell, while insets show the full field profile for selected portions of the L3 coupled-cavity waveguide.

Figure 10

(a) Circles, IPN values of the eigenmodes computed for one disorder realization with $\sigma =0.002a$; blue line, IPN averaged over 500 disorder realizations; red line, average attenuation length of the transmission along the guide. (Inset) Detail in the vicinity of the lower band edge (denoted by a vertical line). (b) The spectral function and the band dispersion are plotted as a reference.

Figure 11

Double logarithmic plot of the attenuation length of the transmission as a function of the nominal group index. The horizontal line indicates the limit set by the finite simulation length, while the diagonal line is proportional to $n_g^{-2}$ and is plotted as a guide to the eye. Blue, W1 waveguide; red, L3 coupled-cavity waveguide neglecting radiation losses; green, L3 coupled-cavity waveguide accounting for radiation losses.