Fano resonances and band structure of two-dimensional photonic structures

We show that the frequency spectrum of two-dimensional photonic crystals is strongly influenced by Fano resonances which can be excited already in the linear array of dielectric cylinders. To support this claim, we calculate the transmission of an electromagnetic wave through a linear array of dielectric cylinders and show that the frequencies of observed Fano resonances coincide with the position of narrow frequency bands found in the spectra of corresponding two-dimensional photonic crystals. The split of the frequency band and the overlap of two bands, observed in the band structure of photonic structures, are also associated with Fano resonances.

Figure 1

(a) Frequency spectrum of the square array of dielectric cylinders with radius $R=0.4a$ ($a$ is the spatial periodicity in the $x$ and $y$ directions) and permittivity $\varepsilon =12$ calculated using the plane-wave expansion method [1, 18]. In the frequency region $a/\lambda <0.9$ we found five $\mathcal{P}$ bands (black solid lines) and five even (dashed red lines) and seven odd (dotted blue lines) $\mathcal{F}$ bands. The inset shows schematically the structure in the $xz$ plane; the red arrow indicates the propagation direction of the electromagnetic wave. (b) Transmission coefficient of the plane wave incident on the linear array of cylinders [the geometry of the problem is shown in (c)]. The incident angles are $\theta =0$ [red (light gray) line] and $\theta =\pi /100$ [blue (dark gray) line]. Note the one-to-one correspondence between resonant frequencies of excited Fano resonances and $\mathcal{F}$ bands observed in the frequency spectrum of the infinite 2D structure. (c) Transmission of the incident electromagnetic wave through the linear array of dielectric cylinders. (d) Geometry of the photonic structure composed from $N$ rows of dielectric cylinders. ($N=24$ throughout this paper). Both structures displayed in (c) and (d) are infinite in the $x$ direction.

Figure 2

(a) The transmission coefficient of the linear chain of the cylinder with radius $R=0.4a$ shown in Fig. 1. Only even resonances are excited since the electromagnetic wave propagates perpendicularly to the cylinder chain [26, 27]. (b) The frequency dependence of parameters ${\beta }_k^{\pm }$. (c) Transmission coefficient for $N=24$ rows of cylinders. Fano resonances observed in the linear chain develop into narrow frequency bands. (d) Detail of the frequency dependence of the transmission coefficients shown in (a) and (c) in the vicinity of the Fano resonance of coefficient ${\beta }_3^{-} \left(a/\lambda =0.562\right)$.

Figure 3

Comparison of the field distribution $E_z$ in the dielectric cylinder for frequencies in the center of (top) the five lowest $\mathcal{P}$ bands and (bottom) five lowest $\mathcal{F}$ bands. The electromagnetic wave propagates through a photonic slab with a thickness of 24 rows of cylinders. For each frequency, only the field inside the tenth cylinder along the $y$ direction is shown.

Figure 4

(a) Linear chain of thin dielectric cylinders $\left(R=0.1a\right)$ and (b) transmission coefficient $T$ and the coefficient ${\beta }_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance, the transmission decreases to zero at $a/\lambda \approx 0.67$. (c) The band structure of a two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges (from top to bottom) ${\mathrm{\Gamma }}_3,{\mathrm{X}}_3,{\mathrm{\Gamma }}_2$, and ${\mathrm{X}}_2$.

Figure 5

(a) Frequency spectrum of a square array of dielectric cylinders with radius $R=0.3a$. Solid lines represent bands with even spatial symmetry; dotted lines correspond, similar to Fig. 1, to odd Fano bands. Note an overlap of the third and fourth bands, shown in detail in the right panel. This overlap results from the coupling of two bands displayed by green dot-dashed and red dashed lines. (b) Frequency dependence of the transmission coefficient for the linear chain of cylinders. Three Fano resonances can easily be identified. (c) The top panel shows the transmission coefficient for 24 rows of cylinders as a function of frequency $a/\lambda$. The two lower panels present the details of the frequency dependence of the transmission coefficient in the vicinity of the Fano resonance [29].

Figure 6

Coefficients $\beta$ for (a) a single dielectric cylinder with radius $R=0.4a$ and (b) a linear row of cylinders. (c) Coefficients ${\beta }_2^{+}$ [red (light gray)] and an inverse determinant ${\mathrm{\Delta }}^{-1}$ (black) for the clusters of $M$ dielectric cylinders $\left(M=1,2,4,24,48,\infty \right)$ along the $x$ axis. The thick blue line shows the position of the $\mathcal{F}$ band $0.4067<a/\lambda <0.4254$ displayed in Fig. 1 by the red dashed line. Multiple points for a given frequency correspond to different intensities of excited resonance at individual cylinders, as shown in (d) for the cluster with $M=48$ cylinders.

Figure 7

(a) Absorption of the electromagnetic wave in the photonic structure discussed in Fig. 5 but with a small imaginary part of the cylinder permittivity. Absorption is large at band edges, where the group velocity is small, and in the region of the Fano bands. (b) Detailed frequency dependence of the absorption in the resonant frequency region for $\text{Im}\varepsilon =0.001$ and 0.01.