Field localization and enhancement near the Dirac point of a finite defectless photonic crystal

We use a rigorous electromagnetic approach to show the existence of strongly localized modes in the stop band of a linear, two-dimensional, finite photonic crystal near its Dirac point. At normal incidence, the crystal exhibits a Dirac point with 100$\%$ transmission. At angles slightly off the normal, where the crystal is 100$\%$ reflective, instead of exponentially decaying fields as in a photonic stop band, the field becomes strongly localized and enhanced inside the crystal. We explain that this anomalous localization is due to guided mode resonances that are the foundation of the Dirac point itself and also shape its adjacent band gap. Besides shedding new light on the physical origin of Dirac points in finite photonic crystals, our results could have applications in many nonlinear light-matter interaction phenomena in which it is crucial to achieve a high degree of light localization.

Figure 1

(a) A two-dimensional photonic crystal made of a square lattice (lattice constant a) of dielectric columns with radius $r$ and electric permittivity ε. The photonic crystal is five periods long in the $z$ direction with length L $=$ Na, where $N$ is the number of rows of columns. A plane electromagnetic wave with the electric field parallel to the axis of the cylinders is incident on the structure with its $k$-vector in the (x, z) plane (in-plane coupling), forming an angle ϑ with respect to the $z$ direction. (b) Transmittance in the (ω, ϑ) plane around the Dirac point numerically calculated by the FMM. The structure's parameters are $N$ $=$ 5, $r$/a $=$ 0.2, ε $=$ 12.5.

Figure 2

Transmittance vs frequency for different incident angles.

Figure 3

(a) Cross-sectional view of the electric field intensity normalized to the incident field intensity at the Dirac point where $T$ $=$ 1. (b) Cross-sectional view of the field localization in the stopband at the transmission minimum for ϑ $=$ 0.1°. The insets show a magnification of the field localization on the columns. The dashed circles indicate the position of the columns. The dashed line parallel to the $z$ axis indicates the section of the field that it is shown in Fig. 3c. (c) Section of the field intensity profiles along the $z$ axis for the two cases.

Figure 4

(a) Localization length vs incident angle for the $N$ $=$ 5 period structure. (b) Average field localization vs incident angle. In the Supplemental Materials (StopBandModes.wmv),[17] we show the field localization for the different angles and the transition from localized to evanescent modes.

Figure 5

(a) Localization length and (b) average field localization compared with number of periods for ϑ $=$ 0.1°. (c) Cross-sectional view of the field localization in the stopband at the transmission minimum for ϑ $=$ 0.1° and $N$ $=$ 15. The inset shows a magnification of the field localization on the columns. The dashed circles indicate the position of the columns. The dashed line parallel to the $z$ axis indicates the section of the field that it is shown in Fig. 5d. (d) Section of the field intensity profile along the $z$ axis.

Figure 6

Transmittance and coupling strength in the (ω, ϑ) plane for different periods. The white dot indicates the position of the Dirac point. The dashed lines represent the dispersion of the GMRs.

Figure 7

(a) A two-dimensional photonic crystal made of a triangular lattice (lattice constant a) of dielectric columns with radius $r$ and electric permittivity ε. The photonic crystal is finite along the $z$ direction with length $L=N\sqrt{3}/2$a$,$ where $N$ is the number of rows of columns. A plane electromagnetic wave with the magnetic field parallel to the axis of the cylinders (TM polarization) is incident on the structure with its $k$-vector in the (x, z) plane (in-plane coupling) forming an angle ϑ with respect to the $z$ direction. (b) Transmittance in the (ω, ϑ) plane around the Dirac point numerically calculated by the FMM. The structure's parameters are $N$ $=$ 5, $r$/a $=$ 0.3, ε $=$ 11.4. (c) Transmittance vs frequency for different incident angles.

Figure 8

Cross-sectional view of the electric field localization normalized to the incident field for the different incident angles. The dashed circles indicate the position of the columns.