Radiation pattern of a classical dipole in a photonic crystal: Photon focusing

An asymptotic analysis of the radiation pattern of a classical dipole in a photonic crystal possessing an incomplete photonic bandgap is presented. The far-field radiation pattern demonstrates a strong modification with respect to the dipole radiation pattern in vacuum. Radiated power is suppressed in the direction of the spatial stop band and strongly enhanced in the direction of the group velocity, which is stationary with respect to a small variation of the wave vector. An effect of radiated power enhancement is explained in terms of photon focusing. A numerical example is given for a square-lattice two-dimensional photonic crystal. Predictions of asymptotic analysis are substantiated with finite-difference time-domain calculations, revealing a reasonable agreement.

Figure 1

Isofrequency and wave contours. Left: the central region of the isofrequency contour for normalized frequency $\Omega =\omega d∕2\pi c=d∕\lambda =0.569$ of an infinite square-lattice $2\mathrm{D}$ photonic crystal made out of dielectric rods placed in vacuum. Rods have the refractive index 2.9 and radius $r=0.15d$, where $d$ is the period of the lattice (see Sec. 6 for details). The stationary points ${\mathbf{k}}^1$, ${\mathbf{k}}^2$, and ${\mathbf{k}}^3$, corresponding to the same observation direction $\widehat{\mathbf{x}}$ are indicated. Right: corresponding wave contour with folds. The shaded and black regions show how two equal solid-angle sections in coordinate space (right) map to widely varying solid-angle sections in $\mathbf{k}$ space (left). The wave and group velocity vectors with numbers illustrate the fold formation of the wave contour.

Figure 2

Diagram showing the relations between $\mathbf{k}$ space and coordinate space quantities. Isofrequency contours for frequencies ${\omega }_{n\mathbf{k}}$ and ${\omega }_{n\mathbf{k}}+d\omega$ are presented.

Figure 3

Diagram to derive the relation between solid angles in $\mathbf{k}$ space and coordinate space. The isofrequency contour for frequency ${\omega }_{n\mathbf{k}}$ is presented. The Jacobian of the transformation, Eqs. (37, 38), is given by the ratio $d\Omega ∕d{\Omega }_{n\mathbf{k}}$. By the definition of the solid angle, the solid angle in $\mathbf{k}$ space is $d{\Omega }_{n\mathbf{k}}=d^2\mathbf{k}\cos \phi ∕{\mid {\mathbf{k}}_n\mid }^2$, while the corresponding solid angle in coordinate space is $d\Omega =d^2\mathbf{k}\mid K_{n\mathbf{k}}\mid$. That gives the Jacobian $J_{n\mathbf{k}}={\mid {\mathbf{k}}_n\mid }^2\mid K_{n\mathbf{k}}\mid ∕\cos \phi$. Here $\phi$ is an angle between the wave vector and the group velocity vector. $d^2\mathbf{k}$ is the surface element of the isofrequency surface.

Figure 4

Photonic band structure of TM modes for the square-lattice photonic crystal with refractive index of the rods $n=2.9$, lattice constant $d$, and radius of the rods $0.15d$. The frequency is normalized to $\Omega =\omega d∕2\pi c=d∕\lambda$. Here $c$ is the speed of light in vacuum. The insets show the first Brillouin zone of the crystal with the irreducible zone shaded light gray (left) and a part of the lattice (right).

Figure 5

Isofrequency contours of the square-lattice photonic crystal for the normalized frequencies $\Omega =0.31$ (dashed line) and $\Omega =0.34$ (solid line). The parabolic points are marked by the black dots. The first Brillouin zone of the lattice is plotted in order to show the spatial relation between zone boundary and isofrequency contours.

Figure 6

Wave contour corresponding to the normalized frequency $\Omega =0.31$.The group velocity is plotted in units of the speed of light in vacuum. High-symmetry directions of the square lattice are specified.

Figure 7

Angular distribution of radiated power corresponding to the normalized frequency $\Omega =0.31$. High-symmetry directions of the square lattice are specified.

Figure 8

Wave contour corresponding to the normalized frequency $\Omega =0.34$. The group velocity is plotted in units of the speed of light in vacuum. The directions corresponding to the folds of the wave contour are shown.

Figure 9

Angular distribution of the radiated power corresponding to the normalized frequency $\Omega =0.34$. The directions of infinite radiated power (caustic) coincide with the directions of the folds of the wave contour (Fig. 8).

Figure 10

(a) FDTD calculation. Map of the modulus of the Poynting vector field for a $50\times 50$ rods photonic crystal excited by a point isotropic source with the normalized frequency $\Omega =0.34$. The location of the crystal in the simulation domain is shown together with asymptotic directions of photon focusing caustics. (b) Isofrequency contours of the square lattice photonic crystal (solid line) and air (dashed line) for normalized frequency $\Omega =0.34$. The dash-dotted line is a construction line. The wave vector corresponding to the parabolic point (black dot), ${\mathbf{k}}^{\mathit{\text{fold}}}$, and the wave vector in air, obtained from the momentum conservation law at the crystal–air interface, ${\mathbf{k}}^{\mathit{\text{air}}}$, are shown.

Figure 11

Isofrequency contours of the square-lattice photonic crystal for the normalized frequencies $\Omega =0.55$ (dotted line), $\Omega =0.565$ (solid and dashed lines) and $\Omega =0.58$ (dash-dotted line). Two branches of the isofrequency contour of $\Omega =0.565$ are plotted as solid and dashed lines. The parabolic points are marked by the black dots. The first Brillouin zone of the lattice is plotted in order to show the spatial relation between the zone boundary and isofrequency contours.

Figure 12

Wave contours corresponding to the normalized frequency $\Omega =0.565$. Solid (dashed) wave contour corresponds to solid (dashed) isofrequency contour in Fig. 11. The group velocity is plotted in the units of the speed of light in vacuum. The directions corresponding to the folds of the wave contour are shown.

Figure 13

Angular distribution of radiated power corresponding to the normalized frequency $\Omega =0.565$. The directions of infinite radiative power (caustic) coincide with the directions of the folds of the wave contour.