What is the Brillouin zone of an anisotropic photonic crystal?

The concept of the Brillouin zone (BZ) in relation to a photonic crystal fabricated in an optically anisotropic material is explored both experimentally and theoretically. In experiment we used femtosecond laser pulses to excite THz polaritons and image their propagation in lithium niobate and lithium tantalate photonic crystal (PhC) slabs. We directly measured the dispersion relation inside PhCs and observed that the lowest band gap expected to form at the BZ boundary forms inside the BZ in the anisotropic lithium niobate PhC. Our analysis shows that in an anisotropic material the BZ—defined as the Wigner-Seitz cell in the reciprocal lattice—is no longer bounded by Bragg planes and thus does not conform to the original definition of the BZ by Brillouin. We construct an alternative Brillouin zone defined by Bragg planes and show its utility in identifying features of the dispersion bands. We show that for an anisotropic two-dimensional PhC without dispersion, the Bragg plane BZ can be constructed by applying the Wigner-Seitz method to a stretched or compressed reciprocal lattice. We also show that in the presence of the dispersion in the underlying material or in a slab waveguide, the Bragg planes are generally represented by curved surfaces rather than planes. The concept of constructing a BZ with Bragg planes should prove useful in understanding the formation of dispersion bands in anisotropic PhCs and in selectively tailoring their optical properties.

Figure 1

Crossing points of the empty lattice dispersion curves identify the Bragg planes. (a) TE dispersion curves for propagation along $\mathrm{\Gamma }\text{−}M\text{−}\mathrm{\Gamma }$ in a 2D empty square lattice with an isotropic refractive index $n=5$ and lattice periodicity $a$. The first two crossing points are labeled as $k_{B\text{−}1}$ and $k_{B\text{−}2}$. (b) Corresponding dispersion curves with a finite perturbation from air holes of radius $r=0.15a$. (c) The first several BZ boundaries in reciprocal space constructed using the WS geometrical method. The propagation direction (red arrow), irreducible Brillouin zone (shaded blue), and reciprocal lattice vectors $b_1$ and $b_2$ are also indicated.

Figure 2

The experimental observation of atypical band gap formation. (a) CCD image of a THz wave obtained by using the polaritonics platform, with the image intensity showing the spatially varying THz E field in the LN PhC slab. The lattice vectors $a_1$ and $a_2$ are oriented at ${45}^{\circ }$ with respect to the principal axes of the LN refractive index. The blue arrows shows propagation along the $y$ axis in real space. Inset shows the experimental geometry, in which a cylindrically focused 800 nm pump pulse passing through the unpatterned part of the LN slab generates THz waves that counterpropagate along the $\pm y$ directions as TE waveguide modes polarized along the $z$ axis. The right-propagating THz wave enters the PhC structure where, at selected times, its E-field spatial distribution is recorded to produce images like the one shown. (b) Space-time plot of the AnPhC in (a), generated by averaging images like the one shown in (a) over the $z$ dimension to produce reduced 1D images and displaying the images one above another in time order. (c) Polariton dispersion in the LN PhC slab obtained by a 2D Fourier transform of (b), with the location of the Bragg plane at $k_{B\text{−}1}$ shown by a dashed line. Inset shows the BZ with a red arrow indicating the wave vector direction along $k_y$. (d) Polariton dispersion in the isotropic LT PhC slab. Thin lines in (c) and (d) show calculated dispersion curves for the first two TE-like bands.

Figure 3

Crossing points of the empty lattice dispersion curves identify Bragg planes and band gaps in a 2D AnPhC. (a) Dispersion curves for propagation along $k_y$ in a 2D empty square lattice with lattice periodicity $a$. The first two crossing points (blue circles) are labeled as $k_{B\text{−}1}$ and $k_{B\text{−}2}$. (b) Corresponding dispersion curves with a finite perturbation from air holes of radius $r=0.15a$. The band gaps that originate from the crossing points $k_B$ are indicated by blue circles. (c) The first several BBZ boundaries in reciprocal space constructed using Eqs. (1), (2), and (6). The propagation direction $k_y$ (red arrow), reciprocal lattice vectors $b_1$ and $b_2$, and corresponding crossing points are also shown.

Figure 4

Isofrequency contour plots demonstrate that the BBZ primitive cell closely reflects the symmetry of EM mode dispersion in an AnPhC. (a) First two numerically solved TE mode IFCs for the Sqr-45-AnPhC with air-hole radius $r=0.1a$. The WSBZ primitive cell and reciprocal lattice vectors $b_1$ and $b_2$ are also shown. (b) Same as (a), but with the BBZ primitive cell shown. (c) The principle axes of refractive index are oriented at $\theta ={24}^{\circ }$ with respect to the reciprocal lattice vectors $b_1$ and $b_2$, which alters the BBZ primitive cell. The associated IBZ (shaded blue) is also shown in all three cases.

Figure 5

An AnPhC slab with curved Bragg surfaces. (a) TE-like $(x$-even) isofrequency surfaces at $\omega a/2\pi c=1.32$, 1.65, and 1.98 for an empty square lattice in a $0.54a$ thick slab. The intersection points of the isofrequency surfaces (blue circles) form the Bragg surface (black line) satisfying Eqs. (1) and (2). (b) The first several curved BBZ boundaries in reciprocal space, constructed using the method illustrated in (a).

Figure 6

Constructing the BBZ of a 2D AnPhC: (a) the original reciprocal lattice; (b) the reciprocal lattice is stretched according to Eqs. (A1) and (A2); (c) the WS cell of the deformed lattice is constructed; and (d) the reciprocal space is deformed back yielding the BBZ of the original reciprocal lattice.