Equi-frequency or isofrequency contour is supplementary in analyzing behaviors of photonic crystals, which is the intersection of a constant frequency ω-plane to a dispersion surface [1]. It quantifies all allowed wavevectors in the photonic crystals and their corresponding frequencies. A perfect circular equi-frequency contour means that light can travel isotropically. A square-like equi-frequency contour can be used to harvest its collimation property (self-collimation), and a star-like contour may be used in designing high-dispersive elements (superprism) that can support many channels in optical communications. In the later two cases the photonics crystal shows strong an-isotropic property. With several such equi-frequency contours at different frequencies, one can analyze whether the photonic crystal can have negative refraction or not, depending on the sign of the dispersion slope around Gamma or around M [3].

Like the band structure construction, in general there are two methods to create the equi-frequency contours. One method uses frequency-domain "Plane Wave Expansion (PWE) " method, and the other uses the time domain method which will be described below.This example shows how to use the time domain method with FDTD. The EFC is obtained at given frequencies by Fourier-transforming the spatial field at steady-state recorded from a frequency-domain field monitor.

Here we use a 2D square photonic crystal similar to the one illustrated in the 2D bandstructure example. It is composed of dielectric rods in air. The lattice constant is ax=ay=0.325 um and the refractive index is 1.583. Since the simulation volume is quite large, and the photonic crystal structures creates strong resonances, the simulation time is set to 10,000fs, which is longer than usual simulation. To speed up simulation, and to excite only modes with a given symmetry, we use symmetry BC's to simulate only one quarter of the photonic crystal. To have accurate spatial Fourier transform, the simulation region is set to be 60 by 60 um, thus an array of 120 by 120 is created. A mesh override region is used to ensure the mesh has the same periodicity as the structure.

To excite all the possible spatial modes, the dipole source is usually not recommended to be located at a high symmetry location (i.e. the center of a hole). We set one dipole source off-center. Due to the symmetry BC's, the dipole source will be mirrored in the other 3 quadrants. A frequency domain field monitor will be used to record the spatial field profile over the entire domain at a number of frequencies. In this case, we simulate TM polarization, so we only need to monitor Ez field.

After the simulation we look at the spatial field intensity at given frequency. The figure below shows |E(x,y)|^2 in log scale at wavelength 1.5um. We then can use a Fourier transform to look at the same data in reciprocal lattice space E(kx,ky) where kx,ky are spatial frequencies or wavenumbers.

Field profile |Ez(x,y)|^2 at 1.5um in log scale

We can either use fast Fourier transform (fft) or chirped Fourier transform (czt). The standard fft function will work, but it will give results for a very large range of wave numbers. The czt function is often more convenient as it allows you to calculate the transform only over the wave numbers of interest. This example uses the czt function to get the transform for Bloch wave numbers from -3*pi/ax to 3*pi/ax. The right figure below shows the Fourier transform of Ez at a wavelength of 1.5um, corresponding to a normalized frequency of f*a/c= 0.217. The high symmetry points Gamma, X and M are illustrated on the figure. Other Brillourin zones are clearly shown, which are not plotted when using the PWE method.The radius of the red circle corresponds to the wavenumber originated from Gamma point.

First Brillouin zone and higher-order Brillouin zones |E(kx,ky)|^2

Below is the bandstructure of this photonic crystal, calculated with FDTD using the technique described in the Bandstructure calculation methodology page. The red line corresponds to the normalized frequency of 0.217. The traditional bandstructure plot only shows modes along the Gamma-X-M-Gamma line. The EFC allows us to see all the possible wavevectors.

Note for Bandstructure plot: It is worth noting that the x-axis in the above figure is not uniformly sampled in k. The figure uses a fixed number of sample points along each arm of the Gamma-X-M-Gamma contour, even though the arms have different lengths. This can be an important detail when trying to estimate the shape of the contour from the bandstructure plot. Of course, if you calculate the EFC, it is easy to see the shape of the contour. A circular region means isotropic propagation, while other shapes indicate anisotropic behaviors.

Users familiar with PWE method may wonder why the equi-frequency contours in the higher-order Brillourin zones (not surrounding Gamma) do not have the same intensity. The reason is that the system was excited by a single real dipole. The coupling of the dipole to each spatial harmonics is not equivalent, which leads to different amplitudes. This can be an advantage of the time domain method: it can analyze the relative amplitude of each spatial harmonics contributed to the Bloch modes. Recall that the Bloch modes are expressed as ref [1]:

$$ \exp (i \vec{k} \vec{r} ) \sum A_{m, n} \exp \left ( i \left ( m\vec{G}_{x} {x}+n \vec{G}_{y} {y} \right ) \vec{r}\right) $$

where \( \vec{k} \) is the fundamental wave vector, Gx and Gy are reciprocal lattice constants, and \({G}_{x} {x}\) =2π, \({G}_{y} {y}\)=2π. Depending on the photonic crystal's structure, one spatial harmonic mode may have higher coupling efficiency than the others.

From other Brillouin zones, one can know the relative amplitude or intensity of other spatial harmonics. The fundamental mode has m=0,n=0 with amplitude A00. So (m,n)=(1,0) is the spatial harmonic mode with \( \vec{k} + \vec{G}_{x} \) and its amplitude A10 is smaller than A00. One can easily identify the modes (0,1),(-1,0),(0,-1) , which have the same amplitude as A10 due to symmetry, and the modes of (1,1),(-1,1),(-1,-1),(1,-1), with smaller amplitude than A10. If necessary, you can estimate quantitatively the amplitudes of those modes [2] (note that the figure shows intensity in log-scale). However, please keep in mind that those modes are not isolated, and all such modes compose of the Bloch waves inside the photonic crystal. In a homogenous medium, the reciprocal lattice constant is zero, so no Bloch modes can be found, although one can get similar EFCs for the fundamental wave vectors at different frequencies arbitrarily depending on the used periodicity.

Most often people are only interested in the EFCs at the first band. Figures below shows EFCs at 1.5um, 1.25um, and 1.0um, or f*a/c=0.216667,0.26 and 0.325, respectively, directly from fft calculation, which are circles in all the cases and all in the first band. This is common for the first band.

Equi-frequency contour at wavelength 1.5um, 1.25um and 1.0um

Equi-frequency contour at 3 different wavelength

It can be useful to plot multiple EFC's on a single figure, to more clearly see how the contour evolves as a function of frequency, as shown above. From the equi-frequency contour plots above, you can see that the radius of the circle increases with frequency, thus the dispersion slope is also positive. This is in consistent (and equivalent) to the bandstructure data obtained by other means.

EFC plots can be a helpful tool when analyzing some aspects of photonic crystals [2], such as negative refraction when the fundamental wave numbers decreases as frequency increases (in higher band [2], or the first band at the partial bandgap [3]), collimation (square-like EFCs [4]), and highly dispersive effects etc.

### References

[1] Dennis W. Prather et al, Photonic crystals: theory, applications, and fabrication, Wiley, 2009

Jean-Michel Lourtioz, Photonic crystals--Towards nanoscale photonic devices, Springer, Berlin, 2003

[2] Guilin Sun and Andrew G. Kirk, "On the relationship between Bloch modes and phase-related refractive index of photonic crystals," Opt. Express 15, 13149-13154 (2007)

http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-13149

[3] Guilin Sun and Andrew G. Kirk,"Analyses of negative refraction in the partial bandgap of photonic crystals," Opt. Express 16, 4330-4336 (2008)

http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-4330

[4] Rafif E. Hamam, Mihai Ibanescu, Steven G. Johnson, J. D. Joannopoulos, and Marin Soljacic, "Broadband super-collimation in a hybrid photonic crystal structure," Opt. Express 17, 8109-8118 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8109