# Bloch wave – MoM method

Bloch wave – MoM is a first principles technique for determining the photonic band structure of triply-periodic electromagnetic media such as photonic crystals. This technique uses the method of moments (MoM) in combination with a Bloch wave expansion of the electromagnetic field in the structure. It is based on the 3-dimensional spectral domain method (Kastner [1987]), specialized to periodic media. This approach is very efficient in terms of the number of plane waves needed for good convergence and is analogous to the spectral domain MoM method commonly used for analyzing 2D periodic structures such as frequency selective surfaces (FSS). In both cases, the field is expanded as a set of eigenfunction modes (either a Bloch wave in 3D or a discrete plane wave - aka Floquet mode - spectrum in 2D), and an integral equation is enforced on the surface of the scatterers in each unit cell. In the FSS case, the unit cell is 2-dimensional and in the photonic crystal case, the unit cell is 3-dimensional.

## Field equations for 3D PEC photonic crystal structures

For perfectly electrically conducting (PEC) structures admitting only electric current sources J, the electric field E is related to the vector magnetic potential A via the well-known relation:

${\displaystyle {\mathbf {E}}~=~-jk\eta \left[{\mathbf {A}}+{\frac {1}{k^{2}}}\nabla (\nabla \bullet {\mathbf {A}})\right]~~~~~~~~~~~~~~(1.1)}$

and the vector magnetic potential is in turn related to the source currents via:

${\displaystyle \nabla ^{2}{\mathbf {A}}+k^{2}{\mathbf {A}}~=~-{\mathbf {J}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.2)}$

where

${\displaystyle k^{2}~=~\omega ^{2}(\mu \epsilon )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.3)}$

## Bloch wave expansion of the fields

To solve equations (1.1) and (1.2) within the infinite periodic volume, we may assume a Bloch wave expansion for all currents, fields and potentials:

${\displaystyle {\mathbf {J}}(x,y,z)~=~\sum _{mnp}~{\mathbf {J}}(\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}~~~~~(2.1a)}$
${\displaystyle {\mathbf {E}}(x,y,z)~=~\sum _{mnp}~{\mathbf {E}}(\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}~~~~(2.1b)}$
${\displaystyle {\mathbf {A}}(x,y,z)~=~\sum _{mnp}~{\mathbf {A}}(\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}~~~(2.1c)}$

where for simplicity, we assume an orthogonal lattice in which α only depends on m, β only depends on n and γ only depends on p. In the equations above,

${\displaystyle \alpha _{m}~=~k_{0}~\sin \theta _{0}~\cos \phi _{0}~+~{\frac {2m\pi }{l_{x}}}~~~~~~~~~~~(2.2a)}$
${\displaystyle \beta _{n}~=~k_{0}~\sin \theta _{0}~\sin \phi _{0}~+~{\frac {2n\pi }{l_{y}}}~~~~~~~~~~~~~(2.2b)}$
${\displaystyle \gamma _{p}~=~k_{0}~\cos \theta _{0}~+~{\frac {2p\pi }{l_{z}}}~~~~~~~~~~~~~~~~~~~~~~(2.2c)}$

and,

${\displaystyle k_{0}~=~2\pi /\lambda ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.3)}$

where lx, ly, lz are the unit cell dimensions in the x,y,z directions respectively, λ is the effective wavelength in the crystal and θ0, φ0 are the directions of propagation in spherical coordinates.

Note that k in equations (1.1) and (1.2) comes from the time derivative in Maxwell's equations and is the free space propagation constant (actually, the propagation constant of whatever medium the metallic scatterers are embedded in), proportional to frequency as we see in equation (1.3). On the other hand, k0 in the equations above comes from our assumed Bloch wave solution given by equations (2.1) & (2.2). As a result, it represents the propagation constant in the periodic medium. These two k's, i.e. the free space propagation constant and the propagation constant of the Bloch wave, are in general different thereby allowing for dispersion in the solution.

The Bloch wave expansions in equations (2.1) are nothing more than exponential Fourier series multiplied by the cell-to-cell propagation factor: ${\displaystyle e^{j(\alpha _{0}x+\beta _{0}y+\gamma _{0}z)}.}$ The Bloch wave expansions are chosen since any field solution within an infinite periodic volume must have the same periodicity as the medium itself, or stated another way, the fields in neighboring cells must be identical up to a (real or complex) propagation factor. In passbands the propagation factor is an exponential function with purely imaginary argument and in the stop bands (or band gaps) it is a decaying exponential function whose argument has a real component.

## Integral equation for PEC media

Substituting equations (2.1) into (1.1) and (1.2) yields the spectral domain Greens function relating the radiated electric field to its source currents:

${\displaystyle {\mathbf {E}}(\alpha _{m},\beta _{n},\gamma _{p})~=~{\frac {jk\eta }{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}~{\mathbf {G}}_{mnp}~{\mathbf {J}}(\alpha _{m},\beta _{n},\gamma _{p})~~~~~~~~~~~~~~~~~~~~~~~~~(3.1)}$

where,

${\displaystyle {\mathbf {G}}_{mnp}~=~\left({\begin{matrix}1-{\frac {\alpha _{m}^{2}}{k^{2}}}&-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}&-{\frac {\alpha _{m}\gamma _{p}}{k^{2}}}\\-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}&1-{\frac {\beta _{n}^{2}}{k^{2}}}&-{\frac {\beta _{n}\gamma _{p}}{k^{2}}}\\-{\frac {\alpha _{m}\gamma _{p}}{k^{2}}}&-{\frac {\beta _{n}\gamma _{p}}{k^{2}}}&1-{\frac {\gamma _{p}^{2}}{k^{2}}}\end{matrix}}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3.2)}$

is the tensor Green's function in the spectral domain. Note that the spatial domain convolution has been transformed into a simple multiplication in the spectral domain, consistent with the convolution theorem for Fourier transforms.

With this, the electric field boundary condition on the surface of PEC material within a unit cell becomes:

${\displaystyle ~\sum _{mnp}~{\frac {1}{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}~{\mathbf {G}}_{mnp}~{\mathbf {J}}(\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}~=~{\mathbf {0}}~~~~~~~~~~~~~~~~(3.3)}$

Since we are seeking characteristic modes (eigenmodes) of the structure, there is no impressed E-field on the RHS of this electric field integral equation (EFIE). Equation (3.3) is not strictly correct since only the tangential components of electric field are actually zero on the surface of the PEC scatterer. This inexactness will be resolved presently when we test with the current basis functions, defined as residing on the surface of the scatterer.

## Method of Moments (MoM) solution

As is usual in the method of moments, the source currents are now expanded in terms of a sum over some known set of basis functions with unknown weighting coefficients Jj :

${\displaystyle ~{\mathbf {J}}(x,y,z)~=~\sum _{j}~J_{j}~{\mathbf {J}}_{j}(x,y,z)~~~~~~~~~~~~~~~~~(4.1)}$

Different structures will have different sets of basis functions for representing the currents on the elements and as in the ordinary spatial domain method of moments, the solution (in this case, the band diagram) is a function of the set of basis functions used.

Substituting (4.1) into (3.3) and then testing the resulting equation with the i-th current basis function (i.e., dotting from the left and integrating over the domain of the i-th current basis function, thereby completing the quadratic form) produces the i-th row of the matrix eigenvalue equation for a 3-dimensional array of PEC scatterers as:

${\displaystyle ~\sum _{j}~J_{j}~\left[~\sum _{mnp}~{\frac {{\mathbf {J}}_{i}(-\alpha _{m},-\beta _{n},-\gamma _{p})~{\mathbf {G}}_{mnp}~{\mathbf {J}}_{j}(\alpha _{m},\beta _{n},\gamma _{p})}{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}\right]~=~{\mathbf {0}}~~~(4.2)}$

This matrix equation is very easy to implement and requires only that the 3D FT of the basis functions be computed, preferably in closed form (see Scott [1998], available on researchgate.net). In fact, computing bands of a 3D photonic crystal with this method is no more difficult than computing reflection and transmission from a 2D periodic surface using the spectral domain method. This is because equation (4.2) is identical to the basic EFIE for a freestanding PEC FSS (Scott [1989] or Frequency selective surface eq. (4.2)), the only difference being the stronger singularity in 3D which significantly accelerates convergence of the triple sums, and of course the fact that the vectors are now 3-dimensional.

It's evident from (4.2) that the EFIE can become singular whenever the free space wavenumber is exactly equal to one of the wave numbers in any of the 3 periodic coordinate directions. This can happen for example when the free space wavelength exactly equals the lattice spacing. This is a statistically rare occurrence in computational practice and corresponds to a propagation anomaly similar to a Wood's reflection anomaly for gratings.

The advantages of this method over both the plane wave expansion method and the Finite element method include mathematical, programming and computational simplicity. The almost instant derivation above required no complex mathematics beyond ordinary vector calculus in rectangular coordinates. Converting an existing spectral domain FSS code to a Bloch wave - MoM code is very straightforward, not that writing one from scratch would be difficult. And computationally, electric current unknowns are only needed on the surface of the scatterers in the unit cell, so the matrix eigenvalue problem can be as small as 1x1 for simple scatterers at low frequencies. As a result, this equation can be easily solved for relatively complex structures on a low-end PC. Plus, the singularity in the sums for each matrix element adds a natural taper which accelerates convergence.

## Computing bands

To compute bands of the crystal (i.e. k-k0 diagrams), we may assume values for (k0, θ0, φ0) and then search for those values of k which drive the determinant of the impedance matrix to zero. Basically we have to try different values of frequency with pre-selected values of propagation constant and propagation direction - through trial and error - until we find a combination that produces the desired zero field condition on the surface of the array of scatterers. Equation (4.2) has been used to compute bands in various types of doped and undoped photonic crystals (Scott[1998], Scott [2002], both available on researchgate.net).

As a matter of computational convenience, whenever the matrix is larger than 2x2 it's vastly more efficient to compute the determinant either by reducing the matrix to upper triangular form using QR decomposition or to compute the determinant by reducing to echelon form using Gaussian elimination, rather than trying to actually compute the determinant of the matrix directly. Then the zeros can be found along the k-axis using a root-finding method such as Newton's method. Oftentimes it's useful to plot the determinant as a function of k, to observe its behavior near the zeros and thereby select the most efficient root-finding method. For 1-dimensional dielectric structures, there's usually a zero crossing (such as typically seen in impedance functions) which makes root-finding easy.