A band structure is expected when a material has spatially periodic structure (21). The range of wavelengths in which propagation of electromagnetic radiation is for- bidden in a photonic band-gap depends on various factors. These factors controlling the presence as well as the size of band-gap depend on the refractive index contrast of the compounds of materials, the shape and the symmetry of lattice structure and the filling fraction, which is defined as the proportion of the material volume to background material in a unit volume. However, the incident wave on a photonic crystals structure is also important, as for a given structure waves are reflected or scattered through the structure and the band-gap can be only obtained for certain frequencies, directions and polarizations. By solving the eigenvalue problem given in the master equation form for all directions, allowed and forbidden frequencies can be determined for a given wave vector. The relation between ω and ~k is called the dispersion relation and defines the band structure of crystal.
The close resemblance of the master equation to Schr¨odinger’s equation govern- ing quantum mechanics helps us to understand the propagation of electromagnetic waves in a periodic medium. As a consequence to the dielectric function in a photonic crystal structure ε(~r) is analogous to the periodic potential V (~r) of elec- trons in a crystal structure. Similar to the electronic energy band-gap that states that certain electronic waves are forbidden in a crystal lattice, photonic crystal structures form photonic band-gaps as the dielectric function varies periodically, whereby electromagnetic wave propagation is forbidden. The direct analogy allows writ- ing the photonic system in ‘Bloch form’, consisting of a plane wave modulated by a function arising from the periodicity of the lattice. This approach becomes particu- larly useful when considering the coupling of light into two and three-dimensional photonic crystals.
2.2.1
The Brillouin Zone and the Reciprocal Lattice
If a dielectric structure has symmetry, this can help to determine properties of elec- tromagnetic system and classify the electromagnetic modes. By studying a certain region that fully characterizes the periodic structure due to the symmetry, the system behaviour can be obtained. This region in reciprocal space presents all the lattice symmetry of the structure and it is called the irreducible Brillouin zone (IBZ). If ad- ditional rotational symmetries are present the first Brillouin zone can be degraded to a fraction of the zone, which can represent all the modes, called irreducible Bril- louin zone. Irreducible Brillouin zone is obtained by reducing the first Brillouin zone to the high-symmetry points, namely Γ, X, K and M for 2D photonic crys- tals where the dispersion characteristics are not related to symmetry. These terms are analogous to the concepts in the solid-state physics and will be explained here briefly.
Instead of searching for the solution for the whole structure which requires a solu- tion of too many plane waves, the calculations are reduced to a primitive unit cell that presents a spatial domain defined by the Wigner-Seitz cell (22). A unit cell depends on the lattice structure, and represents the symmetry of structure. By reg- ularly repeating the unit cell, the whole structure can be duplicated. In real space, everything is known about the solution if the field is known everywhere in the unit cell. Similarly, in reciprocal space everything is known when the solution to the eigen-value problem is obtained at every point within the Brillouin zone.
The Brillouin zone is particularly important for finding the solutions for a periodic system according to the Bloch-Floquet theorem. This is because, in the Brillouin zone, the eigenvalues obey the dispersion relation, which includes the frequency of an eigenfunction ωn(~k), where ~k is the Bloch wave vector associated the modes in
wave number ~k, |~k| = ω/c and shows the dispersive properties of a medium with respect to the frequency. When ωn(~k) is plotted as a function of ~k vectors, the result-
ing figure shows that k-vectors exist for certain frequency intervals or bands while for some other frequency intervals or bands there is no k-vector. These solutions are called band-gap. The band-gap frequencies are the stop band of the structure, and the ~k vectors with frequency solutions are called allowed band or pass band.
The complete information of all modes can be obtained for the values of ~k within the reciprocal lattice. The eigenfunctions of the master equation of a periodic struc- ture correspond to eigenvalues of (ω/c)2.
~
H~k(~r) = ei~k·~r~hn,~k(~r) (2.23)
The above Hermitian eigenproblem over the Brillouin zone eigenvalues ωn(~k) at
each Bloch wave vector ~k defines the dispersion relation, where ~hn,~k(~r) is a peri- odic function that satisfies the master equation. Because of discrete translational symmetry, ωn(~k) is characterized by wave vector and band index n.
The Bloch-Floquet theorem states that, in a periodic medium, the eigenfunctions of a Hermitian eigenvalue problem can be written as the product of plane wave ei~k·~r
and a periodic function u(~r) that has the same periodicity with the lattice vector ~R.
~
H~k(~r) = ei~k·~r~u~k(~r) = ei~k·~r~u~k(~r + ~R) (2.24)
The spatial distribution of unit cell is defined by a set of basis vectors with discrete translational symmetry. The number of the vectors is equal to the number of dimen- sions. These vectors are called primitive lattice vectors. The lattice vector is any
linear combination of the primitive lattice vectors.
The lattice vector can be written in terms of the primitive lattice vectors as:
~
R = l~a1+ m~a2+ n~a3 (2.25)
where (l, m, n) are the integers and ~a1, ~a2, ~a3 are the primitive lattice vectors.
The set of wave vectors exist in reciprocal lattice space, a concept that is funda- mental to solid state physics (23). In order to work in the wave-vector space and derive dielectric function in wave vector representation, real lattice space transforms into reciprocal lattice space through integral transform.
When real-space lattice vectors ~R are known, the reciprocal lattice vectors ~G can be obtained. Since eigen solutions are also periodic functions of ~k and since ~k and ~k + ~G are equivalent: ~G · ~R = 2πN , where N is an integer. In a similar way, the reciprocal lattice vectors can be written as:
~
R = l0~b1+ m0~b2+ n0~b3 (2.26)
Given the lattice vectors, the reciprocal lattice vectors can be calculated by:
~b1 = 2π ~a2 × ~a3 ~a1· (~a2× ~a3) (2.27) ~b2 = 2π ~a3 × ~a1 ~a1· (~a2× ~a3) (2.28) ~b3 = 2π ~a1 × ~a2 ~a1· (~a2× ~a3) (2.29)
satisfying the condition ~ai · ~bj = 2πδij, δij = 1, i = j 0, i 6= j (2.30)
with δij denoting Kronecker symbol. Reciprocal lattices are the inverse transforms
of their crystal lattices multiplied by 2π.
2.2.2
Dispersion relation
The relation between the wave vector ~k and the frequency ω is called dispersion relation. The dispersion of the light in an isotropic dielectric material is given by:
ω(~k) = c~k/n = c~k/√εr (2.31)
where c is the speed of light, ~k is a wave vector, ε is a dielectric constant, n is the refractive index and the relative permeability is µr ≈ 1. n =
√
µrεr =
√ εr
The dispersion relation is a measure of light propagation in a material in comparison to light propagation in vacuum. The speed of light c is reduced by a factor of n in a material defined by phase velocity, vp.
ω(~k) = vp~k (2.32)
The phase velocity is the gradient of a line passing through the origin, intersecting the point (~k, ω). The slope of the dispersion diagram yields the group velocity defined as:
vg =
∂ω
∂~k (2.33)
are often expressed in normalized form.
N ormalised F requency = ωa/2πc = a/λ
where a is the period, namely lattice constant and k is expressed in units of π/a.
k = 2πnλ0 = 2πλ (2.34)
Dispersion relations can be calculated with many numerical techniques by solving the eigenvalue equation. In numerical methods like Finite Difference Time Domain (FDTD) method and Plane Wave Expansion (PWE) method the wave vector is an independent variable and the frequency is an eigenvalue. In Transfer Matrix Method (TMM), Eigen Mode Expansion (EME) method and Finite Element Method (FEM) the frequency is an independent variable and wave vector is an eigenvalue.
In the following section, Brillouin zones are constructed for 1D and 2D photonic crystals and band-gap diagrams are obtained by means of the PWE method, which will be discussed in detailed in the next chapter.