5.2 Symmetries • Symmetries can be used to classify electromagnetic modes and to make statements about the system‘s general behaviour. • Example: inversion symmetry: Consider a 2D metal cavity that looks like this:
J. D. Joannopoulos et al., Photonic Crystals – Molding the flow of light, Princeton University Press (2008). 1 • The shape is somewhat arbitrary, making an analytical solution difficult, but it has inversion symmetry if is a mode with frequency , then must also be a mode with frequency • Unless is a member of a degenerate family of modes, then if has the same frequency, it must be the same mode, i.e. it must be a multiple of : . • If we invert the system twice we obtain or . • A given nondegenerate mode must be of one of the two types, either it is invariant under inversion (even mode) or it becomes its own opposite (odd mode). • Thereby we have classified the modes of the system based on how they respond to one of its symmetry operations. 2 • Let‘s capture this idea in a more abstract language • Suppose is an operator (a 3x3 matrix) that inverts vectors (3x1 matrices), so that
• To invert a vector field, we need an operator that inverts both the vector and ist argument : • For an inversion symmetric system it does not matter if we operate with or if we first invert the coordinates, then operate with , and then change them back, i.e.: (73)
• This equation can be rearranged: =0 • We can define the commutator (74)
which is itself an operator 3 • For our inversion symmetric example system we have ,Θ 0. • If we operate with this commutator on any mode of the system we get ,Θ Θ Θ 0 (75) →Θ Θ • If is a mode with frequency , then must also be a mode with frequency and and can only be different by a multiplicative factor: α • This is an eigenvalue equation for , and the eigenvalues α must be 1 or 1 • We can classify the eigenvectors according to whether they are even or odd under inversion symmetry operation • Generally, when two operators commute, we can construct simultaneous eigenfunctions of both operators • Eigenfunctions and eigenvalues of simple symmetry operators are easily determined, while those of Θ are not
4 5.2.1 Discrete translational symmetry • Photonic crystals are characterized by the periodic modulation of their refractive index they have discrete translational symmetry, i.e. they are invariant under translations by distances that are a multiple of a fixed step length • Example: System with discrete translational symmetry in ‐ direction and contiuous translational symmetry in ‐ direction
• The basic step length is the lattice constant , the basic step vector is the primitive lattice vector, which here is 5 • Because of this discrete symmetry • By repeating this translation where is an integral multiple of , i.e. , where is an integer. • The dielectric unit which is repeated over and over is the unit cell.
• We define a translation operator , which, when operated on a function shifts the argument by • For a system which is translationally invariant under a shift by we have or equivalently and the modes can be classified according to how they behave under . • For our system we have continuous translational symmetry in ‐direction and discrete translational symmetry in ‐ direction • With this knowledge we can identify the modes as simultaneous eigenfunctions of and .
6 • We can proof that these eigenfunctions are plane waves: • We can classify the modes by specifying and • However, modes with wave vector and with wave vector , where is an integer, yield the same eigenvalue of , forming a degenerate set. • We call with the reciprocal lattice vector • Since any linear combination of these degenerate eigenfunctions is itself an eigenfunction with the same eigenvalue, we can take linear combinations of our original modes.
7 • We can put these linear combinations in the form: , ,