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5.2 Symmetries • Symmetries Can Be Used to Classify Electromagnetic Modes and to Make Statements About the System‘S General Behaviour

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5.2 Symmetries • Symmetries Can Be Used to Classify Electromagnetic Modes and to Make Statements About the System‘S General Behaviour 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 ΘΘ 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: , , , • Here, the ‘s are expansion coefficients and is (by construction) a periodic function with • That is, the discrete periodicity in ‐direction leads to a dependence for that is simply the product of a plane wave with a y‐periodic function: (76) • This result is known as Bloch‘s theorem. The form of (76) is known as a Bloch state (sometimes also as Floquet mode) 8 • Visualization example (periodicity in ‐direction): • According to Bloch‘s theorem, the mode frequencies must be periodic in : • Thus, we need only consider to exist in the range • This region of non‐redundant values of is called the (first) Brillouin zone. • Generalization to 3 dimensions: the dielectric is invariant under translations through a multitude of lattice vectors ( are integers) • The ( , , ) give rise to three primitive reciprocal lattice vectors ( , , ) defined so that , spanning the reciprocal lattice which is inhabited by wave vectors. 10 • Brillouin zone example: 2D square lattice (3D is analogous) • The modes of a 3D periodic system are Bloch states that can be labelled by a Bloch wave vector • The eigenvectors take the form with • is a conserved quantity in a periodic system, modulo the addition of reciprocal lattice vectors • Addition of a reciprocal lattice vector does not change the eigenstate or its propagation direction 11 • To solve for , we reformulate the eigenvalue problem by inserting the Bloch state into the master equation: • • • • with the new ‐dependent Hermitian operator 12 • Transversality condition: • Periodicity condition: • Because of the periodicity condition for , we can regard the eigenvalue problem to be restricted to a single unit cell of the PhC • Restricting a Hermitian eigenvalue problem to a finite volume (here: to a single unit cell of the photonic crystal) discrete spectrum of bands • For any value of we can expect to find an infinite set of modes with discretely spaced frequencies, which we can label by an index • enters as a continuous parameter in the frequency of each band should vary continuously as varies. • PhC modes are a family of continuous functions , indexed in order of increasing frequency by the band number. bands of the PhC 13 • Photonic band structure of a PhC: The mode frequencies plotted over the ‐vector • Because of the Bloch theorem, it is sufficient to plot band structures in the first Brillouin zone. • The eigenvalue problem can be solved numerically by an iterative minimization technique for each value of . 14 5.2.2 Other symmetries • When there is an additional symmetry, e.g. rotation, mirror, or inversion symmetry, in the lattice, the frequency bands have that symmetries as well, resulting in redundancies within the Brillouin zone • The smallest region within the Brillouin zone for which the frequency bands are not related by symmetry is called the irreducible Brillouin zone. • Example: Photonic crystal with the symmetry of a simple square lattice: 15 Comparison with quantum mechanics J. D. Joannopoulos et al., Photonic Crystals – Molding the flow of light, Princeton University Press (2008). 16 5.3 Multilayer films: 1D Photonic Crystals Recall FoMO 7.3.3 Periodic multi-layer systems - Bragg-mirrors • The dielectric function varies along one direction (here ) only. We imagine this variation to be periodic. • The system consists of alternating layers of material with different dielectric constants with a spatial period • We imagine that each layer is uniform and extends to infinity along and • Symmetries allow us to classify the modes using ‖, and ‖ • Modes have Bloch form ‖, , ‖, , with zz where is an integral multiple of 1D Photonic Crystals in nature • Mother‐of‐pearl Aragonite [CaCO3] / protein layers http://www.biosbcc.net/ocean/marinesci/06future/abrepro.htm http://www.solids.bnl.gov/~dimasi/bones/abalone/ wikipedia 5.3.1. Band structure of a 1D PhC Consider a homogenous medium … Its dispersion relation is given by … Consider a homogenous medium with artificial periodicity k k m g 2 g a Superposition of two counter propagating plane waves. Consider a homogenous medium with artificial periodicity k k m g 2 g a Bragg condition: 2 λ, 1 2 2/ / Modes have wavelength of 2na For we obtain two standing waves The dispersion relation can be reduced to the 1. Brillouin zone Light line “folds back” at the edge of the Brillouin zone, which corresponds to replacing 2/ by • We now introduce a difference in the dielectric constants of the layers • Structure symmetry allows for two ways to center the modes: ...while these modes, for example, are not compatible with the symmetry of the Photonic Crystal: Thus, the intensity is concentrated either in the high or in the low index material: As a consequence, the two modes “see” different effective materials and we obtain a photonic band gap! Compare (68). The modes are named according to the region where they concentrate most of their energy. The low‐index material is often air. “dielectric mode” “air mode” • As before, the dispersion relation can be reduced to the “air band” 1. Brillouin zone 0 • The bands are named accordingly Photonic band gap Comment: For high index contrast, both modes “dielectric band” concentrate most of their energy in the high‐epsilon material, but to a different degree..
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