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1 2009. 04. Hanjo Lim School of Electrical & Computer Engineering hanjolim@ajou.ac.kr@ajou.ac.kr Lecture 3. Symmetries & Solid State Electromagnetism

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2 Photonic crystals ; the structures having symmetric natures important symm.; translat., rotat., mirror, inversion, time-reversal symmetry of a system => general statements on system’s behavior, classify normal modes. Example : a two-dimensional metal cavity with an inversion symm. If is found, is an eigenmode of freq. even mode odd mode proof) Inversion symm. means that metal pattern; If is nondegenerate, must be the same mode with with the eigenvalue of the inversion operation. Likewise

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3 Formal(mathematical) treatment of inversion symmetry operator that inverts vector, operator that inverts vector field and its argument property of inversion symmetry ; for any operation def) commutator operator of two operators and Symm. system under inversion means with and any

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4 Let then means that is an eigenfunction operator. ∴ If is a harmonic mode, is also a mode with freq If there is no degeneracy, there can only be one mode per frequency General aspect ; With two commuting operators, simultaneous eigenfunctions of both operators can be constructed. Ex) eigenvalue of classify eigenmode parity What if there is degeneracy ? If then?

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5 Continuous Translational Symmetry def) A system with transl. symm.; unchanged by a translation operation through a displacement Let translational operator is for each If a system is invariant under operation, Let our system is translationally invariant, or Then, proof: ∴ The modes of can be classified by how they behave under

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6 def) a system with continuous translat. symm. ; invariant under all ex) If is invariant under all the in the z-direction, for any uniform plane wave propagation in z-direction is the eigenfunction. Then ∴ Eigenvalue of operator ∴ We can classify the waves by Note changes the phase only by ex) If a system is invariant in all three directions (ex: free space), eigenmodes; with any constant vector Note) The eigenfunctions can be classified by their particular values for wave vector Implication of on the plane wave : transversality condition

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7 In conclusion, the plane waves are the solutions of master eq. with 3-dimensional continuous translational symmetry. Proof; ∴ Master eq. becomes

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8 If Holds. ∴ Spectifying ; propagation direction & => how mode behaves ex) An infinite plane of glass with Invariant under the continuous translation operation in the x- or y-direction. The eigenmodes should have the form with the in-plane wave vector and that would not be determined by the symmetry consideration only. But a constraint on exists from the transversality condition

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9 ∴ We can classify the modes by their values of and the band number by if there are many modes for a given Assume a glass plane of width and a mode with the x-polarized field Then master eq. becomes

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11 ∴ Master eq. becomes Let If that is, evanescent wave to the air, i.e., confined wave in the glass => discrete modes(bands) If traveling wave extending both in the glass and the air region. The separation of continuous states and discrete bands at light line.

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12 If is large, i.e, the wave of short or propagation more in z-direction, and the wave is well confined in the glass Let then with These modes decay ever more rapidly as increase, since Discrete translational symmetry Photonic crystals actually have discrete translational symmetry(DTS). ex) 1D PhC : DTS for 1D and CTS for 2D, 2D PhC : DTS for 2D and CTS for 1D, 3D PhC : DTS for 3D. ex) Fig. 4: 2D PhC with primitive lattice vector for with integer and unit cell: xz slab with the width in the y-direction

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13 That is, photonic crystals are composed of repetition of unit cells. Tanslational symmetry means that and with Eigenmodes of simultaneous eigenfunctions of and ∴ Modes can be classified by specifying and values. But not all the values of yield different eigenvalues. With and have the same eigenvalue for if ∴ Any with integer gives identical eigenvqlue of and is a degenerate set. That is, the addition of an integral multiple of on leaves the state unchanged. called as “primitive reciprocal lattice vector”.

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14 Bloch theorem, Bloch ftunction and Brillouin zone Any linear combination of degenerate eigenmodes for and is an eigenfunction. Therefore, the general solution of a system having a DTS(CTS) in the y-direction (x-direction) is periodic ft. in y-dir. Bloch theorem ; a wave propagating through the periodic material in the y-direction can always be expressed as or more generally if dielectrics periodic in 3D. Note) Discrete periodicity in the y-direction gives that is simply the product of a plane wave in the y-direction and a y-periodic ft. and thus, Thus mode frequencies must also be periodic in

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15 so that ∴ Knowl. about (1st BZ) is sufficient. When the dielectric is periodic in 3D, the eigenmodes have the form of with the inside the first BZ and a periodic ft. satisfying for all lattice vector Photonic band structures EM modes of a photonic crystal should have a Bloch form and all the informations about such a mode is given by and To solve for let’s start from from the master eq.

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16 ∴ Master eq. becomes or with ∴ Solving this eigenvalue problem for the unit cell & for each value of => photonic band structure Restricting an eigenvalue problem to a finite volume leads to a discrete spectrum of eigenvlues (ex: nearly free electronics in the 1st BZ). ∴ For each value of an infinite set of modes =>band index has the only as a parameter in it. Thus is a continuously varying ft. with for a given

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17 Rotational Symmetry and Irreducible BZ. - Phonic crystal : usually have rotational, reflection, inversion symmetry. ex) Assume a PhC with a 6-fold rotational symmetry. Let the operator rotates vectors by an angle about the To rotate a vector field we need to transform so that and

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18 def) vector field rotational operator also satisfies the master eq. with the same eigenvalue as Note) State is the Bloch state with Proof; We need to prove (sub proof) Without loss of generality (WLOG), let rotation about the origin through the angle in the xy-plane. Let displacement vector is then with the translation operator

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19 corresponds to operator and to vector Since is the Bloch state with and same eigenvalue as it follows that

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20 In general, whenever a photonic crystal has a rotation, mirro-reflection, or inversion symmetry (point group) have that symmetry as well. Full symmetry of the point group => some regions of BZ have repeated pattern => irreducible BZ( the smallest region not related by symmetry). ex) -Real lattice has 4-fold symmetry and reflection symmetries -Field patterns in real space or in the rest of the BZ is just the copies of the irreducible BZ. Mirror symmetry and Seperation of Modes Mirror reflection symmetry => Separation of the eigenvalue equation for into two separate equations ( to mirror plane) => Provides immediate information about the mode symmetries (ex: Fig. 4).

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21 Mirror reflection in the changes to - and leaves and For a system to have mirror symmetry, it should be invariant under the simultaneous reflection of and Def) mirror reflection operator ex ) Note 1) with and thus Note 2) with the reflected wave vector and an arbitrary phase Proof : Since also satisfies the master eq. with the same eigenvalue as We thus need to prove that

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22 Since transforms to we may take in 2D space. Let then Thus is the Bloch state with the reflected wave vector Note that we can always take a mirror plane so that since our dielectric has CTS in -direction. But only for a certain and If the Bloch wave propagates in the from obeys similar eq. ∴ Both and must be either even or odd under the operation. But, is a vector, is pseudovector. Thus -even mode must be and while the –odd modes must have the components and

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23 Difference of behaviors between vector and pseudovector under inversion operation (coordinate transform) and mirror operation (world transform). In general, for a given mirror operator such that mode separation is possible at the position where for according to the polarization depending on whether or is parallel to mirror (ex: TE and TM modes in 2D PhCs). But this mode separation concept is not so useful for 3D PhCs. Time-Reversal Invariance Since is Hermitian and is real, complex conjugate of master eq. is given by satisfies the same eq. as with same is just the Bloch state at and thus,

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24 holds independent of the photonic crystal structure. Note) If we take such that Thus taking is equivalent to taking time as is a consequence of the time-reversal symmetry of the Maxwell eqs. Electrodynamics in PhCs and electrons in crystals Formation of energy bands and energy gap E g in semiconductors: related to the periodicity of crystals. Schroedinger eq. is with for any translational vector Hamiltonian has the translational property such that since the kinetic term is invariant under any translation.

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