Hanjo Lim School of Electrical & Computer Engineering Lecture 2. Basic Theory of PhCs : EM waves in mixed dielectric media and Eigenvalue approach

2 Maxwell equations are given as; Constitutive relations ; relations between in optical freq. range for most materials and especially for dielectric materials. ∴ Maxwell equations are given as;

3 Then ; complicated functions of time and space. But Maxwell eqs. are linear => time dependence can be expressed by harmonic modes. Then mode profiles of a given frequency are given from Maxwell equations. ; Nonexistence of source or sink Field configurations are build up of transverse EM waves. Transversality : If

4 Take main function as magnetic field Master eq. with condition completely determines * Schrodinger equation : eigenvalue problem => eigenvalue and eigenfunction For a given photonic crystal master equation => eigen modes. If modes for a given

5 Interpretation of Master equation ; Eigenvalue problem operator eigenvalue eigenvector if is allowed. with operation on => eigenvector & eigenvalue eigenvectors ; field patterns of the harmonic modes. Note) operator ; linear operator wave eq.; linear differential eq. ∴ If and are two different solutions of the eq. with same general solution of

6 ∴ Two field patterns that differ only by a multiplier ; same mode. Hermitian property of def) inner product of two vector fields Note that Proof : Note that If called normalized mode, Normalization of with def) Hermitian matrix (self-adjoint) adjoint Hermitian

7 def) Hermitian operator for arbitrary normalizable functions. Properties) 1. If operator is Hermitian are Hermitian. 2. A linear combination of Hermitian operators is a Hermitian operator. 3. The eigenvalues of a Hermitian operator are all real. Proof; Let If Hermitian operator 4. Any operator associated with a physically measurable quantity is Hermitian (postulate).

8 def) Hermitian operator for vector fields and If that is, the inner product of –operated field is independent of which function is operated, Hermitian operator. Proof of is Hermitian operator.

9 Note ; 1) zeros at large distances due to dependence 2) periodic fields in the region of integr. ( ∵ harmonics) After integration,

10 Note ; since is not a constant. General properties of harmonic modes 1) Hermitian operator eigenvalue must be real. Proof) Note that for any operator

11 Proof) Hermitian operator ; Then Note) ; is actually positive => If becomes imaginary in some frequency range, what dose it mean? Proof) From

12 Then positive positive positive If is negative in some frequency range, ; imaginary. Meaning? 2) Operator is Hermitian means that and with different frequencies and are orthogonal. Proof) let than Hermitian

13 Orthogonality & modes Meaning of orthogonality of the scalar functions => normal modes. Meaning of orthogonal vector fields : Meaning of orthogonal vector modes. Degeneracy : related to the rotational symmetry of the modes. Electromagnetic energy & variational principle => qualitative features Def) EM energy functional if is a normal mode. The EM modes are distributed so that the field pattern minimizes the EM energy functional Proof) When

14 Binomial (or Tayler) expansion

15 If is an eigenvector of with an eigenvalue of : stationary with respect to the variations of when is a harmonic mode Lowest EM eigenmode ; minimizes Then next lowest EM eigenmode ; minimizes in the subspace orthogonal to etc.

16 Another property of variational theorem on EM energy functional (why?) is minimized when the displacement field is concentrated in the regions of high dielectric constant (due to with continuous ).

17 Physical energies stored in the electric and magnetic fields Harmonic magnetic field electric field Our approach ; master eq. then and Question ; Can we make up another master eq. for and then calculate from or From

18 ∴ Master eq. should be with the operator and eigenvalue But operator is not Hermitian. Proof) Since is not a constant, is not Hermitian.

19 If we take instead of is equivalent to with and Even if Hermitian operators, it is a numerically difficult task to solve. Scaling properties of the Maxwell eqs./Contra. or expansion of PhCs. Assume an eigenmode in a dielectric configuration then What if we have another configuration of dielectric with a scalling parameter s? If we transform as then ∴ Let’s transform the position vector and operator as

20 Then, But This is just another master eq. with Likewise if dielectric constant is changed by factor of as ∴ Harmonic modes of the new system are unchanged but the mode frequencies are changed so that  Electrodynamics in PhCs and Quantum Mechanics in Solids