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Unformatted text preview: Lecture 6: Mathematical Background • Electromagnetism as an eigenvalue problem • Harmonic modes • Scale invariance and normailzed units M energy and variational principle • EM energy and variational principle • Bloch theorem • Reciprocal lattice Maxwell’s equation in the steady state Timedependent Maxwell’s equation in dielectric media: Time harmonic mode (i.e. steady state): ∇• H r, t ( 29 = ∇• ε E r, t ( 29 = ∇ × H r, t ( 29 ε r ( 29 ∂ ε E r, t ( 29 ( 29 ∂ t = ∇ × E r, t ( 29 + ∂ μ H r, t ( 29 ( 29 ∂ t = H r, t = H r e i ϖ t Maxwell equation for the steady state: ( 29 ( 29 E r, t ( 29 = E r ( 29 e i ϖ t ∇ × E r ( 29 i ϖ μ H r ( 29 ( 29 = ∇ × H r ( 29 + i ϖ ε r ( 29 ε E r ( 29 ( 29 = Master’s equation for steady state in dielectric Expressing the equation in magnetic field only: Thus, the Maxwell’s equation for the steady state can be expressed in rms of an eigenvalue problem, in direct analogy to quantum ∇ × 1 ε r ( 29 ∇ × H r ( 29 = ϖ c 2 H r ( 29 c = 1 ε μ terms of an eigenvalue problem, in direct analogy to quantum mechanics that governs the properties of electrons. Quantum mechanics Electromagnetism Field Eigenvalue problem Operator Ψ r , t ( 29 = Ψ r ( 29 e j ϖ t H r, t ( 29 = H r ( 29 e i ϖ t ˆ H Ψ r ( 29 = E Ψ r ( 29 ˆ H = ℏ 2 ∇ 2 2 m + V r ( 29 Θ H r ( 29 = ϖ 2 c 2 H r ( 29 Θ = ∇ × 1 ε r ( 29 ∇ × Electromagnetism as an eigenvalue problem The master equations define an operator: Importantly, the Θ operator is a Hermitian operator. If we define the inner product of two vector fields F (r) and G (r) as: Θ H r ( 29 ≡ ∇ × 1 ε r ( 29 ∇ × H r ( 29 then F,G ( 29 = d rF * r ( 29 ⋅ G r ( 29 ∫ ( 29 ( 29 ( 29 ( 29 * * * * 1 F, G rF G 1 r F G 1 r F G 1 r F G F,G d d d d ε ε ε ε Θ = ⋅∇ × ∇ × = ∇ × ⋅ ∇ × = ∇ × ⋅ ∇ × = ∇ × ∇ × ⋅ = Θ ∫ ∫ ∫ ∫ General property of the harmonic modes Having the Θ operator to be Hermitian leads to a number of nice properties about the harmonic modes...
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This note was uploaded on 01/10/2011 for the course ECE 695s taught by Professor Staff during the Fall '08 term at Purdue.
 Fall '08
 Staff
 Electromagnet

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