By Luecke W.

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**Extra resources for Introduction to Photonics**

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18 We use standard notation: def a·b = 3 aj bj , def a1 e1 + a2 e2 + a3 e3 = a1 e1 + a2 e2 + a3 e3 , j=1 def |a| = √ a·a≥0 ` 1 ´ ` ´ def ` ´ ` ´ ` ´ a e 1 + a 2 e 2 + a 3 e 3 × b1 e 1 + b2 e 2 + b 3 e 3 = a 2 b3 − a 3 b2 e 1 + a 3 b1 − a 1 b3 e 2 + a 1 b2 − a 2 b1 e 3 . Note that the symmetric bilinear form a·b is an indefinite inner product on C3 and that ℑ (s · s) = 0 =⇒ ℜ(s) · ℑ(s) = 0 . Modes with 0 = e · s ∈ i R for some e ∈ R3 are called evanescent. 16). 31), P⊥s is a ↔ ↔ ↔ projection operator: P⊥s P⊥s = P⊥s .

33) the latter is equivalent to 0 = e1 × N− (ω) s− × E s− (ω) + s′− × E s′− (ω) − N+ (ω) s+ × E s+ (ω) . 31) the splitting30 E s (ω) = Es⊥ (ω) c2 + Es (ω) (c2 × s) for s = s± , s′− . 32) holds. 32) is equivalent to s · E s (ω) = 0 . 31) {s , c2 , c2 × s} is a Basis31 of C3 . Therefore, a splitting of the form E s (ω) = Es⊥ (ω) c2 + Es (ω) (c2 × s) + Esadd (ω) s is possible. 45) holds. 46) N− (ω) (e1 · s− ) Es⊥− (ω) + (e1 · s′− )Es⊥′− (ω) = N+ (ω) (e1 · s+ ) Es⊥+ (ω) . 39) is equivalent to Es⊥− (ω) + Es⊥′− (ω) = Es⊥+ (ω) , (e1 · s− ) Es− (ω) + (e1 · s′− )Es′− (ω) = (e1 · s+ ) Es+ (ω) .

Therefore: For radiation fields inside homogeneous linear media it is sufficient to determine E(x, t) . 9) is — strictly speaking — only relevant for times t with νcr (x, t′ ) = 0 ∀ t′ ≥ t . 11 Compare Exercise 1. 12 Thus allowing for a phase difference between E (x, ω) and D(x, ω) . 1. 1) holds. 2) ω2 ↔ ǫ c (ω) E (x, ω) + i ωµ0 cr (x, ω) , c2 def cr (x, ω) = ex (x, ω) − ind (x, ω) . 10) is equivalent to ω c ∆x + 2↔ ǫ c (ω) E (x, ω) = grad div E (x, ω) − i ω µ0 cr (x, ω) . 7). 8), finally, serves as a definition for ρex respecting the continuity equation.

### Introduction to Photonics by Luecke W.

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