Study of EM waves in Periodic Structures (mathematical details)

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1 Study of EM waves in Periodi Strutures (mathematial details) Massahusetts Institute of Tehnology partial leture notes 1 Introdution: periodi media nomenlature 1. The spae domain is defined by a basis,(a 1, a 2, a 3 ), where any vetor an be written as r = r + R = r + α 1 a 1 + α 2 a 2 + α 3 a 3, (1) where R is the translation vetor, with α 1, α 2, α 3 integers. 2. The spetral domain is defined by a basis, (b 1, b 2, b 3 ), and similarly, the translational vetor is written as = β 1 b 1 + β 2 b 2 + β 3 b 3, (2) where β 1, β 2, β 3 are integers. 3. The two basis are linked sine the funtions (fields, permittivity) are periodi. For example, if we write the permittivity: Fourier expansion: ɛ(r) = ɛ() e i r where ɛ() = 1 V ell dr 3 ɛ(r) e i r. (3) Periodiity: ɛ(r + R) = ɛ() e i (r+r) = ɛ() e i r e i R = ɛ(r) (4) so that e i R = 1 and R = 2mπ where m {..., 1, 0, 1, 2,...}. (5) We an see that ondition (5) is immediately verified if we impose: b j a i = 2πδ ij. (6) 1

2 2 Setion 1. Introdution: periodi media nomenlature 4. Bloh-Floquet theorem: Sine EM fields are periodi, we an write them as a propagating funtion times a funtion with the same periodiity as the medium: ξ k (r) = e ik r ζ k (r) where ζ k (r + R) = ζ k (r), (7) and where ξ an represent either the eletri or magneti fields, E or H. Sine ζ(r) is periodi, we an Fourier expand it: ζ k (r) = ζ e i r, (8) so that we shall write: E k (r) = e e i(k+) r, (9a) H k (r) = h e i(k+) r. (9b) 5. Wave equation in soure-free region: From Maxwell s equation, we an easily obtain the following wave equations in soure-free regions (with ɛ = ɛ(r)): ( ) ω 2 E(r) = µ r ɛ r (r) E(r), (10a) ( ) 1 ω 2 ɛ r (r) H(r) = µ r H(r), (10b) To make these equations more symmetrial, we shall work with 1/ɛ r (r) instead of ɛ r (r) diretly, so that we define κ r (r) = 1 ɛ r (r) = κ r () e i r. (11) The wave equations are rewritten as: ( ) ω 2 κ r (r) E(r) = µ r E(r), (12a) ( ) ω 2 κ r (r) H(r) = µ r H(r). (12b)

3 3 2 Treatment of the E field 2.1 Method 1: diret expansion of the permittivity We want to write Eq. (10a) with the deomposition of Eq. (9a). First, let us ompute the first url (taking as the variable for the expansion): E k (r) = e e i(k+ ) r = i (k + ) e e i(k+ ) r. (13) Taking the url one more time gives E (r) = (k + ) (k + ) e e k i(k+ ) r. (14) Upon using Eq. (3) but hanging the index into, we write ɛ r (r)e(r) = By hanging the variables = + : ɛ r (r)e(r) = ɛ r ( ) e e i(k+ + ) r. (15) The wave equation (see Eq. (10a)) an therefore be rewritten as: ɛ r ( ) e e i(k+) r. (16) ( (k + ) (k + ) e e i(k+ ) r ω = ) 2 µ r We an simplify by exp (ik r) and multiply by exp ( i r) to get: ɛ r ( ) e e i(k+) r. (17) ( (k + ) (k + ) e e i( ) r ω = ) 2 µ r ɛ r ( ) e e i( ) r. (18) If we integrate this equation over the entire spae, we an pull all the terms out of the integral, exept e i( ) r on the left-hand side and. e i( ) r on the right-hand side. Yet, we have V dr 3 e i( ) r = 1 (2π) 3 δ( ), (19) so that Eq. (18) beomes (upon substituting by sine these are dummy variables): (k + ) (k + ) e = ( ) ω 2 µ r ɛ r ( ) e,. (20)

4 4 2.2 Method 2: expansion of the inverse of the permittivity 2.2 Method 2: expansion of the inverse of the permittivity Instead of working with Eq. (10a), we an also use Eq. (12a), whih would need the expansion of Eq. (11). Applying the same method (and transforming the index of Eq. (11) from to ), we get: ( ) κ r ( )(k + ) (k + ) e e i(k+ + ) r ω 2 = µ r e e i(k+ ) r. (21) whih, upon substituting = +, simplifying by exp (ik r), multiplying by exp ( i r), integrating over the whole spae, using Eq. (19) and finally substituting by, beomes: κ r ( )(k + ) (k + ) e = ( ) ω 2 µ r e,. (22) 3 Treatment of the H field The H field is treated in an exatly similar way to eventually obtain very similar equations. However, these equations an still be pushed further by using the fat that H k (r) = 0. Upon using this equality, we see from Eq. (9b) that (using for the expansion of the field): We an therefore define three vetors (ê 1, ê 2, ê 3 ) suh that (k + ) h = 0. (23) k + = k + ê 3, ê 1 ê 3 = ê 2 ê 3 = 0, (24a) (24b) and (ê 1, ê 2, ê 3 ) for an orthonormal tryad. In that ase, we an deompose h = h 1 ê 1 + h 2 ê 2 = =1,2 ê. (25) We need now to introdue this expression into Eq. (12b). First, we ompute H k (r) = i (k + ) ê e i(k+ ) r, (26) so that κ r (r) H (r) = i h k κ r( ) (k + ) ê e i(k+ + ) r = i κ r( ) (k + ) ê e i(k+) r. (27) Taking the next url, we write:

5 5 κ r (r) H (r) = k κ r( ) so that the wave equation (see Eq. (12b)) beomes: (k+) (k+ ) ê e i(k+) r, (28) κ r( ) (k+) (k+ ) ê e i(k+) r ( ) ω 2 ) rê = µ r ei(k+. (29) Always by the same token (multiplying by the proper funtions and integrating over whole spae), we write: κ r( ) (k + ) (k + ) ê = ( ) ω 2 µ r ê. (30) We an further simplify this expression by dot-multiplying the equation by ê that (using C (A B) = B (C A)) (k + ) (k + ) ê ê = (k + ) ê (k + ) ê and noting (31) Therefore, dot-multiplying Eq. (30) by ê, we get the final result: {(k + ) ê } ( ) (k + ) ê κ r ( ω 2 ) = µ r. (32) Upon exhanging and (transformations:,, ), we obtain {(k + ) ê } ( ) (k + ) ê κ r ( ω 2 ) = µ r. (33) whih is the relation given in Joannopoulos et al., 1995, p Upon using the same notation, we rewrite Eq. (33) as: Θ k ( ) ω 2 h = µ r h (),() (), (34a) () where Θ k (),() = κ r( ) (k + ) ê (k + ) ê. (34b)

6 6 3.1 Matrix form 3.1 Matrix form We an ast Eq. (33) in matrix form. First, we rewrite the kernel of the operator of Eq. (34b) as: (k + ) ê (k + ) ê = Remembering that ê 3 ê 1 = ê 2 and ê 3 ê 2 = ê 1, we an write: so that we write the operator as: used in Eq. (34a). ê 3 ê ê 3 ê = (k + ) (k + ) ê 3 ê ê 3 ê. (35) ( ê 2 ê 2 ê 2 ê 1 ê 1 ê 2 ê 1 ê 1 ), (36) Θ k = κ r( ) (k + ) ê (k + ) ê (),() ( ) = κ r ( ) (k + ) (k + ) ê 2 ê 2 ê 2 ê 1, (37) ê 1 ê 2 ê 1 ê 1