DISPERSION GENERALIZED (SLAB)

Transkript

1 2 DISPERSION GENERALIZED (SLAB) Dielectric slab Waveguide (Q finite ) Perfect Conductor waveguide = c / nclad... = k c / n // = k c / n // k // β (Q ) x Finite Q f(x) L Light line definition = c / nclad Above light line: > c / nclad FP modes "fuzzy" modes, "finite Q" Below light line : < c / nclad n clad is index of cladding, n clad =1, or more Discrete Guided modes "Q= FP modes" 21

2 DOS GENERALIZED (SLAB) Dielectric slab Waveguide (Q finite ) overall DOS is smoother = c / nclad Finite Q = k c / n // (Q ) f(x) k // DOS But an important issue is : What is DOS of modes of both types! And the ratio of them! We will come back to this for the emission control / extraction issue Light line definition = c / nclad Above light line: > c / nclad FP modes "fuzzy" modes, "finite Q" Below light line : < c / nclad n clad is index of cladding, n clad =1, or more Discrete Guided modes "Q= FP modes" 22 BASIC OF PERIODIC SYSTEM - SLAB ARRAY - LIGHT LINE 23

3 NEXT SCOPE : slab array π /a? 24 BAND GAP OPENING ~ normal incidence, k z = r ~.2, N~5 ~ reflection 1 N 1 N' 1 N order= phase/2π ~ 2π ~ 4π ~ 6π round-trip phase 25

4 NAIVE ORIGIN OF GAP SIZE (1) phasor of successive reflections ϕ 2 pπ +.25( π / N) N reflections needed to decay by ~1/e N ~1/r where r=reflection of one period ϕ 2 pπ + ( π / N) 26 NAIVE ORIGIN OF GAP SIZE (2) Reflections in one period { r1 r2 Depends on internal round-trip phase (fraction of 1-period round-trip phase) Can be low if rather destructive Can be high if etc. 27

5 EXAMPLE 8 unit cells of dielectric (ε = 5 and µ = 1) embedded in air of filling ratio.4 (case k z =) 1 frequency a/2πc average medium 1 Transmission -π π -π π -2 2 Re(ka) Im(ka) Tr( [T] ) [T]=1period transmission matrix 28 k Periodicity Brillouin zone (Un)folding + symmetry Conventional choice 1st Brillouin Zone: modulus of k minimal -π/a π/a 3π/a reduced Brillouin Zone (takes symmetries into account) 29

6 The "empty lattice" -π/a π/a 3π/a k modes are plane wave, but it is allowed (conventional/...) to represent them shifted by. 3 Dispersion along invariant direction x = c / nclad = k // c / n? Slab ARRAY Single slab k // 31

7 Dispersion along invariant direction f(x) x Now r can be large Bigger gaps, smaller room between or T by evanescence can occur Narrow T windows open instead of «evanescent death» exp(-kx) 32 Slab array in «potential well picture» T modulation becomes the «band & gap» landscape with abrupt cliffs Discrete modes couple together and form bands Examples 1 t 2 ε clad reflection=1 ε clad bands around integer p bands ε core ε core ε effectif ε effectif The whole spectrum becomes a BAND spectrum The nature of waves? Spans from "very localised" to "quasi plane waves" forbidden bands (around halfinteger values of p ) 33

8 Generality of «tight binding» π & π «PARTICLES» ψ ψ 1 ±ψ Ε/ħ 2 «MODES» T 1period ~FP double FP FSR ψ 1,ψ 2,ψ 3 Ε/ħ triple FP ψ 1,ψ 2...ψ n... Ε/ħ k= k=π/a Bloch modes k= k 34 «Simple»... because math is the same (wonkish) * A = * *... * A v = λv ( HΨ = EΨ) * All the eqns then become : v2 + v v N + v N * v N = λv1 * v1 = λv2 λ * v N-2 = v N-1 * v N-1 = λv N -i2πq/n v~ (q)e * ~ + i2πq / N + v (q)e i2πnq / N v~ (q) = v ne n=... N 1 i2πnq / N vn = v~ (q) e q = λv~ (q) E o = Let us define a Fourier transform Hence eigenvalues : λ λ q q -i2πq/n = e + i2πq/n + * e = 2 cos(2πq/n ϕ ) -i2πq/n + ϕ = e + i2πq/n ϕ + e There are only N eigenvalues!! i2πnq / N i2πn v~ (q + N) = vne e n=... N 1 = v~ (q) 35

9 So the overall issue... π /a 36...can be sketched... bands along the periodic direction coupling of guided modes??? how are these two plots compatible bands along invariant directions 37

10 ...can be shown!... very rarely shown (unfortunately in my opinion) Chigrin & Sotomayor-Torres, NATO workshop Proc and brings a third diagram : WAVEVECTOR diagram (for k-space geeks) locus of vector k at constant cut in the planes =cte also called "iso-frequency curves" (analogue to Fermi surfaces) difficulties : several folds! (plusieurs nappes) (i.e., multivalued ) 39

11 The basic point :Structure of k-space... k // π = a k // π = a k // 4