Optical Waveguide Theory (F) Manfred Hammer∗ Theoretical Electrical Engineering Paderborn University, Paderborn, Germany Paderborn University — Winter Semester 2021/2022 ∗Theoretical Electrical Engineering, Paderborn University Phone: +49(0)5251/60-3560 Warburger Straße 100, 33098 Paderborn, Germany E-mail: [email protected] 1 Course overview Optical waveguide theory A Photonics / integrated optics; theory, motto; phenomena, introductory examples. B Brush up on mathematical tools. C Maxwell equations, different formulations, interfaces, energy and power flow. D Classes of simulation tasks: scattering problems, mode analysis, resonance problems. E Normal modes of dielectric optical waveguides, mode interference. F Examples for dielectric optical waveguides. G Waveguide discontinuities & circuits, scattering matrices, reciprocal circuits. H Bent optical waveguides; whispering gallery resonances; circular microresonators. I Coupled mode theory, perturbation theory. • Hybrid analytical / numerical coupled mode theory. J A touch of photonic crystals; a touch of plasmonics. • Oblique semi-guided waves: 2-D integrated optics. • Summary, concluding remarks. 2 2-D waveguide configurations 2 µ = ∼ exp( !t) x R, 1, i (FD) nc nf d • 2-D waveguide, 1-D cross section. z 2 ns • Permittivity = n , refractive index n(x). (1-D waveguide) • @y = 0 @yE = 0, @yH = 0, 2-D TE / TM setting. • @z = 0 Modal solutions that vary harmonically with z : mode profile E¯, H¯ , E E¯ (x; z) = (x) e−iβz, propagation constant β, H H¯ effective index neff = β=k. 3 2-D waveguide configurations 2 µ = ∼ exp( !t) x R, 1, i (FD) nc nf d • 2-D waveguide, 1-D cross section. z 2 ns • Permittivity = n , refractive index n(x). (1-D waveguide) • @y = 0 @yE = 0, @yH = 0, 2-D TE / TM setting. • @z = 0 Modal solutions that vary harmonically with z : mode profile E¯, H¯ , E E¯ (x; z) = (x) e−iβz, propagation constant β, H H¯ effective index neff = β=k. ¯ 2 ¯ 2 2 ¯ (TE): principal component Ey, @x Ey + (k − β )Ey = 0, −β i E¯x = 0; E¯z = 0; H¯x = E¯y; H¯y = 0; H¯z = @xE¯y; !µ0 !µ0 E¯y & @xE¯y continuous at dielectric interfaces. 3 2-D waveguide configurations 2 µ = ∼ exp( !t) x R, 1, i (FD) nc nf d • 2-D waveguide, 1-D cross section. z 2 ns • Permittivity = n , refractive index n(x). (1-D waveguide) • @y = 0 @yE = 0, @yH = 0, 2-D TE / TM setting. • @z = 0 Modal solutions that vary harmonically with z : mode profile E¯, H¯ , E E¯ (x; z) = (x) e−iβz, propagation constant β, H H¯ effective index neff = β=k. 1 (TM): principal component H¯ , ∂ @ H¯ + (k2 − β2)H¯ = 0, y x x y y β −i E¯x = H¯y; E¯y = 0; E¯z = @xH¯y; H¯x = 0; H¯z = 0; !0 !0 −1 H¯y & @xH¯y continuous at dielectric interfaces. 3 Guided 2-D TE / TM modes, orthogonality properties • A set (index m) of guided modes of a 2-D waveguide (), (! Exercise.) p ¯ ¯ p=TE;TM m = (Em; Hm), & βm, βm 6= βl, if l 6= m. 1 Z • (E ; H ; E ; H ) := E∗ H − E∗ H + H∗ E − H∗ E dx : 1 1 2 2 4 1x 2y 1y 2x 1y 2x 1x 2y p • Power Pm per lateral (y) unit length carried by mode m, βm : 8 Z βm 2 Z > jEm;yj dx; if p = TE, p p < 2!µ0 Pm := Sz dx = ( m; m) = Z > βm 1 2 :> jHm;yj dx; if p = TM. 2!0 Z TE TM TE TE βm ∗ ( l ; m ) = 0 ; ( l ; m ) = El;y Em;y dx = δlmPm ; 2!µ0 Z TM TE TM TM βm 1 ∗ ( l ; m ) = 0 ; ( l ; m ) = Hl;y Hm;y dx = δlmPm : 2!0 4 Dielectric multilayer slab waveguide x 2 R, µ = 1, ∼ exp(i !t) (2-D, FD) nN+1 hN nN hN 1 • N interior layers, − piecewise constant = n2: 8 . n h x z < N+1 if N < ; n(x) = nl if hl−1 < x < hl; h1 n h 1 : n if x < h : 0 n0 0 0 • Principal component φ(x) (TE: φ = E¯y, TM: φ = H¯y). 2 2 2 2 • @x φ + (k nl − β )φ = 0, x 2 layer l, l = 0;:::; N + 1 (Half-infinte substrate (l = 0) and cover (l = N + 1) layers.) −2 • φ & η@xφ continuous at x = hl, (TE: η = 1, TM: η = n ). 5 Dielectric multilayer slab waveguide x nN+1 hN • Interior layer l, nN hN 1 − hl−1 < x < hl, local refractive index nl, . z 2 2 2 2 • @x φ = (β − k n )φ. h l 1 n1 h 2 0 n0 • Consider a trial value β 2 R. q • 2 2 2 2 2 2 2 2 β < k nl @x φ = −κl φ, κl := k nl − β , φ(x) = Al sin(κl x) + Bl cos(κl x). q • 2 2 2 2 2 2 2 2 β > k nl @x φ = κl φ, κl := β − k nl , κl x −κl x φ(x) = Al e + Bl e . • Unknowns Al; Bl 2 C. (Local coordinate offsets required to cope with the exponentials.) 6 Dielectric multilayer slab waveguide, guided modes x nN+1 hN • Substrate region, nN hN 1 − x < h0, local refractive index n0, . z 2 2 2 2 • @x φ = (β − k n )φ. h 0 1 n1 h 2 0 n0 • Consider a trial value β 2 R. q • 2 2 2 2 2 2 2 2 β < k n0 @x φ = −κ0 φ, κ0 := k n0 − β , φ(x) = A0 sin(κ0 x) + B0 cos(−κ0 x). q • 2 2 2 2 2 2 2 2 β > k n0 @x φ = κ0 φ, κ0 := β − k n0, κ0 x −κ x φ(x) = A0 e + B0 e 0 . • Unknown A0 2 C. Guided modes: neff = β=k > n0. 7 Dielectric multilayer slab waveguide, guided modes x nN+1 hN • Cover region, nN hN 1 − hN < x, local refractive index nN+1, . z 2 2 2 2 • @x φ = (β − k n )φ. h N+1 1 n1 h 2 0 n0 • Consider a trial value β 2 R. q • 2 2 2 2 2 2 2 2 β < k nN+1 @x φ = −κN+1 φ, κN+1 := k nN+1 − β , φ(x) = AN+1 sin(κN+1 x) + BN+1 cos(−κN+1 x). q • 2 2 2 2 2 2 2 2 β > k nN+1 @x φ = κN+1 φ, κN+1 := β − k nN+1, κN+ x −κN+1 x φ(x) = AN+1 e 1 + BN+1 e . • Unknown BN+1 2 C. Guided modes: neff = β=k > nN+1. 8 Dielectric multilayer slab waveguide x nN+1 hN nN hN 1 − 2 . z Trial value β 2 R, h β=k > n0; nN+1, 1 n1 h0 n0 κl, l = 0;:::; N + 1. 8 B −κN+1 x; h < x; > N+1 e for N <> A sin(κ x) + B cos(κ x); if β2 < k2n2; φ(x) = l l l l l for h < x < h ; κl x −κl x 2 2 2 l−1 l > Al e + Bl e ; if β > k nl ; :> κ x A0 e 0 ; for x < h0: • 2N + 2 unknowns A0; A1; B1;:::; AN; BN; BN+1. • Continuity of φ, η@xφ at N + 1 interfaces 2N + 2 equations. 9 Dielectric multilayer slab waveguide x nN+1 hN nN hN 1 − 2 . z Trial value β 2 R, β=k > n ; n . h1 n 0 N+1 h 1 0 n0 • 2N + 2 unknowns A0; A1; B1;:::; AN; BN; BN+1. • Continuity of φ, η@xφ at N + 1 interfaces 2N + 2 equations. 2 T • Arrange as linear system of equations M(β )(A0;:::; BN+1) = 0. • Identify propagation constants where M(β2) becomes singular. (Equations relate to the series of interfaces $ A transfer-matrix technique can be applied.) • Choose e.g. A0 = 1, fill A1;:::; BN+1, normalize. ( :;: ; :;: ) Guided modes fβm; (E¯m; H¯ m)g. 10 A nonsymmetric 3-layer slab waveguide x nc ns = 1:45, nf = 1:99, nc = 1:0, d = 1:5 µm, λ = 1:55 µm. nf d TE : n = 1:944, TM : n = 1:933, z 0 eff 0 eff ns TE1: neff = 1:804, TM1: neff = 1:759, TE2: neff = 1:562, TM2: neff = 1:490. 25 0.12 20 15 0.08 10 0.04 5 y y 0 0 E H -5 TE0 -0.04 -10 TM0 TE1 -15 TM1 TE2 -0.08 TM2 -20 -25 -0.12 -0.5-1 0 0.5 1 1.5 2 2.5 -0.5-1 0 0.5 1 1.5 2 2.5 x [µm] x [µm] 11 Dielectric multilayer slab waveguide, nodal properties (Fixed polarization, TE / TM.) 2 2 2 @x(@xφ) = −(k n − β )φ. 25 20 k2n2 − β2 determines the rate of change of the slope of φ. 15 10 Imagine a numerical ODE algorithm of “shooting-type”. 5 y 0 E -5• Guided modes with a growing number of nodes (x with φ(x) = 0) TE0 -10 with decreasing effectiveTE1 indices -15 TE2 -20 mode indices = number of nodes in φ. -25 “Quantum numbers”. -0.5-1 0 0.5 1 1.5 2 2.5 x [µm] 12 Dielectric multilayer slab waveguide, nodal properties (Fixed polarization, TE / TM.) 2 2 2 @x(@xφ) = −(k n − β )φ. 25 20 k2n2 − β2 determines the rate of change of the slope of φ. 15 10 Imagine a numerical ODE algorithm of “shooting-type”. 5 y 0 E -5• Guided modes with a growing number of nodes (x with φ(x) = 0) TE0 -10 with decreasing effectiveTE1 indices -15 TE2 -20 mode indices = number of nodes in φ.