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

∈ µ = ∼ exp( ωt) 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

∈ µ = ∼ exp( ωt) 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, ωµ0 ωµ0 E¯y & ∂xE¯y continuous at dielectric interfaces.

3 2-D waveguide conﬁgurations

∈ µ = ∼ exp( ωt) 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, ω0 ω0 −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 ψm = (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 ψm, βm :  Z βm 2 Z  |Em,y| dx, if p = TE, p p  2ωµ0 Pm := Sz dx = (ψm; ψm) = Z  βm 1 2  |Hm,y| dx, if p = TM. 2ω0

Z TE TM TE TE βm ∗ (ψl ; ψm ) = 0 , (ψl ; ψm ) = El,y Em,y dx = δlmPm , 2ωµ0 Z TM TE TM TM βm 1 ∗ (ψl ; ψm ) = 0 , (ψl ; ψm ) = Hl,y Hm,y dx = δlmPm . 2ω0

4 Dielectric multilayer slab waveguide

x ∈ R, µ = 1, ∼ exp(i ωt) (2-D, FD) nN+1 hN nN hN 1 • N interior layers, − piecewise constant = n2: 

...... 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 ∈ 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

nN+1 hN nN • Interior layer l, hN 1 − hl−1 < x < hl, local refractive index nl,

...... z • 2 2 2 2 h ∂x φ = (β − k nl )φ. 1 n1 h0 n 0 • Consider a trial value β2 ∈ 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 ∈ C. (Local coordinate offsets required to cope with the exponentials.)

6 Dielectric multilayer slab waveguide, guided modes

nN+1 hN nN • Substrate region, hN 1 − x < h0, local refractive index n0,

...... z • 2 2 2 2 h ∂x φ = (β − k n0)φ. 1 n1 h0 n 0 • Consider a trial value β2 ∈ 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 ∈ C. Guided modes: neff = β/k > n0.

7 Dielectric multilayer slab waveguide, guided modes

nN+1 hN nN • Cover region, hN 1 − hN < x, local refractive index nN+1,

...... z • 2 2 2 2 h ∂x φ = (β − k nN+1)φ. 1 n1 h0 n 0 • Consider a trial value β2 ∈ 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 ∈ C. Guided modes: neff = β/k > nN+1.

8 Dielectric multilayer slab waveguide

nN+1 hN nN hN 1 −

2 ...... z Trial value β ∈ R, h1 n β/k > n0, nN+1, h 1 0 n0 κl, l = 0,..., N + 1.

 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

nN+1 hN nN hN 1 −

2 ...... z Trial value β ∈ R, h1 n β/k > n0, nN+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 {βm, (E¯m, H¯ m)}. 10 A nonsymmetric 3-layer slab waveguide

x ns = 1.45, nf = 1.99, nc = 1.0, nc d = 1.5 µm, λ = 1.55 µm.

nf d TE0: neff = 1.944, TM0: neff = 1.933, z TE1: neff = 1.804, TM1: neff = 1.759, ns 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

25 (Fixed polarization, TE / TM.) 20 15 10 5

y 0 E -5 TE0 -10 TE1 -15 TE2 -20 2 2 2 -25 ∂ (∂ φ) = −(k n − β )φ. -0.5-1 0 0.5 1 1.5 2 2.5 x x x [µm] k2n2 − β2 determines the rate of change of the slope of φ. Imagine a numerical ODE algorithm of “shooting-type”.

• Guided modes with a growing number of nodes (x with φ(x) = 0) with decreasing effective indices mode indices = number of nodes in φ. “Quantum numbers”.

12 Dielectric multilayer slab waveguide, nodal properties

25 (Fixed polarization, TE / TM.) 20 15 10 5

• Guided modes with a growing number of nodes (x with φ(x) = 0) with decreasing effective indices mode indices = number of nodes in φ. “Quantum numbers”. • A fundamental mode with zero nodes and highest effective index.

12 Dielectric multilayer slab waveguide, nodal properties

25 (Fixed polarization, TE / TM.) 20 15 10 5

12 Dielectric multilayer slab waveguide, nodal properties

25 (Fixed polarization, TE / TM.) 20 15 10 5

• A sign change of ∂xφ is required to form a guided mode There must be some region (layer) with k2n2 − β2 > 0.

Interval for effective indices neff of guided modes:

max{n0, nN+1} < neff < maxl{nl}.

12 3-layer slab waveguide, dispersion curves

x nc Symmetric waveguide, moderate refractive index contrast, z nf d

ns ns = 1.45, nf = 1.99, nc = 1.45.

TE TM λ = 1.55µm TE TM d = 0.4µm 2 n 2 n f f 1.9 0 1.9 1 1.8 2 1.8 3 0 1.7 1.7 1 eff eff n 1.6 n 1.6 2

1.5 1.5 n n s s 1.4 1.4 0 1 2 3 4 5 0.5 1 1.5 2 d [µm] λ [µm]

(Caution: ∂λ = 0 assumed !)

13 3-layer slab waveguide, dispersion curves

x nc Nonsymmetric waveguide,

nf d moderate refractive index contrast, z ns ns = 1.45, nf = 1.99, nc = 1.0.

TE TM λ = 1.55µm TE TM d = 0.4µm 2 n 2 n f f 1.9 0 1.9 1 1.8 2 1.8 3 0 1.7 1.7 eff eff n n 1 1.6 1.6 2 1.5 1.5 n n s s 1.4 1.4 0 1 2 3 4 5 0.5 1 1.5 2 d [µm] λ [µm]

(Caution: ∂λ = 0 assumed !)

13 3-layer slab waveguide, dispersion curves

x nc Symmetric waveguide, high refractive index contrast, z nf d

ns ns = 1.45, nf = 3.45, nc = 1.45.

TE TM d = 0.22µm 3.5 TE TM λ = 1.55µm n 3.5 n f f 0 3 1 3 0 2 1 2.5 2.5 eff eff n 2 n 2 2

1.5 n 1.5 n s s 0 0.5 1 1.5 2 0.5 1 1.5 2 d [µm] λ [µm]

(Caution: ∂λ = 0 assumed !)

13 3-layer slab waveguide, dispersion curves

x nc Nonsymmetric waveguide,

nf d high refractive index contrast, z ns ns = 1.45, nf = 3.45, nc = 1.0.

TE TM d = 0.22µm 3.5 TE TM λ = 1.55µm n 3.5 n f f 0 0 3 1 3 2 1 2.5 2.5 eff eff n 2 n 2 2

1.5 n 1.5 n s s 0 0.5 1 1.5 2 0.5 1 1.5 2 d [µm] λ [µm]

(Caution: ∂λ = 0 assumed !)

13 3-layer slab waveguide, dispersion curves

Remarks / observations: • At large core thicknesses, or short wavelengths, for all modes: neff approaches the level nf of bulk waves in the core material.

• Modes of higher order at the same neff supported by waveguides with thickness increased by specific distances. 2 2 2 2 Guided mode, layer l with κl = (k n − β ) > 0, field φ(x) ∼ cos(κlx + χ) for x ∈ layer l; increase layer thickness by ∆x = π/κl, such that κl(x + ∆x) = κlx + π −→ the thicker waveguide supports a mode of order +1 with the same propagation constant. • Cutoff thicknesses at fixed wavelength. Nonsymmetric 3-layer waveguide ns 6= nc : There exist cutoff thicknesses for all modes. Symmetric 3-layer waveguide ns = nc : Cutoff thicknesses exist for all modes of order ≥ 1, no cutoff thickness for the fundamental TE/TM modes. • λ is the “length-defining” quantity; wavelength scaling, factor a : −1 neff(λ, d) = neff(aλ, ad), β(λ, d) = a β(aλ, ad). • Cutoff wavelengths for waveguides with fixed thickness. For all modes; exception: no cutoff wavelength for the fundamental TE/TM modes in a symmetric 3-layer waveguide.

14 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 1.50 µm, TE0: neff = 1.946.

20 TE0 18 16 14 12 y

E 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 1.25 µm, TE0: neff = 1.932.

22 TE0 20 18 16 14 12 y E 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 1.00 µm, TE0: neff = 1.908.

24 22 TE0 20 18 16 14 y

E 12 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.75 µm, TE0: neff = 1.868.

26 TE0 24 22 20 18 16

y 14 E 12 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.50 µm, TE0: neff = 1.791.

28 TE0

y 16 E 12

0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.25 µm, TE0: neff = 1.630.

32 TE0 28

20 y

E 16

0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.10 µm, TE0: neff = 1.494.

26 TE0 24 22 20 18 16

y 14 E 12 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.05 µm, TE0: neff = 1.462.

20 18 TE0 16 14 12 y

E 10 8 6 4 2 0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.02 µm, TE0: neff = 1.452.

12 TE0

8 y

E 6

0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, mode conﬁnement

x nc Symmetric waveguide, moderate refractive index contrast, z nf d ns = 1.45, nf = 1.99, nc = 1.45, λ = 1.55 µm,

ns d = 0.01 µm, TE0: neff = 1.450.

9 TE0 8

5 y E 4

0 -0.8-1.2 0 0.4 0.8 1.2-0.4 x [µm]

15 3-layer slab waveguide, ray model

x nc θ z nf d θ

Field in the core: −i(κx + βz) −i(−κx + βz) 2 2 2 2 ∼ au e + ad e , k nf = β + κ

propagation angle θ with β = knf cos θ, κ = knf sin θ.

Guided mode formation: • Repleated total internal reﬂection of waves in the core at upper and lower interfaces • Calculate optical phase gain, including phase jumps for reﬂection at interfaces (polarization dependent). • Phase gain of 2π for one “round trip”, “transverse resonance condition” ←→ constructive interference of waves.

(A frequently encountered intuitive model ... of very limited applicability.)

16 3-layer slab waveguide, ray model

x nc

z nf d ?

Field in the core: −i(κx + βz) −i(−κx + βz) 2 2 2 2 ∼ au e + ad e , k nf = β + κ

propagation angle θ with β = knf cos θ, κ = knf sin θ.

(A frequently encountered intuitive model ... of very limited applicability.)

16 3-D waveguides

x n(x, y) y

Cross sections (2-D) of typical integrated-optical waveguides.

17 3-D rectangular waveguides

x No analytical solutions : n(x, y) y • numerical mode solvers. • approximations.

18 Outline: (!)

• Divide into slices ρ = I, II, III : n(x, y) = nρ(x), if y ∈ slice ρ. 00 2 2 2 • Compute polarized modes Xρ(x), βρ, Xρ + (k nρ − βρ)Xρ = 0, Nρ = βρ/k. • Consider a scalar mode equation for the principal component Ψ of the 3-D waveguide 2 2 2 2 2 ∂x Ψ + ∂y Ψ + (k n − β )Ψ = 0, Ψ = Ey (TE), Ψ = Hy (TM). 0 • Ansatz: Ψ(x, y) = Xρ(x) Y(y), if y ∈ slice ρ; require continuity of Y and Y .

• Effective index proﬁle: N(y) := Nρ, if y ∈ slice ρ.

Y00 + (k2 N2 − β2)Y = 0, a 1-D mode equation for Y, β with the effective index proﬁle N in place of the refractive indices.

Effective index method

y n(x, y)

19 Effective index method

I IIIII x

y nI(x) nII(x) nIII(x)

Outline: (!)

• Effective index proﬁle: N(y) := Nρ, if y ∈ slice ρ.

Y00 + (k2 N2 − β2)Y = 0, a 1-D mode equation for Y, β with the effective index proﬁle N in place of the refractive indices.

19 Effective index method, schematically

n(x, y) x

NI NII NIII N(y) β

Remarks / issues: • A popular, quite intuitive method. • Frequently an (often informal) basis for discussion of waveguide properties. • ↔ Relevance of the slab waveguide model. • Manifold variants / ways of improvements exist. • What if a slice does not support a guided slab mode? • What about higher order modes? • How to evaluate modal ﬁelds? What about other than principal components? • ...

20 Outline: (!)

• Identify a reference slice, refractive index proﬁle nr(x).

• Compute polarized guided slab modes (E¯, H¯ )r, βr for the reference slice. • For each each reference slab mode : ... • Choose an ansatz: (VEIM) E E Ex, Ey, Ez 0, E¯r,y(x)Y y (y), E¯r,y(x)Y z (y) (x, y, z) = H H H (TE) Hx, Hy, Hz H¯r,x(x)Y x (y), H¯r,z(x)Y y (y), H¯r,z(x)Y z (y)

E E E Ex, Ey, Ez E¯r,x(x)Y x (y), E¯r,z(x)Y y (y), E¯r,z(x)Y z (y) (x, y, z) = H H (TM) Hx, Hy, Hz 0, H¯r,y(x)Y y (y), H¯r,y(x)Y z (y)

Y·(y) = ?

Variational effective index method

y n(x, y)

21 Variational effective index method

y nr(x)

Outline: (!)

• Identify a reference slice, refractive index proﬁle nr(x).

E E E Ex, Ey, Ez E¯r,x(x)Y x (y), E¯r,z(x)Y y (y), E¯r,z(x)Y z (y) (x, y, z) = H H (TM) Hx, Hy, Hz 0, H¯r,y(x)Y y (y), H¯r,y(x)Y z (y)

Y·(y) = ?

21 A functional for guided modes of 3-D dielectric waveguides

(→ Exercise.) E E¯ • (x, y, z) = (x, y) e−iβz, β ∈ , H H¯ R E¯, H¯ → 0 for x, y → ±∞.

• (C + iβR)E¯ = −iωµ0H¯ , (C + iβR)H¯ = iω0E¯,     0 1 0 0 0 ∂y R =  −1 0 0  , C =  0 0 −∂x  . 0 0 0 −∂y ∂x 0

ω hE, Ei + ωµ hH, Hi + ihE, CHi − ihH, CEi • B(E, H) := 0 0 , ZZ hE, RHi − hH, REi hF, Gi = F∗ · G dx dy.

d B(E¯, H¯ ) = β, B(E¯ + s δE¯, H¯ + s δH¯ ) = 0 ds s=0 at valid mode ﬁelds E¯, H¯ , for arbitrary δE¯, δH¯ .

22 Variational effective index method

y nr(x)

Outline, continued: (!) • Restrict B to the VEIM ansatz, require stationarity with respect to the {Y·} .

1-D mode (“-like”) equations for principal unknowns YHx (TE) and YEx (TM) with effective quantities in place of refractive indices, all other Y· can be computed.

23 Optical ﬁbers

x z

[ Optical Communication A-D ]

24 Circular step index optical ﬁbers

y (FD) r Circular symmetry a θ cylindrical coordinates r, θ, z. ncore x n , r ≤ a, = n2, n(r) = core ncladding, r > a. ncladding Circular and axial symmetry: E E¯ (r, θ, z) = (r) e−ilθ − iβz, l ∈ , β ∈ . H H¯ Z R (Er, Eθ , Ez, Hr, Hθ , Hz) 2 2 Where ∂ = 0: ∆ψ + k n ψ = 0, ψ ∈ {Er,... Hz}. 2 2 1 2 2 2 l ¯ ¯ ∂r φ + ∂rφ + (k n − β − 2 )φ = 0, φ ∈ {Er,... Hz} r r (An ODE of Bessel type.)

& vectorial interface conditions at r = a. (Alternatively: Scalar theory, LP modes.) ( ... ) 25 “Complex” waveguides

∼ exp(i ωt) (FD) Attenuating / gain media, leakage Mode amplitudes change along propagation distance.

∂z = 0, ∂zn = 0, mode ansatz with complex propagation constant: E E¯ (x, y, z) = (x, y) e−iγz, H H¯ E¯, H¯ : mode proﬁle, γ = β − iα ∈ C : propagation constant, neff = γ/k ∈ C, β ∈ R : phase constant, α ∈ R : attenuation constant, −i γz −i βz −αz 2 −2αz ψ(z) ∼ e = e e , |ψ(z)| ∼ e , 1 L = : propagation length, if α > 0. p 2α Applies to all former examples. γ ∈ C: Entire theory needs to be reconsidered, in principle.

26 “Complex” waveguides, loss

x 2-D, nc ns = 1.45, nf = 1.99 − i0.1, nc = 1.0, d = 0.5 µm, λ = 1.55 µm. nf d z Bound modes: ns TE0: neff = 1.767 − i0.093, Lp = 1.32 µm.

30 TE0, Re E 3 y TE0 20 ReE y 2 ImE 10 y 1

y 0 x [µm] E 0

-10 -1

-20 -2

-30 -3 -2-3 0 1 2 3-1 0 1 2 3 4 5 6 7 8 9 x [µm] z [µm]

(Mode attenuation, essentially complex non-plane proﬁles, curved wavefronts, Sx 6= 0.) (Analysis: as before (. . .); boundary conditions: bound ﬁelds, integrability.)

27 “Complex” waveguides, loss

x 2-D, nc ns = 1.45, nf = 1.99 − i0.1, nc = 1.0, d = 0.5 µm, λ = 1.55 µm. nf d z Bound modes: ns TM0: neff = 1.640 − i0.074, Lp = 1.66 µm.

TM0, Re H 0.15 3 y TM0 ReH 0.1 y 2 ImH y 0.05 1

y 0 x [µm] H 0 -0.05 -1 -0.1 -2 -0.15 -3 -2-3 0 1 2 3-1 0 1 2 3 4 5 6 7 8 9 x [µm] z [µm]

(Mode attenuation, essentially complex non-plane proﬁles, curved wavefronts, Sx 6= 0.) (Analysis: as before (. . .); boundary conditions: bound ﬁelds, integrability.)

27 “Complex” waveguides, gain

x 2-D, nc ns = 1.45, nf = 1.99 + i0.1, nc = 1.0, d = 0.5 µm, λ = 1.55 µm. nf d Bound modes: z 1 ns TE0: neff = 1.767 + i0.093, 2|α| = 1.32 µm.

30 TE0, Re E 3 y TE0 20 ReE y 2 ImE 10 y 1

y 0 x [µm] E 0

-10 -1

-20 -2

-30 -3 -2-3 0 1 2 3-1 -8-9 -4-5-6-7 0-1-2-3 x [µm] z [µm]

(Modal gain, essentially complex non-plane proﬁles, curved wavefronts, Sx 6= 0.) (Analysis: as before (. . .); boundary conditions: bound ﬁelds, integrability.)

28 “Complex” waveguides, gain

x 2-D, nc ns = 1.45, nf = 1.99 + i0.1, nc = 1.0, d = 0.5 µm, λ = 1.55 µm. nf d Bound modes: z 1 ns TM0: neff = 1.640+i0.074, 2|α| = 1.66 µm.

TM0, Re H 0.15 3 y TM0 ReH 0.1 y 2 ImH y 0.05 1

y 0 x [µm] H 0 -0.05 -1 -0.1 -2 -0.15 -3 -2-3 0 1 2 3-1 -8-9 -4-5-6-7 0-1-2-3 x [µm] z [µm]

(Modal gain, essentially complex non-plane proﬁles, curved wavefronts, Sx 6= 0.) (Analysis: as before (. . .); boundary conditions: bound ﬁelds, integrability.)

28 “Complex” waveguides, leakage

x nc 2-D, nf d ns = 3.45, nb = 1.45, nf = 3.45, nc = 1.0, d = 0.22 µm, g = 0.5 µm, λ = 1.55 µm. nb g Leaky modes: z TE : n = 2.805 − i2.432 · 10−5, L = 5073 µm. ns 0 eff p

TE(lb), Re E 30 TE(lb) 1.5 y ReE 20 y 1 ImE 10 y 0.5 0 y 0 x [µm] E -0.5 -10 -1 -20 -1.5 -30 -2 -1-2 0 1 2 3 0 1 2 3 4 5 x [µm] z [µm]

(Radiative loss, essentially complex non-plane profiles, curved wavefronts, Sx 6= 0, field growth for x → −∞.) (Analysis: as before (. . .); boundary conditions: outgoing wave for x → −∞, bound field at x → ∞.)

29 “Complex” waveguides, leakage

x nc 2-D, nf d ns = 3.45, nb = 1.45, nf = 3.45, nc = 1.0, d = 0.22 µm, g = 0.5 µm, λ = 1.55 µm. nb g Leaky modes: z TM : n = 1.878−i3.203·10−3, L = 38.51 µm. ns 0 eff p

TM(lb), Re H 0.3 TM(lb) 1.5 y ReH 0.2 y 1 ImH y 0.5 0.1 0 y 0 x [µm] H -0.5 -0.1 -1 -0.2 -1.5 -0.3 -2 -1-2 0 1 2 3 0 1 2 3 4 5 x [µm] z [µm]

29 Upcoming

Next lectures: • Waveguide discontinuities & circuits, scattering matrices, reciprocal circuits. • Bent optical waveguides; whispering gallery resonances; circular microresonators. • Coupled mode theory, perturbation theory.