Optical Waveguide Theory

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Optical Waveguide Theory

Manfred Hammer∗

Theoretical Electrical Engineering Paderborn University, Germany

Paderborn University — Summer Semester 2020

∗_{Theoretical Electrical Engineering, Paderborn University} _{Phone: +49(0)5251/60-3560}

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2 Motto

MMET’08, Mathematical Methods in Electromagnetic Theory Odesa, Ukraine, June 29 – July 2, 2008

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Maxwell equations

SI, in matter, time domain, differential form:

∇·D = ρf,

∇×E = −B,˙

∇·B = 0,

∇×H = Jf+ ˙D,

D = 0E+P, B = µ0(H+M). (+constitutive relations)

E(r,t): electric field,

D(r,t): (di-)electric displacement,

B(r,t): magnetic induction (field, flux density),

H(r,t): magnetic field (. . .),

ρf(r,t): density of free charges,

Jf(r,t): density of free currents,

P(r,t): polarization,

M(r,t): magnetization,

0: free space permittivity, µ0: free space permeability.

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4 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.

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Formalities

Organization of the course: • _{Lectures (}≈14×)

• _{Homework (7}×)

• _{Tutorials, Exercises (13}×₎

• _Exam

Related textbooks (examples):

C. Vassallo,Optical Waveguide Concepts, Elsevier, Amsterdam (1991),

K. Okamoto,Fundamentals of Optical Waveguides, Academic Press, San Diego, USA (2000), R. M¨arz,Integrated Optics: Design and Modeling, Artech House, Norwood, USA (1995), A.W. Snyder, J.D. Love,Optical Waveguide Theory, Chapman and Hall, London, UK (1983); & general introductory texts on classical electrodynamics.

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6 Optical waveguides: phenomena, examples

• _{Beam propagation in free space} • _{Guided light propagation} • _{Waveguide end facet} • _{Crossing of two waveguides}

• _{Modes of 1-D multilayer slab waveguides} • _{Modes of 2-D channel waveguides} • _{Circular step-index optical fibers}

• _{Evanescent coupling between waveguides} • _{Bent waveguides}

• _{Circular microring-resonator} • _{Microdisk resonator}

• _CROW

• _{Waveguide corner}

• _{Photonic crystal waveguide} • _{Exciting TET !}

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Optical waveguide “theory”

Task: solve

∇×E = −B,˙

∇×H = Jf+ ˙D,

∇·D = ρf,

∇·B = 0,

D = 0E+P,

B = µ0(H+M), (& . . .).

In this course:

• _{specialization to problems relevant for integrated optics,} • _{theoretical basis for the — mostly — numerical solution,} • _{approximate concepts,}

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8 Course overview

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Vector calculus, keywords

(here: Cartesian coordinates) Ingredients:

• _{Space and time coordinates:} _r₌





x y z



→(x,y,z), t.

• _{Scalar and vector fields:} φ(r,t), A(r,t), A=





Ax

Ay

Az



.

• _{Inner product:} _A·B=AxBx+AyBy+AzBz.

• _{Vector product:} _A×B=





AyBz−AzBy

AzBx−AxBz

AxBy−AyBx



.

• _{Time derivatives:} ∂φ

∂t, ∂tφ, ˙

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9 Vector calculus, keywords





x y z



→(x,y,z), t.





Ax

Ay

Az



.





AyBz−AzBy

AzBx−AxBz

AxBy−AyBx



.

∂t, ∂tφ, ˙

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



x y z



→(x,y,z), t.





Ax

Ay

Az



.





AyBz−AzBy

AzBx−AxBz

AxBy−AyBx



.

∂t, ∂tφ, ˙

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



x y z



→(x,y,z), t.





Ax

Ay

Az



.





AyBz−AzBy

AzBx−AxBz

AxBy−AyBx



.

∂t, ∂tφ, ˙

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



x y z



→(x,y,z), t.





Ax

Ay

Az



.





AyBz−AzBy

AzBx−AxBz

AxBy−AyBx



.

∂t, ∂tφ, ˙

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• _{Del, nabla:} ∇=



 ∂x

∂y

∂z



.

• _{Gradient: grad}_φ₌∇φ=



 ∂xφ

∂yφ

∂zφ



.

• _{Divergence: div}_A₌∇·A=∂xAx+∂yAy+∂zAz.

• _{Curl: curl}A=rotA=∇×A=





∂yAz−∂zAy

∂zAx−∂xAz

∂xAy−∂yAx



.

• _Laplacian: _{∆ =}∇·∇=∇2,

∆φ=∂x2φ+∂y2φ+∂x2φ, ∆A=





∆Ax

∆Ay

∆Az



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

 ∂x

∂y

∂z



.



 ∂xφ

∂yφ

∂zφ



.





∂yAz−∂zAy

∂zAx−∂xAz

∂xAy−∂yAx



.

• _Laplacian: _{∆ =}∇·∇=∇2,

∆φ=∂x2φ+∂y2φ+∂x2φ, ∆A=





∆Ax

∆Ay

∆Az



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

 ∂x

∂y

∂z



.



 ∂xφ

∂yφ

∂zφ



.





∂yAz−∂zAy

∂zAx−∂xAz

∂xAy−∂yAx



.

• _Laplacian: _{∆ =}∇·∇=∇2,

∆φ=∂x2φ+∂y2φ+∂x2φ, ∆A=





∆Ax

∆Ay

∆Az



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

 ∂x

∂y

∂z



.



 ∂xφ

∂yφ

∂zφ



.





∂yAz−∂zAy

∂zAx−∂xAz

∂xAy−∂yAx



.

• _Laplacian: _{∆ =}∇·∇=∇2,

∆φ=∂x2φ+∂y2φ+∂x2φ, ∆A=





∆Ax

∆Ay

∆Az



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

 ∂x

∂y

∂z



.



 ∂xφ

∂yφ

∂zφ



.





∂yAz−∂zAy

∂zAx−∂xAz

∂xAy−∂yAx



.

• _Laplacian: _{∆ =}∇·∇=∇2,

∆φ=∂2xφ+∂y2φ+∂x2φ, ∆A=





∆Ax

∆Ay

∆Az



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Dirac delta

A linear functional

that extracts the value of a function at one point:

1-D: Z b a

f(x)δ(x−x0)dx=

f(x0), if a<x0<b, 0 otherwise;

δ(x−x0) =0, if x6=x0.

3-D: Z V

f(r)δ(r−r0)dV =

f(r0), if r0∈ V, 0 otherwise;

δ(r−r0) =0, if r6=r0.

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11 Dirac delta

A linear functional

that extracts the value of a function at one point:

1-D: Z b a

f(x)δ(x−x0)dx=

f(x0), if a<x0<b, 0 otherwise;

δ(x−x0) =0, if x6=x0.

3-D: Z V

f(r)δ(r−r0)dV =

f(r0), if r0∈ V, 0 otherwise;

δ(r−r0) =0, if r6=r0. Implications: manifold.

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Fourier transform, 1-D

1-D: A function f(x)∈_C of one variable:

f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• Arbitrary: positioning of factors 1/√2π, signs of exponents.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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12 Fourier transform, 1-D

f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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f(x) = √1

2π

Z ∞

−∞

˜_f(k) _eikx_d_k_, _˜_f_{(k) =}_√1

2π

Z ∞

−∞

f(x)e−ikxdx.

• αf^₁+βf₂ =α˜f₁+β˜f2.

• f(x) =f(−x) ˜f(k) = ˜f(−k).

• f(x) =−f(−x) ˜f(k) =−˜f(−k).

• f ∈_R ˜f(−k) = ˜f∗(k).

• δ(x) = 1

2π

Z ∞

−∞

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13 Fourier transform

3-D: A field φ(r):

φ(r) = √1

2π3 Z

˜

φ(k) eik·rd3k, φ˜(k) = √1

2π3 Z

φ(r) e−ik·rd3r.

4-D: A field φ(r,t):

φ(r,t) = √1

2π4 Z Z

˜

φ(k, ω)ei(k·r−ωt)d3kdω,

˜

φ(k, ω) = √1

2π4 Z Z

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Fourier transform

3-D: A field φ(r):

φ(r) = √1

2π3 Z

˜

φ(k) eik·rd3k, φ˜(k) = √1

2π3 Z

φ(r) e−ik·rd3r.

4-D: A field φ(r,t):

φ(r,t) = √1

2π4 Z Z

˜

φ(k, ω) ei(k·r−ωt)d3kdω,

˜

φ(k, ω) = √1

2π4 Z Z

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14 Directionally constant systems

AlinearPDE in two unknowns

(A∂xx+B∂yy+C∂xy+D∂x+E∂y+F)ψ(x,y) =0, coefficients A(x,y), . . . ,F(x,y).

If the system isconstant inx, ∂xA=. . .=∂xF=0 , • use an ansatz ψ(x,y) = ˜ψ(y)eikx.

B∂yy+ (E+ikC)∂y+ (F+ikD−k2A)

_˜

ψ(y) =0, . . . a DE in one unknown, with parameterk.

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Directionally constant systems

If the system isconstant inx, ∂xA=. . .=∂xF=0 , • writeψas ψ(x,y) =

Z

˜

ψ(k,y) eikxdk.

• _{use an ansatz} ψ(x,y) = ˜ψ(y)eikx.

_˜

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If the system isconstant inx, ∂xA=. . .=∂xF=0 , • writeψas ψ(x,y) =

Z

˜

ψ(k,y) eikxdk. Z

_˜

ψ(k,y) eikxdk=0,

_˜

ψ(k,y) =0, (for allk),

. . . a set of DEs in one unknown.

• use an ansatz ψ(x,y) = ˜ψ(y)eikx.

_˜

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_˜

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_˜

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_˜

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15 General solution of the wave equation

∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

ψ(k, ω)ei(k·r−ωt)dωd3k,

−k2+ ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(k, ω) =a_f(k)δ(ω−ωk) +ab(k)δ(ω+ωk), ωk =c|k|,

ψ(r,t) = 1 (2π)2

Z

af(k)ei(k·r−ωkt) +_a_b₍_k₎ei(k·r+ωkt)d3_k,

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General solution of the wave equation

∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

−k2+ ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(r,t) = 1 (2π)2

Z

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∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

−k2+ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(r,t) = 1 (2π)2

Z

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∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

−k2+ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(r,t) = 1 (2π)2

Z

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∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

−k2+ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(r,t) = 1 (2π)2

Z

a_f(k)ei(k·r−ωkt) +_a

b(k)ei(k·r+ωkt)

d3k,

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∆− 1

c2

∂2 ∂t2

ψ(r,t) =0, ψ(r,0) =ψ0(r), ∂tψ(r,0) =φ0(r),

& ψ(r,t) = 1 (2π)2

Z Z

˜

−k2+ω

2 c2

˜

ψ(k, ω) =0,

˜

ψ(r,t) = 1 (2π)2

Z

a_f(k)ei(k·r−ωkt) +_a

b(k)ei(k·r+ωkt)

d3k,

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16 A touch of variational calculus

• _Functional: L: U −→ _R,C,

u −→ L(u),

a map from a spaceUof functions to real / complex numbers.

• _Stationary_functional: d

dsL(u+s v)

s=0 =0 for all v, the variation ofLatuvanishes for arbitrary directionsv.

• _Restriction_{of a functional:}

. . .to a parametrized family of functions;

extremization with respect to these parameters, approximations of stationary points of the functional.

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A touch of variational calculus

u −→ L(u),

dsL(u+s v)

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u −→ L(u),

dsL(u+s v)

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Example:

U={u: [0, π]→_R|u(0) =u(π) =0},

L:U →_R, L(u) =

Z π

0 (∂xu) 2_d_x

Z π

0

u2dx

.

( . . . )

Lstationary atu,

d

dsL(u+s v)

s=0=0 ∀v.

usatisfies DE & b.c.,

∂x2u=−λu, λ=L(u),

u(0) =u(π) =0.

↓

RestrictL, L(a) =L(u|_a).

Lstationary ata, ∇_aL=0. Approximate solution

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Example:

U={u: [0, π]→_R|u(0) =u(π) =0},

L:U →_R, L(u) =

Z π

0 (∂xu) 2_d_x

Z π

0

u2dx

.

( . . . )

Lstationary atu,

d

dsL(u+s v)

s=0=0 ∀v.

u(0) =u(π) =0.

↓

RestrictL, L(a) =L(u|_a).

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Example:

U={u: [0, π]→_R|u(0) =u(π) =0},

L:U →_R, L(u) =

Z π

0 (∂xu) 2_d_x

Z π

0

u2dx

.

( . . . )

Lstationary atu,

d

dsL(u+s v)

s=0=0 ∀v.

u(0) =u(π) =0.

↓

RestrictL_, L(a) =L(u|_a).

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18 Course overview

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. . . ?

“This concerns time harmonic fields. . .with angular frequency. . .,

for vacuum wavenumber. . ., speed of light. . ., and wavelength. . ..”

“The problem is governed by the Maxwell curl equations in the frequency domain for the electric field. . .and magnetic field. . ., for

(lossless) uncharged dielectric, nonmagnetic linear (isotropic) media with (piecewise constant) relative permittivity. . .:

. . . ( . ) ”

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20 Maxwell equations, Fourier transform

∇·D=ρf, ∇×E=−B˙, ∇·B=0, ∇×H=Jf+ ˙D & F(r,t) = √1

2π Z

˜

F(r, ω)eiωtdω, F˜(r, ω) = √1

2π Z

F(r,t) e−iωtdt

E(r,t), D(r,t), B(r,t), H(r,t), ρf(r,t), Jf(r,t)

˜

E(r, ω), D˜(r, ω), B˜(r, ω), H˜(r, ω), ρ˜f(r, ω), ˜Jf(r, ω),

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD˜

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Maxwell equations, Fourier transform

∇·D=ρf, ∇×E=−B˙, ∇·B=0, ∇×H=Jf+ ˙D & F(r,t) = √1

2π Z

˜

F(r, ω)eiωtdω, F˜(r, ω) = √1

2π Z

F(r,t) e−iωtdt

E(r,t), D(r,t), B(r,t), H(r,t), ρf(r,t), Jf(r,t)

˜

E(r, ω), D˜(r, ω), B˜(r, ω), H˜(r, ω), ρ˜f(r, ω), ˜Jf(r, ω),

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD˜

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21 Maxwell equations, frequency domain

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD.˜

F(r,t)∈_R F˜(r,−ω) = (˜F(r, ω))∗

“at frequencyω0”: F˜(r, ω) =

r π

2F¯(r)δ(ω−ω0) +

r π

2 F¯ ∗

(r)δ(ω+ω0)

F(r,t) = 1

2

n

¯

F(r)eiω0t+ ¯F∗(r)e−iω0to,

F(r,t) =Re

n

¯

F(r)eiω0t

o

, “F(r,t) = 1

2F¯(r)eiω0t+c.c.”.

¯

E(r), D¯(r), B¯(r), H¯(r), ρ¯f(r), ¯Jf(r), ∼exp(iω0t),

∇·D¯ = ¯ρf, ∇×E¯ =−iω₀B,¯ ∇·B¯ =0, ∇×H¯ = ¯Jf+iω0D.¯

(53)

Maxwell equations, frequency domain

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD.˜

F(r,t)∈_R F˜(r,−ω) = (˜F(r, ω))∗

r π

2F¯(r)δ(ω−ω0) +

r π

2 F¯ ∗

(r)δ(ω+ω0)

F(r,t) = 1

2

n

¯

F(r,t) =Re

n

¯

F(r)eiω0t

o

, “F(r,t) = 1

¯

∇·D¯ = ¯ρf, ∇×E¯ =−iω₀B,¯ ∇·B¯ =0, ∇×H¯ = ¯Jf+iω0D.¯

(54)

21 Maxwell equations, frequency domain

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD.˜

F(r,t)∈_R F˜(r,−ω) = (˜F(r, ω))∗

r π

2F¯(r)δ(ω−ω0) +

r π

2 F¯ ∗

(r)δ(ω+ω0)

F(r,t) = 1

2

n

¯

F(r,t) =Re

n

¯

F(r)eiω0t

o

, “F(r,t) = 1

¯

∇·D¯ = ¯ρf, ∇×E¯ =−iω₀B,¯ ∇·B¯ =0, ∇×H¯ = ¯Jf+iω0D.¯

(55)

Maxwell equations, frequency domain

∇·D˜ = ˜ρf, ∇×E˜ =−iωB,˜ ∇·B˜ =0, ∇×H˜ = ˜Jf+iωD.˜

F(r,t)∈_R F˜(r,−ω) = (˜F(r, ω))∗

r π

2F¯(r)δ(ω−ω0) +

r π

2 F¯ ∗

(r)δ(ω+ω0)

F(r,t) = 1

2

n

¯

F(r,t) =Re

n

¯

F(r)eiω0t

o

, “F(r,t) = 1

¯

∇·D¯ = ¯ρf, ∇×E¯ =−iω₀B,¯ ∇·B¯ =0, ∇×H¯ = ¯Jf+iω0D.¯

(56)

22 Polarization

˜

P: density of electric dipole moment (bound charges).

˜

D=0E˜+ ˜P, [ ˜D] = [˜P] = As m

m3 , [˜E] = V m, vacuum permittivity 0=8.854187817. . .·10−12

F

m = As Vm

.

• Local dipoles induced byE˜ P˜(˜E).

• Linear dielectrica:

˜

P=0χˆeE˜, χˆe: dielectric susceptibility, [ ˆχe] = ˆ1.

˜

D=0(ˆ1+ ˆχe)˜E=0ˆE˜, ˆ: relative permittivity, [ˆ] = ˆ1.

• χˆe(r, ω), ˆ(r, ω) are determined in the frequency domain.

• Complications: Im,ˆ(T),ˆ(F),χ(_jkl2)EkEl,χ

(3)

jklmEkElEm,. . . • Simpler cases: ˆ(r), ˆ=ˆ1.

(57)

Polarization

˜

D=0E˜+ ˜P, [ ˜D] = [˜P] = As m

F

m = As Vm

.

˜

(3)

(58)

22 Polarization

˜

D=0E˜+ ˜P, [ ˜D] = [˜P] = As m

F

m = As Vm

.

˜

(3)

(59)

Polarization

˜

D=0E˜+ ˜P, [ ˜D] = [˜P] = As m

F

m = As Vm

.

˜

(3)

(60)

23 Magnetization

˜

M: density of magnetic dipole moments (bound currents).

˜

H= 1

µ0

˜

B−M˜, [ ˜H] = [M] = A m

2

m3 , [˜B] =T= Vs m2, vacuum permeability µ0=4π·10−7

N A2 =

Vs Am

.

• _{Local dipoles induced by}H˜ M˜( ˜H).

• Linear magnetic media:

˜

M= ˆχmH˜, χˆm: magnetic susceptibility, [ ˆχm] = ˆ1.

˜

B=µ0(ˆ1+ ˆχm) ˜H=µ0µˆH˜, µˆ: relative permeability, [ˆµ] = ˆ1. • χˆm(r, ω), µˆ(r, ω) are determined in the frequency domain. • Complications: manifold.

(61)

Magnetization

˜

H= 1

µ0

˜

B−M˜, [ ˜H] = [M] = A m

2

N A2 =

Vs Am

.

• _{Local dipoles induced by}_H˜ _M˜_{( ˜}_H₎_.

˜

B=µ0(ˆ1+ ˆχm) ˜H=µ0µˆH˜, µˆ: relative permeability, [ˆµ] = ˆ1. • χˆm(r, ω), µˆ(r, ω) are determined in the frequency domain. • Complications: manifold.

(62)

23 Magnetization

˜

H= 1

µ0

˜

B−M˜, [ ˜H] = [M] = A m

2

N A2 =

Vs Am

.

˜

B=µ0(ˆ1+ ˆχm) ˜H=µ0µˆH˜, µˆ: relative permeability, [ˆµ] = ˆ1.

• χˆm(r, ω), µˆ(r, ω) are determined in the frequency domain. • Complications: manifold.

(63)

Magnetization

˜

H= 1

µ0

˜

B−M˜, [ ˜H] = [M] = A m

2

N A2 =

Vs Am

.

˜

B=µ0(ˆ1+ ˆχm) ˜H=µ0µˆH˜, µˆ: relative permeability, [ˆµ] = ˆ1. • χˆm(r, ω), µˆ(r, ω) are determined in the frequency domain. • _{Complications: manifold.}

(64)

24 Maxwell equations, dispersion

(Material)dispersion: ˆ(r, ω), µˆ(r, ω) are frequency dependent. ˜

D(r, ω) =0ˆ(r, ω)˜E(r, ω), B˜(r, ω) =µ0µˆ(r, ω) ˜H(r, ω)

D(r,t) =0

Z

ˆ

TD(r,t−t0)E(r,t0)dt0,

B(r,t) =µ0

Z

ˆ

(65)

Maxwell equations, dispersion

(Material)dispersion: ˆ(r, ω), µˆ(r, ω) are frequency dependent. ˜

D(r, ω) =0ˆ(r, ω)˜E(r, ω), B˜(r, ω) =µ0µˆ(r, ω) ˜H(r, ω)

D(r,t) =0

Z

ˆ

TD(r,t−t0)E(r,t0)dt0,

B(r,t) =µ0

Z

ˆ

(66)

25 Helmholtz equations

Linear dielectric media without free charges or currents, time dependence ∼exp(iωt), fields E(r), D(r), B(r), H(r),

material properties ˆ(r), µˆ(r):

∇·D=0, ∇×E=−iωB, ∇·B=0, ∇×H=iωD,

D=0ˆE, B=µ0µˆH.

∇×E=−_iωµ0µˆH, ∇×H=iω0Ê, ∇·Ê=0, ∇·µˆH=0. ∇×(ˆµ−1∇×E) =ω20µ0Ê or ∇×(ˆ−1∇×H) =ω20µ0µˆH.

Where ˆ=ˆ1, ∇=0, µˆ=µˆ1, ∇µ=0: (!) ∆E+ ω

2

c2µE=0 or ∆H+ ω2

c2µH=0, c = 1

√ 0µ0.

(67)

Helmholtz equations

∇·D=0, ∇×E=−iωB, ∇·B=0, ∇×H=iωD,

D=0ˆE, B=µ0µˆH.

∇×E=−_iωµ0µˆH, ∇×H=iω0ˆE, ∇·ˆE=0, ∇·µˆH=0.

∇×(ˆµ−1∇×E) =ω20µ0ˆE or ∇×(ˆ−1∇×H) =ω20µ0µˆH.

Where ˆ=ˆ1, ∇=0, µˆ=µˆ1, ∇µ=0: (!) ∆E+ ω

2

c2µH=0, c = 1

√ 0µ0.

(68)

25 Helmholtz equations

∇·D=0, ∇×E=−iωB, ∇·B=0, ∇×H=iωD,

D=0ˆE, B=µ0µˆH.

Where ˆ=ˆ1, ∇=0, µˆ=µˆ1, ∇µ=0: (!) ∆E+ ω

2

c2µH=0, c = 1

√ 0µ0.

(69)

Helmholtz equations

∇·D=0, ∇×E=−iωB, ∇·B=0, ∇×H=iωD,

D=0ˆE, B=µ0µˆH.

Where ˆ=ˆ1, ∇=0, µˆ=µˆ1, ∇µ=0: (!) ∆E+ ω

2

c2µH=0, c = 1

√ 0µ0.

(70)

26 Plane harmonic waves

Where ˆ=ˆ1, ∇=0, µˆ=µˆ1, ∇µ=0: (!)

Components of E, H satisfy ∆ψ+ω

2

c2µ ψ=0.

ψ(r,t) =ψ0 e−i(km·r−ωt), −k2m+ω 2

c2µ=0.

(Mixture of TD and FD expressions;˜,¯, Re , 1/2, c.c. omitted; sloppy, but common.) • Medium: refractive index: n=√µ

• Periodicity in time: angular frequency: ω,

frequency: f =ω/(2π),

period: T=1/f =2π/ω,

• Spatial periodicity: wave vector: km, km=|km|,

wavenumber: km=ω/cm= (ω/c)n=k n, vacuum wavenumber: k=ω/c,

vacuum wavelength: λ=2π/k=2πc/ω,

wavelength in the medium: λm=2π/k_m=2π/(kn) =λ/n. • Phase velocity: speed of light in vacuum: c=1/√0µ0=λf,

in the medium: cm=c/n=λmf.

(Use of symbols depends highly on context.)

(71)

Interface conditions

n l

σf,Kf

(2) (1)

Surface between media (1) and (2), surface normaln, tangentsl,

surface charge densityσf, surface current densityKf: n·(D1−D2) =σf, l·(E1−E2) =0,

(72)

28 Interface conditions

n l

(2) (1)

surface without free charges or currents: n·(D1−D2) =0, l·(E1−E2) =0, n·(B1−B2) =0, l·(H1−H2) =0 .

(73)

Interface conditions

n l

(1) ǫˆ1, µˆ1

(2) ˆǫ2, µˆ2

linear media with permittivitiesˆ1,ˆ2, and permeabilitiesµˆ1,µˆ2:

n·(ˆ1E1−ˆ2E2) =0, l·(E1−E2) =0, n·(ˆµ1H1−µˆ2H2) =0, l·(H1−H2) =0.

(74)

30 Reflection and transmission of plane waves at dielectric interfaces

y ˆ

ǫ1, µˆ1 (1) (2) ˆǫ2, µˆ2

x

z (FD)

• ∇×E=−_iωµ0µˆH, ∇×H=iω0ˆE

• _ˆ₍_r₎_and_µ_ˆ₍_r₎

are constant alongy,z

E(r) =E0(x)e−i(kyy+kzz), _H₍_r_{) =}_H0₍_x₎e−i(kyy+kzz) 1-D problem for E0, H0. (incoming plane wave at angleθ)

(orient coordinates (ky=0), plane of incidence, distinguish polarizations)

(write ansatz functions for incoming, reflected, and transmitted waves) (interface conditions determine the amplitudes)

(75)

Reflection and transmission of plane waves at dielectric interfaces

y ˆ

ǫ1, µˆ1 (1) (2) ˆǫ2, µˆ2

x

z (FD)

E(r) =E0(x)e−i(kyy+kzz), _H₍_r_{) =}_H0₍_x₎e−i(kyy+kzz) 1-D problem for E0, H0.

(incoming plane wave at angleθ) (orient coordinates (ky=0), plane of incidence, distinguish polarizations)

(76)

30 Reflection and transmission of plane waves at dielectric interfaces

y ˆ

ǫ1, µˆ1 (1) (2) ˆǫ2, µˆ2

x z

θ

(FD)

E(r) =E0(x)e−i(kyy+kzz), _H₍_r_{) =}_H0₍_x₎e−i(kyy+kzz) 1-D problem for E0, H0. (incoming plane wave at angleθ)

(orient coordinates (ky=0), plane of incidence, distinguish polarizations)

(77)

Dielectric multilayer structures

y x z

ˆ

ǫ1, µˆ1 (1) ˆǫ2, µˆ2

(2)

θ

(FD)

• ∇×E=−iωµ0µˆH,

∇×H=iω0ˆE • _ˆ₍_r₎_and_µ_ˆ₍_r₎

E(r) =E0(x)e−i(kyy+kzz)_, _H₍_r_{) =}_H0₍_x₎_e−i(kyy+kzz) 1-D problem for E0, H0. (. . .)

(. . .) (. . .) (. . .)

(78)

32 Energy of electromagnetic fields

(TD) • _{Force on a particle with charge}q, velocityv, in a field E,B:

F=q(E+v×B),

• _{work for shifting the particle by d}_r₌_v_d_t_: dW=F·dr=q(E+v×B)·vdt=qE·vdt,

• _{respective power:} dW

dt =qE·v.

For a charge density ρf(r,t):

force density f =ρf(E+v×B), power density f ·v=ρfE·v=Jf·E,

total work per time unit done inV: dWV

dt =

Z

V

(79)

Energy of electromagnetic fields

F=q(E+v×B),

dt =qE·v.

dt =

Z

V

(80)

32 Energy of electromagnetic fields

F=q(E+v×B),

dt =qE·v.

dt =

Z

V

(81)

Energy of electromagnetic fields

F=q(E+v×B),

dt =qE·v.

dt =

Z

V

(82)

33 Power & energy density, Poynting theorem

(TD) ∇×H=Jf+ ˙D, ∇×E=−B˙

d dtW

mech

V =

Z

V

Jf·EdV =−

Z

V

(E·D˙ +H·B˙)dV − Z

V

∇·(E×H)dV,

• Poynting vector: S=E×H, (energy flux density, power density) • energy density: w= 1

2(E·D+H·B), WVfield =

Z

V

wdV,

• ˆ†= ˆ, ˆ(ω), D=0ˆE, µˆ†= ˆµ, µˆ(ω), B=µ0µˆH (!)

˙

w= (E·D˙ +H·B˙)

Varbitrary

˙

w+∇·S=−J_f·E, d

dt W

mech

V +WVfield

=− I

∂V

(83)

Power & energy density, Poynting theorem

(TD) ∇×H=Jf+ ˙D, ∇×E=−B˙

d dtW

mech

V =

Z

V

Jf·EdV =−

Z

V

(E·D˙ +H·B˙)dV − Z

V

∇·(E×H)dV,

• Poynting vector: S=E×H, (energy flux density, power density)

• energy density: w= 1

Z

V

wdV,

˙

w= (E·D˙ +H·B˙)

Varbitrary

˙

dt W

mech

V +WVfield

=− I

∂V

(84)

(TD) ∇×H=Jf+ ˙D, ∇×E=−B˙

d dtW

mech

V =

Z

V

Jf·EdV =−

Z

V

(E·D˙ +H·B˙)dV − Z

V

∇·(E×H)dV,

Z

V

wdV,

˙

w= (E·D˙ +H·B˙)

Varbitrary

˙

dt W

mech

V +WVfield

=− I

∂V

(85)

Power & energy density, Poynting theorem

(TD) ∇×H=Jf+ ˙D, ∇×E=−B˙

d dtW

mech

V =

Z

V

Jf·EdV =−

Z

V

(E·D˙ +H·B˙)dV − Z

V

∇·(E×H)dV,

Z

V

wdV,

˙

w= (E·D˙ +H·B˙)

Varbitrary

˙

dt W

mech

V +WVfield

=− I

∂V

(86)

(TD) ∇×H=Jf+ ˙D, ∇×E=−B˙

d dtW

mech

V =

Z

V

Jf·EdV =−

Z

V

(E·D˙ +H·B˙)dV − Z

V

∇·(E×H)dV,

Z

V

wdV,

˙

w= (E·D˙ +H·B˙)

Varbitrary

˙

dt W

mech

V +WVfield

=− I

∂V

(87)

Electromagnetic energy, frequency domain Lossless uncharged nondispersive (. . .) linear media:

w= 1

2(0E·ˆE+µ0H·µHˆ ), S=E×H, w˙ +∇·S=0,

E(r,t) =Re E˜(r)eiωt, H(r,t) =ReH˜(r)eiωt

S, w oscillate in time.

(FD) Consider time-averaged quantities: ¯_f_{(t) =} 1

T

Z t+T

t

f(t0)dt0

(exercise)

w= 1

4Re

0E˜∗·ˆE˜ +µ0H˜∗·µˆH˜

, S= 1

2Re

˜

E∗×H˜.

˙

w= ˙w=0, ∇·S=∇·S ∇·S=0,

I

V

S·da=0;

(88)

34 Electromagnetic energy, frequency domain

Lossless uncharged nondispersive (. . .) linear media:

w= 1

2(0E·ˆE+µ0H·µHˆ ), S=E×H, w˙ +∇·S=0,

E(r,t) =Re E˜(r)eiωt, H(r,t) =ReH˜(r)eiωt

S, w oscillate in time.

(FD) Consider time-averaged quantities: ¯_f_{(t) =} 1

T

Z t+T

t

f(t0)dt0

(exercise)

w= 1

4Re

0E˜∗·ˆE˜+µ0H˜∗·µˆH˜

, S= 1

2Re

˜

E∗×H˜.

˙

w= ˙w=0, ∇·S=∇·S ∇·S=0,

I

V

S·da=0;

(89)

Wave propagation in attenuating media

Specifically: homogeneous isotropic conductors, linear media. Electric field drives the free currents:

Ohm’s law Jf=σE, σ: conductivityof the material.

∇·D=ρf, ∇×E=−_B˙_, ∇·B=0, ∇×H=σE+ ˙D.

˙

ρf=− σ

0ρf, ρf(r,t) =ρf(r,t0) exp

− σ

0(t−t0)

, assume ρf(r,t0) =0 ρf(r,t) =0 ∀t.

(90)

35 Wave propagation in attenuating media

∇·D=ρf, ∇×E=−B˙, ∇·B=0, ∇×H=σE+ ˙D.

˙

ρf=− σ

− σ

0(t−t0)

(91)

Wave propagation in attenuating media

∇·D=ρf, ∇×E=−B˙, ∇·B=0, ∇×H=σE+ ˙D.

˙

ρf =− σ

− σ

0(t−t0)

(92)

35 Wave propagation in attenuating media

∇·D=ρf, ∇×E=−B˙, ∇·B=0, ∇×H=σE+ ˙D.

˙

ρf =− σ

− σ

0(t−t0)

(93)

Telegrapher equation

∇·E=0, ∇×E=−µ0µH˙, ∇·H=0, ∇×H=σE+0E˙

∆E−0µ0µE¨−µ0µσE˙ =0, ∆H−0µ0µH¨ −µ0µσH˙ =0, Telegrapher equation.

Frequency domain: E(r,t) = ˜E(r)eiωt, ∆˜E+

ω2

c2µ−iωµ0µσ

˜

E=0.

Nonconducting mediaσ=0, ∆˜E+ ω

2

c2µ

! ˜ E=0.

Define ¯ such that ω

2 c2¯µ=

ω2

c2µ−iωµ0µσ, i.e. ¯=−i

σ 0ω

∆˜E+k2¯µE˜ =0, Helmholtz equation, ¯∈C, k= ω

(94)

36 Telegrapher equation

∇·E=0, ∇×E=−µ0µH˙, ∇·H=0, ∇×H=σE+0E˙

ω2

c2µ−iωµ0µσ

˜

E=0.

2

c2µ

! ˜ E=0.

2 c2¯µ=

ω2

σ 0ω

(95)

Telegrapher equation

∇·E=0, ∇×E=−µ0µH˙, ∇·H=0, ∇×H=σE+0E˙

ω2

c2µ−iωµ0µσ

˜

E=0.

2

c2µ

! ˜ E=0.

2 c2¯µ=

ω2

σ 0ω

(96)

37 Wave attenuation

∆˜E+k2¯µ_E˜ ₌_0, _¯∈_C

solutions ∼ e−i(k¯nz−ωt)

with refractive index ¯n=n0−in00 =√µ¯ ∈C, (!) e−i(k¯nz−ωt_{) =} _e−i(kn0z−ωt) _e−kn00z_,

damped plane wavesolutions

sign ofn00, choice ofexp(±iω).

Issues:

• _{penetration depth,} • _S_and_w_{decay with}_z_, • _{still transverse waves,} • E,Hno longer in phase,

(97)

Wave attenuation

∆˜E+k2¯µ_E˜ ₌_0, _¯∈_C solutions ∼ e−i(k¯nz−ωt)

with refractive index ¯n=n0−in00 =√µ¯ ∈C, (!)

e−i(k¯nz−ωt_{) =} _e−i(kn0z−ωt) _e−kn00z_,

Issues:

(98)

37 Wave attenuation

∆˜E+k2¯µ_E˜ ₌_0, _¯∈_C solutions ∼ e−i(k¯nz−ωt)

with refractive index ¯n=n0−in00 =√µ¯ ∈C, (!) e−i(k¯nz−ωt_{) =} _e−i(kn0z−ωt) _e−kn00z_,

Issues:

(99)

Simulations in integrated optics

A typical setting:

• _{“uncharged dielectric medium”:} qf, Jf. • _{“linear medium”:} _D₌₀_ˆ_E_, _B₌_µ₀_µH_ˆ _. • _{“isotropic medium”:} _ˆ₌ˆ_1, _µ_ˆ₌_µˆ_1.

• _{“nonmagnetic medium”:} _µ_ˆ_{= ˆ}_1.

• _{“lossless medium”:} _ˆ†_{= ˆ}_, _µ_ˆ†_{= ˆ}_µ_{, (}_{, µ}_∈

R).

• _{“piecewise constant”} → “dependent on position”.

• _{“electric and magnetic field”: eliminate}DandB, retainEandH.

• _{“governed by the curl equations”: divergence eqns. are satisfied.} • _{“frequency domain, time harmonic fields, frequency, wavelength”:}

(100)

39 Course overview

(101)

Guided wave scattering problems, schematically

ˆ

ǫ

(

r

)

,

µ

ˆ

(

r

)

P

in

? ?

?

? ?

?

E

=

?

H

=

?

(102)

41 Guided wave scattering problems, schematically

ˆ

ǫ(r),µ(ˆ r) Pin

? ?

?

? ?

?

E=?

H=?

Given ˆ(r),µˆ(r) & external excitation (incoming guided mode),

determine E,H within the computational domain

(103)

Scattering problems, time domain (TD)

E(r,t), H(r,t),

∇×E=−µ0µˆH˙,

∇×H=0ˆE˙.

•





3-D 2-D 1-D



computational domain × time interval.

• _{Initial & boundary conditions} ←→ incident waves.

• _{“Local” time-explicit iterative schemes possible (e.g. FDTD).} • _{Time evolution available; direct modeling of pulse propagation.} • _{Dispersion (}_{. . .}_?).

• _{Guided wave excitation (}_{. . .}_?).

(104)

42 Scattering problems, time domain

(TD)

E(r,t), H(r,t),

∇×E=−µ0µˆH˙,

∇×H=0ˆE˙.

•





3-D 2-D 1-D



computational domain × time interval.

• _{Initial & boundary conditions} ←→ incident waves.

• _{“Local” time-explicit iterative schemes possible (e.g. FDTD).} • _{Time evolution available; direct modeling of pulse propagation.} • _{Dispersion (}_{. . .}_?).

• _{Guided wave excitation (}_{. . .}_?).

(105)

Scattering problems, frequency domain (FD)

E(r), H(r), ∼exp(iωt),

∇×E=−iωµ0µHˆ ,

∇×H=iω0ˆE.

•





3-D 2-D 1-D



computational domain.

• _“M(−−→field) = (−−−−−−→excitation)”;

matrix needs to be determined, stored; system needs to be solved. • _{Spectral information directly available.}

• _{Dispersion — straightforward.}

• _{Guided wave excitation — straightforward.}

(106)

43 Scattering problems, frequency domain

(FD)

E(r), H(r), ∼exp(iωt),

∇×E=−iωµ0µHˆ ,

∇×H=iω0ˆE.

•





3-D 2-D 1-D



computational domain.

• _“M(−−→field) = (−−−−−−→excitation)”;

matrix needs to be determined, stored; system needs to be solved. • _{Spectral information directly available.}

• _{Dispersion — straightforward.}

• _{Guided wave excitation — straightforward.}

(107)

Open problems (TD & FD)

“Open” spatial computational domain boundary conditions need to • _{permit outgoing radiated fields}

& outgoing (reflected) guided modes to exit the domain, • _{launch the incoming external excitation.}

simulate a nonexisting boundary, an unlimited domain.

Keywords: • _{transparent-influx boundary conditions,} • _{absorbing boundary conditions,}

(108)

44 Open problems

(TD & FD)

“Open” spatial computational domain boundary conditions need to • _{permit outgoing radiated fields}

& outgoing (reflected) guided modes to exit the domain, • _{launch the incoming external excitation.}

simulate a nonexisting boundary, an unlimited domain. Keywords: • _{transparent-influx boundary conditions,}

• _{absorbing boundary conditions,} • _{perfectly matched layers (PMLs).}

(109)

2-D problems

ˆ

=ˆ1, µˆ=µˆ1, ∼exp(iωt) (FD)





∂yEz−∂zEy

∂zEx−∂xEz

∂xEy−∂yEx



=−iωµ0µ





Hx

Hy

Hz



, 



∂yHz−∂zHy

∂zHx−∂xHz

∂xHy−∂yHx



=iω0





Ex

Ey

Ez



(110)

45 2-D problems

ˆ

Assume ∂y=0, ∂yµ=0; consider solutions ∂yE=0, ∂yH=0 :





−∂zEy

∂zEx−∂xEz

∂xEy



=−iωµ0µ





Hx

Hy

Hz



, 



−∂zHy

∂zHx−∂xHz

∂xHy



=iω0





Ex

Ey

Ez



(111)

2-D problems

ˆ

Assume ∂y=0, ∂yµ=0; consider solutions ∂yE=0, ∂yH=0 :





−∂zEy

∂zEx−∂xEz

∂xEy



=−iωµ0µ





Hx

Hy

Hz



, 



−∂zHy

∂zHx−∂xHz

∂xHy



=iω0





Ex

Ey

Ez



.

Two decoupled sets of equations:

• {Ey,Hx,Hz}: transverse electric (TE)fields, E⊥ x-z-plane. • {Hy,Ex,Ez}: transverse magnetic (TM)fields, H⊥ x-z-plane.

(Different conventions on the use of TE, TM.) (Applies also to the TD.)

(112)

46 2-D TE waves

k2=ω2/c2=ω20µ0 (FD)

• _{Principal component}Ey, Hx=

−i

ωµ0µ∂zEy, Hz= i

ωµ0µ∂xEy, iω0Ey =∂zHx−∂xHz

∂x 1

µ∂xEy+∂z

1

µ∂zEy+k

2_E

y =0. (∗) • _{Continuity of} _E_y, 1

µ∂nEy required at interfaces with normaln.

• _If_µ₌_{1 :} ₍x,z) (!)

∂x2Ey+∂z2Ey+k2Ey =0, (∗∗)

scalar2-D (TE) Helmholtz equation (Ey, ∂nEy continuous).

(113)

2-D TE waves

k2=ω2/c2=ω20µ0 (FD)

• _{Principal component}Ey, Hx=

−i

ωµ0µ∂zEy, Hz= i

ωµ0µ∂xEy, iω0Ey =∂zHx−∂xHz

∂x1

µ∂xEy+∂z

1

µ∂zEy+k

2_E

y =0. (∗)

• _{Continuity of} _E_y, 1

µ∂nEy required at interfaces with normaln.

• _If_µ₌_{1 :} ₍x,z) (!)

∂x2Ey+∂z2Ey+k2Ey =0, (∗∗)

scalar2-D (TE) Helmholtz equation (Ey, ∂nEy continuous).