1

Optical Waveguide and Resonator

2

Free Space and Plane Wave

• Propagating wave in free space – plane wave

• Free space without dispersion – modulated plane wave can still propagate without any distortion, since the slow time-varying of the EM wave on its envelope, frequency, or phase won’t bring any effect to the propagation

• Free space with dispersion – modulated plane wave has its envelope gradually expanded in time-domain in propagation, eventually converts itself to harmonic plane wave

• How can we force the EM wave to propagate along a specific direction in the 3D world?

3

Concept of Waveguide

• The wave has to localized in certain directions

• How to localize the wave? – Convert the traveling wave into the standing wave

• Introduce the transverse resonance

12

1 1| |k

1xk

zx

2 2 21 1| | xk k

2 2| |k

2xk

2 2 22 2| | xk k

s-wave reflection at the boundary:

1 2

1 2

x x

x x

k kR

k k

standing wave is formed underneaththe boundary

4

Concept of Waveguide

1

2

| | 1R

1 (2 )20

xjk dE R e

zx

1 1(2 ) (2 )2 20 0 1x xjk d jk dE E R e R e

0E

The resonance condition for standingwave in transverse direction (x):

A necessary condition is:

d

How to make it possible?

TIR – dielectric waveguide

Conductor reflection – metallic waveguide

Photonic crystal – Bragg waveguide

Plasma reflection – plasmonic polariton waveguide

5

Dielectric Waveguide

2 22

2 2 2 21 1

1

tan

2 2 2 22 22 2 1 2 2

1 12 2 2 21 1

2 tan tan( )2 2

d md m

2xk2 2 2 2 2 2 2

2 2 2 2 2| | | |xk k k j

2 222 1 11 2 1 2 2

2 21 2 1 2 1 1

| | | |, tan tan

| |jx x x x x

x x x x x

k k k j k kR e

k k k j k k

If becomes purely imaginary, or:

The resonance condition becomes: 12411 2 4 2xjk djxe e k d m or:

2 222 2

1 2 21

tan

Even mode Odd mode

Dispersion relation for thedielectric slab waveguide

Obviously, we have: 12 With definition effn 0

we find: 1122 reffr nnn effn - waveguide effective index

6

Dielectric Waveguide

11 //1 nc

22 //1 nc

effeff ncn //1 0

Dispersion relation E-field )(0 )( tzjexE

2/,

2/2/),cos(

2/,

)()2/(

1

)2/(

0

2

2

dxCe

dxdxkB

dxAe

xEdxk

x

dxk

x

x

Symmetric (even mode):

Anti-symmetric (odd mode):

2/,

2/2/),sin(

2/,

)()2/(

1

)2/(

0

2

2

dxCe

dxdxkB

dxAe

xEdxk

x

dxk

x

x

A, B, C – given by the tangential boundary condition

H-field is given by the Faraday’s law, with x and zcomponents only – that’s the TE wave

(y-component)

7

Dielectric Waveguide

Similarly, the guided TM wave solution can be derived from the reflection ofthe p-wave at the boundary

The E-field of the TM wave has abrupt changeat the boundary!

Hence, the effective index of the TM wave is smaller than that of the TE wave.

Application examples:

Single mode waveguide – higher order mode cut-off

2D waveguide – no analytical solution

Slot waveguide – utilizing the abrupt change of the E-field normal to the boundary, for TM wave guidance only

8

Metallic Waveguide

By letting 2xk j in previous derivations, we will be able to obtain the

EM wave solution in metallic waveguide.

1D (slab) or 2D dielectric waveguide – support TE and TM waves, not TEM wave

1D (slab) metallic waveguide – support all TEM, TE, and TM waves

2D (hollow) metallic waveguide – support TE and TM waves, not TEM wave

9

Hollow Metallic Waveguide – How to Treat Wave Equations

2 2 2 2 20 00 ( , ) ( ) ( , ) 0TE E E x y k E x y

a

b( )

0 ( , )j z tE x y e

xy

Once the propagation along z is identified

Substitute it back into the wave equation

For any component, we have2 2

2 22 2

( ) ( , ) ( ) ( , ) 0u x y k u x yx y

Variable separation

where 2 2 2| |k k

1 1 2 2( , ) ( ) ( ) [ cos( ) sin( )][ cos( ) sin( )]x x y yu x y X x Y y C k x D k x C k y D k y

where2 2 2 2x yk k k

By imposing boundary conditions

000 0 0 00 ( 0, ) 0 ( 0, )yxy z x z

EEE E x a E E y b

x y

10

Hollow Metallic Waveguide – Characteristics

, , 0,1,2,...x y

m nk k m n

a b

0 1

0 2

0 3

cos( )sin( )

sin( )cos( )

sin( )sin( )

x x y

y x y

z x y

E A k x k y

E A k x k y

E A k x k y

We find: where

For a given set of (m, n) there are two independent modes.

and (from the divergence-free condition)

1 2 3 0x yk A k A j A

For the 3 coefficients, only 2 of them are independent. For a fixed set of (m, n),if one independent mode (A) is chosen as the transverse E-field, the other independent mode (B) must have the E-field with a non-zero z component, and mode A must have its H-field with a non-zero z component. Therefore:

Similar to the dielectric waveguide, a hollow metallic waveguide doesn’t support the TEM wave. All its supported modes can be classified as the TE wave (mode A) and the TM wave (mode B).

2 2 2 22

2 2

m nk

a b

Once the E-field is obtained, the H-field can be found by Faraday’s law.

11

Hollow Metallic Waveguide – Characteristics

Cut-off frequency

Applications:

- HPF

- Single mode waveguide

2 2

min max2 20 2

m nna

a b a

12

Transmission Line – TEM Wave

For TEM wave: k Hence:

0),(

),()(

0

0

2

2

2

2

yxE

yxE

yx y

x

TEM solution is the same as the static electric and magnetic field solution.

I

I

E

Hparallel lines ortwisted pair

coaxial cable

EH

13

Transmission Line – TEM Wave

Reason: in the source-less cross-sectional region perpendicular to thepropagation direction, both E- and H- fields have to be curl-free and divergence-free – the same as the static E- and M- fields, all field componentsare described by the Laplace equation.

Therefore, only for a cross-sectional structure that supports both static E- andH- fields, it will be able to support the TEM wave. A hollow metallic waveguidedoesn’t support a static H-field, hence it cannot support the TEM wave.

A necessary condition for a TEM waveguide: at least two pieces of disconnectedconductors in the cross-sectional region for supporting the static H-field.

Why we emphasize this point? – pre-knowledge on field types will help us to construct the solution: identify non-zero components and the function form,by substituting such constructed solution back into the wave equation, we willget the problem solved as the wave equation will usually be greatly simplified.

14

TEM Wave - Characteristics

• TEM wave – “localized” plane wave with k, E, H mutually orthogonal: k is along the direction in which the waveguide (transmission line) is extended; E- and H- fields are restricted in the 2D cross-section, with their longitudinal dependence identical to the plane wave, and transverse dependence identical to the static E- and H- fields with the same boundary condition.

• Propagation of the TEM wave relies on the free charge and conduction current on the metal (conductor) – dielectric surface. Namely, the TEM wave is a resonance between the EM fields and the free charge distribution.

• For the TEM wave, we can readily introduce the voltage and current concept to turn a field problem into a circuit problem.

• The TEM wave can be supported by the dual conductor transmission line:

– Parallel lines or twisted pair– Coaxial cable– Printed metal stripe lines (on PCB or other substrates)

• The TEM wave has no cut-off frequency, it can send DC power through.

15

Resonator

• In its propagation along the waveguide, the EM wave oscillates in the cross-section to form a standing wave. For a given cross-sectional area and material refractive index distribution, there is a cut-off frequency associated with.

• A waveguide cannot support a EM wave with its frequency below the cut-off frequency, simply because the phase matching condition cannot possibly be satisfied.

• For the EM wave with its frequency higher than the cut-off frequency, there exists a real propagation constant given by the dispersion relation. Hence the wave can propagate along the waveguide.

• Simply because the waveguide is open in its propagation direction, there is no constraint imposed on the phase of the guided EM wave. Hence once the frequency is beyond the cut-off, there is a solution for the guided EM wave with a real propagation constant.

• In this sense, the waveguide is actually a high-pass filter.

16

Resonator

• Once the waveguide is even “closed” along the propagation direction, extra boundary matching conditions are imposed on both ends. Since the only degree of freedom is the phase of the guided EM wave, the phase will therefore be fixed by the “closing” of the waveguide. As such, the associated frequency will be fixed.

• In conclusion, for a resonator formed by a “closed” waveguide, it only supports EM waves with a discrete set of specific frequencies.

• That’s why a closed cavity (with constraints imposed on every dimension in the 3D space) is essential to form a resonator that supports discrete frequencies.

• Examples

17

Resonator

• More specifically, once we force the guided EM wave to reflect at the two ends of a waveguide, the EM wave will form a standing wave in all dimensions in the closed waveguide (or cavity), a EM wave resonator is therefore formed for the creation or selection of a single or a discrete set of frequencies.

18

Resonator as an Oscillator

E-field inside the cavity ])[( )()(0

tzjtzj BeAexE

RR

L

Boundary condition at the two ends 122

LjLjLj eR

BeRAe

RBA

For cavity with zero-gain 1|| R

Meaning less since no energy can be collected from the outside.

When gain is available ir j the resonance condition becomes:

||ln1|| RLeR iLi mLe r

Lj r 12Amplitude Phase

19

Resonator as a Filter

inLj

Lj

AeR

eTxE

22

2

0 1)(

inLj

Lj

inLj

LjLjin

AeR

TB

AeR

TA

BeRAe

TARBA

22

2

22

1

Re1

RR

L

If there is wave incidence (from left)

At the right hand side E-field transmissivity Lj

Lj

eR

eT

22

2

1

inLj

Lj

AReR

TxE )1

Re)((

22

22

0

At the left hand side E-field reflectivity Lj

Lj

eR

eR

22

2

1

)1(