9.5 Guided Waves

9.5 Guided Waves

slab strip Coaxial (fiber)

strip embedded strip rib or ridge strip loaded

Symmetric & Asymmetric waveguides

Cladding (or, cover) : n_c

Core (or, film) : n_f

Cladding (or, substrate) : n_s x

y

9.5 Guided Waves

Consider electromagnetic waves confined to the interior of a hollow pipe, or wave guide.

Assume the wave guide is a perfect conductor, E = 0 and B = 0 inside the material itself.

The boundary conditions at the inner wall are

9.5.1 Perfect-conductor (or, perfect mirror) waveguides

|| 0, 0 E  B^ 

For monochromatic waves that propagate down the tube in z direction,

Confined waves are not (in general) transverse; they can include longitudinal components.

Putting this into Maxwell' s equations;

If E_z = 0 we call these TE ("transverse electric") waves if B_z = 0 they are called TM ("transverse magnetic") waves

if both E_z = 0 and B_z = 0, we call them TEM waves

Problem 9.26

Show that TEM waves cannot occur in a hollow wave guide.

Prove!

9.5.2 TE waves (E

2

2 2 2 2

( )

x y z

k k k k

c

     

  

at x = 0 and x = a

/ ( 0,1, 2, ) kx  m a m  

0 a

b

The general solution is

The boundary conditions require that B^  0 B_x 0

vanishes at x = 0 and x = a.

0 & sin( _x ) 0

A  B k a 

/ ( 0,1, 2, ) ky  n b n   ( ) ( )

Bz  X x Y y

( ) ( ) 0 cos( / ) cos( / ) Bz  X x Y y  B m x a n y b





/ /

x y

k m a k m b















 



k  kz   c  ^ m a  n b ^ 0

a

b



 



mn c m a n b

   

 cutoff frequency of TE_mn mode

The wave (phase) velocity is The wave number is

The group velocity is

> c In a rectangular waveguide

[Note] Planar perfect-conductor waveguides

(for the case of 1-dimensional waveguides)

Waveguide modes

Condition of self-consistency

2 2

cos 2 (1 2 sin ) - 2 sin 2 sin AB AC   AC    AC AB AC   d 

Bounce angles

Transverse Component of the wavevector

A propagation ray picture of wave guidance by multiple reflections

/ /

x y

k m a k m b









Propagation constants

Bounce angle

Propagation constant

z

2 2 2

m k kym

  

2 2

2 2 2 2 2 2

x y z z x y

k k k k k k

c c

 

         

   

   

c k

 _

Field distributions : TE modes

The complex amplitude of the total field in the waveguide is the superposition of the two bouncing TEM plane waves :

upward wave + downward wave

= +

: symmetric modes, odd modes

: antisymmetric modes, even modes

are normalized

are orthogonal in [-d/2. d/2] interval

Assume that the bouncing TEM plane wave is polarized in the x direction,

the guided wave is a transverse-electric (TE) wave.

TE mode

Each mode can be view as a standing waves in the y direction, traveling in the z direction.

Modes of large m vary in the transverse plane at a greater rate k_y, and travel with a smaller propagation constant .

The field vanishes at y = +d/2 for all modes, so that the boundary conditions at the surface of the mirrors are always satisfied.

[ TE guided waves ]

Number of modes

( d < /2 ) 

 ( /2 < d <  )  single-mode waveguide



 



mn c m a n b

   

Group velocities

Group velocity of mode m :

More oblique modes travel with a smaller group velocity

since they are delayed by the longer path of the zigzaging process.

Geometrically,

Field distributions : TM modes

upward wave + downward wave Magnetic field is in the x direction,

the guided wave is a transverse-magnetic (TM) wave.

TM mode

Since the z component of the electric field is parallel to the mirror, it must behave like the x component of the TE mode :

The y components of the electric field of these waves are

_m E_z

E_y E

Multimode fields

( m = 1 )

( m = 2 )

( m = 1 & 2 )

7.2 Planar dielectric waveguides

Cladding (or, cover) : n_c

Core (or, film) : n_f

Cladding (or, substrate) : n_s x

y

Let’s first consider a symmetric waveguide.

Transverse Component of the wavevector

: The ray-optics picture of light guidance by multiple reflections

Self-consistency condition

Self-consistency condition : TE modes

Self-consistency condition (TE mode)

From the TIR of TE modes,

Self-consistency condition : TE modes

(open circles)

Number of modes : TE modes

Single-mode waveguide (TE mode)

In a dielectric waveguide, there is at least one TE mode, since the fundamental mode m = 0 is always allowed.

No cutt-off frequency

Propagation constants : TE modes

: propagation constant ( the z-component of wavevector)

Field distributions : TE modes

: Extinction coefficient

Confinement factor

 the ratio of power in the slab to the total power

Dispersion relation

2 2 2 2

1 2 1 2

2 2

2 2

1 1

2 tan

c

n n n n

m n n

n n



 





 

         

/

n c

  

/ 2

/(2 )

c

c d NA

  

(n : effective index)

Group velocities

The group velocities lie between c1 and c2 (the phase velocities in the slab and substrate).

At a given ,

the lowest-order mode (the least oblique mode, m = 0) travels with a group velocity closest to c1.

The most oblique mode m = M has a group velocity ~ c2.

Group velocities

1

cos c z

 

 



More oblique modes travel this lateral distance at a fast speed than less oblique modes

Goose-Hanchen shift

Goose-Hanchen effect  Evanescent field