9.5 Guided Waves

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9.5 Guided Waves

slab strip Coaxial (fiber)

strip embedded strip rib or ridge strip loaded

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Symmetric & Asymmetric waveguides

Cladding (or, cover) : n_c

Core (or, film) : n_f

Cladding (or, substrate) : n_s x

y

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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;

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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!

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9.5.2 TE waves (E

_z = 0 ) in a rectangular waveguide

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

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

 ^/c



²  kx² ky² k²⁽ kz²⁾

/ /

x y

k m a k m b







^/



² ²



^/

 

² ^/



²

k  kz   c  ^ m a  n b ^ 0

a

b



^/

 

² ^/



²

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

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@ Planar perfect-conductor waveguides (1-dimensional waveguides)

Waveguide modes

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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

: The propagation ray picture of wave guidance by multiple reflections

/ /

x y

k m a k m b





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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

 _

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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

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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 ]

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Number of modes

( d < /2 ) 

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



^/

 

² ^/



²

mn c m a n b

   

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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,

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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

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Multimode fields

( m = 1 )

( m = 2 )

( m = 1 & 2 )