DIELECTRIC WAVE GUIDE

KEY CONCEPTS:-

Introduction Why it is used???TypesTM waves along a dielectric slabTE waves along a dielectric slabHE ModeApplication

D i e l e c t r i c w a v e g u i d e s a r e t h e s t r u c t u r e s t h a t a r e u s e d t o c o n fi n e a n d g u i d e t h e l i g h t i n t h e g u i d e d - w a v e d e v i c e s a n d c i r c u i t s o f i n t e g r a t e d o p ti c s . A w e l l - k n o w n d i e l e c t r i c w a v e g u i d e i s , o f c o u r s e , t h e o p ti c a l fi b e r w h i c h u s u a l l y h a s a c i r c u l a r c r o s s - s e c ti o n .

I n c o n t r a s t , t h e g u i d e s o f i n t e r e s t t o i n t e g r a t e d o p ti c s t h e s i m p l e s t d i e l e c t r i c g u i d e i s t h e p l a n a r s l a b g u i d e s h o w n i n fi g .

INTRODUCTION: DIELECTRIC WAVE GUIDE

• Where a planar film of refractive index nf is sandwiched between a substrate and a cover material with lower refractive indices ns and nc (nf>n1>nc). Often the cover material is air, in which case n¢=1.

Typical differences between the indices of the film and the substrate range from 10 .3 to 10 -1, and a typical film thickness is 1 gm.

The light is confined by total internal reflection at the film-substrate and film-cover interfaces.At high frequencies (optical frequencies) the loss associated with the induced current in the metal walls is too high.

A transmission line filled with dielectric material but without conducting walls is another structure that may be used to guide electromagnetic waves. This dielectric slab waveguide eliminates themetallic absorption loss.

In Dielectric Waveguide, They Can Be Used To Study Transmission Of Light Waves Along Quartz Or Glass Fibers That From Optical Waveguides.

Main Importance As Transmission Media For Communication Or Control Systems Because Of Low Attenuation And Large Bandwidth Properties.

They Extremely Compact And Flexible.

Why it is used?????

Consider the dielectric s lab as in fig. ,

Bouncing wave interpretation of propagating waves along dielectric waveguide:

If We Know A Plane Wave In The Slab With PermittivityIncident Obliquely On The Lower Boundary At An Angle Of >

The Reflected Wave Will Be Incident On Upper Region At Same Angle Of Incidence And will Be Totally Reflected So It Gives 2 Sets Of Multiply Reflected Waves:

(I)going From Upper Boundary Toward The Lower Boundary(Ii)lower Boundary Toward The Upper Boundary.

F o r T M w a v e s , h z = 0 , s i n c e t h e r e i s n o x - d e p e n d e n c e s o e q . ,

w e h a v e ,

+ h 2 ( y ) = 0 … … … . . ( 1 )w h e r e ,

h 2 = 2 + 2 Μ … … … ( 2 )

 

TM waves along a dielectric slab:

00

00 dd

S o l u t i o n O f E q . ( 1 ) m u s t B e C o n s i d e r e d I n B o t h S l a b A n d F r e e S p a c e R e g i o n W h i c h M a t c h A t B o u n d a r i e s .

I n S l a b R e g i o n We A s s u m e T h a t Wa v e P r o p a g a t e I n + Z D i r e c t i o n Wi t h o u t A t t e n u a t i o n S o ,

= j β … … . ( 3 )

S o l u t i o n o f e q . I n d i e l e c t r i c s l a b m a y c o n t a i n b o t h s i n e & c o s i n e t e r m , w h i c h a r e a n o d d & e v e n f u n c t i o n , r e s p e c t i v e l y o f y :

( y ) = s i n y + = y , | y | ≤ … … ( 4 )

w h e r e , = 2 - = … … … ( 5 )

I n f r e e s p a c e r e g i o n ( y > & y < ) w a v e m u s t d e c a y e x p o n e n t i a l l y s o t h e y g u i d e d a l o n g s l a b & d o n o t r a d i a t e a w a y f r o m i t . T h e n w e h a v e ,( y ) = … … … . ( 6 )

where , == - ……….(7 )

eq . (7 )& (5 ) a re d i spers ion re la t i ons because i t shows non l inear dependence o f phase cons tan t & .For f o l l ow ing ampl i tudes , we w i l l cons ider odd & even te rm fo r TM modes .

[A] Odd TM mode :In Th i s Mode ( y ) D e s c r i b e Wi t h S i n e F u n c t i o n T h a t I s A n t i - S y m m e t r i c w i t h R e s p e c t Y = 0 p l a n e .

i ) i n t h e d i e l e c t r i c re g i o n , | y | ≤ :

F ro m e q . ( 4 ) , a n d = t h e n ,

(y ) = s i ny ……(8 )

(y ) = ( - y … … … . . ( 9 )(y )= = y ………(10)

i i ) In upper f ree - space reg ion , y≥

(y ) = s in ) ………(11)

( y ) = s i n ) ……….(12)

(y ) = s in )……(13)

where in eq . 6 has been set to equa l s i n w h i c h i s v a l u e o f e q . ( 8 ) .

i i ) I n l ower space reg ion , y

( y ) = s i n ) ………(14)

( y ) = s i n ) … … … . ( 1 5 )

( y ) = s i n ) ……(16)

where i n eq . 6 h as been s e t t o equa l s i n w h i c h i s v a l u e o f e q . ( 8 ) a t l o w e r i n t e r f e r e n c e y .

w e m u s t d e t e r m i n e f o r g i v e n f r e q . , s o f r o m e q .( 1 0 ) & ( 1 3 ) , t a k i n g r a ti o o f t h e m ,

= tan [ODD TM MODE]

I n T h i s M o d e ( y ) D e s c r i b e W i t h C o s i n e F u n c ti o n T h a t I s A n ti - S y m m e t r i c W i t h R e s p e c t Y = 0 P l a n e .

( y ) = y, | y | … … . . ( 1 7 )

a n d s i m i l a r l y a s p e r e a r l i e r d i s c u s s i o n r e l a ti o n o f b e c o m e s ,

= - cot [Even TM MODE]

[B] Even TM mode:

F o r T E w a v e s , E z = 0 , s i n c e t h e r e i s n o x - d e p e n d e n c e s o e q . ,

+ h 2 ( y ) = 0 … … . . ( 1 8 )

w h e r e h 2 = 2 + 2 Μ … … … ( 1 9 )

s o l u t i o n o f e q . 1 8 g i v e s a s ,( y ) = s i n y + = y , | y | ≤ … ( 2 0 )

TE waves along a dielectric slab:

I n f r e e s p a c e r e g i o n ( y > & y < ) w a v e m u s t d e c a y e x p o n e n t i a l l y s o t h e y g u i d e d a l o n g s l a b & d o n o t r a d i a t e a w a y f r o m i t . T h e n w e h a v e ,

( y ) = ………(21)

[A]odd TE Mode :( i ) In the die lectr ic region ,|y|≤:

(y) =siny …(22)

(y) =(- y ….. (23)

(y)= =(- ) y …(24)

ii) In upper free-space region , y≥

(y ) =s in ) … … … ( 2 5 )

( y ) = s i n ) … … … . ( 2 6 )

(y ) s in) … … ( 2 7 )

ii) In lower space region , y

(y ) = s in ) ………(14)

( y ) = s i n ) ……….(15)

(y ) = - s in ) ……(16)

R E L AT I O N B E T W E E N o f b e c o m e s , f ro m e q . ( 2 4 ) & ( 2 7 ) ta k i n g rati o o f t h e m t h u s ,

= tan [ODD TE MODE]

I n T h i s M o d e ( y ) D e s c r i b e W i t h C o s i n e F u n c ti o n T h a t I s A n ti - S y m m e t r i c W i t h R e s p e c t Y = 0 P l a n e .

( y ) = y, | y | … … . . ( 2 8 )

a n d s i m i l a r l y a s p e r e a r l i e r d i s c u s s i o n r e l a ti o n o f b e c o m e s ,

= - cot [Even TE MODE]

[B] Even TE Mode:

In Dielectric Waveguide ,Generally It Leads To Bessel's Differential Eq.,Because Study Is Complicated By The Fact Pure TM Or TE Modes Are Possible Only If The Fields Are Circularly Symmetrical Which Means Fields AreIndependentOf Angle Φ.

But When Fields Are Dependent On Φ, Separation Into TM And TE Modes Is No Longer Possible,& It Is Necessary The Existence Of Both & Simultaneously Which Is Known As Hybrid Mode.

HE mode:

Cons ider an example tm mode for roundd ie lec t r ic rod o f rad ius a and permi t t i v i t y in a i r. From eq . ( y ) = n Φ By se t t ing n=0,

where , …. (29)

T h e c o r r e s p o n d i n g i s , f r o m e q . I t g i v e s ,

= - … … … … … ( 3 0 )

O u t s i d e t h e d i e l e c t r i c r o d , t h e fi e l d s a r e r e q u i r e d t o b e e v a n e s c e n t a n d m u s t d e c r e a s e e x p o n e n ti a l l y w i t h d i s t a n c e . A n a p p r o p r i a t e c h o i c e o f & ( ζ r ) B e s s e l f u n . G i v e s ,

= … … … … ( 3 1 )

Where, = 2 -2 …..(32)

and

= ( ζ r ) , r ………(33)

and the field component &must be continuous at r=a which requires,

F R O M E Q . ( 2 9 ) & ( 3 1 ) C O M B I N I N G B O T H O F T H E M W E H AV E

( h a ) , = ( ζ a ) … . . ( 3 4 )

a n d f ro m e q . ( 3 0 ) & ( 3 3 )w e g e t b y c o m b i n i n g b o t h o f t h e m a s ,

=- ( ζ r ) , . . ( 35 )

C o m b i n a ti o n o f a b o v e e q . ( 3 4 ) & ( 3 5 ) f o r c i r c u l a r l y s y m m e t r i c a l T M m o d e s :

=-

w h e r e a n d h a r e r e l a t e d t h r o u g h e q . ( 2 9 ) A n d ( 3 2 )

+ = 2

AB OVE EQ. CAN BE SOLVED IF THE EIGENVALUES FOUND ,THE CUTOFF FREQUENCYS AND OTHER PROPERTIES CAN BE DETERMINED.

Above eq. Can be solved i f the eigenvalues found ,the cutoff frequency's and other properties can be determined.