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Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
b
w bx
zy
n2
d n1
n2
y=0
y=d/2
y=-d/2
y
z
2-D waveguides
3-D waveguides ?
- TE, TM modes
- TE: Ez = 0, Hz non-zero
- TM: non-zero Ez, Hz = 0
- ky is qunatized
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
x
y
za
b
)0,0,0(
Rectangular metallic waveguide
22
2 0E Et
( , , , ) ( , ) j z j tE x y z t E x y e e
2 22 2
2 2 ( , ) ( ) ( , ) 0E x y k E x yx y
22
2 0H Ht
( , , , ) ( , ) j z j tH x y z t H x y e e
2 22 2
2 2 ( , ) ( ) ( , ) 0H x y k H x yx y
Solve for Ez with Hz=0 Solve for Hz with Ez=0
(TM) (TE)
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
x
y
za
b
)0,0,0(
Rectangular Metallic Waveguide: TE modes
2 22 2
2 2 ( , ) ( ) ( , ) 0z zH x y k H x yx y
( , ) ( ) ( )zH x y X x Y y
2 22 2
2 2( ) ( ) ( )( ) 0XY XY k XYx y
Separation of variables
2 22 2
2 2
1 1 ( ) 0X Y kX x Y x
22
2 0xd X k Xdx
2 22 2
2 2
1 1 ( ) 0x yX Y k kX x Y x
22
2 0yd Y k Ydx
2 2 2 2x yk k k
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
( , ) ( ) ( ),zH x y X x Y y
0 cos cosz x yH (x, y) H (k x) (k y)
x
y
za
b
)0,0,0(
sin cosy yY C (k y) D (k y)
00
xX x
0x a
Xx
0
y b
Yy
xmπk ,a
22
2 0xd X k Xdx
sin cosx xX A (k x) B (k x)
00
y
Y y
22
2 0yd Y k Ydx
2
0 A
,x k a m xmπka
cos xX B (k x)
0 C
,yk b n ynπkb
cos yY D (k y)
,ynπkb
2 2 2( ) ( )mπ nπk
a b
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
2 2 2( ) mπ nπh ( )a b
y)b
nπ(x)a
mπ()Hb
nπ(hj(x,y)Ex sincos02
02( , ) cos sinyj nπ mπ nπH x y ( )H ( x) ( y)h b a b
y)b
nπ(x)a
mπ()Ha
mπ(hjy(xEy cossin), 02
02( , ) sin cosxj mπ mπ nπH x y ( )H ( x) ( y)h a a b
2 2 2x yk k k
2 21 ( ) ( )2c
m nfa b
0From cos cos , it can be derivedzmπ nπH (x, y) H ( x) ( y)a b
Dominant TE mode?01 10TE or TE mode
2 2 2( )mπ nπ( )a b
mn(TE mode)
(Smallest fc, or largest )
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
x
y
za
b
)0,0,0(
TM mode
22
2 0E Et
( , , , ) ( , ) j z j tE x y z t E x y e e
2 22 2
2 2 ( , ) ( ) ( , ) 0E x y k E x yx y
Solve for Ez with Hz=0
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
0 sin sinzmπ nπE (x, y) E ( x) ( y)a b
02 cos sinxj mπ mπ nπE (x, y) ( )E ( x) ( y)h a a b
02( , ) sin cosxj nπ mπ nπH x y ( )E ( x) ( y)h b a b
02, ) sin cosyj nπ mπ nπE (x y ( )E ( x) ( y)h b a b
y)b
nπ(x)a
mπ()Ea
mπ(hjyxH y sincos),( 02
2 2 2x yk k k
2 21 ( ) ( )2c
m nfa b
mnTM Mode
11TM ModeDominant TM mode?
2 2 2( )mπ nπh ( )a b
2 2 2( )mπ nπ( )a b
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
TM11 TE10
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
Wave propagation for TE10 and TM11 modes in rectangular waveguide
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
y
z
z
x
y
x
10
02
02
0
TE
, , ) sin sin( )
( , , ) sin sin( )
, cos cos( )
y
x
z
π πE (x y t ( )H ( x) t zh a a
π πH x y t ( )H ( x) t zh a a
πH (x, y t) H ( x) t za
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
y
z
z
x
y
x
11
02
02
0
02
02
TM
, cos sin sin( )
, , ) sin cos sin( )
, sin sin cos( )
( , , ) sin cos sin( )
( , , ) c
x
y
z
x
y
π π πE (x, y t) ( )E ( x) ( y) t zh a a b
π π πE (x y t ( )E ( x) ( y) t zh b a b
π πE (x, y t) E ( x) ( y) t za bπ π πH x y t ( )E ( x) ( y) t z
h b a bπH x y t ( )E
h a
os sin sin( )π π( x) ( y) t za b
Lect. 14 Rectangular Metallic Waveguide
W.-Y. ChoiOptoelectronics (2016/2)
