Waveguide coupling by apertures: (from Collin)
The foregoing formulation of the radiation from currents in a waveguide in terms
of radiation from equivalent electric and magnetic dipoles is directly applicable to the
coupling of waveguides by small apertures, or holes in a common wall. To a first
approximation a small aperture in a conducting wall is equivalent to an electric dipole
normal to the aperture and having a strength proportional to the normal component of
exciting electric field, plus a magnetic dipole in the plane of the aperture and having a
strength proportional to the exciting tangential magnetic field.
The constants of
proportionality are parameters that depend on the aperture size and shape. These
constants are called the electric and magnetic polarizabilites of the aperture and
characterize the coupling or radiating properties of the aperture. A qualitative argument
to demonstrate the physical reasonableness of these properties of an aperture is given
below.
Figure D.III.2.1.a (4.32a) illustrates the normal electric field of strength  at a
conducting surface without an aperture. When an aperture is cut in the screen, the electric
field lines fringe lines fringe through the aperture in the manner indicated in Figure
D.III.2.1.b (4.32b). But this field distribution is essentially that produced by an equivalent
electric dipole as shown in Figure D.III.2.1.c. Note that the dipole is oriented normal to
the aperture.
In a similar manner the tangential magnetic field, lines shown in Figure
D.III.2.1.d. will fringe through the aperture as in Figure D.III.2.1.e. These fringing field
lines are equivalent to those produced by a magnetic dipole located in the plane of the
aperture.
(4.32)
Figure D.III.2.1 Aperture in a conducting wall.
For a small circular aperture of radius ro<<o the dipole moments are related to
the incident electromagnetic field, in the absence of the aperture, as follows:
P    o  e n  n
(D.III.2.1.a)
   m  t
(D.III.2.1.b)
where n   is the normal electric field and t is the tangential magnetic field at the center
of the aperture. The electric polarizability e is given by
e  
2 3
r0
3
(D.III.2.2.a)
and the magnetic polarizability m is given by
4
 m  r o3
3
(D.III.2.2.b)
The presence of an aperture also perturbs the field on the incident side of the
screen. This perturbed field is that radiated by equivalent dipoles which are the negative
of those given by Equation (D.III.2.1) and located on the input side of the screen. It is
important to note that when the aperture is now closed by a conducting wall. The
equivalent dipoles correctly account for the field coupled through the aperture in the
conducting screen. It should also be pointed out that the theory is an approximate one
valid for small apertures only. The examples that follow will clarify the application of the
theory.
Aperture in a transverse wall:
Figure D.III.2.2.a (4.27a) illustrates a small aperture in a transverse wall in a
rectangular waveguide. To determine the exciting field, assume that the aperture is
closed. A TE10 mode incident from z  0 is reflected by the conducting wall at z  0 to
produce a standing wave field in the region z  0 . This field is


 v  C e  j z  e j z sin

x
a

 z   CY e  j z  e j z sin
(D.III.2.3.a)
x
a
(D.III.2.3.b)
plus z component of magnetic field which is not required to be known for the present
problem.
(4.33)
Figure D.III.2.2 Aperture in a transverse waveguide wall
The normal electric field at the aperture is zero; so no induced electric dipole is produced.
The tangential magnetic field at the center of the aperture is, from Equation (D.III.2.3.b)
 z   2CY
and hence an induced x-directed magnetic dipole M is produced and is given by :
8
    m  x  2CY m  r o 3 CY
3
The field radiated in to the region z  0 is that radiated by the magnetic dipole M, as
illustrated in Figure D.III.2.2.b. This dipole is equivalent to a half circular current loop in
the yz plane as illustrated. To find the field radiated by this dipole in the presence of the
conducting transverse wall, image theory may be used. Since the image of the half
circular current loop in the transverse wall is the other half of the current loop, the image
of M is another magnetic dipole of moment M. The effect of the transverse wall is
equivalent to removing the wall and doubling the strength of the dipole, as depicted in
Figure D.III.2.2.c. If the field radiated in to the region z  0 is
 v  Ae  j z sin
x
 A e v e  j z
a
 z   AY e  j z sin
x
 Ah z e  j z
a
application of formula (4.95a) gives
A
j
 z  2 
10

j 0  16 3

  r o CY 
10
3

 x 
since the field  n  is  o h z  o  sin   in the present case. The constant 10 is
 a 
given by
a b
10   2  e y hx dx dy
o o
a b
2   sin 2
x
a
dx dy  ab
o o
Hence we obtain:
A
16 3 j  o
ro
3
ab
(D.III.2.4)
Since the incident TE10 mode has an amplitude C, the aperture has a transmission
coefficient
T
k 
A 16 3 o
16
 jr o
 j ro3 o o
C 3
a b 
3
ab 
The presence of the aperture causes a field to be scattered into the region z  0
also. For radiation into this region the effective magnetic dipole moment is the negative
of that given by Equation (D.III.2.4). Application of (4.95b) now gives
 y  A sin
x
a
e j z
with A given by Equation (D.III.2.4) for the perturbed electric field in the region z < 0.
The total electric field for z < 0 in now
 x

 y  C e  j  z  A C e j z sin
a 



Therefore the aperture in a transverse wall produces a reflection coefficient

AC A
16
β
  1  j r o3
1
C
C
3
ab
(D.III.2.5)
ko  o
in Equation (D.III.2.4). A normalized shunt susceptance
β
upon replacing ω by

jB would produce a reflection coefficient

1  in 1 1  jB   jB


1  in 1  1  jB  2  jB

when connected across a transmission line. When B is very large, we have
1
 jB 
2 
2
1 
   1  j

jB 
jB 
B
Comparing with Equation (D.III.2.5) shows that the aperture is equivalent to a
normalized inductive susceptance
jB   j
3ab
8r o 3
(D.III.2.6)
Example D.III.2.1 Aperture in transverse wall of a rectangular waveguide.
Aperture in broad wall of a waveguide:
Figure D.III.2.3 illustrates a circular aperture of radius r0 placed in the broad wall
separating two rectangular waveguides. The incident field is a TE10 in the lower guide is
given by
 y  C sin
π x  jβ z
e
a
 z   CYω sin
π x  j βz
e
a
(D.III.2.7.a)
(D.III.2.7.b)
z  j
πω
π x  jβ z
C cos
e
β a
a
(D.III.2.7.c)
Figure D.III.2.3 Aperture in a broad wall separating two waveguides.
At the center of the aperture located at z  0, x  d , the exciting field is
 y  C sin
πd
a
πd
π 

  CYω   az sin
j

a
β a

Example D.III.2.2 Aperture in broad wall of a rectangular waveguide.
Example D.III.2.3 A single-hole directional coupler.