The practical usefulness of ferrites arises from the property that the
permeability tensor is not symmetric: 
, where T
denotes matrix transpose. The reason this is significant is that the Lorentz reciprocity theorem predicts that if a material is characterized by
either a permittivity tensor 
with 
or
a permeability tensor 
with 
, then it is not
reciprocal. In particular, isotropic materials (with scalar and ) are necessaril
y reciprocal.
What is reciprocity? It basically says if we apply an ``input'' 
at one end of a material, the wave propagates through the
material and exits at the ``output'' as 
, then if we send
back in through the material at the output end emerges at the input. To cla
rify this, we explain two
particular types of nonreciprocal devices: an isolator and a circulator, the symbols of which are shown in Figure 3.1.
An isolator is basically a one-way device, in which a signal applied at one
end, say port A, will pass through and emerge at the output port B; but
if a signal is applied at port B, nothing will emerge back at port A. A
circulator is a three-port device, in which inputs at A are sent to B,
inputs at B to C, and inputs at C to A. For a circulator, for
example, reciprocity would force if A goes to B then B goes to A.
Thus, to construct isolators or circulators, we must use nonreciprocal
materials, and in particular simple materials (with scalar , ) cannot work! An iso
lator would be useful, for example, at a transmitter
antenna, where we wish to take a signal out from a generator to the
transmitting antenna, but we do not want any reflections or noise signals
picked up by the antenna to be fed back to the signal generation circuit. In
a circulator, we can use a single antenna for transmission and reception.
Place the transmitter circuit at port A, the antenna at port B, and the
receiver circuit at port C. The transmitted signal at A is sent out to
the antenna at B, incoming signals at the antenna B are sent to the
receiver C.
The implementation of isolators and circulators with ferrites is based upon Faraday rotation, in which a linearly polarized plane wave propagating through a ferrite undergoes a rotation of its polarized direction independently of whether it is propagating in a forward or backward direction. First, let us mathematically define a linearly polarized plane wave as one whose phasor form is:
where 
is a constant real number, 
is a unit vector
indicating the direction of polarization, and 
is the wavenumber vector. In particular, the direction of is the
direction of propagation (direction of power flow), and the wavelength is 
. At all points in space, a linearly
polarized wave has a magnetic field with constant magnitude oriented in the
same direction (namely ). A plane wave must have the property that
.
For simplicity, consider a plane wave propagating in the 
direction,
so linearly polarized waves in the x- or y-directions are permitted. A
circularly polarized wave has the form:
The time-domain form of a circularly polarized wave is:
Thus, a circularly polarized wave has constant magnitude at all times at all points in space, and the direction of orientation rotates with time in a circular fashion. The rotation is counterclockwise, in the first case, or clockwise, in the second case.
Given a linearly polarized wave with 
in the x-direction and
propagating in the positive z-direction, that is:
we can express it as the sum of two circularly polarized waves:
where:
If this wave propagates through a ferrite, the total 
field is the
sum of the fields 
and 
corresponding to each of the circularly polarized fields, respectively.
However, if we compute 
and 
we get:
where 
and 
are scalars. Specifically:
We assume (as is the usual case) that the ferrite behaves like a linear,
homogeneous isotropic dielectric, having constant scalar . Then a
circularly polarized field propagates through the ferrite as if the medium
were linear homogeneous isotropic; either 
or 
characterize the material, depending
upon the direction of rotation of the circularly polarized wave. A more
careful study indicates the effective permeability, 
depends upon the orientation of the polarization with respect to the bias
field (here assumed in the +z-direction), not whether the plane wave is
propagating forward (+z) or backward (-z).
A plane wave propagating through a distance L through a linear, isotropic
homogeneous medium characterized by 
is
unchanged except for a phase shift 
where 
. Then if we apply the linearly pol
arized wave (3.19) at the input, we get at the output:
This can be simplified to:
where 
is the
effective phase constant of the material (that is, the field undergoes a
phase shift governed by 
), and is a unit
vector rotated in the counterclockwise direction from 
through an
angle 
.
To summarize, a linearly polarized wave propagating though a ferrite through
a fixed distance L will emerge as linearly polarized but where the
direction of polarization is rotated through a fixed angle 
.
Moreover, this rotation does not depend on the direction of propagation
(forward or backward). Suppose, for example, a forward wave undergoes a 
rotation; if we reflect the wave and put it back in at a orientation, it emerges with a 
orientation, not
the original one. This illustrates the nonreciprocal behavior of the
ferrite. This phenomenon is called Faraday rotation.
To construct an isolator or circulator using ferrites, we first need to
ensure the waves at the ports are polarized in certain directions. This can
be accomplished, for example, with a rectangular waveguide, which is a
``pipe'' comprised of a dielectric with rectangular cross-section bounded by
four conducting walls. For a certain range of frequencies, the only type of
wave that can propagate (with a real power flow) through the waveguide must
be linearly polarized, specifically with the magnetic field aligned along
the broadside of the waveguide; this field is denoted the TE 
mode.
Now, let us design an isolator as follows. Arrange the input waveguide at a 
orientation and the output waveguide at a
orientation, with the ferrite set up to establish a Faraday rotation of . A wave going from the input to the output will emerge with the
appropriate orientation to successfully couple power into the output port.
However, if a wave comes back in at the output port, it emerges at the input
port with a orientation, and hence couples no energy to
that port!
For a circulator, assume the three ports A,B,C are arranged at , and
, respectively, with the ferrite inducing
a Faraday rotation through 
. Then an input at port A couples
into B, and input at B couples into C, and an input at C couples
into A.
