Microwave Ferrites
6.2 Basic Properties of Ferrite Materials
6.2.4 Faraday Rotation
When a linearly polarized (LP) wave propagates through ferrite in the same direction as an applied DC magnetic field, the plane of polarization of the wave will be rotated. This phenomenon is known as Faraday rotation, and can be explained through consideration of the behaviour of circularly polarized (CP) waves within the material.
Any LP wave can be decomposed into the sum of two CP waves with opposite hands of rotation, as depicted in Figure 6.5.
Consider an LP wave applied to a cylinder of ferrite material together with an axial DC magnetic field, of strengthHo, as shown in Figure 6.6.
We know that the spinning electrons within the ferrite will be precessing around the direction of the DC magnetic field.
With the direction ofHoshown in Figure 6.6, the electrons will be precessing in a clockwise direction. Also, we know that the LP wave can be decomposed into two CP waves rotating in opposite directions. Thus, in the ferrite the CP wave with clock- wise rotation will be closely coupled to the precessional motion of the electrons, whilst the CP wave with counter-clockwise
k k
E/2 E/2
≡
+
Figure 6.5 Linearly polarized wave decomposed into the sum of two circularly polarized waves.
E
E Ho
θ
Ferrite
Figure 6.6 Linearly polarized wave applied to ferrite.
x
y
z
(a)
H
Hy = −jHx
(b)
Hy = jHx
H
(c)
Figure 6.7 Magnetic field vector rotation inx−yplane: (a) coordinate system, (b) clockwise rotation, and (c) counter-clockwise rotation.
rotation will be essentially unaffected by the precessional motion of the electrons. This means that each of the CP waves will have a different velocity of propagation within the ferrite, and hence a different transmission phase. Thus, when the two CP waves recombine at the output of the ferrite, the resulting LP wave will appear to have been rotated through some angle𝜃1, as shown in Figure 6.6. The value of𝜃will depend on the properties of the ferrite material, the strength of the axial magnetic field, and the length of the ferrite cylinder. (This rotation is called Faraday rotation, after Michael Faraday, who first observed this phenomenon through experiments in optics.)
We can obtain an expression for𝜃in terms of the elements of the permeability tensor, by first comparing Eq. (6.5) with the magnetic field relationships for a CP wave. Figure 6.7 shows a magnetic field vector,H, propagating in thez-direction and rotating in thex−yplane.
For clockwise, or positive rotation, theycomponent of magnetic field will always lead thexcomponent by 90∘, and so Hy=jHx. Substituting this relationship into Eq. (6.5) gives, for a loss-less ferrite material
Bx=𝜇Hx−jkHy
=𝜇Hx−kHx
= (𝜇−k)Hx (6.13)
and so the effective permeability of a clockwise rotating CP wave is
(𝜇eff)+=𝜇−k. (6.14)
k k
6.2 Basic Properties of Ferrite Materials 177
Similarly, substitutingHy= −jHxinto Eq. (6.5) to represent a counter-clockwise rotating CP wave gives
Bx= (𝜇+k)Hx (6.15)
and
(𝜇eff)−= (𝜇+k), (6.16)
where in Eqs. (6.13) and (6.15) we have used the accepted convention of subscripts+and−to represent clockwise and counter-clockwise CP waves, respectively.
The phase propagation constants for the clockwise and counter-clockwise CP waves are then 𝛽+=2𝜋
𝜆+
= 2𝜋f (vp)+ =𝜔
c
√ 𝜀r
(𝜇eff)+ 𝜇o
= 𝜔 c
√ 𝜀r
𝜇−k 𝜇o
, i.e.
𝛽+=𝜔√ 𝜀√
𝜇−k (6.17)
and similarly
𝛽−=𝜔√ 𝜀√
𝜇+k (6.18)
where𝜀and𝜀rare the permittivity and relative permittivity, respectively, of the ferrite material, and the other symbols have their usual meanings. Thus, through a length,l, of ferrite the phases of the clockwise and counter-clockwise CP waves will have changed by𝛽+land𝛽−l, respectively.
When the two CP waves recombine after transmission through the ferrite the plane of polarization of the LP wave will appear to have been rotated by
𝜃= 𝛽+l−𝛽−l
2 = 𝜔√
𝜀 2
(√𝜇−k−√ 𝜇+k
)l. (6.19)
The sign of𝜃, i.e. whether the plane of polarization is rotated to the right or left, will depend on the value ofk, which can be positive or negative depending on whether the frequency of operation is below or above the resonant frequency of the ferrite material, i.e. the Lamor frequency, which was defined in Section 6.2.2.
0 1 2 3 4
20 40 60 80 100
–60 –40 –20 0 20 40 60
20 40 60 80 100
Ho (kA/m)
Ho (kA/m) (μeff)+ /μo
(μeff)– /μo
Resonance
Figure 6.8 Variation of effective permeabilities of CP waves in ferrite as a function of magnetizing field at 2 GHz. Ferrite is magnetized in the direction of propagation (𝛾=35 kHz.m/A 𝜔m=2𝜋×5.6×109rad/s).
k k 𝜙=𝛽+l+𝛽−l
2 = 𝜔√
𝜀 2
(√𝜇−k+√ 𝜇+k
)
l, (6.20)
i.e.
𝜙=𝜔√ 𝜀
2 [(𝜇eff)++ (𝜇eff)−]l. (6.21)
The values of the effective permeabilities (𝜇eff)+and (𝜇eff)−are dependent on the frequency of operation, the value of the DC magnetic field, and the parameters of the ferrite material. As an illustration, Figure 6.8 shows the variation of these permeabilities as a function of magnetizing field strength, above saturation, in the region of resonance. It can be seen that the effective permeability of CP waves with clockwise rotation remains substantially constant, but those waves with counter-clockwise rotation experience significant changes in permeability on either side of the resonance position.
In particular, it can be seen that there is a change of sign of the effective permeability of counter-clockwise rotating waves on either side of resonance.
Example 6.2 The permeability tensor at 6 GHz of a cylinder of ferrite magnetized in the axial direction (i.e.z-direction) is given by
𝝁=𝜇o⎡
⎢⎢
⎣
2.43 −j1.02 0 j1.02 2.43 0
0 0 1
⎤⎥
⎥⎦ .
Determine the Faraday rotation in a 1 cm length of the cylinder at 6 GHz, given that the ferrite has a dielectric constant of 12.
Solution
Comparing the given permeability tensor with Eq. (6.12) gives
𝜇=2.43𝜇o=2.43×4𝜋×10−7 H∕m=30.54×10−7 H∕m, k=1.02𝜇o=1.02×4𝜋×10−7 H∕m=12.82×10−7 H∕m, 𝜇−k= (30.54−12.82) ×10−7 H∕m=17.72×10−7 H∕m, 𝜇+k= (30.54+12.82) ×10−7 H∕m=43.36×10−7 H∕m.
Using Eq. (6.19):
𝜃= 𝜔√ 𝜀o𝜀 2 (√
𝜇−k−√ 𝜇+k)l
= 2𝜋×6×109×√
8.84×10−12×12
2 ×
(√
17.72×10−7−
√
43.36×10−7 )
×0.01 rad
= −1.46 rad
≡−83.64∘.
Comment: The negative sign indicates that the rotation is counter-clockwise when viewed in the direction of magnetization.