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Lecture 10. Plane Wave Propagating in a Ferrite Medium Propagation in a Ferrite-Loaded Rectangular Waveguide Faraday Rotation Birefringence Isolators Circulators. Plane Wave Propagating in a Ferrite Medium.

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Lecture 10 l.jpg
Lecture 10

  • Plane Wave Propagating in a Ferrite Medium

  • Propagation in a Ferrite-Loaded Rectangular Waveguide

    • Faraday Rotation

    • Birefringence

  • Isolators

  • Circulators

Microwave Technique


Plane wave propagating in a ferrite medium l.jpg
Plane Wave Propagating in a Ferrite Medium

  • in an isotropic medium, a circularly polarized (CP) wave remains a CP wave when it propagates

  • in an anisotropic medium, a CP may become a linearly polarized (LP) depending on the medium properties

Microwave Technique


Plane wave propagating in a ferrite medium3 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • one example of an anisotropic medium is a quarter-wave plate in which a CP is converted to a LP

  • another example is the ferrite medium with a bias field

  • we will look at Faraday rotation, ordinary wave and extrordinary wave, and birefringence

Microwave Technique


Plane wave propagating in a ferrite medium4 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • in an infinite ferrite-filled region with a DC bias field give by , the magnetic properties of the medium are governed by a permeability tensor reads

    , [U] is an identity matrix and [c] is a susceptibility tensor

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • Plane Wave Propagating in a Ferrite Medium

  • in a source-free region, we have

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • assume a plane wave propagating in the z-direction, we have

  • with

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • therefore, we have

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • Since both and are zero, we have a TEM wave

  • in matrix form, we have

    ,

Microwave Technique



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Plane Wave Propagating in a Ferrite Medium

  • note that this is an eigenvalue problem, eigenvalues are obtained by setting the determinant of the matrix to zero, the obtained electric field for each eigenvalue is an eigenvector

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • the eigenvalues are obtained from

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • let us consider the first mode corresponds to

Microwave Technique


Plane wave propagating in a ferrite medium13 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • Recall that

    , we have

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • this is a right-hand circularly polarized wave (RHCP), note that the angle between Ey and Ex,

  • i.e.,

  • keep changing with time

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • the wave impedance is

  • as

    ,

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • for the second mode corresponds to

  • We have

Microwave Technique



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Plane Wave Propagating in a Ferrite Medium

  • this is a LPCP wave,

  • two modes are propagating with different propagation constant

    and

Microwave Technique


Plane wave propagating in a ferrite medium19 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • consider a linear polarized (in x) wave at z = 0 given by E =

  • Note that propagates with

  • While propagates with

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • along z-direction, the electric field will be represented by

Microwave Technique


Plane wave propagating in a ferrite medium21 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • Take a command factor

  • note that the two components and have equal amplitude and the phase between them is given by

Microwave Technique


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Plane Wave Propagating in a Ferrite Medium

  • this is still a LP but the phase is different at each location of z, i.e., the angle between Ey and Ex rotates as the wave propagates in z

  • this is called the Faraday rotation

Microwave Technique


Plane wave propagating in a ferrite medium23 l.jpg
Plane Wave Propagating in a Ferrite Medium

  • for a round trip, the Faraday rotation is 2f

  • for +z going wave

  • for -z going wave

Microwave Technique


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Birefringence

  • now let us consider the DC bias is in the x direction with the wave still propagating in the z-direction

  • the permeability tensor is given by

Microwave Technique


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Birefringence

  • we have

  • With

  • Therefore, we have

Microwave Technique


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Birefringence

  • for the component, we have

Microwave Technique


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Birefringence

  • for the component, we have

Microwave Technique


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Birefringence

  • there are two propagating modes,

    and

  • For mode,

    leading to

Microwave Technique


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Birefringence

  • When ,

  • We have

  • ,

Microwave Technique


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Birefringence

  • this is called the ordinary wave which is unaffected by the magnetization of the ferrite medium

  • this happens when the magnetic fields transverse to the bias direction are zero, i.e.,

Microwave Technique


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Birefringence

  • For mode,

Microwave Technique


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Birefringence

  • When , and

  • the field components are

    ,

  • the phase between the E and H field is a constant

Microwave Technique


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Birefringence

  • the magnetic field has a longitudinal component, i.e., in the direction of propagation, this is called the extraordinary wave

  • a linear polarized wave in y direction propagates with (ordinary wave) and the wave linear in the x direction propagates with

Microwave Technique


Birefringence34 l.jpg
Birefringence

  • the fact that propagation constant depends on the polarization direction is called birefringence

  • note that

    , ,

  • if m < k, is negative imaginary

Microwave Technique


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Birefringence

  • therefore, a circularly polarized wave passing through the bias ferrite medium will becomes a linear polarized wave as the x component will be rapidly diminishing while the y component is unaffected

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • assuming the ferrite slab is biased in the y direction

Microwave Technique


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TEm0 mode of Waveguide with a Single Ferrite Slab

  • Writing

  • And , we have

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab38 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab39 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • for TEm0 mode,

    as

  • Let us look for

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab40 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • we can equate hz and hx from the first two equations and then substitute the results into the third equation

Microwave Technique


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TEm0 mode of Waveguide with a Single Ferrite Slab

  • Similarly, we have

  • Therefore, we hvae

Microwave Technique


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TEm0 mode of Waveguide with a Single Ferrite Slab

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab43 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • this is the equation in the ferrite slab region, in the air region, we can replace by

Microwave Technique


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TEm0 mode of Waveguide with a Single Ferrite Slab

  • the field equations in the waveguide regions are given by

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab45 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • we need the match the tangential fields, namely, at each of the interfaces

    ,

    in the air region k = 0 and m = mo

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab46 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • matching the boundary conditions and eliminating all the unknowns one obtains

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab47 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • note that the equation contains odd power of

  • when the bias field changes in direction the propagation constant will have a different solution

  • changing the bias field is equivalent to the changing direction of propagation as the magnetic field component changes sign when propagates in the opposite direction

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab48 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • therefore, one can obtain for the forward going wave and for the backward wave

  • by choosing the parameters such that is purely real, the wave is unattenuated while is a large positive imaginary number which attenuates rapidly

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab49 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • this would work very well as an isolator

  • the isolator has the scattering matrix of the form

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab50 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • transmission is from Port 1 to Port 2, both ports are matched

  • S is not unitary and must be lossy, it is not symmetric and therefore nonreciprocal

  • this is often placed between a high-power source and a load to prevent possible reflections from damaging the source

Microwave Technique


Te m0 mode of waveguide with a single ferrite slab51 l.jpg
TEm0 mode of Waveguide with a Single Ferrite Slab

  • ferrite phase shifters can be designed by controlling the phase, phase shifters have applications in phased arrays in which antenna beams can be scanned electronically

Microwave Technique


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Ferrite Circulators

  • three port device, lossless and matched at all ports, nonreciprocal

Microwave Technique


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Ferrite Circulators

  • if the ports are not perfectly matched, we have

Microwave Technique


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Ferrite Circulators

  • here G and b are small while a is close to 1

  • for a lossless system, the scattering matrix must be unitary

Microwave Technique


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Ferrite Circulators

  • since both G and b are small, we can drop their product term leading to

Microwave Technique


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Ferrite Circulators

  • we can add to the right hand side without affecting the solution

Microwave Technique


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Ferrite Circulators

  • ignoring the phase, we can see how the scattering changes with port mismatch

  • the isolation and transmission deteriorate as the mismatch increases

Microwave Technique


Junction circulator l.jpg
Junction Circulator

  • we can use a thin cavity model in which we put electric walls on top and bottom and magnetic wall on the side

Microwave Technique


Junction circulator59 l.jpg
Junction Circulator

  • , TM modes

  • is antisymmetric, we can reduce the problem to 1 ferrite disks

  • the problem is best solved in cylindrical coordinates

Microwave Technique


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Junction Circulator

  • we first determine the permeability tensor in cylindrical coordinates

Microwave Technique


Junction circulator61 l.jpg
Junction Circulator

Microwave Technique


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Junction Circulator

  • Similarly,

  • the tensor has the same form in the cylindrical coordinates as in the Cartesian coordinates

Microwave Technique


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Junction Circulator

  • we can write Maxwell’s equations in cylindrical coordinates withfor a very thin disk

Microwave Technique


Junction circulator64 l.jpg
Junction Circulator

  • Similarly,

Microwave Technique


Junction circulator65 l.jpg
Junction Circulator

  • we follow the same approach as before, i.e., find the magnetic field components in terms of and its derivative

Microwave Technique


Junction circulator66 l.jpg
Junction Circulator

  • substituting the results in the third equation leading to

  • ,

Microwave Technique


Junction circulator67 l.jpg
Junction Circulator

  • use the method of separation of variables, we obtain

  • we also need which is given by

Microwave Technique


Junction circulator68 l.jpg
Junction Circulator

  • at the magnetic wall

  • when the ferrite disk is not magnetized, k = 0 and

  • Resonance occurs at

  • ,

Microwave Technique


Junction circulator69 l.jpg
Junction Circulator

  • when the ferrite is magnetized,

  • the +n mode is obtained by

  • while the -n mode is obtained by replacing the + sign with a - sign

Microwave Technique


Junction circulator70 l.jpg
Junction Circulator

  • for n = 1, we define = and

  • for k/m is small, we have

  • As

Microwave Technique


Junction circulator71 l.jpg
Junction Circulator

  • the resonance frequencies are then given by

Microwave Technique


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Junction Circulator

  • these two modes are used to design the circulator

  • the amplitude of these modes are adjusted to provide the coupling from Port 1 to 2 and to provide cancellation at the isolated port

Microwave Technique


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Junction Circulator

  • the operating frequency is at so that the is not zero at

  • assume the following and

  • at

Microwave Technique


Junction circulator74 l.jpg
Junction Circulator

  • Since

  • and

Microwave Technique


Junction circulator75 l.jpg
Junction Circulator

  • the electric and magnetic fields are therefore given by

Microwave Technique


Junction circulator76 l.jpg
Junction Circulator

  • note that the derived H field is not yet matches our design

  • Let us expand in a Fourier series in

Microwave Technique


Junction circulator77 l.jpg
Junction Circulator

  • the n=1 term gives

Microwave Technique


Junction circulator78 l.jpg
Junction Circulator

  • compare this with the previous equation equate at r = a, we conclude that

Microwave Technique


Junction circulator79 l.jpg
Junction Circulator

  • the first condition corresponds to nonmagnetized case, i.e., the operating frequency should be the resonance frequency for the

    k = 0, m =

Microwave Technique


Junction circulator80 l.jpg
Junction Circulator

  • we can use this condition to determine the disk radius for a given resonance frequency

Microwave Technique


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Junction Circulator

  • the second condition can be used to determine the wave impedance as

  • as

Microwave Technique


Junction circulator82 l.jpg
Junction Circulator

  • finally, the power flows at the three ports are:

Microwave Technique


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