Lecture 10

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

2. 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

3. 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

4. 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

5. Plane Wave Propagating in a Ferrite Medium • Plane Wave Propagating in a Ferrite Medium • in a source-free region, we have Microwave Technique

6. Plane Wave Propagating in a Ferrite Medium • assume a plane wave propagating in the z-direction, we have • with Microwave Technique

7. Plane Wave Propagating in a Ferrite Medium • therefore, we have Microwave Technique

8. Plane Wave Propagating in a Ferrite Medium • Since both and are zero, we have a TEM wave • in matrix form, we have , Microwave Technique

9. Plane Wave Propagating in a Ferrite Medium Microwave Technique

10. 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

11. Plane Wave Propagating in a Ferrite Medium • the eigenvalues are obtained from Microwave Technique

12. Plane Wave Propagating in a Ferrite Medium • let us consider the first mode corresponds to Microwave Technique

13. Plane Wave Propagating in a Ferrite Medium • Recall that , we have Microwave Technique

14. 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

15. Plane Wave Propagating in a Ferrite Medium • the wave impedance is • as , Microwave Technique

16. Plane Wave Propagating in a Ferrite Medium • for the second mode corresponds to • We have Microwave Technique

17. Plane Wave Propagating in a Ferrite Medium Microwave Technique

18. Plane Wave Propagating in a Ferrite Medium • this is a LPCP wave, • two modes are propagating with different propagation constant and Microwave Technique

19. 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

20. Plane Wave Propagating in a Ferrite Medium • along z-direction, the electric field will be represented by Microwave Technique

21. 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

22. 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

23. 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

24. 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

25. Birefringence • we have • With • Therefore, we have Microwave Technique

26. Birefringence • for the component, we have Microwave Technique

27. Birefringence • for the component, we have Microwave Technique

28. Birefringence • there are two propagating modes, and • For mode, leading to Microwave Technique

29. Birefringence • When , • We have • , Microwave Technique

30. 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

31. Birefringence • For mode, Microwave Technique

32. Birefringence • When , and • the field components are , • the phase between the E and H field is a constant Microwave Technique

33. 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

34. Birefringence • the fact that propagation constant depends on the polarization direction is called birefringence • note that , , • if m < k, is negative imaginary Microwave Technique

35. 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

36. TEm0 mode of Waveguide with a Single Ferrite Slab • assuming the ferrite slab is biased in the y direction Microwave Technique

37. TEm0 mode of Waveguide with a Single Ferrite Slab • Writing • And , we have Microwave Technique

38. TEm0 mode of Waveguide with a Single Ferrite Slab Microwave Technique

39. TEm0 mode of Waveguide with a Single Ferrite Slab • for TEm0 mode, as • Let us look for Microwave Technique

40. 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

41. TEm0 mode of Waveguide with a Single Ferrite Slab • Similarly, we have • Therefore, we hvae Microwave Technique

42. TEm0 mode of Waveguide with a Single Ferrite Slab Microwave Technique

43. 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

44. TEm0 mode of Waveguide with a Single Ferrite Slab • the field equations in the waveguide regions are given by Microwave Technique

45. 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

46. TEm0 mode of Waveguide with a Single Ferrite Slab • matching the boundary conditions and eliminating all the unknowns one obtains Microwave Technique

47. 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

48. 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

49. 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

50. 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