1. 8. Wave Guides and Cavities 8A. Wave Guides Suppose we have a region bounded by a conductor We want to consider oscillating fields in the non-conducting region For oscillating fields, changing B would imply non-zero E But E must vanish in the conductor On the surface of the conductors,  and J are present Which appear in two of Maxwell’s equations Therefore, D  and H || need not be continuous But E || and B  must be continuous So the correct boundary conditions must be Boundary Conditions at Perfect Conductors

2. Cylindrical Wave Guides Consider now a hollow infinite cylinder of arbitrary cross-section –We’ll make it along the z-direction We want to find solutions moving along the cylinder Assume it is filled with a linear material: We will use complex notation –Time-dependence will look like e -i  t Maxwell’s equations with no sources Use linearity plus time dependence Take curl of either of the last two equations: Use the double cross-product rule And therefore

3. Transverse and Longitudinal Dependence Since we have translation dependence along z, it makes sense to look for solutions that go like e  ikz –Moving in the  z direction Divide any derivative into transverse and longitudinal parts Then we have, for example Also break up fields into longitudinal and transverse parts: We now want to write our Maxwell’s equations broken up this way

4. Breaking Up Some Maxwell’s Equations Let’s look at some Maxwell’s equations: But we assume fields look like Therefore the last equation becomes We similarly have

5. TEM Modes Can we find solutions with E z = B z = 0? Such modes are called TEM modes –Because both E and B are transverse Then we would have Multiplying the first by But the left side is We would therefore have –Implies phase velocity equal to free waves Recall also And also This tells us finding E t is a 2D electrostatic problem Recall also that potential is constant on surfaces Only get non-trivial solutions if there are at least two conducting surfaces

6. Sample Problem 8.1 A coaxial cable has inner radius a and outer radius b, and is filled with a material with electric permittivity  and magnetic susceptibility . Find exact electric and magnetic field solutions for TEM modes. We will start by finding the electric field, which is transverse and satisfies Since it has no curl, it is derivable from a potential Potential must be constant on the inner and outer surfaces Symmetry implies that E t must be radial: No divergence tells us: Put back in the z- and t-dependence Where: The magnetic field is then Take real part to get actual fields

7. All We Need is the Z-Direction of the Fields Now consider non-TEM modes –E z or B z are non-zero Multiply second equation by i  and substitute the first one For a transverse vector, We therefore have Solve for E t : Also recall Normally you have just B z or E z –Modes with E z = 0 are called TE modes (transverse electric) –Modes with B z = 0 are called TM modes (transverse magnetic)

8. Finding Non-TEM modes All modes of E and B satisfy Let us define Then our equations become We must also satisfy our boundary conditions

9. TE Modes Case 1: TE modes (transverse electric) Search for solutions with E z = 0, so everything comes from B z The transverse fields are then We must also satisfy boundary conditions These imply  t B z must be parallel to the walls of the cylinder Solve the eigenvalue equation subject to the boundary conditions

10. TM modes Case 2: TM modes (transverse magnetic) Search for solutions with B z = 0, so everything comes from E z The transverse fields are then We must also satisfy the boundary conditions These imply that E z must vanish on the walls Solve the eigenvalue equation subject to the boundary conditions

11. Sample Problem 8.2 (1) A hollow cylindrical waveguide has circular cross-section of radius a. Find the relationship between the frequency and wave number for the lowest frequency modes for the TE and TM modes. The frequencies are given by The  2 values are eigenvalues of the equation –Where  is B z (TE) or E z (TM) modes Makes sense to work in cylindrical coordinates Rotational symmetry implies solutions of the form Substitute in: Let Then we have

12. Sample Problem 8.2 (2) A hollow cylindrical waveguide has circular cross-section of radius a. Find the relationship between the frequency and wave number for the lowest frequency modes for the TE and TM modes. This is Bessel’s Equation Solutions are Bessel functions Recall that  represents B z or E z And we have boundary conditions We therefore must have Let x mn be the roots of J m and let y mn the roots of its derivative Then the formula for the frequencies will be:

13. Sample Problem 8.2 (3) A hollow cylindrical waveguide has circular cross-section of radius a. Find the relationship between the frequency and wave number for the lowest frequency modes for the TE and TM modes. Maple is happy to find roots of Bessel’s equation With a little coaxing we can also get it to find the y’s > for m from 0 to 3 do evalf(BesselJZeros(m,1..3)) end do; > for m from 0 to 3 do seq(fsolve(diff(BesselJ(m,x),x), x=(n+m/2-3/4)*Pi),n=1..3) end do;

14. Comments on Modes Note that for each mode, there is a minimum frequency If you are at a limited , only a finite number of modes are possible –Usually a good thing –Ideally, want only one mode The lowest mode is usually a TE mode Note that the lowest modes often have m > 1 –Actually represent two modes because modes can be e  im 

15. 8B. Rectangular Wave Guides Let’s now consider a rectangular wave guide –Dimension a  b with a  b We will work in Cartesian coordinates Boundary conditions: For TE modes, we want waves of the form For TM modes, we want waves of the form In each case, we have Working out the Modes a b

16. For the TM modes (E z ), we must have m > 0 and n > 0 For the TE modes (B z ), we can’t have m = 0 and n = 0 –Since then B z has no transverse derivative Hence TE modes have m > 0 OR n > 0 Degeneracy between TE mode and TM mode if both positive Lowest frequency mode: TE mode with (m,n) = (1,0) Second lowest modes: modes with (m,n) = (2,0) or (0,1) If a  2b, then  factor of two difference between lowest and next lowest modes Normally make a  2b Restrictions on Modes a b

17. We can work out all the fields for the TE 10 mode –Put back in the z and t dependence We then work out all the other pieces using We therefore have The factors of i represent a phase shift between the various modes The TE 10 mode a b

18. 8C. Cavities Let’s now cap off the cylindrical cavity, make the ends conducting Let it be along z with z from 0 to d On the end, we must have This means at z = 0 and z = d, we have We have up until now used modes with We no longer want e  ikz, instead we want We will have to take linear combinations of our previous solutions Cylindrical Cavities d

19. For TE modes, we had But we also need We therefore must have B z to be like To get it to vanish at z = d, must have So B z is given by To get the transverse components, take same linear combinations: TE Modes in Cavities d

20. For TM modes, we had But we also need We have to make the two contributions to E t cancel at z = 0, d This can be done if E z is like To get it to work at z = d, must have So E z is given by To get the transverse components, take same linear combinations: TM Modes in Cavities d

21. For both TE and TM modes, we have –k > 0 for TE, k  0 for TM We have to solve the equation –  represents B z for TA and E z for TM Boundary conditions for TE: Boundary conditions for TM: For TM modes, we then have Rules for Cavities Summarized d

22. Consider a cavity of dimensions a  b  d Frequency is given be Where k is given by –p = 1, 2, 3, … for TE –p = 0, 1, 2, … for TM In both cases, we had –m > 0 OR n > 0 for TE –m > 0 AND n > 0 for TM Therefore, our frequencies are: At least two of m, n, p must be non-zero –If m = 0 or n = 0, it is a TE mode –If p = 0, it is a TM modes –If all three are non-zero, it can be a TE or TM mode Box-Shaped Cavities d a b

23. Sample Problem 8.2 (1) A conducting box-shaped cavity has dimensions a  b  d. Write explicitly the form of all fields for the TE 1,0,1 mode, and find the energy in electric and magnetic fields as a function of time. For TE modes, we first solve –Where  is B z The 1,0 solution is –I arbitrarily included  2 /  2  is given by Next find B z : Where k is given by So we have d a b

24. Sample Problem 8.2 (2)... Write explicitly the form of all fields for the TE modes, and find the energy in electric and magnetic fields as a function of time. Now let’s get B t : And finally, let’s go for E t : We will also need the frequencies, which are given by

25. Sample Problem 8.2 (3)... find the energy in electric and magnetic fields as a function of time. Keep in mind that we need to take the real part of these expressions The electric energy density is: Total electric energy is

26. Sample Problem 8.2 (4)... find the energy in electric and magnetic fields as a function of time. Again, take the real part Now let’s go for the magnetic energy density:

27. Sample Problem 8.2 (5) A conducting box-shaped cavity has dimensions a  b  d. Write explicitly the form of all fields for the TE 1,0,1 mode, and find the energy in electric and magnetic fields as a function of time. It is pretty easy to see that Of course, this is just conservation of energy d a b