8. Wave Guides and Cavities

1. 8. Wave Guides and Cavities 8A. Wave Guides Boundary Conditions at Perfect Conductors • 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

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, itmakes sense to look forsolutions that go like eikz • Moving in the z direction • Divide any derivative intotransverse and longitudinal parts • Then we have, for example • Also break up fields intolongitudinal 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 Ez = Bz = 0? • Such modes are called TEM modes • Because both E and B are transverse • Then wewould 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 Et 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 electricfield, 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 Et must be radial: • No divergencetells us: • Put back in the z- and t-dependence • Where: • The magneticfield 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 • Ez or Bz are non-zero • Multiply second equation by i and substitute the first one • For a transverse vector, • We therefore have • Solve for Et: • Also recall • Normally you have just Bz or Ez • Modes with Ez= 0 are called TE modes (transverse electric) • Modes with Bz = 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 Ez = 0, so everything comes from Bz • The transverse fields are then • We must also satisfy boundary conditions • These imply tBz must be parallelto 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 Bz = 0, so everything comes from Ez • The transverse fields are then • We must also satisfy the boundary conditions • These imply that Ez 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 2values are eigenvalues of the equation • Where  is Bz (TE) or Ez (TM) modes • Makes sense to work incylindrical coordinates • Rotational symmetryimplies 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 Bz or Ez • And we have boundary conditions • We therefore must have • Let xmn be the roots of Jm and let ymn the roots of its derivative • Then the formula forthe 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, thereis 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 Working out the Modes • 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, wewant waves of the form • For TM modes, wewant waves of the form • In each case, we have b a

16. Restrictions on Modes • For the TM modes (Ez), we must have m > 0 and n > 0 • For the TE modes (Bz), we can’t have m = 0 and n = 0 • Since then Bz has no transverse derivative • Hence TE modes have m > 0 ORn > 0 • Degeneracy between TE mode and TM mode if both positive • Lowest frequency mode: TE modewith (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 b a

17. The TE10 mode • We can work out all the fields for the TE10 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 thevarious modes b a

18. 8C. Cavities Cylindrical 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 combinationsof our previous solutions d

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

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

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

22. Box-Shaped Cavities • 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 d b a

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 TE1,0,1 mode, and find the energy in electric and magnetic fields as a function of time. • For TE modes, we first solve • Where  is Bz • The 1,0 solution is • I arbitrarilyincluded 2/2 •  is given by • Next find Bz: • Where k is given by • So we have d b a

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 Bt: • And finally, let’s go for Et: • We will also need thefrequencies, 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 energydensity 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 TE1,0,1 mode, and find the energy in electric and magnetic fields as a function of time. d • It is pretty easy to see that • Of course, this is just conservation of energy b a