8 Wave Guides and Cavities 8 A Wave

8. Wave Guides and Cavities 8 A. 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

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

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

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

TEM Modes • Can we find solutions with Ez = Bz = 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 Et is a 2 D electrostatic problem Recall also that potential is constant on surfaces Only get non-trivial solutions if there at least two conducting surfaces

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 Et 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

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)

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

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 t. Bz must be parallel to the walls of the cylinder • Solve the eigenvalue equation subject to the boundary conditions

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

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 Bz (TE) or Ez (TM) modes • Makes sense to work in cylindrical coordinates • Rotational symmetry implies solutions of the form • Substitute in: • Let • Then we have

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 for the frequencies will be:

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 > for m from 0 to 3 do evalf(Bessel. JZeros(m, 1. . 3)) end do; • With a little coaxing we can also get it to find the y’s > for m from 0 to 3 do seq(fsolve(diff(Bessel. J(m, x), x=(n+m/2 -3/4)*Pi), n=1. . 3) end do;

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

8 B. 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, we want waves of the form • For TM modes, we want waves of the form • In each case, we have a b

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 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 b a (m, n) = (2, 0) or (0, 1) • If a 2 b, then factor of two difference between lowest and next lowest modes • Normally make a 2 b

The TE 10 mode • 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 a b

8 C. 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 combinations of our previous solutions d

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

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

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 Bz for TA and Ez for TM • Boundary conditions for TE: • Boundary conditions for TM: • For TM modes, we then have d

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 a b

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 Bz • The 1, 0 solution is – I arbitrarily included 2/ 2 • is given by • Next find Bz: • Where k is given by • So we have d a b

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 the frequencies, which are given by

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

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:

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. d • It is pretty easy to see that • Of course, this is just conservation of energy a b