TM Modes in Rectangular Wave Guide

A rectangular wave guide of sides a = 7.21 cm and b = 3.40 cm is used in the transverse magnetic (TM) mode (see Figure 1.1). TM modes are modes in which the magnetic field is perpendicular to the direction of propagation, here z. Assume that the walls are perfect conductors.

Figure 1.1

a) By calculating the lowest cutoff frequency, determine whether TM radiation of angular frequency ω = 6.1 × 10¹⁰ s^-1 will propagate in the wave guide.

b) What is the dispersion relation (i.e., the relationship between ω and the wave vector k) for this guide.

c) Find the attenuation length, i.e., the distance over which the power drops to e^-1 of its starting value, for a frequency ω that is half the cutoff frequency.

SOLUTION

a) We can express all the fields in terms of a single longitudinal component. In this problem, we are considering TM waves so B_z = 0, and we use E_z instead. We find for the field components

(1)

where again

The wave equation for the E_z component is

(2)

where

The solution to (2) with the boundary condition E × n|_s = 0 is given by

(3)

where n and m are integers. So

(4)

The frequency ω is given by

(5)

The cutoff frequency corresponds to k = 0, so

(6)

For TM waves, we cannot take any of the n or m = 0 modes because that would make E_z = 0 So the lowest cutoff frequency corresponds to m = n = 1

So the TM radiation with frequency 6.1 × 10¹⁰ s^-1 will propagate in the guide.

b) The dispersion relation was given in (a):

c) The wave number as a function of the cutoff frequency ω_λ can be written in the form

The wave of frequency ω < ω_λ cannot propagate (k becomes imaginary), and in fact the attenuation of the field will be given by k_λ. In our case,

We may write E_z in the form

The power dissipation will be proportional to |E_z|²:

We wish to find the point where P(z)/P(0) = 1/e. Hence,

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