Question

A parallel-plate waveguide is known to have a cutoff wavelength for the \(m=1 \mathrm{TE}\) and TM modes of \(\lambda_{c 1}=4.1 \mathrm{~mm}\). The guide is operated at wavelength \(\lambda=1.0 \mathrm{~mm}\). How many modes propagate?

Short answer

Answer: 8 modes (4 TE and 4 TM modes).

Step-by-step solution

\(a = \dfrac{1}{2} \cdot 4.1 \text{mm} = 2.05 \text{mm}\) #tag_step2#Calculate the maximum mode number Now we have to calculate the maximum mode number for the given operating wavelength \(\lambda=1.0 \mathrm{~mm}\) using the formula: $$\hat{m} = \dfrac{2a}{\lambda}$$

\( \hat{m} = \dfrac{2 \cdot 2.05 \text{mm}}{1.0 \text{mm}} = 4.1\) Here, we get the maximum mode number as \(\hat{m} = 4.1\). However, the mode number must be an integer, so we must round it down to the nearest integer. #tag_step3#Find the nearest integer Round down the maximum mode number to the nearest integer.

\(\lfloor \hat{m} \rfloor = \lfloor 4.1 \rfloor = 4\) #tag_step4#Count the propagating modes Since we have found the maximum mode number for the given operating wavelength, we can count the propagating modes by considering both TE and TM modes for each mode number up to 4:

For m=1: TE and TM (2 modes) For m=2: TE and TM (2 modes) For m=3: TE and TM (2 modes) For m=4: TE and TM (2 modes) Total modes: 8 So, there are 8 modes (4 TE and 4 TM modes) propagating in the parallel-plate waveguide at the operating wavelength of 1.0 mm.

Key concepts

Cutoff Wavelength

The cutoff wavelength in a parallel-plate waveguide is a critical parameter that determines whether certain modes can propagate within the waveguide structure. In simple terms, it is the longest wavelength at which a particular mode can still travel through the waveguide.

For instance, in our original exercise, the cutoff wavelength for the first transverse electric (TE) and transverse magnetic (TM) modes is given as \( \lambda_{c1} = 4.1 \,\mathrm{mm}\). This means that any wavelength shorter than 4.1 mm will allow these modes to propagate, while any wavelength longer will not support them. The cutoff wavelength is inversely proportional to the waveguide's dimensions and depends on the mode numbers and the intrinsic impedance of the medium within the waveguide.

TE and TM Modes

TE (Transverse Electric) and TM (Transverse Magnetic) are two fundamental mode types in waveguides. TE modes are characterized by having no electric field component (E-field) in the direction of wave propagation, whereas TM modes have no magnetic field component (H-field) in that direction.

The existence of multiple modes, TE and TM, of various mode numbers in a waveguide allows for the propagation of multiple waves simultaneously. This characteristic can be employed in applications like multiplexing, where different signals are carried over different modes. In our discussed waveguide, at an operating wavelength of 1.0 mm, a total of 4 TE and 4 TM modes would propagate.

Operating Wavelength

Operating wavelength refers to the actual wavelength of the signal that is being transmitted through the waveguide. It plays a significant role in determining how many modes a waveguide can support. Modes are only supported when the operating wavelength is shorter than the cutoff wavelength for those modes.

The operating wavelength in the exercise is 1.0 mm, which is significantly shorter than the cutoff wavelength, hence allowing multiple modes to propagate. It is worth noting that the operating wavelength affects the waveguide's dispersion characteristics, affecting the signal integrity over longer distances.

Waveguide Mode Number

The waveguide mode number, often denoted with \( m \) for TE and TM modes, indicates the order of the mode. Each mode number corresponds to a different field pattern within the waveguide. The mode number affects the cutoff frequency and therefore the cutoff wavelength for that particular mode.

The mode number in our problem was calculated using the formula \( \hat{m} = \dfrac{2a}{\lambda} \), where \( a \) is half the cutoff wavelength, and we had to round down to the nearest whole number to find the maximum number of modes. For a given operating wavelength, a waveguide can support a range of modes based on their mode numbers, affecting the capacity and efficiency of the waveguide.