TE and TM Mode Patterns in Rectangular Waveguides

An infinite number of TE and TM modes can exist in a rectangular waveguide. Each one consists of a characteristic combination of E and H field distributions. These modes are designated as TEmn and TMmn modes, where m and n are digits that identify each of the modes. In general, an infinite number of these TE and TM modes can be present simul-taneously in a waveguide, depending on the method of excitation, the size and shape of the waveguide, and the frequency of the waves, as will be explained later. The variable m specifies how many peaks occur in the mode pattern across the width of the waveguide (the x direction in Figure 3.38), and the n specifies how many peaks occur in the mode pat-tern across the height of the waveguide (the y direction in Figure 3.38).

x

y

z

Rectangular Cylindrical

Figure 3.38

Two commonly used waveguides, rectangular and cylindrical.

Figure 3.39 shows the E and H vector field patterns at one instant of time for the TE10

mode, in which m = 1 and n = 0. The pattern propagates down the waveguide as time goes on. m = 1 indicates that there is one peak across the waveguide. This corresponds to the Ey

x z

x z x z

y

E fields

E fields

H fields TE₁₀ mode

m = 1: one peak across the waveguide in the x direction

n= 0: no variation in the y direction

Figure 3.39

The TE10 mode patterns in three different planes in a rectangular waveguide at one instant of time. As time progresses, the patterns move in the +z direction.

(the component of E in the y direction) having a maximum in the center and being zero on each side wall. Note that in all cases, the boundary conditions (see Section 1.12) require that the tangential component of the E field be zero at the metallic walls (assuming that they are perfectly conducting) of the waveguide. Or, in other words, the E field must be normal to the perfectly conducting walls. Because Ey is tangential to the side walls, it must be zero there, but because it is normal to the top and bottom walls, it need not be zero there.

n = 0 indicates that there is no variation in the y direction; that is, at a given value of x, all three vectors shown in the diagram have the same length in y. As another example, patterns for the TE₂₀ mode are shown in Figure 3.40. Again, there is no variation in the y direction, but m = 2 indicates that two peaks occur across the waveguide. In this case, E_y is zero in the center as well as at the two side walls.

The peaks in the mode patterns are more clearly displayed in terms of the envelopes (see discussion in connection with Figure 3.9 in Section 3.3.1) of E, as shown in Figure 3.41 for three modes, TE₁₀, TE₂₀, and TE₁₁. The TE₁₀ mode has one peak across the waveguide, and the TE₂₀ mode has two peaks across the waveguide. The TE₁₁ mode pattern is more compli-cated than the other two in Figure 3.41 because in it there is one peak across the guide in both the x direction and the y direction, as shown in Figure 3.41(c). Again, the E fields are normal to the metallic walls at the walls.

These examples illustrate the nature of mode patterns. Other patterns in rectangular waveguide are similar in nature and behavior. Mode patterns in other kinds of waveguide,

rESOnAnCE EFFECTS In THE HuMAn BODy

The resonance phenomenon also describes waves that partially reflect back and forth in some sections of the body, “reverberating” until they are absorbed rather than simply passing through the body. This causes much more power to be absorbed at specific frequencies than would be predicted simply using an analysis of attenuation.

Attenuation effects alone usually absorb more power at high frequencies than low frequencies, but resonance causes the body to absorb higher power within a small band of frequencies, with ranges on either side that have less absorption. Prediction of the whole-body resonance can be approximated reasonably well by treating the body as a half-wave dipole (where the body height is one-half of a wavelength) if the feet are ungrounded, or as a monopole (where the height is one-quarter of a wave-length, due to the presence of a ground plane) if the feet are grounded. Thus, a 6-foot-tall human should resonate around 75 MHz if ungrounded and 38 MHz if grounded.

This effect is indeed seen.

This does not tell the whole story, though. Individual parts of the body, such as the torso, can resonate at higher frequencies than the whole body. Also, animals of different sizes and shapes will have different resonant frequencies than a human, for instance. This can cause some difficulties when doing biological tests of electro-magnetic fields. If a rat is being treated at its resonant frequency, the dose of depos-ited electromagnetic power at that frequency can be far higher than it would be in a human (who is not resonant at the same frequency). It is nontrivial but important to assess these effects.

Thus, electromagnetic absorption in humans is strongly dependent on frequency and is controlled not only by the properties of the tissue, but also by the shape and

such as the cylindrical waveguide, are similar in characteristic behavior, but differ in detail because of the round shape of the waveguide. Now that we have discussed what modes are, we next discuss how they are excited and how they exist in combinations in waveguides.