Lecture 21

Resonators

21.1 Cavity Resonators

21.1.1 Transmission Line Model

The simplest cavity resonator is formed by using a transmission line. The source end can

be terminated by ZS and the load end can be terminated by ZL. When ZS and ZL are

non-dissipative, such as when they are reactive loads, then no energy is dissipitated as a wave

is reflected off them. Therefore, if the wave can bounce constructively between the two ends,

a coherent solution can exist due to constructive inference, or a resonance solution can exist.

Figure 21.1: A simple resonator can be made by terminating a transmission line with two

reactive loads at its two ends, the source end with ZS and the load end with ZL.

The transverse resonance condition for 1D problem can be used to derive the resonance

condition, namely that

1 = ΓS ΓLe−2jβz d (21.1.1)

where ΓS and ΓL are the reflection coefficients at the source and the load ends, respectively,

βz the the wave number of the wave traveling in the z direction, and d is the length of

the transmission line. For a TEM mode in the transmission line, as in a coax filled with

203

204 Electromagnetic Field Theory

homogeneous medium, then βz = β, where β is the wavenumber for the homogeneous medium.

Otherwise, for a quasi-TEM mode, βz = βe where βe is some effective wavenumber for a z-

propagating wave in a mixed medium. In general,

βe = ω/ve (21.1.2)

where ve is the effective phase velocity of the wave in a heterogeneous structure.

When the source and load impedances are replaced by short or open circuits, then the

reflection coefficients are −1 for a short, and +1 for an open circuit. The above then becomes

±1 = e−2jβed (21.1.3)

When a “+” sign is chosen, the resonance condition is such that

βed = pπ, p = 0, 1, 2, . . . , or integer (21.1.4)

For a TEM or a quasi-TEM mode in a transmission line, p = 0 is not allowed as the voltage will

be uniformly zero on the transmisson line. The lowest mode then is when p = 1 corresponding

to a half wavelength on the transmission line.

Whereas when the line is open at one end, and shorted at the other end in (21.1.1), the

resonance condition corresponds to the “−” sign in (21.1.3), which gives rise to

βed = pπ/2, p odd (21.1.5)

The lowest mode is when p = 1 corresponding to a quarter wavelength on the transmission

line, which is smaller than that of the short terminated transmission line. Designing a small

resonator is a prerogative in modern day electronic design. For example, miniaturization in

cell phones calls for smaller components that can be packed into smaller spaces.

A quarter wavelength resonator made with a coax is shown in Figure 21.2. It is easier to

make a short indicated at the left end, but it is hard to make a true open circuit as shown at

the right end. A true open circuit means that the current has to be zero. But when a coax is

terminated with an open, the electric current does end abruptly. The fringing field at the right

end gives rise to stray capacitance through which displacement current can flow in accordance

to the generalized Ampere’s law. Hence, we have to model the right end termination with a

small stray or fringing field capacitance as shown in Figure 21.2.

Figure 21.2: A short and open circuited transmission line can be a resonator, but the open

end has to be modeled with a fringing field capacitance Cf since there is no exact open circuit.

Resonators 205

21.1.2 Cylindrical Waveguide Resonators

Since a cylindrical waveguide is homomorphic to a transmission line, we can model a mode

in this waveguide as a transmission line. Then the termination of the waveguide with either

a short or an open circuit at its end makes it into a resonator.

Again, there is no true open circuit in an open ended waveguide, as there will be fringing

fields at its open ends. If the aperture is large enough, the open end of the waveguide radiates

and may be used as an antenna as shown in Figure 21.3.

Figure 21.3: A rectangular waveguide terminated with a short at one end, and an open circuit

at the other end. The open end can also act as an antenna as it also radiates (courtesy of

RFcurrent.com).

As previously shown, single-section waveguide resonators can be modeled with a transmis-

sion line model using homomorphism with the appropriately chosen βz . Then, βz = √β2 − β2

s

where βs can be found by first solving a 2D waveguide problem corresponding to the reduced-

wave equation.

For a rectangular waveguide, for example,

βz =

√

β2 −

( mπ

a

)2

−

( nπ

b

)2

(21.1.6)

If the waveguide is terminated with two shorts (which is easy to make) at its ends, then the

resonance condition is that

βz = pπ/d, p integer (21.1.7)

Together, using (21.1.6), we have the condition that

β2 = ω2

c2 =

( mπ

a

)2

+

( nπ

b

)2

+

( pπ

d

)2

(21.1.8)

206 Electromagnetic Field Theory

The above can only be satisfied by certain select frequencies, and these frequencies are the

resonant frequencies of the cavity. The corresponding mode is called the TEmnp mode or the

TMmnp mode depending on if these modes are TE to z or TM to z.

The entire electromagnetic fields of the cavity can be found from the scalar potentials

previously defined, namely that

E = ∇ × ˆzΨh, H = ∇ × E/(−jωμH) (21.1.9)

H = ∇ × ˆzΨe, E = ∇ × H/(jωεH) (21.1.10)

Figure 21.4: A waveguide filled with layered dielectrics can also become a resonator. The

transverse resonance condition can be used to find the resonant modes.

Since the layered medium problem in a waveguide is the same as the layered medium

problem in open space, we can use the generalized transverse resonance condition to find the

resonant modes of a waveguide cavity loaded with layered medium as shown in Figure 21.4.

This condition is repeated below as:

˜ R− ˜ R+e−2jβz d = 1 (21.1.11)

where d is the length of the waveguide section where the above is applied, and ˜ R− and ˜ R+ are

the generalized reflection coefficient to the left and right of the waveguide section. The above

is similar to the resonant condition using the transmission line model in (21.1.1), except that

now, we have replaced the transmission line reflection coefficient with TE or TM generalized

reflection coefficients.

Consider now a single section waveguide terminated with metallic shorts at its two ends.

Then RT E = −1 and RT M = 1. Right at cutoff of the cylindrical waveguide, βz = 0 implying

no z variation in the field. When the two ends of the waveguide is terminated with shorts

implying that RT E = −1, even though (21.1.11) is satisfied, the electric field is uniformly zero

in the waveguide, so is the magnetic field. Thus this mode is not interesting. But for TM

modes in the waveguide, RT M = 1, and the magnetic field is not zeroed out in the waveguide,

when βz = 0.

The lowest TM mode in a rectanglar waveguide is the TM11 mode. At the cutoff of this

mode, the βz = 0 or p = 0, implying no variation of the field in the z direction. When the

two ends are terminated with metallic shorts, the tangential magnetic field is not shorted

out. Even though the tangential electric field is shorted to zero in the entire cavity but the

longitudinal electric still exists (see Figures 21.5 and 21.6). As such, for the TM mode, m = 1,

n = 1 and p = 0 is possible giving a non-zero field in the cavity. This is the TM110 mode

of the resonant cavity, which is the lowest mode in the cavity if a > b > d. The top and

Resonators 207

side views of the fields of this mode is shown in Figures 21.5 and 21.6. The corresponding

resonant frequency of this mode satisfies the equation

ω2

110

c2 =

( π

a

)2

+

( π

b

)2

(21.1.12)

Figure 21.5: The top view of the E and H fields of a rectangular resonant cavity.

Figure 21.6: The side view of the E and H fields of a rectangular resonant cavity (courtesy

of J.A. Kong [31]).

For the TE modes, it is required that p 6 = 0, otherwise, the field is zero in the cavity. For

example, it is possible to have the TE101 mode.

ω2

101

c2 =

( π

a

)2

+

( π

d

)2

(21.1.13)

Clearly, this mode has a higher resonant frequency compared to the TM110 mode if d < b.

208 Electromagnetic Field Theory

The above analysis can be applied to circular and other cylindrical waveguides with βs

determined differently. For instance, for a circular waveguide, βs is determined differently

using Bessel functions, and for a general arbitrarily shaped waveguide, βs may be determined

numerically.

Figure 21.7: A circular resonant cavity made by terminating a circular waveguide (courtesy

of Kong [31]).

For a spherical cavity, one would have to analyze the problem in spherical coordinates.

The equations will have to be solved by separation of variables using spherical harmonics.

Details are given on p. 468 of Kong [31].

21.2 Some Applications of Resonators

Resonators in microwaves and optics can be used for designing filters, energy trapping devices,

and antennas. As filters, they are used like LC resonators in circuit theory. A concatenation

of them can be used to narrow or broaden the bandwidth of a filter. As an energy trapping

device, a resonator can build up a strong field inside the cavity if it is excited with energy

close to its resonance frequency. They can be used in klystrons and magnetrons as microwave

sources, a laser cavity for optical sources, or as a wavemeter to measure the frequency of

the electromagnetic field at microwave frequencies. An antenna is a radiator that we will

discuss more fully later. The use of a resonator can help in resonance tunneling to enhance

the radiation efficiency of an antenna.

Resonators 209

21.2.1 Filters

Microstrip line resonators are often used to make filters. Transmission lines are often used to

model microstrip lines in a microwave integrated circuits (MIC). In MIC, due to the etching

process, it is a lot easier to make an open circuit rather than a short circuit. But a true open

circuit is hard to make as an open ended microstrip line has fringing field at its end as shown

in Figure 21.8 [114, 115]. The fringing field gives rise to fringing field capacitance as shown

in Figure 21.2. Then the appropriate ΓS and ΓL can be used to model the effect of fringing

field capacitance. Figure 21.9 shows a concatenation of two microstrip resonators to make

a microstrip filter. This is like using a concatenation of LC tank circuits to design filters in

circuit theory.

Figure 21.8: End effects and junction effects in a microwave integrated circuit [114, 115]

(courtesy of Microwave Journal).

Figure 21.9: A microstrip filter designed using concatenated resonators. The connectors to

the coax cable are the SMA (sub-miniature type A) connectors (courtesy of aginas.fe.up.pt).

210 Electromagnetic Field Theory

Optical filters can be made with optical etalon as in a Fabry-Perot resonator, or concate-

nation of them. This is shown in Figure 21.10.

Figure 21.10: Design of a Fabry-Perot resonator [50, 75, 116, 117].

21.2.2 Electromagnetic Sources

Microwave sources are often made by transferring kinetic energy from an electron beam

to microwave energy. Klystrons, magnetrons, and traveling wave tubes are such devices.

However, the cavity resonator in a klystron enhances the interaction of the electrons with the

microwave field, causing the field to grow in amplitude as shown in Figure 21.11.

Resonators 211

Figure 21.11: A klystron works by converting the kinetic energy of an electron beam into the

energy of a traveling microwave next to the beam (courtesy of Wiki [118]).

Magnetron cavity works also by transferring the kinetic energy of the electron into the

microwave energy. By injecting hot electrons into the magnetron cavity, the cavity resonance

is magnified by the kinetic energy from the hot electrons, giving rise to microwave energy.

Figure 21.12: A magnetron works by having a high-Q microwave cavity resonator. When the

cavity is injected with energetic electrons from the cathode to the anode, the kinetic energy

of the electron feeds into the energy of the microwave (courtesy of Wiki [119]).

Figure 21.13 shows laser cavity resonator to enhance of light wave interaction with material

212 Electromagnetic Field Theory

media. By using stimulated emission of electronic transition, light energy can be produced.

Figure 21.13: A simple view of the physical principle behind the working of the laser (courtesy

of www.optique-ingenieur.org).

Energy trapping of a waveguide or a resonator can be used to enhance the efficiency of

a semiconductor laser as shown in Figure 21.14. The trapping of the light energy by the

heterojunctions as well as the index profile allows the light to interact more strongly with

the lasing medium or the active medium of the laser. This enables a semiconductor laser

to work at room temperature. In 2000, Z. I. Alferov and H. Kroemer, together with J.S.

Kilby, were awarded the Nobel Prize for information and communication technology. Alferov

and Kroemer for the invention of room-temperature semiconductor laser, and Kilby for the

invention of electronic integrated circuit (IC) or the chip.

Figure 21.14: A semiconductor laser at work. Room temperature lasing is possible due to

both the tight confinement of light and carriers (courtesy of Photonics.com).

Resonators 213

21.2.3 Frequency Sensor

Because a cavity resonator can be used as an energy trap, it will siphon off energy from

a microwave waveguide when it hits the resonance frequency of the passing wave in the

waveguide. This can be used to determine the frequency of the passing wave. Wavemeters

are shown in Figure 21.15 and 21.16.

Figure 21.15: An absorption wave meter can be used to measure the frequency of microwave

(courtesy of Wiki [120]).

214 Electromagnetic Field Theory

Figure 21.16: The innards of a wavemeter (courtesy of eeeguide.com).

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