Dressel.Gruner.Electrodynamics of Solids.2003

a

(a)

b

Fig. 9.3. (a) Cross-section through a microstripline; the line of width a is usually made from copper and is separated from the metallic ground plate by an dielectric spacer of thickness b. (b) Design of a stripline where the metallic line is embedded in the dielectric material with the ground plates at the bottom and on the top; in most cases b1 = b2 d.

A ≈ b/a. The characteristic impedance is then evaluated as

The parallel plate transmission line is the model for microstriplines and striplines (Fig. 9.3) [Bha91, Gar94].

Coaxial cable

Coaxial cables are the preferred transmission lines in the microwave and in the lower millimeter wave spectral range. The geometrical constant of a lossless coaxial cable, with a and b the radii of the inner and the outer conductor as displayed in Fig. 9.2c, is A = (1/2π ) ln {b/a}. The characteristic impedance of a lossless

Note, that – as this solution implies – coaxial cables can have different dimensions with the same impedance. Besides the losses of dielectric material (σ1 = 0), the attenuation is determined by the metal; in the latter case the resistance per unit length is given by

which sets the higher frequency limitation for the use of coaxial cables. A second restriction is the occurrence of higher modes as seen in Section 9.1.2. A commonly used coaxial cable has an impedance of 50 .

Rectangular waveguide at TE10 mode

The most important application of transverse electrical waves is in standard rect-

the cutoff frequency fc = c/λc = c/(2a). As usual a is the larger side of the rectangular cross-section of the waveguide as displayed in Fig. 9.2d; it is about half the wavelength. The length of the guided wave λ is defined as the distance between two equal phase planes along the waveguide,

where λ0 is the wavelength in free space. In the most common case of an empty waveguide ( 1 = µ1 = 1), the relations for ZTE and λ reduce to:

(9.1.27) The frequency dependences of the phase velocity and group velocity behave similarly. Below the cutoff frequency ωc = π c/a, no wave propagation is possible: the attenuation α, the impedance ZTE, and the phase velocity vph diverge, while the group velocity vgr and β = Re{qˆ } (the real part of the wavevector) drop to zero.

230 9 Propagation and scattering of waves

and increases with frequency.

9.2 Scattering at boundaries

As we have discussed in the previous section, the propagation of electromagnetic radiation through transmission lines (and even free space) is fully described by the characteristic impedance of this structure. Hence we can disregard the particular arrangement of a transmission line and only consider its impedance. As we have seen, the concept of the characteristic impedance is quite general, and it allows us to present a unified description of wave propagation and scattering for low frequency problems as well as for typical optical arrangements. Any change in the characteristic impedance leads not only to a variation in the propagation parameter γˆ as we have discussed in the previous section, but also to a partial reflection of the electromagnetic wave at the boundary or interfaces of media with different Zˆ S. The scattering of the wave at the interface of two impedances is the subject of this section.

A large number of treatises deal with changes of the geometry of transmission lines or obstacles in the transmission line which modify the characteristic impedance [Mar48, Sch68]. Here we are less interested in a variation of the geometry than in the effects associated with specimens placed within the structures. We consider the change in the characteristic impedance if a specimen (of dielectric properties under consideration) is placed at an appropriate position in the transmission structure. The simplest case is light reflected off a mirror: the propagation in free space is described by the impedance Z0, and the mirror by the surface impedance Zˆ S. From these two quantities we can immediately evaluate the complex reflection coefficient as demonstrated in Section 2.4.5. Similarly, the sample could be a thin rod inside a waveguide, some material used to replace the dielectric of a coaxial cable, or a device under test connected to the ports of a network analyzer. In these examples the simple change of the material properties described by the surface impedance Zˆ S has to be supplemented by geometrical considerations leading to the concept of the load impedance Zˆ L.

In many practical cases, a specimen of finite size is placed inside a transmission line, implying that there is a second boundary at the back of the sample after

Zc

z = 0

Fig. 9.4. Circuit representation of a transmission line with characteristic impedance Zˆ c

which is terminated at the position z = 0 by the load impedance Zˆ L. VL denotes the voltage across the load impedance.

which the transmission line continues. A thin, semitransparent film in the path of light propagating in free space is the simplest example. If the wave is not fully absorbed by the sample, part of the incident power will be transmitted through both boundaries and continues to propagate in the rear transmission line. Transparent samples of finite thickness are discussed in Appendix B at length, here we will present only the main ideas pertaining to the scattering problem.

9.2.1 Single bounce

Let us start with the circuit representation introduced in Section 9.1.1 to illustrate the approach taken. The equivalent circuit of this general scattering problem is shown in Fig. 9.4, where z is the direction along the transmission line. In the case of no attenuation of the undisturbed transmission line (α = 0), the solutions of the second order differential equations (9.1.2) for the voltage and current are the following (considering the spatial variation only):

as defined in the previous section, β is the phase constant and Zˆ c is the characteristic impedance of the transmission line which also depends on the mode which is excited. Assuming that we terminate one end of the line (at the position z = 0) with a load impedance Zˆ L = RL + iXL, part of the electromagnetic wave is reflected

back, and the ratio of the amplitude of the reflected V− and incident waves V+ is called the reflection coefficient r = V−/ V+. The voltage at the load resistance is VL = V+ + V−, and the current going through is IL = I+ − I−. Together with VL = Zˆ L IL this yields the complex reflection coefficient

which also includes the phase difference between the incident and reflected waves. The transmission coefficient

describes the ratio of the voltage (or electric field) which passes the boundary to the incident one; it also takes the phase change into account. These two are the main equations which fully characterize the behavior of a wave as it hits a boundary. These equations also imply that the electromagnetic wave is sensitive only to the impedance in the guiding structure, meaning that changes in geometry or material cannot be observed if the complex impedances on both sides of the interface match. Note that we assumed implicitly that the same wave type can propagate on both sides of the boundary. This, for example, does not hold if a metallic waveguide where a TE mode propagates is terminated by an open end, since only TEM waves are possible in free space; in this case the reflection coefficient rˆ = 1.

In practice, the approach is the opposite: from the measurement of both components of the complex reflection (or transmission) coefficient and knowing the characteristic line impedance of the transmission line, it is straightforward to calculate the complex load impedance Zˆ L of the specimen under test:

Zˆ L depends on the geometry used and on the optical parameters of the material forming the load. For more complicated and multiple scattering events, the representation by scattering matrices S is advantageous and is widely used in engineering textbooks [Ell93, Gar84, Poz90]. The impedance is the ratio of the total voltage to the total current at one point; the scattering matrix relates the electric field of the incident wave to the field of the wave reflected from this point. If the S matrix is known, the impedance Zˆ can be evaluated, and vice versa.

This general concept holds for scattering on any kind of impedance mismatch experienced by the traveling wave in a transmission line; it is equally appropriate when we terminate a coaxial line by some material of interest, if we fill the waveguide with the sample, or replace a wire or plate in a Lecher line or parallel

plate transmission line by the unknown metal. In our simple example of a large (compared to the wavelength and beam diameter) and thick (d δ0) mirror placed in the optical path of a light beam, it is the surface impedance of the mirror Zˆ S which solely determines the reflection coefficient according to the equation

In this case, and also when the material acts as a short for a coaxial line or waveguide (where the free space impedance Z0 is replaced by the characteristic impedance Zˆ c), the load impedance Zˆ L is equal to the surface impedance Zˆ S. In general a geometrical factor enters the evaluation; this is, as a rule, difficult to calculate. The main problem is the following: if we know the load impedance Zˆ L which causes the scattering of the wave as it propagates along the transmission line, it is no trivial task to evaluate the surface impedance Zˆ S of the material from which the obstacle is made because of depolarization effects. Having obtained the surface impedance, we then evaluate the complex conductivity or the complex dielectric constant of the specimen as discussed in Chapter 2.

A remark on the difference between measuring conductivity (or admittance) and resistivity (or impedance) is in order here. In the first case, the applied voltage is kept constant and the induced current is measured; or the applied electric field is constant and the current is evaluated by observing the dissipated power due to losses (Joule heat) within the material. In the second case when the impedance is probed, the current flowing through the device is kept constant and the voltage drop studied: a typical resistance measurement. This implies that absorption measurements look for dissipation and thus the admittance Yˆ = 1/Zˆ ; the same holds for reflection experiments off a thick (d δ0) material which probe the power not absorbed R = 1 − A. Optical transmission experiments through thin films, on the other hand, actually measure the impedance Zˆ .

9.2.2 Two interfaces

In general it is not possible to evaluate the transmission of the electromagnetic wave through a single interface, but we can measure the transmission through a sample of finite thickness; thus we have to consider two interfaces. This can be done, for instance, if we replace a certain part (length d) of a copper waveguide (characterized by an impedance Zˆ c) by a metal (of the same geometry) with a load impedance Zˆ L and the transmission line continues with its characteristic impedance Zˆ c. Alternatively we can fill a microstripline with some unknown dielectric over a length d, or we can shine light through a slab of material of thickness d. The latter case can be simplified for thick samples: when the skin depth δ0 is much smaller

than the thickness d, the problem is reduced to a single bounce. Hence the load impedance Zˆ L is equivalent to the surface impedance Zˆ S of the material as defined in Eq. (2.4.23); no power is transmitted through the material: it is either reflected or absorbed. However, if this is not the case, the second impedance mismatch (at the back of the sample) also causes reflection of the wave and thus reduces the power transmitted into the rear part of the transmission line; the reflection coefficients for both sides of the sample are the same (though with opposite sign) if the line continues with Zˆ c. For arbitrary systems (but assuming local and linear electrodynamics) the load impedance Zˆ L can be calculated by

if we assume that the surrounding structure has the same characteristic impedance

no transmission and are left with a single bounce. In all other cases part of the wave is reflected off the front surface, and part is reflected at the rear surface of the material. This wave, however, is again partially reflected at the front, and the process repeats itself ad infinitum, as depicted in Fig. 9.5. Thus we have to take all these contributions to the totally reflected and totally transmitted wave into account and sum them up using the proper phase. The optical properties of media with finite thickness (and also of systems with many layers) are discussed in Appendix B in more detail. The main features of the reflected and transmitted radiation are as follows. Whenever the thickness d of the slab is a multiple of half the wavelength in the material λ, we observe a maximum in the transmitted power. Hence, the absorption is enhanced at certain frequencies (and suppressed at others) due to multireflection; an effect which will be utilized by resonant techniques discussed below.

If the material does not fill the entire waveguide or coaxial line, and hence part of the radiation is transmitted beyond the obstacle, the analysis is more complicated because geometrical factors enter the appropriate relations. Many simple cases, however, such as a rod of a certain diameter in a waveguide or a coaxial cable (see Section 11.1) can be analyzed analytically [Joo94, Kim88, Mar91, Sri85].

9.3 Resonant structures

In the previous sections we have discussed the propagation of electromagnetic waves in various structures (transmission lines) together with the reflection off

Ei

Fig. 9.5. Reflection off and transmission through a thin semitransparent material with thickness d and optical parameters 1, σ1, and µ1. The multiple reflections cause interference. Ei, Et, and Er indicate the incident, transmitted, and reflected electric fields, respectively. The optical properties of the vacuum are given by 1 = µ1 = 1 and σ1 = 0.

boundaries. If a transmission line contains two impedance mismatches at a certain distance (a distance of the order of the wavelength, as will be discussed below), as sketched in Fig. 9.6, the electromagnetic wave can be partially trapped between these discontinuities by successive reflections. If the distance is roughly a multiple of half the wavelength, the fields for each cycle add with the proper phase relations and thus are enhanced: we call the structure to be at resonance. In any real system, losses cannot be avoided and a (generally small) fraction of the energy per cycle dissipates.

The resonant system becomes useful for measurements of materials when part of a well characterized structure can be replaced by a specimen of interest. If the resonance characteristics are modified only weakly by the specimen, this modification can be considered to be a perturbation of the resonant structure and analyzed accordingly. If the geometry is known, the change in the width and shift in resonant frequency allow the evaluation of the impedance and finally the complex conductivity of the material under consideration. Crudely speaking, the losses of the material (for example given by the surface resistance) determine the quality factor Q or the width of the resonance, whereas the refractive index

(a)

(b)

(c)

d

Fig. 9.6. Transformation of a transmission line to a resonant structure by introducing two impedance mismatches (discontinuities) at a certain distance d. (a) Parallel plate transmission line, (b) rectangular waveguide, and (c) free space becoming an open resonator.

(or surface reactance) determines the resonance frequency. The advantage when compared with a simple reflection or transmission experiment is the fact that the electromagnetic wave bounces off the material which forms the resonance structure many times (roughly of the order of Q); this then enhances the interaction, and thus the sensitivity, significantly.

9.3.1 Circuit representation

In Section 9.1 we discussed the characteristics of transmission lines in terms of an electrical circuit analogy and have established the relevant electrical parameters such as the characteristic impedance. A similar approach is also useful for discussing the various resonant structures which are employed to study the electrodynamics of solids. We consider a series RLC circuit as shown in Fig. 9.7a; similar considerations hold for parallel RLC circuits shown in Fig. 9.7b. The impedance Zˆ of a resonant structure is in general given by

If the system is lossless (R = 0), the impedance is purely imaginary and the phase angle φ = arctan{(1/ωC − ωL)/R} = π/2. The structure resonates at the angular frequency