Dressel.Gruner.Electrodynamics of Solids.2003

thus which are weakly coupled to the exterior. This includes different arrangements such as RLC circuits, enclosed cavities, or Fabry–Perot resonators.

9.1 Propagation of electromagnetic radiation

In the absence of free charge and current, the propagation of electromagnetic radiation in free space or in a homogeneous medium is described by the wave equations (2.1.19) and (2.2.20), respectively. In the presence of conducting material, such as a wire or a waveguide, the appropriate boundary conditions (2.4.4) have to be taken into account; i.e. the tangential component of the electric field E and the normal component of the magnetic induction B are zero at the surface of a good conductor: ns × E = 0 and ns · B = 0, where ns denotes the unit vector perpendicular to the boundary. The solutions for many simple, mainly symmetrical, configurations (a pair of parallel wires, microstriplines, coaxial cables, and rectangular waveguides, for example) are well documented, and we do not elaborate on them here.

The important overall concept which emerges is that the structure which is used to propagate the electromagnetic radiation can be characterized by a quantity called the impedance Zˆ c, a complex quantity which depends on the particular structure used. The parameter also appears in the equation which describes the reflection from or transmission through a test structure (the sample to be measured) which is placed in the path of the electromagnetic radiation; the structure is again described by the characteristic impedance.

9.1.1 Circuit representation

Instead of solving Maxwell’s equations for the description of wave propagation, for cables and wires it is more appropriate to use the circuit representation, eventually leading to the telegraphist’s equation. This approach is common to electrical engineering, and is a convenient way of describing and calculating the wave propagation in transmission lines. Although the description can be applied to all waveguiding structures, it is best explained if we look at a pair of wires or a coaxial cable. Let us assume that the leads have a certain resistance and inductance, and that between the two wires there is some capacitance and maybe even losses (expressed as conductance) due to a not perfectly insulating dielectric. The circuit which includes these four components can be used to write down the relationship between currents and voltages at both ends of a transmission line segment displayed in Fig. 9.1. We define the wave to be traveling in the z direction and therefore expand V (z + z, t) and I (z + z, t) in a Taylor series ignoring the

dz

Fig. 9.1. Circuit representing a transmission line: the losses in the wires are given by the resistance Rl dz, the inductance of the wires is Ll dz, the capacitance between them is given by Cldz, and the losses of the dielectric between the wires are described by the conductance Gldz. If a voltage V is applied between the two contacts on the left hand side of the circuit,

V + ∂∂Vz dz is obtained at the right hand side. A current I on one side becomes I + ∂∂ zI dz.

where Rl, Ll, Gl, and Cl are the resistance, inductance, conductance, and capacitance per unit length, respectively. Combining the two equations, the propagation of electromagnetic waves in a transmission line can now be described by the so-called telegraphist’s equations of voltage V (z, t) and current I (z, t)

One set of solutions of these equations is given by traveling attenuated waves1

(a similar equation exists to describe the current), where we obtain for the propagation constant

1In engineering textbooks generally exp{jωt} is used, leading to equations apparently different from the one given here. In cases where the time dependence of waves is concerned, the replacement of i by −j leads to identical results; there is never any difference in the physical results.

where the parameter α is the attenuation constant and β is the phase constant. The complex propagation constant γˆ is the common replacement for the wavevector q = ωc Nˆ nq as defined in Eq. (2.3.2), where qˆ = iγˆ ; it fully characterizes the wave propagation. Hence the solution of the telegraphist’s equation is reduced to a static problem which has to be calculated for each specific cross-section. Knowing the parameters of our circuit, we can therefore calculate how the electromagnetic waves propagate in the transmission line. From Eq. (9.1.1a) we find the voltage decay

and the decay of the current follows from Eq. (9.1.1),

where the electromagnetic fields propagating to the left (positive γˆ in Eq. (9.1.3)) are neglected, and Yˆc is called the admittance. The ratio of voltage to current

is called the characteristic impedance of the transmission line; it has the units of a resistance and is in accordance with the definition of the characteristic wave impedance as the ratio of electric field and magnetic field given by Eq. (2.3.27). This impedance fully characterizes the wave propagation in a transmission line and contains all information necessary for applications. It is the basic parameter used in the next section to calculate the reflectivity off or transmission through a material placed in the path of the electromagnetic wave.

Losses along the line and between the leads are assumed to be relatively small, i.e. Rl < ωLl and Gl < ωCl. This condition allows (roughly speaking) the propagation of waves. The phase constant β becomes

describes the exponential decay of the electric field maxima; i.e. the power dissipation along the line P(z) = P0 exp{−αz} according to Lambert–Beer’s law (2.3.17). Without dissipation Gl = Rl = 0, α = 0, and β = ω(LlCl)1/2. The characteristic impedance Zˆ c is then always a real quantity with Zˆ c = Rc = (Ll/Cl)1/2. The losses

of a transmission line are determined by the conductance Gl and the resistance Rl per unit length; in most common cases, Gl ≈ 0 and Rl is small. The phase velocity vph of an electromagnetic wave is given by vph = ω/β ≈ (LlCl)−1/2 for a lossless

without damping by the line (evanescent waves).

9.1.2 Electromagnetic waves

Whereas in free space only transverse electric and magnetic (TEM) waves propagate, transmission lines may also support the propagation of transverse electric (TE) and transverse magnetic (TM) modes. As a point of fact this is not always disturbing: the modes excited in the transmission line influence the attenuation and may also be important when these structures are used for investigations of condensed matter.

TEM waves

As we have seen in Section 2.1.2, the electric and magnetic field components of electromagnetic waves in free space are perpendicular to the direction of propagation. Due to the boundary conditions this does not hold in the case of conducting media. However, with ohmic losses along the line negligible, guiding structures which contain two or more conductors are in general capable of supporting electromagnetic waves that are entirely transverse to the direction of propagation; the electric and the magnetic field have no longitudinal components (EL = Ez = 0 and HL = Hz = 0). They are called transverse electromagnetic (TEM) waves. The transverse electric field is ET = − TΦ exp{i(qˆ z − ωt)}; we want to suppress the time dependence in the following. The propagation can be expressed as

[( T)2 + γˆ 2 + qˆ 2]ETΦ exp{−γˆ z} = 0 .

The magnetic field is given by Eq. (2.2.21), and with nq the unit vector along the propagation direction q:

we find that the characteristic impedance of the TEM waves is given by

where γˆ is determined by the particular propagating mode. This implies that the characteristic impedance of a transmission line is different for different modes. Although not commonly done, in principle TM waves can be used if the longitudinal response of a material is studied. If we place a material of interest between two plates, we can excite a TM wave which has also a longitudinal component of the electric field. Thus σ1L and 1L can be measured and evaluated at finite frequencies, which is not possible for free space wave propagation.

TE waves

In analogy to TM waves, lossless guiding structures that contain one or more conductors and a homogeneous dielectric are also capable of supporting electromagnetic waves in which the electric field is entirely transverse to the direction of propagation (EL = Ez = 0) but the magnetic field may also have a longitudinal component (HL = Hz = 0). These waves are called TE or H waves. The propagation of electric fields E = ET exp{−γˆ z} and magnetic fields H = (HT + HL) exp{−γˆ z}. The spatial propagation can be written as

[( T)2 + γˆ 2 + qˆ 2](HT + HL) exp{−γˆ z} = 0 ,

where the longitudinal component is [( T)2 + qˆc2]HL = 0. The equations for TE waves are:

As in the case of TM waves the characteristic impedance of the guiding structure is defined in analogy to the wave impedance of free space (Eqs (2.3.27) to (2.3.29)). The characteristic impedance describes the resistance of the transmission line; in contrast to TM waves, the impedance of TE waves does not depend on the dielectric material the structure is filled with. The cutoff wavenumber is defined in the same way as for TM waves: qˆc2 = γˆ 2 + qˆ 2 = (2π /λc)2; for smaller wavevectors no wave propagation is possible.

9.1.3 Transmission line structures

Next we evaluate Zˆ c, the parameter which is needed to describe the scattering problem: the scattering of electromagnetic radiation on a sample which, as we will see, is described by an impedance Zˆ S.

In the case that metallic boundaries are present, the characteristic impedance Zˆ c depends not only on the medium, but also on the geometry and the mode which is excited. The task at hand now is to evaluate the geometrical factor for the transmission lines of interest and then see how the propagation parameters are modified compared to free space propagation.

The best known examples of transmission lines (Fig. 9.2) are two parallel wires (Lecher line), parallel plates (microstripline and stripline), coaxial cables, and hollow (rectangular) waveguides; the respective circuit representations are given in [Ell93, Gar84, Poz90, Ram93] and in numerous handbooks [Dix91, Mag92, Mar48, Smi93]. There is a simple concept which applies in general: the calculation of the four circuit parameters (capacitance, inductance, conductance, and resistance) is reduced to two geometrical parameters (called A and A ) and the knowledge of the material parameters which form the transmission line.2 In this chapter we denote the material parameters of the surrounding dielectric by a prime, and unprimed symbols refer to the properties of the actual transmission line. The transmission line is assumed to be filled by a material with the dielectric constant1 and permeability µ1; the ohmic losses, due to the conductance of the material, are described by σ1. The capacitance per unit length of a system is given by

where the integral is taken over the cross-section S of the transmission line. A is a constant which solely depends on the geometry of the particular setup; we will give some examples below. In a similar way the inductance, impedance, and conductance can be calculated:

The ohmic losses per unit length R are proportional to the surface resistance RS of the guiding material and can be calculated by integrating over the conductor

2Optical fibers commonly used in the visible and near-infrared range of frequency are not covered by the description due to different boundary conditions since there is no conducting material used, but total reflection at oblique incidence is used [Ada81, Mar91]. Of course total reflection is also covered by the impedance mismatch approach.

226 9 Propagation and scattering of waves

boundaries:

where the proportionality constant A also depends on the geometry of the transmission line. σ1 describes the conductivity of the metal which constitutes the line, f = ω/2π is the frequency of the transmitted waves, and δ0 is the skin depth. From this equation we see that the losses of a transmission line increase with frequency f . In the infrared range this becomes a problem, in particular since σ1(ω) of the metal decreases and σ1(ω) of the dielectric in general increases. This then makes the use of free space propagation or the utilization of optical fibers advantageous. The power losses per unit length are in general due to both the finite conductivity of the metallic conductors σ1 and the lossy dielectric σ1 with which the line is filled. The attenuation α along the line is defined by the power dissipation in direction z; it can easily be calculated by Eq. (9.1.7), if it is only due to the metallic surface

Integrating the Poynting vector over the cross-section yields the power propagation along the line:

For low-loss transmission lines A should be large and A should be small. From Eq. (9.1.6) we see that for a transmission line with losses Rl = Gl = 0 the phase velocity vph = (LlCl)−1/2 = c/(µ1 1)1/2 independent of the particular geometry.

The problem of wave propagation in a transmission line is now reduced to finding the geometrical factors A and A for special cases of interest; this means evaluating the field arrangement and solving the integrals in the above equations. Note that the particular field distribution does not only depend on the geometry, but also on the mode which is excited, as discussed in Section 9.1.2.

Free space and medium

Before we discuss special configurations of metallic transmission lines, let us recall the propagation of a plane wave along the z direction in an isotropic medium (with1, µ1 but no losses) or free space ( 1 = µ1 = 1), in order to demonstrate that our approach of circuit representation also covers this case. Since H = ( 1/µ1)1/2nz × E and nz is the unit vector along z, we find that only transverse electromagnetic waves propagate. The wavevector q = 2π/λ is given by

q = ωc (µ1 1)1/2 = iγˆ = −β ,

and a plane wave is not attenuated in free space. Obviously there are no losses associated with free space propagation. The impedance of free space can be evaluated to be

we have derived in Section 2.3.2. If the wave propagates in a dielectric medium, characterized by ˆ and µ1, without boundaries we obtain

for the impedance according to Eq. (2.3.30). Depending on the imaginary part of the dielectric constant ˆ of the medium, losses become important.

Two wire transmission line

Perhaps the simplest transmission line, the so-called Lecher system shown in Fig. 9.2a, consists of two parallel wires separated by an insulating material ( 1, µ1)

– which might also be air. Let us assume two perfectly conducting wires of diameter 2a spaced a distance b apart. The geometrical constant A of this system can be calculated, and we find A = (1/π )arccosh {2b/a} ≈ (1/π ) ln {b/a} where the approximation is valid for a b; typical values are b/a ≈ 10. Assuming Rl → 0 and Gl → 0 the characteristic impedance Zc of a two wire transmission line is given by:

where σ1 is the conductivity of the (usually copper) wires.

Parallel plate transmission line

Similar considerations hold for a parallel plate transmission line of spacing b and width a (Fig. 9.2b) which in general may be filled with a dielectric, to first approximation lossless material ( 1 and µ1, σ1 = 0). For wide transmission lines a b, when the effect at the edges is negligible, the geometrical factor is approximately