Dressel.Gruner.Electrodynamics of Solids.2003

The power absorption, as well as being dependent on k, is also dependent on n because the electromagnetic wave travels in the medium at a reduced velocity c/n. We see that the absorption coefficient α is the power absorbed in the unit volume, which we write as σ1 E2, divided by the flux, i.e. the energy density times the energy velocity

where v = c/n is the velocity of light within the medium of the index of refraction n.

2.3.2 Impedance

The refractive index Nˆ characterizes the propagation of waves in the medium; it is related to a modified wavevector q compared to the free space value ω/c. Let us now introduce the impedance as another optical constant in order to describe the relationship between the electric and magnetic fields and how it changes upon the wave traveling into matter.

The ratio of the electric field E to the magnetic field H at position z defines the load presented to the wave by the medium beyond the point z, and is the impedance

with units of resistance. The impedance is a response function which determines the relationship between the electric and magnetic fields in a medium. Using Eqs (2.2.14) we can write the complex impedance as Zˆ S = |Zˆ S| exp{iφ}. The absolute value of |Zˆ S| indicates the ratio of the electric and magnetic field amplitudes, while the phase difference between the fields equals the phase angle φ. Since E and H are perpendicular to each other, using Eq. (2.2.17) we obtain

if a harmonic spatial and time dependence of the fields is assumed as given by Eqs (2.2.14). This leads to the following expressions for the impedance:

If σ1 = 0, the dielectric constant ˆ and thus the impedance are real; in this case the electric displacement and magnetic induction fields are equal (Dˆ = Bˆ ) and in phase (φ = 0). The above equation is then reduced to ZS = (4π/c)(µ1/ 1)1/2. In the case of free space (µ1 = 1, 1 = 1), we obtain the impedance of a vacuum:

The impedance fully characterizes the propagation of the electromagnetic wave as we will discuss in more detail in Section 9.1. The presence of non-magnetic matter (µ1 = 1) in general leads to a decrease of the electric field compared with the magnetic field, implying a reduction of |Zˆ S|; for a metal, |Zˆ S| is small because |Hˆ | |Eˆ |. Depending on the context, Zˆ S is also called the characteristic impedance or wave impedance. For reasons we discuss in Section 2.4.4, the expression surface impedance is commonly used for Zˆ S when metals are considered.

In the case of conducting matter a current is induced in the material, thus the electric and magnetic fields experience a phase shift. The impedance of the wave has a non-vanishing imaginary part and can be evaluated by substituting the full definition (2.2.18) of the complex wavevector qˆ into Eq. (2.3.28). The final result for the complex impedance is:

The real part RS is called the surface resistance and the imaginary part XS the surface reactance. If we assume that | 1| 1, which is certainly true in the case of metals at low temperatures and low frequencies but may also be fulfilled for dielectrics, we obtain from Eq. (2.3.30) the well known relation for the surface impedance

30 2 The interaction of radiation with matter

the expressions for the real and imaginary parts, RS and XS, are11

find RS = −XS; meaning that the surface resistance and (the absolute value of) the reactance are equal in the case of a metal.

In cases where both the surface resistance and surface reactance are measured,

In Eq. (2.3.28) we have shown that the impedance is just proportional to the inverse wavevector and to the inverse complex refractive index

This allows us to write the surface resistance RS and surface reactance XS as :12

where Z0 = 4π /c is the wave impedance of a vacuum as derived above. Obviously XS = 0 for materials with no losses (k = 0). The ratio of the two components of the surface impedance

gives the phase difference φ between magnetic and electric fields, already expressed by Eq. (2.3.8).

11 In the case of a vacuum, for example, σ1 = 0 and the negative root of σ22 has to be taken, since σ2 stands

for − 1ω/4π . The impedance then is purely real (XS = 0) but the real part is not zero, RS = Z0 = 4π/c. It becomes identical to the wave impedance derived in Eq. (2.3.29). In the case of metals at low frequencies,1 < 0 and σ2 is positive.

12The negative sign of XS is a result of the definition of the surface impedance as Zˆ S = RS + iXS and thus purely conventional; it is often neglected or suppressed in the literature.

2.4 Changes of electromagnetic radiation at the interface

Next we address the question of how the propagation of electromagnetic radiation changes at the boundary between free space and a medium, or in general at the interface of two materials with different optical constants Nˆ and Nˆ . Note, we assume the medium to be infinitely thick; materials of finite thickness will be discussed in detail in Appendix B. The description of the phenomena leads to the optical parameters, such as the reflectivity R, the absorptivity A, and the transmission (or transmissivity) T of the electromagnetic radiation. Optical parameters as understood here are properties of the interface and they depend upon the boundary for their definition. All these quantities are directly accessible to experiments, and it just depends on the experimental configuration used and on the spectral range of interest that one is more useful for the description than the other. In this section we define these parameters, discuss their applicability, and establish the relationship between them. In general, experiments for both the amplitude and phase of the reflected and transmitted radiation can be conducted, but most often only quantities related to the intensity of the electromagnetic radiation (e.g. I = |E|2 ) are of practical interest or only these are accessible.

2.4.1 Fresnel’s formulas for reflection and transmission

Let us consider the propagation of a plane electromagnetic wave from a vacuum ( 1 = µ1 = 1, σ1 = 0) into a material with finite 1 and σ1. The surface lies in the x y plane while the z axis is positive in the direction toward the bulk of the medium; the surface plane is described by the unit vector ns normal to the surface. We further suppose that the direction of propagation is the xz plane (plane of incidence), as displayed in Fig. 2.3. The incident waves

arrive at the surface at an angle ψi, which is the angle between wavevector qi and ns in the xz plane; nqi is the unit vector along qi. One part of the electric and magnetic fields enters the material, and we write this portion as

the other part is reflected off the surface and is written as

Fig. 2.3. (a) Reflection and refraction of an electromagnetic wave with the electric field E perpendicular to the plane of incidence; the magnetic field H lies in the plane of incidence.

(b) Reflection and refraction of the electromagnetic wave with E parallel to the plane of incidence; the magnetic field H is directed perpendicular to the plane of incidence. In the x y plane at z = 0 there is an interface between two media. The first medium (left side) is characterized by the parameters 1, σ1, and µ1, the second medium by 1, σ1, and µ1. ψi, ψr, and ψt are the angles of incidence, reflection, and transmission, respectively.

By using the subscripts i, t, and r we make explicitly clear that all parameters may have changed upon interaction with the material. At the boundary, the normal components of D and B, as well as the tangential components of E and H, have to be continuous:

At the surface (z = 0) the spatial and time variation of all fields have to obey these boundary conditions; the frequency13 and the phase factors have to be the same for the three waves:

ωi = ωt = ωr = ω

(qi · r)z=0 = (qt · r)z=0 = (qr · r)z=0 .

From the latter equation we see that both the reflected and the refracted waves lie

which states that the angle of the radiation transmitted into the medium ψt becomes smaller as the refractive index increases.14 For σ1 = 0, we see that formally sin ψt is complex because the refractive index Nˆ has an imaginary part, indicating that the wave gets attenuated. Although the angle φt is solely determined by the real part of the refractive indices, in addition the complex notation expresses the fact that the wave experiences a different attenuation upon passing through the interface. As mentioned above, we assume that n = 1 and k = 0, i.e. Nˆ = 1 (the wave travels in a vacuum before it hits the material), thus Snell’s law can be reduced to

There are two cases to be distinguished. First, the electric field is normal to the

13Non-linear processes may lead to higher harmonics; also (inelastic) Raman and Brillouin scattering cause a change in frequency. These effects are not considered in this book.

14The angle is measured with respect to normal incidence, in contrast to the Bragg equation where the angle is commonly measured with respect to the surface.

plane of incidence (Fig. 2.3a), and therefore parallel to the surface of the material; second, the electric field is in the plane of incidence15 (Fig. 2.3b). We want to point out that we use the Verdet convention which relates the coordinate system to the wavevectors.16 In the case when the electric vector is perpendicular to the plane of incidence, Eqs (2.4.4c) and (2.4.4d) give E0i+E0r−E0t = 0 and (E0i−E0r) cos ψi− ( 1/µ1)1/2 E0t cos ψt = 0. This yields Fresnel’s formulas for Ei perpendicular to the plane of incidence; the complex transmission and reflection coefficients are:

If the electric vector Ei is in the plane of incidence, the Eqs (2.4.4a), (2.4.4c), and

(2.4.4d) lead to (E0i − E0r) cos ψi − E0t cos ψt = 0 and (E0i + E0r) − ( 1/µ1)1/2 E0t = 0. We then obtain Fresnel’s formulas for Ei parallel to the plane of incidence:

These formulas are valid for Nˆ complex. To cover the general case of an interface between two media (the material parameters of the first medium are indicated by a prime: 1 = 1, µ1 = 1, σ1 = 0; and the second medium without: 1 = 1, µ1 = 1, σ1 = 0), the following replacements in Fresnel’s formulas are sufficient: Nˆ →

Nˆ /Nˆ and µ1 → µ1/µ1.

2.4.2 Reflectivity and transmissivity by normal incidence

In the special configuration of normal incidence (ψi = ψt = ψr = 0), the distinction between the two cases, parallel and perpendicular to the plane of incidence, becomes irrelevant (Fig. 2.3). The wavevector q is perpendicular to the surface while E and H point in the x and y directions, respectively. In order to satisfy

15From the German words ‘senkrecht’ and ‘parallel’, they are often called s-component and p-component.

16A different perspective (Fresnel convention) relates the coordinate system to the sample surface instead of to the field vectors [Hol91, Mul69, Sok67].

the boundary conditions, i.e. the tangential components of E and H must be continuous across the boundary, the amplitudes of the incident (E0i, H0i), transmitted (E0t, H0t), and reflected (E0r, H0r) waves are

Starting with Eqs (2.2.14) and Eq. (2.3.2), on the right hand side of the x y plane, the electric and magnetic fields in the dielectric medium (µ1 = 1) are given by

for the complex reflection coefficient rˆ, in agreement with Eqs (2.4.7b) and (2.4.7d) on setting ψ = 0. Note, we only consider µ1 = µ1 = 1. The phase shift φr is the difference between the phases of the reflected and the incident waves. More generally, at the interface between two media (Nˆ = 1 and Nˆ = 1) we obtain

Et

Ei

Er

Fig. 2.4. Incident Ei, reflected Er, and transmitted electric wave Et traveling normal to the interface between two media. The first medium has the parameters 1, σ1, and µ1, the second medium 1, σ1, and µ1; the longer wavelength λ > λ indicates that 1 < 1 assuming µ1 = µ2. The amplitudes of the reflected and transmitted waves as well as the phase difference depend on the optical properties of the media.

Here φr → π if (crudely speaking) Nˆ > Nˆ ; this means that for normal incidence the wave suffers a phase shift of 180◦ upon reflection on an optically denser medium (defined by the refractive index); φr → 0 if n 2 + k 2 > n2 + k2.

During an experiment the reflected power is usually observed and the phase information is lost by taking the absolute value of the complex quantity |Eˆ |2 = E02. The reflectivity R is defined as the ratio of the time averaged energy flux reflected

from the surface Sr = (c/4π )|E0r ×H0r| to the incident flux Si = (c/4π )|E0i ×H0i|. Substituting the electric and magnetic fields given by Eqs (2.4.1) and (2.4.3) into

the definition of the Poynting vector Eq. (2.3.22) yields the simple expression

For a dielectric material without losses (k → 0), the reflectivity is solely determined by the refractive index:

and it can approach unity if n is large; then tan φr = 0. Both parameters R and φr can also be expressed in terms of the complex conductivity σˆ = σ1 + iσ2 by using the relations listed in Table 2.1; and then

The transmission coefficient tˆ for an electromagnetic wave passing through the

is the phase shift of the transmitted to incident wave; again we have assumed the case Nˆ = 1 for the second transformation. The power transmitted into the medium is given by the ratio of the time averaged transmitted energy flux St = (c/4π )|E0t × H0t| to the incident flux Si. With Eqs (2.4.1) and (2.4.2) we obtain the so-called transmissivity (often simply called transmission)

It is important to note that |tˆ|2 = 1 − |rˆ|2 due to the modified fields and thus energy density inside the material.