Dressel.Gruner.Electrodynamics of Solids.2003

2.4.3 Reflectivity and transmissivity for oblique incidence

We now want to consider the general case of light hitting the interface at arbitrary angles. As derived in Fresnel’s equations (2.4.7a)–(2.4.7d) the result depends upon whether the electric fields are oriented parallel or perpendicular to the plane of incidence. In general the applicability of Fresnel’s equations is not restricted to certain angles of incidence ψi or limited in frequency ω. However, if the frequency ω is comparable to the plasma frequency ωp, (longitudinal) plasma waves (socalled plasmons) may occur for ψi = 0 and parallel polarization [Bec64]. This is the case in thin metal films [McA63], anisotropic conducting materials [Bru75], but also in bulk metals [Mel70]. More details on oblique incidence can be found in several textbooks on optics [Bor99, Hec98, Kle86].

In Figs 2.5–2.7 we show the angular dependence of the reflection and transmission coefficients together with the change in phase for both polarizations parallel and perpendicular to the plane of incidence. Fig. 2.5a was calculated with n = 1.5 and k = 0 by using Eqs (2.4.7a)–(2.4.7d). The reflection coefficients are real and |r | ≤ |r | in the entire range; rˆ shows a zero-crossing at the Brewster angle

rˆ = −rˆ ; as mentioned before the change of sign is a question of definition. Although for ψB only one direction of polarization is reflected, the transmitted light is just slightly polarized. The features displayed in Fig. 2.5 are smeared out if the dielectric becomes lossy, i.e. if k > 0. The coefficients rˆ and rˆ become complex quantities, indicating the attenuation of the wave. As an example, in Fig. 2.6 we plot the same parameters for n = 1.5 and k = 1.5. The absolute value |rˆ | still shows a minimum, but there is no well defined Brewster angle. The properties of oblique incident radiation are utilized in ellipsometry to determine the complex reflection coefficient (Section 11.1.4). Grazing incidence is also used to enhance the sensitivity of reflection measurements off metals because the absorptivity A = 1 − |rˆ |2 increases approximately as 1/ cos ψi. Using oblique incidence with parallel polarization, the optical properties perpendicular to the surface can be probed.

By traveling from an optically denser medium to a medium with smaller n, as shown in Fig. 2.7, the wave is totally reflected if the angle of incidence exceeds ψT > ψB, the angle of total reflection (tan ψB = sin ψr = n). The example calculated corresponds to the case where the wave moves from a medium with n = 1.5 to free space n = 1 (corresponding to n = 1 and n = 1/1.5). Again |r | ≥ |r | in the entire range. Interestingly the reflectivity for ψi = 0 is the same

for n = 1.5 and n = 1/1.5 due to the reciprocity of the optical properties if k = 0. For ψi > ψT, surface waves develop which lead to a spatial offset of the reflected wave [Goo47]. The range of total reflection is characterized by a phase differenceof the polarizations E and E in time. If we write E = E n + E n with E = E cos φ exp{i } and E = E sin φ we find

Ex (t) = |E | cos{ωt + } and Ey (t) = |E | sin{ωt} ,

respectively, which characterize an elliptically polarized light. Although the light is totally reflected, the fields extend beyond the interface by a distance of the order of the wavelength and their amplitude decays exponentially (evanescent waves). This fact is widely used by a technique called attenuated total reflection, for instance to study lattice and molecular vibrations at surfaces [Spr91]. The fact that the transmission coefficient tˆ is larger than 1 does not violate the energy conservation because the transmitted energy is given by T = 1µ1/µ1 1 1/2 |tˆ|2(cos ψt/ cos ψi) as a generalization of Eq. (2.4.21) for oblique incidence; this parameter is always smaller than unity.

In general, the angle of incidence is not very critical for most experimental purposes; if it is chosen sufficiently near to zero, the value of the reflectance coefficients for the two states of polarization rˆ and rˆ will differ from rˆ by an amount less than the precision of the measurement; e.g. less than 0.001 for angles of incidence as large as 10◦.

2.4.4 Surface impedance

As already mentioned above, the impedance of the wave is called surface impedance in the case of conducting matter because originally Zˆ S was defined as the ratio of the electric field E normal to the surface of a metal to the total current density J induced in the material:

where Jˆ(z) is the spatially dependent current per unit area which decays exponentially with increasing distance from the surface z. The electric field Eˆ z=+0 refers to the transmitted wave at the interface upon entering the material. Here we see that, for the definition of Zˆ S, it is not relevant that the wave approaches the material coming from the vacuum because the material prior to z = 0 does not enter the constituent equation (2.4.23). The surface impedance only depends on the optical properties of the material beyond z; hence Zˆ S is defined at any point in a material, as long as it extends to infinity. In Appendix B we explain how even this drawback of the definition can be overcome.

The real part RS, the surface resistance, determines the power absorption in the metal; the surface reactance XS accounts for the phase difference between E and J. Since the ratio of electric field to current density corresponds to a resistivity, the surface impedance can be interpreted as a resistance. Although mainly used in the characterization of the high frequency properties of metals, the concept of surface impedance is more general and is particularly useful if the thickness of the medium d is much larger than the skin depth δ0.

We can regain the above definition of the impedance (2.3.27) by using Maxwell’s third equation (2.2.7c) which relates the magnetic field H and the current density J. For ordinary metals up to room temperature the displacement current ω 1 E/(4π i)

where Eˆ and Hˆ refer to the electric field and the magnetic field as complex quantities defined in Eqs (2.2.14) with a phase shift φ between Eˆ and Hˆ . The surface impedance Zˆ S does not depend on the interface of two materials but rather is an optical parameter like the refractive index. Another approach used to calculate the surface impedance from Eq. (2.4.23) utilizes Eq. (2.3.24) to obtain the current density

leading to the total current in the surface layer

This brings us back to Eq. (2.3.28) which connects the impedance Zˆ S to the complex refractive index and the wavevector.

Since the surface impedance relates the electric field to the induced current in a surface layer, RS and XS must also be connected to the skin depth δ0. Combining Eq. (2.3.32b) with Eq. (2.3.15b), we find

for σ1 |σ2|, which is the low frequency limit of good conductors. The last

approximation can be easily understood by replacing the integration in Eq. (2.4.23) by the characteristic length scale of the penetration. Also we can write

if RS = −XS upon penetrating the material. Since 2π c/ω = λ0 is the wavelength of the radiation in a vacuum,

i.e. in the limit σ1 |σ2| the surface impedance is determined by the ratio of the skin depth to the wavelength. We again want to point out that, in spite of the common names, both the surface impedance and the skin depth have been defined without referring to the surface of the material. We just consider the decay of the E and H fields and the change in phase between them.

2.4.5 Relationship between the surface impedance and the reflectivity

Finally we want to present the relationships between the surface impedance and the optical properties such as the reflectivity or transmissivity. The specular reflection for normal incidence (ψi = ψt = 0) at the boundary between two media has been derived in Eq. (2.4.13) assuming a wave traveling from one medium (Nˆ , µ1) to another (Nˆ , µ1). We can combine this with the relation for the surface impedance (2.3.28):

The reflectivity (i.e. reflected power) is then given by the very general expression

As we discuss in more detail in Chapter 9, a reflection of waves also occurs if the impedance of a guiding structure (waveguide, cable) changes; hence the previous equation holds for any impedance mismatch.

The reflectivity at the interface does not depend on whether the wave travels from medium 1 to medium 2 or from medium 2 to medium 1; the reflectivity of an interface is reciprocal. The phase φr, however, changes by π . In the case that the first medium is a vacuum (Zˆ S = Z0 = 377 ) the reflectivity can be written as

The transmission coefficient as a function of the surface impedances is

In the special case of transmission from a vacuum into the medium with finite absorption:

In the case of an infinite thickness, which can be realized by making the material thicker than the skin depth δ0, all the power transmitted through the interface is absorbed, and the absorptivity A = T . From Eqs (2.4.31) and (2.4.33),

in this limit.

References

[Bec64] R. Becker, Electromagnetic Fields and Interaction (Bluisdell Publisher, New York, 1964)

[Ber78] D. Bergman, Phys. Rep. 43, 377 (1978)

[Blo65] N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965)

[Bor99] M. Born and E. Wolf, Principles of Optics, 6th edition (Cambridge University Press, Cambridge, 1999)

[Bru75] P. Bruesch,¨ Optical Properties of the One-Dimensional Pt-Complex Compounds, in: One-Dimensional Conductors, edited by H.G. Schuster, Lecture Notes in Physics 34 (Springer-Verlag, Berlin, 1975), p. 194

[But91] P.N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge, 1991)

[Gay67] P. Gay, An Introduction to Crystal Optics (Longman, London, 1967)

[Goo47] F. Goos and H. Hanchen,¨ Ann. Phys. Lpz. 1, 333 (1947); F. Goos and H. Lindberg-Hanchen,¨ Ann. Phys. Lpz. 5, 251 (1949)

[Hec98] E. Hecht, Optics, 3rd edition (Addison-Wesley, Reading, MA, 1998)

[Hol91] R.T. Holm, Convention Confusions, in: Handbook of Optical Constants of Solids, Vol. II, edited by E.D. Palik (Academic Press, Boston, MA, 1991), p. 21

[Jac75] J. D. Jackson, Classical Electrodynamics, 2nd edition (John Wiley & Sons, New York, 1975); 3rd edition (John Wiley & Sons, New York, 1998)

[Kle86] M.V. Klein, Optics, 2nd edition (John Wiley & Sons, New York, 1986)

[Lan84] L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition (Butterworth-Heinemann, Oxford, 1984)

[Lan78] R. Landauer, AIP Conference Proceedings 40, 2 (1978) [McA63] A.J. McAlister and E.A. Stern, Phys. Rev. 132, 1599 (1963) [Mel70] A.R. Melnyk and M.J. Harrison, Phys. Rev. B 2, 835 (1970) [Mil91] P.L. Mills, Nonlinear Optics (Springer-Verlag, Berlin, 1991)

[Mul69] R.H. Muller,¨ Surf. Sci. 16, 14 (1969)

[Nye57] J.F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1957)

[Por92] A.M. Portis, Electrodynamics of High-Temperature Superconductors (World Scientific, Singapore, 1992)

[Rik96] G.L.J.A. Rikken and B.A. van Tiggelen, Nature 381, 54 (1996); Phys. Rev. Lett. 78, 847 (1997)

[She84] Y.R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984)

[Sok67] A.V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967)

[Spr91] G.J. Sprokel and J.D. Swalen, The Attenuated Total Reflection Method, in:

Handbook of Optical Constants of Solids, Vol. II, edited by E.D. Palik (Academic Press, Orlando, FL, 1991), p. 75

[Ste63] F. Stern, Elementary Theory of the Optical Properties of Solids, in: Solid State Physics 15, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1963), p. 299

[Woo72] F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972)

Further reading

[Bri60] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960)

[Cle66] P.C. Clemmov, The Plane Wave Spectrum Representation of Electrodynamic Fields (Pergamon Press, Oxford, 1966)

[Fey63] R.P. Feynman, R.B. Leighton, and M. Sand, The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, MA, 1963)

[For86] F. Forstmann and R.R. Gerhardts, Metal Optics Near the Plasma Frequency, Springer Tracts in Modern Physics, 109 (Springer-Verlag, Berlin, 1987)

[Gro79] P. Grosse, Freie Elektronen in Festkorpern¨ (Springer-Verlag, Berlin, 1979) [Lek87] J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, 1987)