Dressel.Gruner.Electrodynamics of Solids.2003

This page intentionally left blank

Table A.1. Some important functions f (t) and their Fourier transforms F(ω). F(ω) = !−∞∞ f (t) exp{−iωt} dt for the transformation and

f (t) = 21π !−∞∞ F(ω) exp{iωt} dω for the retransformation.

since the convolution theorem states that the Fourier transform of a product of two functions f and g equals the convolution product of the individual spectra, and vice versa:

Applied to the autocorrelation we arrive at Parseval’s identity, which expresses the fact that the total energy of a time dependent function, measured as the integral over | f (t)|2, is equal to the total energy of its spectrum |F(ω)|2 (the so-called Wiener– Khinchine theorem). From these formulas we can easily derive the Nyquist criterion, which states that any waveform (which can be composed by harmonic functions) can be sampled unambiguously and without any loss of information using a sampling frequency greater than or equal to twice the bandwidth of the system (Shannon’s sampling theorem). In a Fourier transform spectrometer the data points have to be taken at a distance of mirror displacement shorter than λ/2 of the maximum frequency f = c/λ which should be obtained. These considerations also limit the resolution of a Fourier transform spectrometer due to the maximum length of the path difference. According to Rayleigh’s criterion the interferogram

has to be measured up to a path length of at least δmax in order to resolve two spectral lines separated by a frequency c/δmax. The scaling of the function in t in

the form f (at) leads to the inverse scaling of the Fourier transform: |a1| F ωa . The differentiation ddt f (t) becomes a multiplication in the Fourier transform: iω F(ω).

In the following we want to give some examples of the Fourier transformations depicted in Fig. A.1. A box function, for instance,

leads to a sinc function (Fig. A.1a)

well known from the diffraction pattern of a slit. The function f (t) = t−1/2 remains

as displayed in Fig. A.1b. The width of the spectrum ω is related to the width of its Fourier transform by t = 1/ ω. A Lorentzian curve (Fig. A.1c) is obtained by transforming the exponential decay f (t) = exp{− ω|t|}:

the line shape of atomic transitions, for instance. A delta function f (t) = δ{t} leads to a flat response F(ω) = 1 as depicted in Fig. A.1d; for the converse transformation of f (ω) = δ{ω} we obtain F(t) = 1/2π since δ{ω/2π } = 2π δ{ω}. Fig. A.1e shows that from a harmonic wave f (t) = exp{iωt} we get a δ-function at the frequency ω0 as the Fourier transform:

Table A.1 summarizes the most important examples of the Fourier transformation. The Fourier transformation is a powerful tool which, besides fast signal analysis, can lead to deep insight into the properties of time or space dependent phenomena. Although we have only discussed the one-dimensional case, the Fourier transformation can be extended to two and three dimensions, which can be useful for the

description of problems on surfaces or in crystals, for example.

A.2 Laplace transformation

The attempt to apply the Fourier transformation in a straightforward manner in order to obtain the Fourier transform of a step function

fails because the integral in Eq. (A.1a) does not converge. The problem can be avoided by multiplying f (t) by a convergency factor exp{−ηt}, which then allows the integral to be solved by finally taking the limη→0. Since the function is odd, we can express it in terms of the sinc function

This can be expressed more elegantly by the Laplace transformation. The definition of the Laplace transform and its retransform is

where ω is complex (Fig. A.2). The Fourier transform is the degenerate form of the Laplace transform if the latter has purely imaginary arguments (c → 0).

The transformation is linear and the convolution becomes a multiplication

References

[Cha73] D.C. Champeney, Fourier Transforms and Their Physical Applications

(Academic Press, London, 1973)

[Doe74] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation (Springer-Verlag, Berlin, 1974)

[Duf83] P.M. Duffieux, The Fourier Transform and Its Applications to Optics (John Wiley & Sons, New York, 1983)

[Fra49] P. Franklin, An Introduction to Fourier Methods and the Laplace Transformation

(Dover, New York, 1949)

[Mer65] L. Mertz, Transformations in Optics (John Wiley & Sons, New York, 1965) [Ste83] E.G. Steward, Fourier Optics: An Introduction (Halsted Press, New York, 1983)

Ei

Fig. B.1. Reflection off and transmission through a dielectric slab with thickness d and optical parameters 1, σ1, and µ1. The multireflections cause interference. Ei, Et, and Er indicate the incident, transmitted, and reflected electric fields, respectively. The optical properties of vacuum are given by 1 = µ1 = 1 and σ1 = 0.

films (d δ0) the film impedance is given by

if we assume that the current density Jˆ is uniform throughout the film. Hˆ1 and Hˆ2 are the magnetic fields at the two sides of the film; Eˆ denotes the electric field. For intermediate thickness d ≈ δ0, we have to integrate over the actual field distribution in the film, leading to [Sch94]:

√

where the wavevector is given by qˆ = ωc ˆ. In the general case of a thin film of thickness d, width b, and length l, the film impedance is given by [Sch75]

These finite thickness corrections are especially important in the radiofrequency range since for these frequencies the skin depth δ0 is of the order of the typical film thickness d ≈ 1 µm.