Selected Topics in Advanced Experimental Physics
Subject: Physics
Author or source: Markus Lippitz
Updated:
Complete 101-page English university lecture notes by Markus Lippitz. Covers Fourier transformations, X-ray scattering, Fourier optics, hybridisation, quantum optics and plasmonic particle lattices. Original author PDF dated 5 November 2026.
Contents from the author
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Contents I Concept: Fourier Transformation 5 1 Fourier transformation 7 2 X-Ray Scattering 17 3 Fourier Optics 25 4 Spatial Light Modulator 33 5 Mössbauer Spectroscopy 41 II Concept: Hybridization 51 6 Hybridization in classical systems 53 7 Hybridization of quantum mechanical systems 63 8 Hybridization of quantum optical systems 73 9 Digression: Bloch formalism of spin-1/2 systems 79 III Example: All Together 87 10 Lattice of plasmonic particles 89 This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license.
Original lecture notes: Fourier transformation
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Chapter 1 Fourier transformation Markus Lippitz November 5, 2026 Overview It is useful and helpful to have an intuitive approach to the Fourier transform. The bottom line is that in experimental physics one rarely needs to actually calculate a Fourier transform. Very often it is sufficient to know a few fre- quently occurring Fourier pairs and to combine them with simple rules. This is what I want to present here. A very nice and much more detailed presenta- tion can be found in Butz, 2015. I will follow his notation here. Before we get to Fourier pairs, however, we need to lay down some foun- dations. Fourier series: a periodic function and its Fourier coefficients We first consider everything here in one dimension in time or frequency space with the variables t and ω = 2πν. Let the function f(t) be periodic in time with period T, i.e. f(t) = f(t + T) . (1.1) Then this can be written as a Fourier series f(t) = ∞ ∑︂ k=−∞ Ck ei ωk t with ωk = 2π k T (1.2) and the Fourier coefficients Ck = 1 T ∫︂T/2 −T/2 f(t) e−i ωk t dt . (1.3) Note the negative sign in the exponential function in contrast to the equation before. For real-valued functions f(t), ’opposite’ Ck are conjugate-complex, so Ck = C⋆ −k. For k < 0 the frequencies ωk are negative, but this is not a problem.1 Thus, the zeroth coefficient C0 is just the time average of the 1 One could alternatively require k ≥0 and apply a sin and cos series. function f(t). This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license.
Original lecture notes: Fourier transformation
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8 Advanced Experimental Physics An arbitrary function and its Fourier transform Now we remove the restriction to periodic functions f(t) by letting the pe- riod T go to infinity. This turns the sum into an integral and the discrete ωk become continuous. Thus F(ω) = ∫︂+∞ −∞ f(t) e−iω t dt (1.4) f(t) = 1 2π ∫︂+∞ −∞ F(ω) e+iω t dω . (1.5) Here, the first equation is the forward transformation (minus sign in the exponent), and the second is the reverse transformation (plus sign in the exponent). The symmetry is broken by the 2π. But this is necessary if one wants to keep F(ω = 0) as mean2. Alternatively, we could formulate all 2 F(0) = ∫︁ f(t) dt without 1/T in front of it is meant here by Butz as mean! this with ν instead of ω, but then we would have a 2π in many more places, though not before the integral. Sidenote: Delta Function The delta function can be written as δ(x) = lim a→0 fa(x) with fa(x) = {︄ a if |x| < 1 2a 0 other (1.6) or as δ(x) = 1 2π ∫︂+∞ −∞ e+i x y dy . (1.7) An important property is that the delta function selects a value, i.e. ∫︂+∞ −∞ δ(x) f(x) dx = f(0) . (1.8) Important Fourier pairs It is very often sufficient to know the following pairs of functions and their Fourier transforms. I write them here, following Butz, as pairs in t and ω (not ν = ω/(2π)). In the same way, one could have written pairs in x and k. The important question is whether a 2π appears in the exponential function of the plane wave or not. So eiωt and eikx , but ei2πνt . (1.9) Further, I follow here the convention made above about the asymmetric distribution of the 2π between forward and reverse transformations. If you distribute them differently, then of course the prefactors change. A good overview of many more Fourier pairs in various ’2π’ conventions can be found in the English Wikipedia under ’Fourier transform’. In their nomenclature, the Butz convention used here is ’non-unitary, angular frequency’. constant and delta function f(t) = a becomes F(ω) = a 2π δ(ω) and f(t) = a δ(t) becomes F(ω) = a. This is again the asymmetric 2π.