Abstract
Apart from the direction of propagation, the electromagnetic radiation is characterised by two other fundamental properties, typical of waves: the spectrum and the polarisation. These properties are of crucial importance to obtain the physical characteristics of the body that is emitting the radiation that we observe, be it an atom or a star. These properties are encoded in the variation of the electric and magnetic field vectors as a function of time. This chapter deals with the mathematical concepts underlying the definition of the spectrum and the polarisation of the electromagnetic radiation. We will also discuss the physical measurements of these properties as performed with appropriate instruments, such as grating spectrometers and polarimeters, of which we will illustrate the principles of operation.
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Notes
- 1.
In principle, the Dirac delta is not a function but rather a distribution, i.e. a functional that associates to any real function f(x) a real number F[f(x)]. The main properties of the Dirac delta are collected in Sect. 16.3.
- 2.
The flux density of a given physical quantity (e.g. energy, charge, mass, etc.) is generally defined as the amount of such quantity that flows across the unit area in unit time. However, this definition is not universally accepted and sometimes flux is defined as the quantity which crosses an area (not necessarily an unit area) per unit time. In mathematical physics, the flux of a vector across a closed surface is defined in a different way (recall Gauss theorem), without any reference to the unit of time.
- 3.
In practice, spectral measurements are usually given in terms of relative units, the function \(|\hat{E}(\omega)|^{2}\) being measured apart from a constant of proportionality. More precisely, we should therefore say that the spectrum of the electromagnetic radiation is proportional to \(|\hat{E}(\omega)|^{2}\).
- 4.
Since we have assumed that the function f(t) varies over timescales much shorter than \(\mathcal{T}\), the integral in dt can be extended between −∞ and ∞ instead of extending it from \({-\mathcal{T}/2}\) to \({\mathcal{T}/2}\).
- 5.
The plane transmission grating is the most simple type of diffraction grating. In practice, many other types of gratings can be used (reflection, echelle, saw-tooth, phase-transparency, circular, concave, etc.).
- 6.
For an ideal transmission grating it can be shown that k is given by acosθ/λ, where a is the size of the transmission area of each rule and λ is the wavelength (see, e.g., Toraldo di Francia 1958). In practice, the constant k depends on how the grating is actually built.
- 7.
We refer here to the case where the radiation has a stationary behaviour, so we need to refer to the Fourier transform relative to a sampling time \(\mathcal{T}\).
- 8.
Nowadays, the detector is generally a CCD camera or a series of photo-multipliers. Previously, photographic plates were commonly used.
- 9.
Note that by substituting the angular frequency with the wavelength (ω=2πc/λ), Eq. (2.12) can be written in the form dsinθ m =mλ, which is the equation resulting from the elementary theory of the diffraction grating.
- 10.
This quantity was denoted with the symbol F in Sect. 2.1, where we neglected the polarisation properties.
References
Toraldo di Francia, G.: La Diffrazione della Luce. Edizioni Scientifiche Einaudi. Boringhieri, Torino (1958)
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Landi Degl’Innocenti, E. (2014). Spectrum and Polarisation. In: Atomic Spectroscopy and Radiative Processes. UNITEXT for Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-2808-1_2
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DOI: https://doi.org/10.1007/978-88-470-2808-1_2
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