Limited optical path difference and apodization

S(˜ν) cos(2πνδ)d˜˜ ν . (3.2)

This is one part of a Fourier transform pair. The spectrum S(˜ν) is calculated from the interferogram through the inverse transform:

S(˜ν) = Z ∞

−∞I(δ) cos(2πνδ)dδ˜ . (3.3)

In theory, a spectrum can be calculated with infinitely high resolution and accuracy using Eq. 3.3. In reality, this is not possible because of practical limitations, including the following three: (i) the optical path difference is limited by the finite scan length of the traveling mirror; (ii) the beam through the interferometer is not perfectly collimated because of an extended source; and (iii) the exact position of the zero path difference is often not known introducing a phase shift. In the following sections, I discuss some aspects of these three limitations with applications to the Göttingen FTS . A much more detailed and in-depth discussion can again be found in the already mentioned textbooks on Fourier transform spectroscopy (see Sect. 3.1).

3.2.2 Limited optical path difference and apodization

A real-world interferometer has a finite size and therefore a finite scan length limiting the achievable resolution. As a result, the theoretical spectrum of a monochromatic light source is no longer a sharp δ-peak, but the spectral line has a non-zero line width. In addition, the truncation of the interferogram leads to artifacts in the spectrum in form of sidelobes near the spectral lines. A way to reduce these sidelobes is the application of an apodization function which will be discussed later in this section.

Mathematically, the finite maximum optical path difference ∆ modifies the integration limits in Eq. 3.3:

S_∆(˜ν) = Z _∆

−∆

I(δ)·cos(2πνδ)dδ˜ .

Instead of modifying the integration limits, we can also multiply the interferogramI(δ) with a boxcar functionD(δ)≡Π(δ/2∆) having the following properties:

D(δ) =

(1, if|δ| ≤∆

0, if|δ|>∆ . (3.4)

The convolution theorem states that the Fourier transform of two multiplied functions is equal to the convolution of the two individually Fourier transformed functions (e.g., Woan,

3.2 Principles of Fourier transform spectroscopy

2000). The Fourier transform of the interferogramI(δ) is of course the spectrumS(˜ν), and the Fourier transform of the boxcar function is designated with F(˜ν):

S_∆(˜ν) =

The sinc functionF(˜ν) is the natural instrumental line shape (ILS) function of the FTS, i.e., the broadened line profile of a monochromatic light source due to the limited optical path difference. This is inevitably connected to a fundamental question in spectroscopy, which is the definition of a resolution criterion: how far do two spectral features need to be separated in wavelength to be individually measurable? Different resolution criteria, such as Rayleigh or full width at half maximum (FWHM), have been defined (Griffiths

& de Haseth, 2007, Sect. 2.3). The width of the ILS becomes a crucial parameter in this context. To calculate the FWHM of the sinc function, we find the wavenumber ˜ν¹

2

where the profile has half the flux compared to its center at wavenumber ˜ν0:

F(˜ν¹ We solve numerically for ˜ν¹

2

and obtain ˜ν¹

2

=±0.60335/(2∆). Consequently, the FWHM of the sinc profile is given by:

FWHM_sinc = 2· |˜ν1

2|= 0.60335

∆ .

The left panel of Fig. 3.2 shows one sinc function centered around wavenumber ˜ν0 (blue curve) and a second sinc function shifted by FWHM_sinc = 0.6/∆. The dashed red curve represents the sum of the two sinc profiles. One can see that the two peaks are not resolved if they are separated by their FWHM, as the red dashed curve does only show one peak at its center. In addition, one can infer from the left panel of Fig. 3.2 that the convolution with a sinc function does not only broaden a spectral line but does also introduce strong sidelobes. These are artefacts from the abrupt truncation of the interferogram at its end and do not have a physical meaning. The sidelobes can be reduced by multiplying the interferogram with an apodization function. The apodization modifies the interferogram to ensure a smoother transition toward zero at its ends. The case where no apodization is applied is also called boxcar apodization because of the shape of function D(δ).

The triangular function is a special apodization profile when working with the Bruker IFS 125 HR FTS. The Fourier transform of the triangular function is the function sinc²(πν∆)˜ and its profile is shown in the right panel of Fig. 3.2 (e.g., Woan, 2000). The sidelobes are much smaller for this type of apodization and the two spectral lines can be resolved if separated byFWHM_sinc2 = 0.9/∆. For the Bruker IFS 125 HR FTS, the resolution and the maximum OPD are related through

∆ = 0.9

(∆˜ν)_OPUS , (3.6)

−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0 (˜ν−ν˜0) / OPD

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

normalizedflux

−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0 (˜ν−˜ν0) / OPD

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

normalizedflux

Figure 3.2: Left panel (boxcar apodization): The functionsinc(2π˜ν∆)centered at wavenumberν˜0

(blue curve) and the same function shifted by one FWHM (green curve). The red dashed curve is the sum of the two sinc profiles. Right panel (triangular apodization): Same as left panel, but with function sinc²(π˜ν∆).

where (∆˜ν)_OPUSis the resolution set in the OPUS software which controls the FTS (Bruker Optics, private communication and IFS 120 HR user’s manual). The resolution (∆˜ν)_OPUS corresponds to the FWHM of the ILS using a triangular apodization. The reference to the triangular apodization has an additional motivation: most conventional spectrographs use an entrance slit that determines the amount of light entering the instrument, and the diffraction pattern behind a slit follows the profile of a sinc² function (see, e.g., Demtröder, 2006). Beside the triangular apodization, other apodization function have been developed and Table 3.1 lists the methods that are available in the OPUS software. The usage of apodization comes at the cost of a wider ILS function and a reduced spectral resolution.

The stronger the apodization, i.e., the more the sidelobes are suppressed, the higher the degradation of the resolution. The selection of a particular type of apodization is therefore always a compromise between these two aspects. In practice, the observer specifies a desired resolution in the OPUS software. This resolution assumes the usage of the triangular apodization function. If a different apodization is used, the corresponding observed resolution needs to be calculated by comparing the change in the width of the ILS function. The last column in Table 3.1 lists the conversion factors between (∆˜ν)_OPUS and the observed resolution ∆˜ν for each apodization function.

The nominal resolution ∆˜ν = 1/∆ is generally used if the exact value is not required (Griffiths & de Haseth, 2007). It is a good approximation and independent of the applied

apodization and resolution criterion.

3.2 Principles of Fourier transform spectroscopy

apodization FWHM ∆˜ν/(∆˜ν)_OPUS

Boxcar / none 0.61/∆ 0.68

Triangular 0.9/∆ 1

Four points / Trapezoidal between boxcar and triangular

Happ-Genzel 1.1/∆ 1.23

Blackman-Harris, 3-term 0.92/∆ 1.02 Blackman-Harris, 4-term 1.4/∆ 1.52

Norton-Beer, weak 0.73/∆ 0.81

Norton-Beer, medium 0.85/∆ 0.95

Norton-Beer, strong 0.98/∆ 1.08

Table 3.1: Apodization functions provided by the OPUS software with the FWHM of the respective Fourier transforms (ILS functions). The last column gives the conversion factor between the resolution value in the OPUS software and the actual FWHM of the ILS. Reference: Bruker Optics, IFS 120 HR user’s manual; see also Wartewig (2003).