Finite size of entrance aperture

A perfectly collimated beam was assumed for the discussion of the ideal spectrometer in Sect. 3.2.1. However, only a point source can in theory be perfectly collimated, while an extended source will always slowly diverge. For the FTS, the size of the light source is set by the circular entrance aperture. It has a finite opening which introduces two effects: the wavelength scale is shifted and the resolution is degraded. Both effects are discussed in this section.

The light rays in a not perfectly collimated beam travel different paths through the interferometer and interfere after different optical path differences. If α is the divergence half angle of the beam andl is the distance that the moving mirror travels, the generalized optical path difference is given by the following relation (Brault, 1985, Sect. 3.2):

δ = 2lcos(α) . (3.7)

This results inδ= 2l for the central ray traveling on the optical axis (α= 0) and a smaller OPD for the outer rays (α > 0). The diverging beam creates a light cone with a solid angle Ω that is related to the divergence half angle through the relation Ω =πα² =πr²/f². Here,r is the radius of the cone at the (focal) distancef, and α is assumed to be small enough to apply the small angle approximation. The solid angle extends to Ω_max for the outer rays.

For simplicity, a monochromatic light source is assumed for the following calculation. We

substitute Eq. 3.7 into Eq. 3.1 and integrate over the solid angle:

See Brault (1985, Sect. 3.2 and Appendix I) for the evaluation of the integral. In case of a polychromatic light source, one would still need to integrate over all wavenumbers.

The modified interferogram I_Ω contains additional terms compared to the interferogram given in Eq. 3.1. These terms represent the two already mentioned effects: the modified argument of the cosine function is related to the wavelength shift and the additional factor before the cosine function changes the resolution. I discuss both effect now in more detail.

For a diverging beam, the measured wavenumbers ˜νmeasured are shifted compared to the wavenumbers ˜ν_centerthat would be recorded from the central ray traveling at the optical axis of the instrument. The result is a redshift of the spectrum, i.e., the wavenumbers appear to be smaller than the true values:

ν˜_measured= ˜ν_center

The internal HeNe laser is another component that adds a wavenumber shift. It does not take the exact same path through the interferometer as the science light beam, and also has a different beam divergence (Griffiths & de Haseth, 2007, Sect. 2.6). The total wavenumber shift is linear and can be corrected with a single correction factor κ:

ν˜corrected= ˜νmeasured(1 +κ) . (3.9)

Salit et al. (2004) showed that a single multiplicative correction factor κ is accurate to 6·10⁻⁹ (corresponds to about 2 m/s) for their FTS at optical wavelengths. For the Göttingen FTS, wavelength stability measurements with the discharge spectra (compare Sect. 5.3) have also not revealed a noticeable wavelength dependency.

The expression for the interferogram I_Ω also includes a new factor Ω_maxsinc(˜νlΩ_max/2) as compared to Eq. 3.1. This results in a circular fringe pattern at the detector plane, even for a monochromatic light source as it can be seen in Fig. 2.13 in Griffiths & de Haseth (2007). With other words, the divergence angle of the beam is mapped into radii of the fringe pattern (see also Fig. 5.2 in Davis et al., 2001). In the spectral domain, the Fourier transform of the function sinc(˜ν0lΩmax/2) becomes the boxcar function Π(2πν/Ω˜ maxν˜0) with a width of w = Ω_maxν˜₀/2π (compare Eqs. 3.4 and 3.5). Thus, the ILS is further broadened through a convolution with a boxcar function. To minimize this degradation in resolution, only light from the central fringe must be recorded at the detector. It can be shown (Davis et al., 2001, Sect. 5.2) that this can be achieved by limiting the diameter of the entrance aperture: there is a maximum aperture diameterd_maxfor a chosen

3.2 Principles of Fourier transform spectroscopy

resolving powerR= ˜ν/∆˜ν, or a maximum wavenumber ˜ν_maxthat can be recorded without a significant loss of resolution for a given diameter d:

d_max=f r8

R ⇔ ν˜_max= ∆˜ν·8f²

d² . (3.10)

I made the experience, that this criterion is used by the OPUS software to signal the user the maximum recommended aperture size for a given wavelength range.

We can estimate the fraction of the additional broadening of the ILS function relative to its width caused by the limited scan length. The maximum effect occurs for the largest aperture size at the largest wavenumber. We calculate the width of the boxcar function for the maximum aperture dmax:

w= ˜ν0 Thus, in the case of maximum aperture size, the maximum width of the boxcar function (at ˜ν₀ = ˜ν_max) is exactly one resolution element. In the case of no further apodization,

this equals the width of the sinc function w_max= ∆˜ν = 0.6/∆. I show in the left panel of Fig. 3.3 the additional broadening of the ILS function caused by the convolution with the boxcar function of width w_max. The solid blue curve shows a sinc profile which is the ILS function for the boxcar apodization and the dashed blue curve is the result after convolution. The width of the line profile increases by 10.9%. For comparison, if the boxcar has only half the width of the sinc function, the profile increases only by about 2% after convolution. The effect is therefore usually neglectable if the aperture size is limited as described above (Davis et al., 2001, Sect. 5.2).

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0

4000 6000 8000 10000 12000 14000 16000 18000 20000 wavenumber [cm⁻¹]

Figure 3.3: Left panel: Self-apodization due to finite entrance aperture. The solid blue curve represents a sinc profile. The dashed curve represents the profile after convolution with the boxcar function. Right panel: Maximum resolving power R of the FTS as a function of wavenumber considering limited scan length and finite aperture.

For preparation of measurements, one often needs to know the maximum resolving powerR that can be achieved for a certain maximum OPD or a maximum aperture size. The

right panel of Fig. 3.3 shows the achievable resolving power R of the FTS as a func-tion of wavenumber ˜ν. The solid blue and green lines mark the instrumentation limits that are caused by the finite scan lengths for two different modes of operation: single-sided and double-single-sided interferograms. The curves follow the relationR = ˜ν/∆˜ν, with

∆˜ν = 0.0044 cm⁻¹ for single-sided measurements and ∆˜ν = 0.018 cm⁻¹ for double-sided measurements. The dashed red and cyan lines mark the limits due to a finite aperture for two different diameters of d= 0.8 mm andd= 1.3 mm, respectively (see Eq. 3.10). In practice, a resolving power above one million is hardly reached for the IFS 125 HR (Bruker Optics, personal communication and instrument brochure).