# What measurement errors occur in an inteferogram?

This question is bordering on physics, but it is the signal side that I am interested in so I hope this is the correct forum.

I am very familiar with instrumental noise sources in CCD based spectrographs and what their signal properties are. I have come to realise how important understanding noise sources are in order to effectively mitigate for them.

Now I am taking an interest in FTIR which uses an interferometer to do the measurement and would like to understand how measurement errors manifest in the interferogram and how they manifest after the FT.

My suspicion is that the variations due to the technical act of measurement will share some aspects with CCD instruments but also present some sources that are very different. For example, spectrograph-CCD systems are affected by e.g. pixel-to-pixel shot noise, fixed pattern response, read out noise, spectra of optics (blank signal), dark signal, vibrations, alignment etc.

How do the following present in both the interferogram and the FT?

• random noise (i'm guessing its very different to shot noise in CCDs, not only pixel-to-pixel)
• vibrations (physical or thermal)
• mechanical whiplash

As I am not familiar with the technological errors associated with an interferometer I'm sure I've not anticipated some sources, so please educate me on any other sources of variation due to the technology (not the sample or sample-technology interface)

I've tried some searches but not found any useful resources so any good resources would be appreciated

• Assuming a conventional moving mirror FTIR, some error sourses are scan rate jitter, scan reproducibility (because averaging is SOP) and source noise gets modulated by the interferometer and then subsequently demodulated by the FFT. So the multiplex disadvantage can arise, i.e., a noisy strong peak can spread noise, thereby impacting resolution of small peaks. This can be problematic in regard to spectrally resolving small peaks that are near large peaks, if the noise is at low frequencies.
– Ed V
Aug 9 '19 at 14:55

You did an excellent job in presenting the background and context of your question and your listing of error sources is fine. But FTIR also presents some other issues, particularly given the relatively poor SNR performance of IR detectors compared to UV-Vis regime detectors. So two things. First, I recommend this book: P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectrometry, 2nd Ed., Wiley-Interscience, John Wiley & Sons, Inc., NY, ©2007.

Second, when I simulate FTIR using my optical calculus software, I generally include light source temperature fluctuations, jitter on the moving mirror and, of course, detector and preamplifier noises. The next few figures show the simulation model (with the 3 noise souces I stated), which looks somewhat like a block diagram, but is actually the simulation program, and some results for ethyl acrylate in the 400 to 4000 wavenumber range. Figure 1 follows:

Above is Fig. 1, the FTIR simulation program with ethyl acrylate as the test substance. The simulation conditions are shown in Fig. 1.

Figure 2 is the last of the ten 16k interferograms generated. It takes about 2.5 minutes per interferogram on my 2006 iMac 24. (It would be much faster on a modern Windows-based PC, but speed isn't everything and I am currently holding off buying a new PC.) Figure 2 follows:

The next figure (Fig. 3) is an expanded view of the center of the interferogram:

Now Fig. 4 shows the pseudo-transmittance of the ethyl acrylate test substance:

Then Fig. 5 shows the pseudo-transmittance of the background:

Finally, dividing the two pseudo-transmittances and taking the negative log yields the absorbance spectrum of ethyl acrylate, as shown in Fig. 6:

My software runs on PCs (and my real old Macs) and is free, with full commented source code. However, it is an add-on to a commercial simulation program called ExtendSim® (with which I am not financially associated: I am just a very early user).

If I can find any relevant and detailed papers discussing FTIR considerations in particular, I will edit this answer to include them. And it would benefit me as well, even though the spectroscopy I used to do was not in the IR.

• Looks like very useful simulations, but the licence fee would need more than justification to explore that further hand on. So what I'm gathering is that measurement noise includes bandwidth modulation, not just independent modulation of pixel response as shot noise does in CCDs. In figure 5 is that the instrument background arising from the technological factors you list? I'm guessing from the reference beam of a split beam setup? Aug 10 '19 at 13:09
• Lots that can be said, but to start, the background in Fig. 5 is due to the graybody light source: 3400K color temperature, with 0.1K standard deviation, and 0.4 emissivity. In Fig. 1, I state that I used a white responsivity for the InSb detector. That was because I did not have the responsivities vs. either wavenumber or wavelength. So there is simply not much light at low wavenumbers. This is apparent in Fig. 6: below 800 wavenumbers, there simply is not much light. Of course, I can increase the source intensity in the sim, but the idea is to match the real system as close as possible.
– Ed V
Aug 10 '19 at 13:29
• Shot noise is present, of course, but the detector and preamplifier noises are crucial: the throughput and multiplex advantages are integral to what makes FTIR possible. Everything must be done to get as much light to the detector as possible, since IR detectors are poor (in several ways) compared to CCDs, etc. As for modulation, each color of light, with its wavenumber value, is modulated at a frequency equal to the product of twice the moving mirror's speed and the wavenumber value. So the FFT does the decoding/demodulation. There is also the multiplex disadvantage and lots more.
– Ed V
Aug 10 '19 at 13:36
• Sorry to keep pestering you, I'm trying to understand the implications of this to check I've grasped it. So if there was random jitter in the interferometer that will modulate the spatial frequency of the features in the interferogram (fig 2-3)? And when FT is applied will this translate into peak width changes centred around a frequency defined by the exact nature of the jitter? Or would it impact across the full spectrum by affecting the convolution of the different frequencies and therefore a bit more complicated? Aug 12 '19 at 11:02
• No problem! The answer to your first question is yes: the interferogram is adversely affected by the jitter and this translates, after the FFT, to peak broadening in the IR spectrum. To mitigate this, a He-Ne laser is used for reference registration: it has a wavelength of 632.82 nm (= 15802.3 $cm^{-1}$), so its true frequency component in the interferogram is at 2V x 15802.3 $cm^{-1}$, where V is the scan speed. For a few more details, see my answer here: chemistry.stackexchange.com/a/116326/79678 . This addresses the source noise issue and provides a literature reference.
– Ed V
Aug 12 '19 at 13:34