Published: 2018-12-09 | Categories: [»] Engineeringand[»] Optics.

Resolution is an important property that describes a spectrometer performances. I have found that, however, resolution is often a misunderstood factor. I will therefore dedicate this post to detail a little bit what resolution is and what it is not.

Before we move on to more technical details, I would like to define some terminology that is sometimes mixed up:

- Dispersion or dispersive power, usually expressed in nm/px, nm/mm or eventually nm/deg, is the ability of the spectrometer to spread light. For instance, a 10 nm/mm spectrometer means that two rays separated by 10 nm will be imaged distant by 1 mm on the detector. It is remotely connected to resolution but it is not resolution because it does not take into account the optical limitation of the system (see next).

- Full-Width-at-Half-Maximum (FWHM), expressed in either µm on the detector or converted to nm through the dispersive power of the system, is the actual appearance of an infinitely thin rays as it is imaged on the detector, including any optical limitation on the system. It corresponds to the waist of the peak taken at 50% of the peak maximum intensity. Optician likes this quantity because it is easily predicted by optical simulation softwares.

- Resolution, expressed in nm or equivalent unit, is the ability of the device to resolve two separates rays (hence the name “resolution”). Please note that this is the strict definition of resolution, as defined by the IUPAC.

Now that we have defined the terms, I will cover the two common mistakes that people make when talking about resolution in spectrometers.

1. Confusing dispersive power and resolution. Dispersive power is not a measure of resolution. It is like having a 50-hp car with a speed-o-meter that goes up to 200 mph; it does not mean that the car can ever reach 200 mph! Similarly, a 10 pm/px spectrometer does not mean that the spectrometer can actually resolve two peaks separated by 10 pm. This confusion is sometimes even heard from large spectrometer suppliers which advertise the dispersive power of their spectrometer but not their actual resolution, leading the user into believing that they are the same thing when they are not.

2. Confusing FWHM with resolution. This one is trickier and is the major reason I started this post. Opticians like the FWHM quantity because it is easily predictable by software simulation but is often abused by system engineers who use it as a measure of resolution without even questioning the actual meaning of the term “resolution”. Let us dig a little bit further into that one now.

So, if they are not the same thing, why do many opticians confuse FWHM and resolution?

The answer is actually easy: they were trained for that. Any grad student in optics will be able to tell you about the Rayleigh or the Sparrow criterion for resolution which relates the resolution power of a system to its FWHM. This is illustrated in Figure 1 and illustrate that resolution ≈ FWHM. If you are interestedin the 0.85, it is coming from the Sparrow criterion which gives the exact moment where the second derivative cancels and where the two peaks start to be resolved.

Figure 1 – Sparrow criterion on two gaussian peaks

Then, is the 1 FWHM rule wrong? No, but it lays on a hypothesis that is often forgotten: it is true only for peaks of the same height. This is generally not true in spectroscopy where large peaks can be close to smaller ones. This is illustrated in Figure 2 with two peaks that have a height ratio of 1:10. Although the peaks are separated by 1 FWHM, they cannot be resolved anymore.

Figure 2 – 1 FWHM rule on two gaussian peaks with a 1:10 ratio

Under these circumstances, the resolution becomes a function of the peak height ratio. To study this dependence, I have run simulations on two gaussian curves with the same FWHM by changing the peak height ratio and checking at how many FWHM they could be resolved by the Sparrow criterion. The results are given in Figure 3. I have limited the simulations to a maximum ratio of 1:1000 because it is unlikely that the dynamic range of your spectrometer will be much larger than that unless you are making very long exposures.

Figure 3 – Adapted Sparrow criterion for varying peaks ratio

The conclusion we can draw from Figure 3 is that resolution is bounded by about two FWHM at the upper range and gets better for peaks of equivalent heights. This is an important conclusion because it allows designing spectrometers safely without knowing too much about the actual user experiments.

I have however omitted to talk about two important hypothesis that I made for these simulations:

1. I have considered spectral rays that were infinitely thin. This yield peaks whose FWHM match the system PSF FWHM as generated by the optical simulation software. In the real world, rays are never infinitely thin and always have some thicknesses. Atomic emission lines in the gaseous state is usually the thinnest lines you will observe but are still on the order of ~0.1 pm large (they can however be even smaller under vacuum!). The worst case will usually be absorption lines in water which can be as broad as 100 nm or more. For that we cannot do anything but let the user select the spectrometer with a resolution that is at maximum on the order of resolution as the thickness of his lines. To check if a system is limited by the spectrometer itself or the experimental lines thickness, the only way is to switch to a more performant spectrometer (or increase its performance by decreasing its slit size or numerical aperture) and to check if the lines widths have decreased or not.

2. I have considered peaks that are gaussians. This is an important decision that I would like to cover.

The fact that I chose gaussians over an Airy function means that I am considering aberrated systems that are not achieving diffraction-limited performances. “Booh, aberrated systems are just poor design” you will hear (or say!). There are actually two reasons you do not want a diffraction-limited system for spectroscopy.

First, diffraction-limited design means that you are dominated by diffraction effects and not by your RMS spot size. This means that you can open up your numerical aperture a little bit to reduce the diffraction effects until you reach a point where aberrations start to increase again and lower overall performances. This will not only improve your spectrometer resolution but will also allow more light to enter the spectrometer which is good for the SNR.

Second, diffraction-limited systems have PSFs that are Airy disks. This is actually a big problem for spectrometers because Airy disks have ripples (side lobes) that can be relatively high. These sides lobes can mask secondary peaks that have low amplitudes. This is illustrated in Figure 4 with a 1% secondary peak. The only way to overcome this is to change the shape of the PSF such that it looks more like a gaussian, since gaussians do not have these sides lobes. Note that gaussians spreads too on the side, but they do not ripple as Airy disk does. I don’t like Airy disk ripples because they can mask automatic detection based on curvature change, such as in the Sparrow criterion. Gaussians just don’t do that.

Figure 4 – Airy disk masking a secondary peak of low amplitude

Then, to be correct, you should run the simulation of Figure 3 with the final PSF of your system. Actually, to be fully correct, you should also include the effect of the slit width.

So far, I have only considered infinitely thin slits. However, what we read on the detector is actually the convolution between the PSF and the slit (with eventually a magnification depending on the optical design). This is illustrated in Figure 5. A next question is then: how does the slit size affect resolution?

Figure 5 – Convoluted slit images

This is the part where designers usually becomes lazy. They have the PSF of their system, the slit size and they just sum the two. Sometimes, they refine the approximation when the slit is of comparable size to the PSF, and say the final FWHM goes as the square root of the sum of the squared terms just as if they were summing gaussians:

Many have committed this crime and I plead guilty too. As a system engineer and optical designer, I did use that approximation quite often just because I was too lazy computing the exact convolution. But is this such a bad approximation?

Simulations can help us see clearer here again. Figure 6 shows how the actual FWHM of a full system increases as the slit size increase. The plot is normalized with the PSF FWHM (so the slit width is expressed as how many times as large it is compared to the PSF).

Figure 6 – FWHM as a function of slit size

From Figure 6 we see that the simplified law follows relatively well the simulation with an error below 20% at maximum. We also see that the simplified law actually underestimate the spectrometer maximum resolution and is then relatively safe to use. So the lazy stuff does actually work! That will help me sleep tonight.

To conclude this post on resolution I would like to address one more important part which affects how dispersive power may affect a system performance.

Often, scientists are not only interested in seeing the spectral lines but also in determining their actual centre position and height. Height of peaks are of particular importance in quantitative analysis and we must therefore be sure to get it right.

This is where the dispersive power of your system may ruin everything. You may have built the most performant spectrometer in terms of resolution, it may get the actual peaks height completely wrong if you did not care about the dispersive power at the design stage. This can happen when you over-optimize your resolution or when you try to make as many lines as possible to enter on your detector with the best resolution possible.

To understand how dispersive power can affect height estimation, let us look at three gaussians with the same FWHM but sampled differently in Figure 7.

Figure 7 – Same gaussian with different sampling

When the sampling gets too low, the peak height starts getting completely inaccurate. To correctly estimates peak height, one will have to fit either a gaussian or a Voigt profile (depending on the experiment) on the data. This is only possible if you have at the very least 3 pixels of data to capture the peak. Put differently, the dispersive power of the spectrometer must be at least three times as large as its resolution.

Note that I said “at least”, not “at best”. This is a general rule of thumb so if you can build a system with a five-times factor, or even a ten-times factor, it will be better. However, this is often to be traded off with the spectral range that you would like to cover as detector sizes are finite and building optical systems that are performant on large fields is complicated too.

Finally, you should also note that secondary peaks will tend to shift neighbouring peaks closer to them. You can notice this effect in Figure 1. This is something that high-end software compensated as they fit the peaks.

And that is all! I hope you enjoyed this post and understand a little bit better how tricky resolution can be when applied to spectroscopy!

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