Published: 2018-08-26 | Categories: [»] Engineeringand[»] Optics.

A few months ago, I presented a [»] 400-800 nm spectrometer with a resolution that was tested to be better than 2 nm at 640 nm and that had almost no aberrations through all the lateral field. There were a few questions remaining however such as (1) what is the true resolution at any wavelength? (2) What are the effects on resolution of using a slit rather than a pinhole? (3) Up to which Numerical Aperture (NA) can the spectrometer be pushed?

Testing the resolution of the spectrometer at all wavelengths could be done by measuring the FWHM of very sharp peaks (elemental lines) of a source like an Argon-Mercury lamp but, unfortunately, I do not have one (or I would have already done this test months ago…). Another way to proceed, inferior to experimentation but still very meaningful, is to go through simulation using Ray Tracing softwares. This is the solution I have chosen here.

I have performed two measurement in such a simulated system: measuring the spot size in µm at wavelength varying from 400 nm to 800 nm, and measuring the spot size in µm at fixed wavelength (600 nm, arbitrarily) and varying field height to see how the resolution changes when we use a slit. From the answer in µm, I have converted the results in an expected resolution in nm by multiplying by the dispersion of the system in nm/px.

Concerning the spot size, there are several ways to implement it. Here I have chosen the width that enclose 90% of the diffracted energy. If you are familiar with the Rayleigh criterion for resolution, this is the same but with a factor 1.6 instead of the conventional 1.22, so the width is about 30% larger (Rayleigh criterion is the width that enclose 86% of the diffracted energy). This is just a definition choice and I could just have used the conventional Rayleigh criterion but I decided to do not.

Figure 1 presents the results obtained at NA=0.10 assuming a slit of 10 µm. The left figure shows how the resolution changed at centre of the field (pinhole case) when the wavelength changes and the figure on the right shows how the resolution changes with field height (slit case) at a fixed 600 nm wavelength. The first figure also shows the diffraction limited resolution (assuming the 90% encircled energy criterion) and the figure on the right shows the relative SNR gain by integrating the slit at larger heights (using more pixels will reduce the noise in the measurement).

Figure 1 – Expected resolution with 10 µm slit at NA=0.10. (left) at field centre with varying centre, (right) at 600 nm with varying field height.

From the results, we see that the system is not so far from diffraction limit and that the expected resolution changes (at centre) from 0.7 nm to max 1.4 nm depending on the wavelength. This is in strong agreement with the actual test performed at 640 nm that shows a peak width on the order of 1 nm. On the second figure, we see that the resolution quickly drops to 1.2 nm at 0.2 mm field height before decreasing again to the initial resolution at maximum slit height (slits from Thorlabs are 3 mm long). This may be due to the relatively strong astigmatism introduced by the diffraction grating although I do not have an explanation yet on why the resolution goes back to normal at larger field heights.

You may be tempted to say that it is better to use a pinhole since the resolution is better at field centre but a slit has the advantage of bringing more energy into the system. For instance, integrating the full 3 mm of the slit will boost the SNR by a factor 17 when compared to just using the pinhole area. That is, the system will be about 20 times more sensitive with a 10 µm slit of 3 mm than with a 10 µm pinhole. The downside is that the resolution is decreased to about 1.2 nm (actually a bit less than that because of averaging at different field heights). Extrapolating this result to the other wavelengths would mean that we have a resolution that is at worse 2 nm when using a slit which is not so bad for such a broad range of wavelengths.

The next thing of interest that we can try with the simulations is to increase the numerical aperture to see how it affects the performances of the spectrometer. If you are wondering why we should bother maximizing the numerical aperture, just have a look at Figure 2 which shows how the signal will increase as we open the system NA. Going from NA=0.10 to NA=0.2 equals 4 times more light hitting the sensor (vignetting taken apart) !

Figure 2 – Relative energy gain with varying Numerical Aperture

I have run the same simulations at varying NA to see how it would affect the system performances. The results are summarized in Figure 3.

Figure 3 – Expected resolution with a 10 µm slit

We see that the system performances remaining acceptable up to NA=0.14 before the aberrations start to strongly affect the system at NA=0.16 and greater.

From Figure 3, I would therefore be tempted to increase the system numerical aperture up to 0.14~0.15 to get a 2× energy boost without compromising the system performances too much.

To have a better idea of the actual effects on the system, I have run the same encircled energy study as in Figure 1 but with NA=0.14. The results are presented in Figure 4.

Figure 4 – Expected resolution with 10 µm slit at NA=0.14. (left) at field centre with varying centre, (right) at 600 nm with varying field height.

We see that the behaviour at NA=0.14 is almost the same as measured at NA=0.10 expects that the resolution is scaled by some penalty factor due to the increased aberrations. The maximum expected resolution goes up to 1.8 nm at field centre and decreases by about 60% a 0.2 mm field height. Extrapolating the values to all wavelengths would means that the resolution would be at worse 2.9 nm when using a slit. This is not as good as with NA=0.10 but remember that we have a 2× energy boost.

This concludes the theoretical analysis on the performances of the [»] spectrometer presented recently. We see that the predicted resolution when used as in the experimental tests is at worse 1.4 nm at field centre and that the value predicted at 640 nm matches perfectly the experimentally measured peak width (1 nm). Furthermore, we have predicted that a slit would decrease the resolution slightly but would also provide a 17× SNR boost. Finally, we have seen that the system numerical aperture could be increased to about NA=0.14 to benefit from a 2× signal boost without compromising the resolution too much.

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