THE PULSAR Engineering
Published: 2018-01-27 | Categories: [»] Engineeringand[»] Optics.

Building a dispersive spectrometer is not a hard task as we have shown in [»] previous posts. However, as you try to resolve finer and finer spectral lines, problems start to accumulate.

All dispersive spectrometers (i.e. grating and prism based spectrometers) are built on the same concept: the light to analyse is fed into a slit or pinhole, collimated with a first optical system, sent to the dispersive element (grating or prism) which can be seen as a mirror whose output angle is colour sensitive and the output beam is refocused by a second optical system on a linear sensor or on a camera. I have put the monochromator design (another pinhole in the refocusing part) away from the discussion because it obeys to slightly different rules. A generic schematic design is shown in Figure 1. Please note that just like the dispersive element can be a grating or a prism, the optical systems can be a refractive (lens) or reflective (concave mirrors) design. The exact nature of the elements does not change the outcome of the discussion that I will address here.

Figure 1 - Dispersive spectrometers generic design

If we were to replace the dispersive element by a mirror, the setup would simply be a relay imaging system of the input pinhole where the image of the pinhole would be shifted laterally based on the exact angle of the mirror and the position of the optical systems. The fact that the dispersive element is colour-sensitive will produce an image that is actually convolved with the spectrum of the input light. So, if our light source contains two lines colour, one 650 nm red and one 460 nm blue, we will get two images of the slit shifted laterally from each other by the angular dispersion between the red and blue colours multiplied by the focal length of the imaging system. A very important consequence of this convolution operating system is that even if the light spectral lines are infinitely thin (e.g. ultra pure single mode laser with 0.00001 nm line width), we would get an image that has the width of the input slit and not a zero width. Put differently, the slit size in a spectrometer acts s a low-pass filter on the spectrum to resolve and will smooth out any small details.

Based on a system analysis of Figure 1 there are several options that you can use to increase the spectral resolution of our spectrometer: (1) increase the angular dispersive power of the dispersive element by taking a prism with higher dispersion (e.g. N-SF11 prisms vs. N-BK7 prisms in the visible range) or take a grating with higher groove density, (2) increase the focal length of the collimating and imaging optical systems to convert the same angular dispersion into a larger lateral dispersion or, (3) use a smaller pinhole to decrease the low-pass effect that we discussed.

I will put (1) on the side as it can be considered to be the first thing you optimised and I will focus on (2) and (3).

Increasing the focal length of the system can be seen as a very good way to increase the resolution. This is true and many high-resolution spectrometers such as laser profiler have focal lengths of several meters. But this is high-resolution, not high-performance. To understand the difference let’s have a look at the energy efficiency of the system.

Imagine we have a 10 µm fiber as an input pinhole with a numerical aperture of 0.10 (typical). That means the light spread out of the fiber as a cone with a half-angle of about 0.10 radians (that is 5.74°). If we use a focal length of 1 meter, we would need an optical system of 200 mm to collect all the light (D=2*f*NA). The collimated beam exiting the first lens system will also be 200 mm in diameter and so we would have to use a dispersive element of more than 200 mm diameter to avoid clipping photons. Now wait… when was the last time you saw a 200 mm grating in your optical supplier catalog? And that is the problem, they are pretty expensive and that is why the very high-resolution spectrometers also costs a ton.

Most of the time you will work with 25 mm elements or at maximum 50 mm if you have the money. That means you will limit the actual numerical aperture of your setup by the diameter of the smallest element in your system. With a 25 mm grating, the actual numerical aperture will be smaller than 0.0125 with a 1 m focal length (which I will round to 0.01). Now if we check the amount of energy transferred in the system from the optical fiber, we see that most of the light is not collected by the spectrometer. In terms of efficiency, we have (NAsystem/NAsource)² (ratio of the apertures area, the focal length term disappear and only the NA remains) which is only 1% here! When analysing very faint light sources, this is clearly unacceptable. If you look at commercial spectrometer, they give the f-number of the spectrometer instead of the NA (f/#=(2NA)-1) and you should always try to match the f-number of your source to keep a good efficiency. As an order of magnitude, most commercial spectrometers are f/4 spectrometers but you may find some that goes down to f/1.3 (at a much different prices of course…).

And that answer the question on the difference between high-resolution and high-performance. With our 1 meter focal length and 25 mm grating, we achieved a high-resolution due to the very long focal length but we did not achieved a good efficiency unless we use massive elements. When looking for high-performance we want both the resolution and the efficiency.

So unless you can afford the money expense for a really large dispersive element, increasing the focal length is not the good way to achieve high-performance spectroscopy.

That leaves option (3) which is to reduce the slit width. Indeed, a smaller slit/pinhole/fiber will decrease the low-pass effect on the spectral convolution. At the same lateral dispersion, a smaller slit will then usually produce a higher resolution. I say usually because what we should look at is the size of the image of the slit and not the size of the slit itself. That may sound like the same but it is not.

The image of the slit is actually the convolution between the physical slit and the point spread function (PSF) of the system. A poor optical system will have a poor (large) PSF and a good optical system will have a good (small) PSF. As a consequence, the same physical slit will produce a larger image with a poor optical system than with a good one. This is exactly what you can see in [»] this post where I compared different lenses system that had a 50 mm focal length (a plano-convex lens, a camera objective and a microscopy objective). Camera and microscopy objectives had much better performances than the plano-convex lens because they are made of several optical components that aims at reducing the aberrations to a maximum and therefore increase the quality of the PSF. Please have a look at [»] this post if you are not comfortable with the concept of optical aberrations.

To illustrate this, I have performed a simple experiment with a 50 mm doublet lens (that already has a good image quality when compared to a singlet lens) and a 10±3 µm fiber. I assembled a spectrometer-like setup where I replaced the dispersive element by a mirror. I have then recorded the image quality using different apertures value. Indeed, closing the aperture will reduce the optical aberrations in the system (mostly spherical aberrations) and increase the image quality. I took the measurement at different places in the field of view, from centre to the edge of the camera sensor, and compared this to the result obtained with an Olympus Plan Achromat 4× 0.10 NA (45 mm focal length) which is known the be diffraction-limited on a field of 5.5 mm. The results are presented in Figure 2.

Figure 2 - Experimental results

As it could have been expected, the performances are much better for the smallest apertures.

The size of the spot was evaluated at centre for the Olympus microscopy objective using a threshold of 63% (stdev value). The result is shown in Figure 3 and is 15 µm which is consistent with the size of the diffraction spot (3 µm) and the size of the fiber (10±3 µm).

Figure 3 - Size of the fiber image using the Olympus objective

The same procedure is repeated for all the spots of Figure 2. The results are shown in Figure 4. Clearly, the system performs better at apertures of 4 mm and 6 mm.

Figure 4 - Spot sizes as a function of aperture diameter

To build a spectrometer, one can then choose to use either the microscopy objective or to close the aperture of the doublet to about 4 mm. This is true for the image size but we have not look at the efficiency yet. Indeed, by closing the aperture we are also limiting the light throughput of our system from f/2 (25 mm aperture) to f/12.5 (4 mm aperture). This has a consequence on the efficiency. A plot of the efficiency ratio versus the microscopy objective collection power is shown in Figure 5.

Figure 5 - Relative efficiency as a function of aperture diameter

The system is therefore not so efficient for the same image quality as the microscopy objective but becomes much more efficient if we release a little bit our resolution requirements. There is then a trade-off to make here between efficiency and resolution. Does that mean that both systems are equivalent? No because, as we have seen, it depends on what we would like to achieve. Does that mean that all systems are potentially good as long as you use it properly? Once again, the answer is no. To understand why we need to make a performance analysis of the different systems that we are looking at.

Performance analysis consist of comparing a set of solution in all their aspects (strong points and weak points) and see if any solution outperforms the others. This analysis has to be performed in the early phases of the design and optical design software comes very handy so that we do not have to make actual experiments as I did here but simulate them instead.

I have then plotted each measured on a graph that is shown in Figure 6. Normally, the exercise would be done through a simulation process in a software like Zemax OpticsStudio or OSLO.

Figure 6 - Performance analysis of the experimental results

To have the highest performance spectrometer we would like to have the f-number as low as possible (high numerical aperture, therefore high signal throughput) while maintaining the highest image quality and so having the smallest PSF possible. We see that some solutions clearly outperform others in Figure 6. A solution A is said to outperform another solution B if all the attributes of A are better than the attributes of B. All outperformed solutions can be removed from the analysis as they give no additional value. Which solution you pick from the remaining set will depend on the importance that you give to each attribute. The choice will be very different if you need more efficiency than if you need more resolution. A common solution to this, but far from being optimal, is to assign a weight to each attribute and to sum the weighted attributes to give an overall score for each solution. Then, you just pick the solution that has the smallest overall score. Each weight should be related to how important an attribute is to you. And if you are not happy with the final solution, it means the weights are not correct and you should consider revising them. That is the part where I don’t like the methodology because you go from a very objective approach (performance analysis) to a very subjective one (weight-based approach). But that is a different story and I don’t have enough space to cover it here.

In Figure 6, outperformed solutions are shown by a cross mark and outperforming solutions by a circle. A dashed grey line connects all the outperforming solutions found. When looking at Figure 6, it is relatively obvious that there are only two acceptable solutions: the microscopy objective and the full aperture doublet. Strictly speaking, the 12 mm aperture is not outperformed by the other solutions because it has a slightly better PSF than the full aperture and a slightly smaller f-number than the microscopy objective. However, none of these improvements are significant and we can disregard this solution in profit of the two remaining ones.

You should remember that your performance analysis only reflects your knowledge of the system as there might always be an unknown solution that will outperforms all the others. Also, it is impossible to list all solutions because that would be a time-consuming operation (you cannot simulate thousands of optical systems to populate your solution set). Finally, you should check if you are considering all the attributes that are meaningful. For instance, here I have limited myself to f-number (efficiency) and PSF (resolution) but I did not consider price that we could add as a third attribute of interest. Also, you should always have the performance that goes in the same way such that you have to always maximize (or inversely minimize) all the attributes.

To play a little bit I can recommend that you add to Figure 6 a third axis for the price and the results for different optical systems such as: a concave spherical mirror, a parabolic off-axis mirror, one plano-convex lens, two doublet lenses, commercial objectives etc. For the exercise, keep using 50 mm focal length systems or change the second axis from PSF to angular resolution in PSF/focal length units.

Don’t hesitate to e-mail me if you find a promising solution that is extra performant!

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