High-Resolution Spectroscopy

In the previous posts on spectroscopy, I focused on large bands spectroscopy such as 400-800 nm spectral analysis with resolutions limited to about 1 nm. Here, I would try to address the topic of small bands, high resolution spectroscopy. When I say “high resolution”, I mean a discrimination factor (λ/∆λ) on the order of 100,000 typically. The concept behind high resolution spectroscopy is not so different than in conventional spectroscopy but it has a major drawback in its efficiency that I will cover later in this post.

The system I have built is based on a Littrow configuration which is represented on Figure 1. If you have read carefully my previous posts, you will also recognize the basis of the setup as the [»] angular detection setup built recently but with a fiber output as the light source this time. This is for sure not a coincidence since I wanted to demonstrate how generic the autocollimator setup can be in an optical lab.

If you remember the angular measurement setup, you will recall that light escaping the source is first collimated by the lens before being reflected by a mirror with a tilt angle ∆θ. After reflection by the mirror, the light is then focalized on the camera sensor with a lateral shift of f∆θ (for small angles), where f is the focal length of the lens.

Gratings are, however, not mirrors and act slightly differently. Because of the groove pattern on the grating surface, each wavelength composing the light source will be deflected by a slightly different angle. The autocollimator will then translate this angular shift to a lateral shift on the camera sensor and we will see duplicates of the fiber head convoluted with the light spectra on the camera image.

I also said that the spectroscope built here is based on the Littrow configuration. This corresponds to a special spectroscopy setup where the incoming light beam is reflected almost directly back to the source. This works only for one specific wavelength which can be computed using the diffraction grating law:

with

this becomes

For instance, at 1,200 lines/mm and first diffraction order (m=1), the wavelength 500 nm has a Littrow angle of 17.46°. Other wavelengths angle can then be found using the diffraction law:

A small variation of wavelength ∆λ will produce a small angular shift ∆θ of (by differentiation):

Littrow configuration is so popular that a special type of gratings (the blazed gratings) have been optimized to concentrate energy into a given order when used in this specific configuration. This is the case of the visible gratings that I have purchased from Thorlabs previously, even if I did not used them in the Littrow configuration at that moment.

Now, to make a high-resolution spectroscope there are two things to do: (1) select a grating with enough wavelength resolution (see below), (2) design an optical imaging setup such that it is not diffraction limiting. For the latter point, this means that we need to select a very small fiber head (typically, twice the size of a camera pixel) and a very long focal length with high numerical aperture to achieve a good lateral separation of the small angular variations ∆θ. There is indeed little benefit in reaching a ∆θ of ~0.01 µrad if the focal length is so small that it will be re-imaged on the same pixel! Similarly, there will be little benefit in having a good separation of the ∆θ on the camera sensor but have everything masked by the big blob of the image of the fiber head.

In this setup I have chosen a 10 µm multimode fiber (0.1 NA) because my camera has 5.2 µm pixels. This gives an Airy spot of 3 µm so we will not be diffraction limited. For the lens, I have used a 2” diameter 200 mm doublet achromat lens for the wavelength separation, it is not really long and will actually be the limiting factor in this setup. I have used a 200 mm lens because I do not have a sufficiently long optical table at this time and I have not yet finished working on my telephoto lens setup (oops sorry, spoiler alert ;-)). If you can afford it, go up to 500 mm to be comfortable. Take care however that with longer lenses you will also reach the diffraction limit more quickly due to the limited numerical aperture of the lens. Typically, a 500 mm lens with 25 mm clear aperture will give an Airy spot of ~12 µm. Going further than 500 mm has then no interest for us unless you can buy larger gratings and lenses. The clear aperture here is only 25 mm (and not 2”, the diameter of the lens) because the grating has smaller dimension than the lens. As you can see, there are a lot of things to consider when building such type of setups.

Now, concerning the resolution of the grating itself, it will be limited by the total number of grooves illuminated and the diffraction order at which we are working. There are two ways to deal with that limit. Either you are interested in medium-resolution spectroscopy and you go for a 1,200 lines/mm or 1,800 lines/mm grating used in the first diffraction order (m=1), or, you use special “Echelle” grating which have much lower grooves density but which are designed to be used at high diffraction orders. This last solution is made possible by the optimization of the blaze angle which can reach extremely high values as the groove density is low. Typically, you can find 79 lines/mm gratings designed to operate at 63° Littrow angle at Thorlabs. Such a grating would be equivalent to a grating of 3300 lines/mm used in the first diffraction order. When used with 500 mm lenses, I have seen people obtaining resolution of a factor of 150,000 with these Echelle gratings! This is truly high resolution. Here, because of the limited focal length, we can reach a 0.013 nm resolution at 532 nm (resolution factor of 41,000).

By the way, when using high focal length to resolve small angles, you will definitively want to use some kinematic adjuster to adapt the tilt of your grating. It also takes some time to locate the correct diffraction order so I would recommend to align first using a 200 µm multimode fiber before swapping with the 10 µm multimode.

These Echelle grating may look like the panacea but it comes with a drawback that is due to the spectral sensitivity. Because we use it at high diffraction orders, some other wavelengths will mix because their “mλ” quantity will have the same value. For instance, 532 nm at order 42 will also be mixed with 520 nm at order 43, 510 nm at order 44, 545 nm at order 41, 559 nm at order 40 and so on…

With our setup we have an angular range of ±1° which will give the overlapping of diffraction orders of Figure 2. I have limited the plot to the wavelengths ranging from 350 nm to 750 nm. You can see in the figure that there are a lot of wavelengths mixing near to the UV region because of the higher diffraction orders. Of course, due to the blazed nature of the grating, one mode will be strongly enhanced relatively to the others and so you may not notice the mixing but you cannot take this effect for granted because tiny signals in the blazed order may be mixed with strong signals that are in higher (or lower) diffraction orders.

To avoid overlapping the various diffraction orders, we will have to use a second grating (or prism) to separate the orders orthogonally with the fine division. That way, we make the full usage of the camera 2d area because we will have the fine resolution along “x” (let’s say) and the coarse/orders division along “y”. A second way it to cascade two spectrometers and to feed the high-resolution spectrometer with a signal already filtered by a low-resolution monochromator. I will not cover this here and leave that for a future post.

One way to use the setup without orders separation is to use an almost monochromatic source. Here, I have used the setup to study the longitudinal modes of my 532 nm NdYAG:2 laser. The results are shown in Figure 3.

You can clearly see that there are several distinct lasing modes that are well separated. These longitudinal modes are responsible for the limited coherence length that you may experience with such lasers. The two modes on the left of the figure allows for an estimation of the resolution accuracy. These modes are separated by 0.010 nm which is the theoretical resolution of our spectrometer when limited by the imaging resolution. Also, when I recorded the data of Figure 3 I had to saturate a little bit the centre mode to view the side modes and so the peak at 532 nm is actually double (you may indeed notice a small bump on the side). The distance between these two mode is also ~0.015 nm although it does not appear frankly here.

The setup therefore behaves very satisfactorily and we have plenty of signal with a SNR above 500. What could we ask more?

But now comes the real drawback of the system: it has a very poor efficiency with non laser-like sources. The efficiency of the setup is mainly limited by the étendue of the system and you will not notice it with laser sources because their étendue is not the limiting element. Let us dig in a bit of theory first.

Etendue is an illumination quantifier that tells how much light you can put into an element. The overall setup will always be limited by the element having the smallest étendue. Mathematically speaking, when you have an element of area A and numerical aperture NA, its étendue will be:

Once the étendues are set, there is nothing you can do but accept the photon lost. A good optical system is then one where there is no clear bottleneck in étendue. Poor étendue matching is the best way to kill your SNR.

The limiting étendue in our setup is the fiber because it has a low numerical aperture (0.10 NA) and a very small surface (10 µm diameter disk). We did not notice it with the laser because the laser has an even smaller étendue because its output is an almost perfectly parallel beam that has an extremely low equivalent numerical aperture.

But now imagine you want to send the light from a light bulb into the 10 µm fiber. The light bulb emitting element is about 10 mm wide and it spreads at typically ±45° for a desk lamp. If we image the emitting element on the fiber head, we need a very small magnification ratio to transform the 10 mm element into a 10 µm image. But if we do that, we will produce much higher angles than the fiber angular acceptance and almost all the light will directly escape the fiber. If, on the other hand, we try to decrease the ±45° such that it enters the 0.10 NA, we will have to increase the size of the image of the light bulb emitting element and all the light will be around the fiber head, but not at its core. In both scenarios, we lose a lot of light.

Speaking in terms of étendues, we have

The étendue of the fiber is clearly limiting and the ratio between the two will give an efficiency of 6.10^-9. That means, if the light bulb emits 1 Watt energy, in the best configuration possible we will only achieve putting 6 nWatts into the fiber. We did not notice it with the laser because its étendue was about 10^-5 mm².sr and on the 5 mW entering the setup, we managed to couple 260 µWatts, way enough to produce a clear image on the camera.

This is clearly due to the very small core of the fiber that is required for the high resolution. When working with larger cores (e.g.: 200 µm, 0.22 NA), we can easily couple 6,500 times more light from the light bulb into the fiber! This makes a big difference in terms of detection unless you can work with 10 kWatts lamps at home (which are pretty uncommon by the way!).

When looking more closely at the setup of Figure 1 we also see that we are throwing away some efficiency with the effective numerical aperture of the lens because it does not match the output fiber. Indeed, because the grating elements typically limits the returned light to a surface of about 25 mm, the resulting offered aperture for the fiber with a 200 mm focal length lens is only 0.062 which is less than the 0.1 NA of the fiber, reducing the efficiency by a factor 2.6. Also, each pass through the beam splitter will spoil 50% of the light. Adding to that the efficiency of the grating, we lose about 95% of the photons who made it through the fiber entrance! With the efficiency of the light coupling, this makes the overall efficiency drops to about 3.10^-10 for our light bulb! For the laser, we still have about 15 µWatts (50,000 times more signal).

Despite all of that, I did however succeed in measuring the spectrum of a fluocompact light bulb. Sure, it required tons of blackout material around the setup, the longest expositions available on my camera and the gain set to 100 but it worked. The results are presented in Figure 4.