Published: 2021-04-11 | Category: [»] Optics.

Up to now, we have performed unconstrained [»] paraxial raytracing in our system. This means that any ray at any height and any orientation can make its way through the system up to the image plane. In most (if not all) cases, this is not a desired behaviour and will almost always result in poor image quality.

All optical systems must include an aperture stop whose role is to control which rays are accepted into the system. It is either fixed as a thin blade or tuneable as an iris. An alternative to the fixed aperture stop is to machine the stop directly into the mount that hold the lenses. When done that way, it is important to chamfer the stop so that it presents a sharp edge to the rays. Also, remember that iris are not perfect apertures because they are made of distinct blades that do not overlap perfectly. Always favor a fixed aperture stop when possible.

Figure 1 shows a lens with an aperture stop in front of it and three bunches of collimated rays passing through the system at different field angles.

Figure 1 – Simple lens with aperture stop

Although not all systems are as simple as the one of Figure 1, it helps seeing how the aperture stop controls the light rays entering the system. Note that all the different fields (red, green and blue) pass through the aperture stop and have their diameter limited by it.

As I already mentioned, one of the roles of the aperture stop (often written “STOP” in optical design software) is to control image quality. Indeed, apart from the distortion aberration, all third order primary aberrations increase as the size of the aperture stop increases. A small STOP therefore usually yield a better image quality than a larger STOP. Also, manufactured lenses rarely have good optical properties on the edges and it is recommended that the last 10% is not used. This is referred to as the clear aperture of the lens in the manufacturing phase. When a system performs badly, one trivial way to increase the quality is often to “stop down” the system by using a smaller aperture. Photographers know this very well because even high quality photographic lenses perform better when they do not operate at full aperture. You will often hear them say “this lenses performs best at f/4” which means that an aperture of f/4 (more on this below) gives the best image quality for that lens.

The second usage of the aperture stop is to control the quantity of light that enters a system. Imagine you have a point source emitting light rays in all directions. Each ray bring a fraction of the total energy emitted by the source. A large aperture stop will let more rays enter the system and will therefore collect more energy. More energy means better signal-to-noise ratio on the sensor, which means better overall image definition at the end. This is (one of) the reason manufacturers make photographic lenses that open wider than their best image spot quality. A wide-open lens is better at making image in low light conditions than a lens that has been stopped down. A lot of consideration obviously enters into the balance such as camera shake which blurs the image etc. This is beyond the topic of this post.

In addition to the aperture stop, it is essential to discuss about pupils too. Taking an assembled lens system (with its STOP), the entrance pupil is the image of the STOP when looking from the front of the lens system. Similarly, the exit pupil is the image of the STOP when looking from the rear of the lens system.

Figure 2 illustrates this with a more complex lens system.

Figure 2 – A more complex lens system with aperture stop and entrance pupil

If we ignore pupil aberrations, we are free to place the aperture stop in any of its conjugated planes which are called pupils. The entrance and exit pupils being therefore only two particular pupils of the system. Note that every lens in the system produces its own image of the STOP but the images are not always real and can also be virtual. When the image is virtual, it is not possible to place a physical aperture there.

Figure 2 shows a system that take collimated rays and image them on a sensor plane (at the right of the figure). The red rays simulate the image forming rays. The green rays passe through the center of the aperture stop and we can therefore locate pupils by looking for intersection of green rays. As you can see in Figure 2, the system possesses a real entrance pupil but no exit pupil because it is located at infinity. When the system entrance or exit pupil are at infinity, the system is said to be telecentric. Telecentric lenses have many advantages that I will cover in a later post.

The concept of entrance and exit pupils are extremely important in optical design because when you mate two optical systems together, you have to match their STOPs to avoid vignetting which is the clipping of rays. Since pupils are equivalent to STOPs because they are its image, matching STOPs means that the exit pupil of a system A must be super-imposed on the entrance pupil of a system B when you want to mate system A and B together. You should at least match the pupil positions but you don’t necessarily have to match the pupil diameter. The overall system aperture will be limited by the smallest pupil diameter but this may leave some room for positioning.

A typical example of pupil matching is placing your eye in front of a rifle scope. Every good quality rifle scope will give you both its exit pupil position (commonly called eye-relief) and diameter. When placing your eye behind the rifle scope, you need to match the entrance pupil of your eye to the rifle scope exit pupil. If you don’t, you will see a black disk around the scene when looking through the scope. The more you move your head away from the correct position, the larger the black disk becomes. Also, shooters will tell you that large scopes with small magnification are easier to “lock” for the eye. This is normal because these scopes have an exit pupil diameter much larger than the entrance pupil of your eye and you will acquire a good image of the scene as long as your eye ends up somewhere in the exit pupil disk. At contrario, large magnification scopes have very small exit pupil diameters which force you to have your head in a very precise alignment with the scope to get an image of the scene.

To easily compute the location of the entrance and exit pupils let us divide our system into two concatenated matrices: Mf will represent all the lenses in front of the STOP and Mr will represent all the lenses at the rear of the STOP. We obviously have the relation M=Mr*Mf with M the complete matrix of the system.

We can find the location of the entrance pupil by solving the problem (remember the green rays of Figure 2)

which we can put in equations as

where ABCD is the representation of the Mf matrix.

We find

Remember however that if you would like to draw the entrance pupil relative to the first lens you will have to flip the sign of L because it is located at -B/A from the first interface of the system.

The magnification of the system is u/u’ which gives

Since AD-BC=1 for a system composed only of thin lenses and air gap as we have shown in our [»] previous post.

For a given stop diameter S, the entrance pupil diameter will be S*u/u’.

Note that if the aperture stop is located in front of the system we have Mf=I (the identity matrix) and therefore L=0 and u/u’=1.

Similarly, we can find the position and diameter of the exit pupil by solving the system

Which gives

and

Note that here we have considered u’/u and not u/u’ to get the magnification for the exit pupil. Also, this time you do not have to flip the sign of L to draw the exit pupil because you add the quantity to the last interface of the system.

Finally, there are two commons ways to discuss about “how wide open” a lens system is:

(1) The f-number is the ratio between the equivalent focal length of the system and the entrance pupil diameter. For instance, a f/1.8 35 mm photographic lens means a lens with a 35 mm focal length and a 35/1.8 mm = 19.4 mm entrance pupil.

(2) The numerical aperture (NA) is the sine of the half angle of an on-axis point source that completely fills the STOP of the system. Although this formulation looks more complex at first sight, many physical properties linked to the optical system directly relates to the numerical aperture.

The size of the STOP has important consequence on the maximum achievable resolution of an optical system but I will have to address this in a later post when we will start discussing diffraction effects in optical systems. We have many more things to discuss about aperture stop but I wanted to keep this post light in maths to let you breath a bit.

This is therefore all for today and we will continue our journey in the #DevOptical series. We still have very important topics to discuss about the sketching up of system that I’m really excited to talk about so be sure to stay tuned in for updates!

I would like to give a big thanks to James, Daniel, Naif, Lilith, Cam and Samuel who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!

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