Published: 2021-05-02 | Category: [»] Optics.

In [»] part 2 of the series I have shown that you can represent any combination of thin-lenses and air-gaps as the matrix

so that the system is entirely described by its effective focal length (EFFL), front focal length (FFL) and back focal length (BFL).

If you feel a bit lost with this definition, you can always fall down to a single thin-lens of focal length f with EFFL=BFL=FFL=f. Although most of the time you will do these derivations with a single thin-lens, I will keep it generic here to show that it can be implemented with any systems.

Let’s have a look on what happens when you put an air-gap of length BFL after the lens system M. This corresponds to

We can do the same with an air-gap of distance FFL placed before the lens system:

And we can do both as well

What is interesting with these systems is that they bring some zeros into the matrices. Zeros are convenient because they simplify the behaviour of systems.

As an example, let’s have a look on what the system M3 does to a ray (y,u):

This is rather an odd behaviour because y’ is solely based on u and u’ is solely based on y. You can then either transform of position into an angle or an angle into a position using this system.

We can infer two important rules from the expression of (y’,u’):

1/ Any ray originating from the optical axis (y=0) outputs parallel to the optical axis with a height proportional to its initial slope.

2/ Any ray parallel to the optical axis (u=0) output will originate on axis (y’=0) and have a slope proportional to its initial height.

Some refractometers are based on the first property but you can do much more. For instance, the angular intensity distribution generated by light scattering of a small particle is size-dependent (see [∞] Mie scattering). By projecting the scattered light on a camera sensor using system M3 you can visualize the angular distribution and determine the average particle size in your sample.

Rule #1 is interesting because if the input of the system corresponds to a pupil location (see [»] part 3 for a discussion on aperture stop and pupils), we know that all chief rays will pass by coordinate y=0 (by definition of what a chief ray is).

Figure 1 illustrates this for a single thin-lens with the aperture stop placed at the FFL distance of the lens. See how all chief rays exit with u’=0.

Figure 1 – Example of a telecentric lens

Notice how all fields (red, green and blue bundles of rays) strike the image plane identically with an average angle of zero. This condition occurs as soon as the aperture stop or a pupil is located at the FFL distance of a lens system. It is called a telecentric system.

Telecentric systems are widely used in optics. They offer homogeneous illumination independent on the field position and they have orthographic projections meaning that the magnification does not dependant on the depth of the sample. They are widely used in optical metrology instrument but also became some sort of standard for DSLR cameras because the camera sensor behaves better when the rays strike them perpendicular. Edmunds optics has a nice [∞] webpage on telecentric lenses that you can check. I personally work with telecentric lenses every single day at the office!

Now, let’s check what happens when we put two systems like M3 with effective focal lengths EFFL1 and EFFL2 back-to-back:


This system acts as a magnifier. If we follow a ray (y,u) we find (y’,u’) such that

We see that heights are magnified by K and slopes are magnified by 1/K. Note that the system also inverts the sign of both the ray height and slopes if the effective focal lengths are of the same sign. If you would like to have non-inverting system, you can either use two inverting systems or one system with both a positive and a negative focal length. In this last scenario, it is not possible to use a physical aperture between the two sub-systems however.

The system described by matrix M is known as a 4f system. The name originates from using two lenses of focal length f, each surrounded by air gap of f in both direction; the total length of such system being then 4*f. You don’t have to use the same focal lengths to call such system a 4f system however.

Figure 2 shows a simple 4f system with a threefold magnification. Note that here I’m using single thin-lenses but in practice you will use many more lenses to overcome aberrations.

Figure 2 – A simple 4f system with K=3

The system described by matrix M has however an odd behaviour: its C terms equals to zero which means it does not have an EFFL, or, more precisely, it would have an infinite one. By consequence, it does not have a back nor front focal length either even though its A and D terms are non-zero (recall that BFL=-A/C and FFL=-D/C from our former [»] derivations). Because it has no focal lengths, it cannot be represented by our generic definition introduced before and belongs to a different class of systems: afocal systems.

The 4f system is not the only afocal system that exists. For instance, if you put the system M2 after the system M1 you obtain

still with

If we send rays through this system, they exit as

The ray direction is magnified by a factor -1/K while the ray heights are magnified by a factor –K and shifted by a factor proportional to the ray direction. The behaviour is the same as the 4f system except that we have some offset factor for the exit height position and we break the pure y to u’ and u to y’ relationship.

Such systems are however widely used as well to magnify slopes or expand beams. They are known as telescopes. In the inverting configuration (same EFFLs signs), they are referred to as Keplerian telescopes. In the non-inverting configuration (EFFLs of different signs) they are referred to as Galilean telescopes.

Keplerian telescopes form an intermediate image while Galilean telescopes don’t. This can have consequences in some systems, such as when enlarging a laser beam, as focusing a high-power laser at an intermediate image plane can lead to serious safety concerns. Galilean telescopes are also more compact than Keplerian telescopes because the back (or front) focal length is negative, thus shortening the distance between the two elements.

Figure 3 illustrates a Galilean vs Keplerian telescope with the same focal lengths of the sub-systems (here simple thin-lenses).

Figure 3 – Galilean and Keplerian telescopes

Finally, it is worth noting that telescopes systems do not generally include an aperture stop like the 4f systems or the clear aperture the lenses makes the aperture stop. This is very common in mirror-based telescopes where the larger mirror (called primary mirror) defines the stop of the system. Most of the time, the telescope is part of a more complex system that will define the stop position.

One of the consequences of all the development that we have derived here is the conclusion that system of thin-lenses as described by their equivalent focal length, back and front focals are only a part of optical systems and that there exist other systems that have to be described differently, such as what we have obtained here with our afocal systems (generic, pure magnifier, inverting or non-inverting). This is an important concept to consider for the #DevOptical methodology that we are building. We have also introduced the very important notion of telecentric systems, that we will use many times in the following posts.

In the next post, we will use an afocal system to make our [»] air-spaced doublet system more robust to positioning tolerances!

I would like to give a big thanks to James, Daniel, Naif, Lilith, Ian, Cam, Samuel, Themulticaster and Sivaraman 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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You may also like:

[»] #DevOptical Part 7: Replacing Thin-Lenses by Real Lenses

[»] #DevOptical Part 3: Aperture STOP and Pupils

[»] #DevOptical Part 9: Geometrical Image Simulation

[»] #DevOptical Part 5: Replacing a Thin-Lens by More

[»] #DevOptical Part 4: The Air-Spaced Doublet