**[»] Optics**.

In [»] part #2 of the series I said that any system made exclusively of thin-lenses and air gaps can be described by three quantities only: the *effective focal length* (EFFL), the *back focal length* (BFL) and the *front focal length* (FFL). The ABCD matrix representation of such lens systems was proven to be

In theory, any thin-lens system with 3 degrees of freedom should therefore be enough to be decomposed in the here-above matrix. Although it is not the only possible solution, a compact representation is the air-spaced doublet made with two lenses of focal length *f _{1}* and

*f*separated by a distance

_{2}*L*. Its matrix representation is

and we therefore have the following relationships

which solves to

This obviously imposes the constrains

Note that if

we find

which therefore reduces to a single thin-lens of focal-length *EFFL*.

Also, since negative lengths do not make sense, we would like to restrict the solutions to L>0 which impose the condition

for positive *EFFL* and

for negative *EFFL* (more on this below).

Also, in some cases you might want to specify the air gap between the two lenses instead of either the *BFL*, *FFL* or *EFFL*. In this case you can use one of the three following relations

Note that the solution exists only when

in case either the *FFL* or *BFL* is negative and the other one positive.

Similarly, in case you would like to specify a telecentric system with the aperture stop located at the front focal point, the total track *T=L+FFL* of the lens might be of interest. Converting the total track to other values is given using the set of equations

Finally, it may also be interesting to get direct expressions for the *BFL*, *FFL* and *EFFL* of a given system to study its tolerance. The expressions are derived from the same quantities:

The sensitivity for a change in distance *L* is obtained by differentiating the equations

Similarly, the sensitivity for a change in focal *f _{1}* gives

And for *f _{2}*

The total expected error for each quantity (EFFL, BFL and FFL) is therefore

Note that I considered here each source of error as an independent statistical variable and therefore summed the variances of the different equations. I will have the occasion to come back on this in a later post when we talk about tolerances in optical systems.

For instance, the system that has BFL=300 mm, FFL=40 mm and EFFL=135 mm consists of two thin lenses of focal lengths -37.7 mm and 65.5 mm separated by a distance of 46.1 mm. With a positioning tolerance of 50 µm and a tolerance of 1% on the focal lengths, we expect the final EFFL to be 135±6.3 mm, the final BFL to be 300±14.6 mm and the final FFL to be 40±2.8 mm. This is a relatively large error set when you compare to the 1% tolerance on the focal lengths! Either we can live with them or we will need some form of alignment procedure. The system is represented in Figure 1.

If we choose this time the same BFL and FFL but an effective focal length of 75 mm, we break the condition EFFL²>BFL*FFL and have to flip the sign of the EFFL in the computations. This results in a ray that cross the optical axis as represented in Figure 2. Note that the negative sign is only a mathematical artefact due to the fact that the definition of the effective focal length is *y/u’* and hence becomes negative when the ray crosses the optical axis.

This is all for today! I understand that this post was rather mathematical but we are finally reaching some very important stuff that will help us composing system of lenses. In the next post, I will show how we can replace thin-lenses by a more complex lens system (*i.e.* splitting of lenses).

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!

**You may also like:**

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

[»] #DevOptical Part 1: The Real, the Thin and the Thick Lenses

[»] #DevOptical Part 2: Paraxial Raytracing and the ABCD Matrix