In [part #2](/article/-devoptical-part-2--paraxial-raytracing-and-the-abcd-matrix/) 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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image001.png)

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~2~* separated by a distance *L*. Its matrix representation is

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image002.png)

and we therefore have the following relationships

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image003.png)

which solves to

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image004.png)

This obviously imposes the constrains

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image005.png)

Note that if

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image006.png)

we find

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image007.png)

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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image008.png)

for positive *EFFL* and

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image009.png)

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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image010.png)

Note that the solution exists only when

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image011.png)

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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image012.png)

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:

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image013.png)

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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image014.png)

Similarly, the sensitivity for a change in focal *f~1~* gives

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image015.png)

And for *f~2~*

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image016.png)

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

![](/images/articles/-devoptical-part-4--the-air-spaced-doublet/image017.png)

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.

![Figure 1 – Example system with air-spaced doublet](/images/articles/-devoptical-part-4--the-air-spaced-doublet/figure1.png)

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.

![Figure 2 – Negative equivalent focal length system](/images/articles/-devoptical-part-4--the-air-spaced-doublet/figure2.png)

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](https://www.patreon.com/thepulsar). 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!