**[»] Optics**.

In my [»] previous post I described how an air-spaced doublet can be used to represent any system of thin-lenses by using only two focal lengths and one air distance as degree of freedom to get customized front focal length (FFL), back focal length (BFL) and effective focal length (EFFL) at once. I also mentioned that it was not the only combination that could represent thin-lenses system.

At that time, I skipped on purpose the trivial solution that consists of only one thin-lens with two air gaps. Yes, you read me right: one thin-lens is enough to represent any system made of several thin lenses. This is not completely true because some of the systems are not physically accessible because they would require negative air gaps which only exists as mathematical curiosities. Nonetheless, the derivations that we will do today are of uttermost importance for the conception of systems (more on this in a few minutes!).

The matrix representation of a thin-lens of focal *f* surrounded by two air-gaps *L _{1}* and

*L*is

_{2}Which we have proven [»] here to correspond to the generic thin-lens system matrix described by

And therefore

for which a physical solution exists only if the distance *L _{1}* and

*L*are non-negatives

_{2}Now comes the interesting part. Let us multiply the matrix *M* by the inverse matrices of air-gaps. We obtain

But since *M* also has the form

And that the inverse of an air-gap matrix is the matrix of a negative air-gap of the same length

We can state that any system of thin-lenses is equivalent to a thin-lens of the same focal length provided we pad the previous system by air-gaps -*L _{1}* and -

*L*as derived here-above

_{2}The interest part in this formula is to be able to compute how we can replace a single thin-lens by a system containing more lenses. More lenses mean more money but it also means less aberrations because you gain extra degree of freedom. We will cover aberration and aberration reduction later, but keep in mind that sometimes having more lenses is better.

As an illustration, imagine you have a thin-lens of focal-length *f* that you wish to replace by an air-spaced doublet. The matrix representation of the [»] airspace doublet is

And we have the relations

When replacing a thin-lens, we usually select the smallest distance that is physically possible to have two lenses almost in contact. Imagine that we also say that we want *f _{1}=f_{2}=f’* for convenience. The first thing to do is to find

*f’*using the equation of

*EFFL*

Then we compute the quantities *L _{1}* and

*L*from the back and front focal lengths using that value

_{2}*f’*

We can then replace the thin-lens *f* by two thin-lenses *f _{1}=f’* and

*f*separated by distance

_{2}=f’*L*and with air-gaps of

*–L*and

_{1}*–L*around it

_{2}Figure 1 shows a raytracing plot of such replacement for a 1:1 imager system of a f=100 mm lens.

Here we replaced a thin-lens by an air-spaced doublet but we could have done the same with any system of thin-lenses. In fact, we can also replace a thin-lens by a thick-lens using the same procedure. Since I did not define what is the ABCD matrix of a thick-lens, we will have to keep this for later. I believe that having the ability to replace a thin-lens by a custom lens-system (either simple or complex) is a must-have feature in a software such as what I would like to achieve for the #DevOptical concept!

I would like to give a big thanks to **James**, **Daniel**, **Naif**, **Lilith**, **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!

**You may also like:**

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

[»] #DevOptical Part 6: Telecentric, 4f and Afocal Systems

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

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