#DevOptical Part 16: Partition Ratios Revisited

In a [»] former post, I quickly introduced the concept of partition ratios of thinlenses as well as a formula to compute the best shape of a singlet lens to minimize [»] spherical aberration. The formula read as

where R₁ and R₂ were the front and back radii of curvature of the thinlens.

The partition ratio therefore represent how much of the curvature is put on the front part of the lens compared to its rear and ultimately gives the shape of the lens (meniscus, plano or biconvex and all the flavors in-between). So far, so good.

Recently, while studying the composition of doublet lenses, I felt the need of introducing a similar concept for the combination of two lenses. The idea was to keep the total power of the doublet constant but to vary the amount of power placed on the front element compared to the rear one. Since each element could take any power value, real or negative, up to infinity, I first came up with a transposition of the thinlens formula:

This however introduced a problem in the case P₁=-P₂. This might look irrelevant but it is possible (not to say frequent) to have two lenses of opposite power having a total non-zero power. This is because the total power also depends on the distance, L, between the elements:

I then updated the formula to reflect what I actually meant by partition ratio: how much of the total power is placed on the front and rear elements. The formula now reads as

which has no solution only when the total power is null.

The individual powers can be retrieved by solving the quadratic equation

with

As with the [»] thinlens, only one solution makes physical sense.

Note that in the context of a thin-doublet (L=0) we have

for which we can verify that when X=0 we get P₁=P₂=P_tot/2.

Having stated the partition ratio like this, we can also update our formula for singlet lenses to handle thick-lenses as well. The formula becomes:

It represents how much of the power is placed on the front curvature, P₁=(n-1)*c₁, and how much is placed on the back curvature, P₂=-(n-1)*c₂, compared to the total power of the lens P. From our post on [»] thick lenses we also have

with T the thickness of the lens.

Again, the individual curvatures can be found by solving the quadratic equation

with

and

where still only one solution makes physical sense.

Note that the case T=0 (thinlens) gives back

which is therefore a particular case of a more general formula.

You may wonder what’s all this fuzz about partition ratios. The reason is best explained with an example.

Figure 1 shows four different doublets. They all have an equivalent focal length of 100 mm and are optimized to minimize chromatic aberration and to cancel both spherical aberration and coma (more on that later). The two first doublets (X_p=+2.78) have the first lens made of N-BK7 glass while the second glass is N-SF64. For the two lasts (X_p=-2.78), it is the opposite. It is exactly the kind of elements you want to use in your optical designs! Note that although the two sets seem to be mirror image but they are not as you can see with the individual lenses partition ratios (X₁ and X₂).

The great potential of partition ratios is that they allow to separate the construction parameters from the correction of aberrations. When you change X_p, you don’t change the overall doublet power but you will affect more or less of this power to the front element. Similarly, when you change X₁ and X₂ you don’t modify the individual lens powers but only how much of that power is placed on the front surface. As we have seen in our [»] previous post, X₁ and X₂ affects coma and spherical aberration. I haven’t touched about chromatic aberration yet but it is affected by X_p.

Changing the partition ratio allows you to smoothly affect aberration in a much less cumbersome way because you have only 3 parameters here to deal with (X_p, X₁ and X₂). Also, when increasing the distance L between the lenses, or the thickness T of each lens, the neat formula we have just set allows to keep the same bending ratio and overall power of the doublet without having to tune anything.

Once your mind has been freed of all the unnecessary things, you can focus on finding the values of X_p, X₁ and X₂ that will optimize the performance of your doublet. This however goes beyond the scope of this post but we will unravel the mystery one piece at a time!

I would like to give a big thanks to Young, Samuel, Arif, Mehmet, James , Lilith, Vaclav, Hitesh, Jesse, Sivaraman, Jon, Themulticaster, Eric, Cory, Karel, Kewei, Marcel and Alex 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!