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The thin-doublet is an important step forward in our journey because it is one of the most used optical design elements out there. If you check suppliers like Thorlabs or Edmund Optics, you will see that they offer as much, if not more, doublet lenses than singlet elements.

But if the thin-doublet is an important step forward, it is also because we need to develop some kind of strategy this time to derive its equations, compared to what we have done previously.

If we recap, we have seen how aberrations are affected by [»] lens bending and [»] STOP position. I then presented two different methods to cancel coma and astigmatism using either the [»] thin-lenses aberration theory or the [»] Seidel equations directly. We then focused on how to cancel the remaining aberrations by presenting the [»] aplanatic aspheric lens, the [»] Schmidt corrector and, more recently, the [»] Perfect Dialyte.

All these corrections were relatively easy in the sense that they are kind of orthogonal: correcting one does not influence the other. In the aplanatic aspheric lens, we first corrected the lens for coma and then adjusted the asphericity to cancel the remaining spherical aberration. In the Schmidt corrector, we introduced a second element to cancel the remaining spherical aberration of the system. And in the Perfect Dialyte, we draw very simple equations to cancel astigmatism and Petzval provided that both lenses were aplanatic.

Here, things are different, and we will have to cancel aberrations using elements that influence each other.

In this post, we study systems of two thinlenses with no air gap between them and how we can control aberrations with them. They are known as **thin doublets**.

The first order study of the system is pretty straightforward because two thinlenses of power *P _{1}* and

*P*without an airgap can be represented by the matrix

_{2}which is equivalent to a thinlens of power *P _{1}+P_{2}*.

The total power of the thin doublet will be noted *P*. In a first order study, it is therefore very convenient to express the thin doublet in terms of total power and [»] partition ratio which tell us how much of the power is attributed to the front and rear elements:

The interesting part comes with the third-order aberration study.

Each of the two elements will contribute to the total aberrations of the system:

and we can compute the individual aberrations using the [»] thinlens aberration theory and the [»] stop-shift theory.

The aberrations of a single thinlens at the STOP are given by

where the subscripts *a* stand for each individual lens.

Each of these quantities then need to be corrected by the stop-shift equations to take into account the actual position of the STOP

where *Q* is the eccentricity factor given by the ratio between the interception height of the chief ray and the marginal ray.

This is where things simplify a bit because thin doublets have no airgap and therefore shares the same eccentricity factor

and therefore

Most usages of the thin doublet focus on achieving no spherical aberration (*S _{1}=0*) and no coma (

*S*) such that we have

_{2}=0where I neglected distortion because it only affects the metric and does not blur the image.

A thin doublet corrected for spherical aberration and coma will therefore always have astigmatism and Petzval. The case *P _{1}+P_{2}=0* produces zero astigmatism but is rarely useful because that would mean a doublet with no optical power. On the other hand, the case

*P*is already more interesting because it will cancel Petzval provided we use two glasses of different material such that

_{1}/n_{1}+P_{2}/n_{2}*n*. The condition is met with a partition ratio of

_{1}≠n_{2}In practice it is rarely useful and requires using glasses with very different indices of refraction to avoid having very large partition ratios which would require extremely powerful elements that would bring higher order aberrations. For instance, using N-SF66 with N-BK7 yields *X _{p}=8.47* which means a 10D doublet would require using

*P*and

_{1}=47.35D*P*. These are already extremely powerful lenses for a moderate overall power of 10D.

_{2}=-37.35DI will now focus on the spherical aberration and coma only, considering that the power partition ratio is fixed. In later posts, I will show other methods to select the propre value for *X _{p}*.

Considering that the powers of each element are fixed (because we chose a value for *X _{p}*), we need to find the bending of the two lenses such that we cancel spherical aberration and coma at the same time. This is not easy because the bending of each lens affect both the total spherical aberration and coma and we cannot use the orthogonal method we used so far to first cancel one aberration and then use a second element or parameter to cancel the remaining aberrations.

The only solution is to solve both aberration at the same time using a system of two equations with two variables:

Because each of the thinlens aberration function is a relatively complex formula, I will rewrite the equations for spherical aberration and coma as polynomial functions of their respective bending *X _{a}*

where the coefficients are

we therefore have

which is a system of a one second order and one first order equations.

We will solve this system just like any other first order system by using one equation to link *X _{1}* to

*X*and solve the value for the remaining parameter using the second equation.

_{2}Because the coma is a linear equation, I will use this equation to link *X _{1}* to

*X*

_{2}that I can now reinject in the equation of spherical aberration to yield a new polynomial of the form

with

which, just like any other second order equations, admits either two, one or no solutions.

Combining the two equations, we therefore get one, two or no solutions that have no spherical aberration and no coma.

While there are complex cases involving the values of *P _{1}*,

*P*,

_{2}*n*,

_{1}*n*,

_{2}*M*and

_{1}*M*, a relatively straightforward analysis shows that combining a strong negative element with a strong positive element (or vice-versa), can yield a satisfactory condition. Indeed, we see that spherical aberration is proportional to the cube of the power of the lens and to some other factor that is a function of the other variables but independent of the power:

_{2}Using the sign of the power, we can therefore control the sign of the spherical aberration by looking at the sign of the other term. And since the total spherical aberration is the sum of the two separate spherical aberrations, we should be able to find bending that satisfy a complete compensation of the spherical aberration of the first lens by the second one.

For coma, things are easier because it’s a linear relationship so there will always be a bending for the second lens that produces a compensation of the first lens.

I will present techniques to select the proper power distributions in a later post but, for now, remember that using partition ratios whose magnitude are larger than 1 should produce solvable cases for collimated rays.

Figure 1 shows an example of the two solutions for a 20D doublet made of the same glasses. The first solution is referred to as the **Gauss form** while the second solution is known as either the **Fraunhoffer** or the **Steinheil form** depending on the choice of the glasses when we use different glasses. Note that although we talk about thin-doublets, I reoptimized the case using a thin air gap for manufacturability purposes but the initial conditions were obtained using the formula here-above.

When using two different types of glass, we double the number of solutions because we can choose to either put the first glass at front or at rear of the doublet. The four solutions are shown in Figure 2.

All these forms have zero 3^{rd} order contributions but have different amounts of residual, zonal and chromatic aberrations. These topics will be left for another post as they are more advanced.

We can use any of the forms of Figure 2 as a replacement for an [»] aspheric lens of the same power. It is usually more economical to use doublets but leads to more mass and footprint which is not always acceptable.

Before we conclude this post, there is one important case that we need to discuss: **cemented doublets**.

If you watch closely Figure 2 you will see that the second and third solutions have lenses that looks like they want to mate each other. Mathematically speaking, the radii of curvatures of the rear surface of the first lens almost matches the front one of the second lens. It is tempting to force the two to be exactly equal such that we can remove the air-gap between them.

One advantage of doing so is that we can ask the optical manufacturing shop to perfectly align the two lenses and “glue” (**cement** is the proper word) them such that we now have only one lens assembly to deal with. Most doublet lenses sold by Thorlabs or Edmund Optics are in fact cemented doublets!

This however comes at the condition that the two lenses are made of different materials (*n _{1}≠n_{2}* or there is no refraction at the interface) and also at the expense that we lose one degree of freedom and can therefore not correct anymore both coma and spherical aberration at the same time. This later remark seems to kill all practical interest in cemented doublets but it is possible to achieve zero coma and very low remaining spherical aberration through a smart choice of glass materials.

The two corresponding cemented doublets of Figure 2 optimized for coma are shown in Figure 3. Note that it is common to have the negative element having a larger diameter.

Now that we understand the concept of a cemented doublet, let’s dig into the associated first and third order maths.

The first order matrix of the thick cemented doublet lens is

This formula can be used to find the value of *c _{3}* once

*c*and

_{1}*c*are chosen since the total power of the doublet is fixed. The thin cemented doublet equations can be obtained by setting the thicknesses

_{2}*T*and

_{1}*T*equal to zero.

_{2}As mentioned, once the glasses and the power partition ratio are chosen it is not possible to have both equations to control spherical aberration and coma. What is usually done is that the lens is corrected for coma and a suitable pair of glasses is chosen such that spherical aberration is almost completely cancelled. Alternatively, but less often used to my knowledge, the power partition ratio can also be used to cancel spherical aberration as well.

To cancel coma, we will rely on different forms of the aberration formula that we presented in our [»] thinlens aberration post:

where *c _{a}* and

*c*are the total curvatures of the front and rear element and

_{b}*c*is the curvature of the cemented interface

_{2}Since

we find a linear equation

with

which gives a solution for *c _{2}=-k_{1}/k_{2}*.

Once *c _{2}* is fixed we find

*c*and

_{1}*c*from

_{3}*c*and

_{a}*c*. The total spherical aberration can then be computed and adjustment made to either the glasses choice or the power partition ratio to cancel it.

_{b}For a given glass choice, we therefore have only one solution. And since we can swap the glasses, we get the two solutions of Figure 3.

As a conclusion, we saw that there are two aplanatic solutions for thin doublets of the same glass, four aplanatic solutions if we use different glasses and two more nearly aplanatic solutions by cementing elements.

The total six solutions to our 20D doublet problem are given in Figure 4.

In my next post, I will revisit my Perfect Dialyte using our newly discovered aplanatic lenses!

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I would also like to give a big thanks to **Young**, **Naif**,** Samuel**,** James**,** ****Sebastian**,** Lilith**,** ****Alex**,** ****Stephen**,** ****Jesse**, **Jon**, **Sivaraman**,** Cory**,** Karel**,** ****Themulticaster**,** Tayyab**,** Marcel**, **Kirk **and **Dennis** 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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