**[»] Tutorials**and

**[»] Optics**.

So far we have seen how to compute the [»] third-order aberrations coefficients for a given thinlens and how to compute the effect of a [»] shift of the stop on that lens but we have no direct equation to direct our design in the first place. In my post on [»] thick-lenses, I however gave a formula to estimate the required shape of a lens to minimize spherical aberration.

In this post, I present a series of equations to compute the Seidel coefficients for any thinlens based on either its partition factor, its front, or, its rear curvature. These equations will help us selecting initial starting points during our optical design and bring some important bits of knowledge about lenses and their aberration. This post is a bit theoretical (again) but will lay the important formula required for the case study that will follow.

In its essence, computing the Seidel aberrations of a thinlens is not difficult. All it requires is computing the seidel aberration coefficients for the first and second interface whose curvature are known and labelled *c _{1}* and

*c*respectively. In practice, you quickly end up gargling in loads of pages of mathematical expansions of the neat and compact Seidel formulas. Computing the sums for the stop placed at the element, such that

_{2}*, already ease-up the situation but does not remove all the complexity. I will therefore limit myself here to the final formula adapted from the textbooks I have. I validated all of them using Zemax which was not necessarily trivial because the conventions used by the various authors were not immediately compatible!*

__h__=_{1}__h__=0_{2}In all cases, astigmatism (*S _{3}*), Petzval (

*S*) and distortion (

_{4}*S*) are easily obtained from the Lagrange invariant,

_{5}*L*, the lens power,

*P*, and its refractive index,

*n*. They are

These are given for the stop placed at the lens. When the stop is at a different position, you can adjust using the [»] stop-shift formulas.

This is already an important piece of knowledge because, **when the stop is placed on a thinlens, it is impossible to have no astigmatism**. On the bright side, **with the stop at the lens, distortion will be zero**. And since we know that we cannot affect Petzval through stop-shifting, **it is impossible to have no Petzval for a single thinlens lens, independently of the stop position**. Field curvature, on the other side, which is proportional to *S _{3}+S_{4}* can be cancelled by a proper stop position but only for one meridian, sagittal or tangential. The other meridian will always suffer from the astigmatism required to cancel the never-zero Petzval. Remember that this applies to single thinlenses only.

This leaves spherical aberration and coma which depends on the lens shape.

The *Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems* gives formula for them based on the [»] partition ratio, *X*, and the marginal ray intercept height, *h*. They are

with the magnification number, *M*,

and the thin-lens partition ratio, *X*,

where *u’* and *u* are the marginal ray angles after and before the lens and *c _{1}* and

*c*are the front and rear curvature of the lens respectively.

_{2}It is worth analyzing a bit the formula for spherical aberration and coma.

In the case of coma, we see that there is always a bending of the lens that will cancel it:

which is around *X~1* for a glass with *n~1.5* and *M=1* (collimated beam).

This is for the stop placed at the lens. When moving the stop, you can create new coma due to spherical aberration that might not be null for that bending.

The case of spherical aberration is a bit more complex because it is a quadratic equation and the existence of a solution depends on both the values of *M* and *n*.

Indeed, we have

with

A solution to the problem *S _{1}=0* exists only when

Figure 1 shows how to select a glass for a given *M* value through its index of refraction. The plot gives the minimum index of refraction to use for a given magnification number, *M*, and also displays N-BK7 and N-SF66 glasses. Only the values to the right of the black curves allows zero spherical aberration.

N-SF66 is the Schott glass with the highest refractive index in the visible and can only achieve a solution for *M>3*. N-BK7 is a very common glass and only achieve a solution for *M>4,5*. **No known glasses can produce zero spherical aberration for collimated beams ( M=±1).** Even Germanium in the infrared with its index of

*n=4*can only reach a solution for

*M>1,6*.

In all cases however, the equation admits a minimum/maximum at

which is the formula I gave in my post about [»] replacing thinlenses by thick ones.

It is worth noting that the value at which the spherical aberration reaches a minimum/maximum is very close to the one that gives zero coma for practical values of *n* (between *n=1* and *n=4*).

For collimated beams (*M=±1*), the solution is

which is a planoconvex lens (*X**=±1*) when *n~1,69* (typically, N-SF8 glass).

Most optical engineers know that a planoconvex lens with the curved side oriented towards infinity gives the least spherical aberration, but few knows that the actual best shape is planoconvex only at *n~1,69*. For other refractive indices, the best shape is either shifted towards a meniscus (*n<1.69*) or a biconvex (*n>1.69*) shape.

Orienting the curved side of a planoconvex lens towards infinity is commonly referred as the “good” orientation of a lens because placing it the other way around (the “bad” way) will produce more spherical aberration. The exact amount can be computed using the here-above formula:

In the visible (*n~1.4* to *n~1.9*) the order of magnitude is about

**When designing a lens, we are therefore constrained in optimizing either for coma or for spherical aberration.** In addition, only few cases allow to reach zero spherical aberration which means that, when shifting the stop, coma will be generated. **When optimizing the lens for zero coma, it is therefore important to fix the stop position and to target a coma value of -Q*S_{1}**

Although we will frequently rely on the partition factor to find the shape of the lens, some cases like cemented doublets are best solved by expressing the spherical aberration and coma from either the front or the rear curvature of the lens.

After adapting the formula of Kingslake in *Fundamentals of Lens Design* (themselves taken from Conrady but I don’t have a copy of his books), we obtain

with *c _{1}* the front curvature,

*c*the rear curvature,

_{2}*(h,u)* the marginal ray before the thinlens,

and the so-called **g-sums**

Similar equations exist for the rear curvature

with

That’s all for today! You should now be able to solve my [»] challenge and build an acomatic, anastigmat, singlet lens using only its shape and the position of the stop!

I will present the result in the next post. You shall not miss it because the results will most probably shock you!

From now on, we will also gradually shift toward more practical lens design features but you will see that the thinlens formulas given here-above, as well as the stop-shift equations, become very handy to understand what we are doing and direct our design forms. Without this knowledge, you cannot be a real optical designer!

I would like to give a big thanks to **Young**, **Samuel**, **Mehmet**, **Arif**, **James**, **Lilith**, **Vaclav**, **Hitesh**,** ****Jesse**, **Sivaraman**, **Jon**, **Sebastian,** **Eric**, **Themulticaster**, **Cory**, **Karel**, **Alex**, and **Marcel** 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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[»] #DevOptical Part 17: The Stop-Shift Equations

[»] #DevOptical Part 20: The Anastigmat Singlet

[»] #DevOptical Part 16: Partition Ratios Revisited