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In a [»] previous post, I challenged the reader to find how to produce a system with no coma and no astigmatism using only a singlet lens (did you find how?). Here, I present the solution to the problem. Prepare to be shocked by the conclusions! :-)

The derivations presented here heavily rely on the [»] optical aberrations of tinlenses and [»] the stop-shift formulas, so be sure to have a good understanding of these before proceeding.

The [»] stop-shift equations give the aberration of thin elements as you move the stop based on the known aberrations at a given stop position. They are expressed as

where the starred values are those after the shift of the stop.

Furthermore, [»] the aberrations of a thinlens with the stop at the lens are given by

where *P* is the total power of the lens, *h* the ray intercept height at the lens, *L* the Lagrange invariant, *n* the refraction index, *X* the [»] partition ratio and *M* the magnification number. *S _{1}* to

*S*are the [»] Seidel aberrations.

_{5}If you aren’t so sure about the meaning of any of these terms, check back our previous posts until you are confident you understand them all.

Looking at all these equations we can state that the shape (bending) of the lens will affect both the coma and spherical for a given power and marginal ray intercept height when the stop is at the lens. On the other hand, moving the stop will affect coma, astigmatism and distortion.

Provided there is some spherical aberration, there exists a position of the stop that will cancel coma independently of the bending of the lens.

Indeed,

when

Knowing that

we can convert *Q* to a position for the stop using the chief-ray angle, *θ*,

At this stop position, we can also compute the value of astigmatism

This means that it may exists a configuration where astigmatism cancels

And since we have the expression of the thinlenses here-above for the non-starred terms *S _{1}*,

*S*and

_{2}*S*, we can develop the expression into

_{3}which expand to the following quadratic equation after a lot of work

where

As a consequence, there exists one, two or no bending of a thinlens that will provide zero coma and zero astigmatism and the solutions are given by the quadratic function here-above and depends only on refractive index, *n*, and magnification number, *M*.

Well, that’s nice but you aren’t shocked so far and I promised some of you would fall from their chair in front of the results. So, what’s the big deal?

Let solve the equation for the common case of an incoming collimated beam. In such condition, *M=+1*.

The equation becomes

It admits two solutions *X _{1}* and

*X*

_{2}The first solution is a plano-convex lens with the plano side oriented towards infinity and the second is a meniscus lens.

To compute the position of the stop, we first need to compute the values of Q using our former formula to cancel coma

We find

which we convert to a distance *d* using the chief ray slope *θ*

using the fact that (by definition)

Conversely, we find

and

The two solutions are given in Figure 1.

If you are a bit in the optical engineering world, the top solution (plano-convex) should trigger your attention: the lens is placed **“with the wrong orientation”**! All junior optical engineers know that plano-convexes lenses shall always be oriented with the curvy side towards infinity to minimize aberrations! In fact, this is even a question that my employer used to ask during job interviews. At that time, like most people, I answered like the textbooks: curvy side forward! And now I’m telling you to do the exact opposite.

So, where’s the catch?

First, and this is important to emphasize, the maths presented here are correct. A plano-convex lens placed with the plano-side towards infinity will give zero coma and zero astigmatism provided the stop is at the correct position. This is true at any refractive index and therefore at any wavelength (UV, visible, deep infrared *etc.*).

Second, what is currently referred to as “the correction orientation” is entirely defined through the perspective of spherical aberration. In my [»] previous post I showed that a plano-convex lens will have less spherical aberration with the convex side towards infinity than the other way around.

The key difference here is that one solution focuses on **off-axis performances** while the other one focuses on **on-axis performances**. By minimizing spherical aberration, you emphasize on-axis, but, by cancelling astigmatism and coma you emphasize off-axis. Also, and that worth to be mentioned, the plano-convex shape gives only a true minimum for spherical aberration when *n~1.69*. Below that you move to a meniscus lens and above to a biconvex one. Concerning the anastigmat configuration this time, the plano-convex shape is the optimal one independently of the refractive index, so it is always the best solution.

Finally, we may ask ourselves which of the two anastigmat solutions has the best optical performances.

We know that in both cases coma and astigmatism are zero by design. Petzval is unaffected by shape and stop position and will be the same for both solutions. Stays spherical aberration.

For the plano-convex configuration we have

And for the meniscus configuration we have

Since the refractive index, *n*, is always larger than 1, we know that the plano-convex configuration will have the least aberrations of the two. That is a relatively good news because any COTS supplier proposes plano-convex elements for all kind of focal lengths and you just boosted your application range through these off-the-shelves elements!

This is confirmed by a MTF analysis. The results are displayed in Figure 2 and Figure 3 for the plano-convex configuration and meniscus configuration respectively. MTFs are given for a 4° maximum field-angle and a speed of f/10.

As a comparison, the MTF of the same plano-convex lens but with the conventional orientation (on-axis configuration) is given in Figure 4. The STOP position was optimized for coma as well.

As expected, the first configuration yields an overall better MTF due to its reduced spherical aberration compared to the second one. We can also see that the conventional lens orientation offers a very good on-axis performance but quite poor off-axis ones *at contrario* to our off-axis optimized configurations. One important aspect here corresponds to the amount of astigmatism in the various MTF plots. Indeed, our two anastigmat configurations show no (or very little) astigmatism while the conventional lens orientation has plenty of it (as can be seen in the separation between the S and T curves). Also, both configurations show very consistent image quality throughout field which is, again, not the case of the conventional orientation.

So, the next time people ask you what is the good orientation for a planoconvex lens, make them fall off their chair too! :D

That’s all for today! I hope that you enjoyed the twist of this post and the unexpected conclusion that there is more to the good and wrong orientation of a plano-convex lens than first meets the eye! We will now tackle a new challenge: cancelling spherical aberration of our plano-convex singlet lens as well!

Want to discuss this further? Check out our new [∞] community board!

I would also like to give a big thanks to **Naif**,** Young**,** Samuel**,** Eric**,** James**,** Lilith**,** Andrew**,** Hitesh**,** Sebastian**,** Jesse**,** Sivaraman**,** Jon**,** Cory**,** Karel**,** ****Themulticaster**, **Alex**,** Tayyab**,** Stephen** 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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