Today we continue our journey into the exciting world of [third-order aberration theory](/article/-devoptical-part-14--third-order-aberration-theory/) by deriving the extremely important set of **stop-shift equations**.

The stop-shift equations allow computing the seidel aberration of a surface when moving the stop in the system using their values at the former stop position. I must confess that, when I was first exposed to them, back in 2017, I completely disregarded their usage thinking that I would not need them because I had the computational power to directly access the seidel terms everytime I updated the system. I was wrong, terribly wrong! The stop-shift equations play a central role in optical system design and are mandatory to understand the origin of aberrations in a system and how we can tailor the system to minimize or even cancel them!

Now that I have all your attention, we can start digging into some maths :)

In my previous post, I introduced the Seidel aberration contribution of a surface as

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image001.png)

where *h* is the ray intercept height, *c* the surface curvature, *u* the incoming ray angle, *u’* the exit ray angle, *n* the initial refractive index, *n’* the refractive index after the interface and

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image002.png)

where *(h,u)* refers to the marginal ray and *(h,u)* to the chief ray.

The quantity

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image003.png)

represents the change of *x* as it goes through the interface.

I also shown that the following relations holds directly from the previous representation

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image004.png)

Recall that *S~1~* is the (seidel) spherical aberration, *S~2~* is the (seidel) coma, *S~3~* is the (seidel) astigmatism, *S~4~* is the Petzval and *S~5~* is the (seidel) distortion. I put “seidel” in front of each because these coefficients need to be scaled to convert to wavefront. You can come back to my [initial post](/article/-devoptical-part-14--third-order-aberration-theory/) to read more on that.

Let’s now re-arrange a bit these last equations.

We can compute the coma for two different chief rays *(h~0~,u~0~)* and *(h~1~,u~1~)* but the same marginal ray *(h,u)*:

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image005.png)

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image006.png)

Since *S~1~* does not depend on the chief ray *(h,u)* we have

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image007.png)

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image008.png)

 

The added coma due to the change in chief ray is

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image009.png)

Knowing that, from the Lagrange invariant, we have

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image010.png)

we have the relation

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image011.png)

leading to

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image012.png)

with *Q* the **eccentricity factor**

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image013.png)

This means that, knowing the seidel coma of the surface at some chief ray *(h~0~,u~0~)*, we can compute its value for another chief ray position *(h~1~,u~1~)* using the eccentricity factor as

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image014.png)

In practice, we affect the chief ray by moving the stop in the system. This is represented in Figure 1 where you can see that moving the stop lower the chief ray interception height with the surface.

![Figure 1 – Stop-shifting a lens](/images/articles/-devoptical-part-17--the-stop-shift-equations/figure.png)

Let’s analyze a bit what is going on exactly in Figure 1. You can see a lens in black and a STOP that is moved from the lens to some other position in front of the lens. The initial chief ray, in purple, crosses the optical axis at the lens, so *h~0~=0*. This is because of the definition of what a chief ray is. By moving the STOP, we get the green chief ray which intercept the optical axis at the STOP position (again, by definition), and now intercept the lens at some higher position *h~1~*. You can obtain the value of *Q* by using the here-above formula. In Figure 1, since the marginal ray intercept the lens at about the same height as the new chief ray and the initial chief ray intercept height is zero, we get *Q**≈**1*.

We can derive similar equations for all the Seidel aberrations:

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image016.png)

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

It is frequent to apply the stop-shift equations to an initial situation where the stop is located at the element in which case *h~0~**=0* but this is by no mean compulsory.

So why are the stop-shit equations so important in optical design? This is because they allow predicting the aberration of a system as you move the stop. For example, it allows to state that **if an interface has spherical aberration, there will always be a position of the stop that will cancel coma**:

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image017.png)

which will occur when

![](/images/articles/-devoptical-part-17--the-stop-shift-equations/image018.png)

*At contrario*, we know that **it is impossible to correct the coma of a system by moving the stop if the interface has no spherical aberration**.

Once you have identified the eccentricity value you would like to achieve, you can compute the amount of shift of the stop to apply knowing *h~0~*, *u~0~* and *u~1~* (or indirectly *via* *L*).

Similarly, **an interface that does not have coma nor spherical cannot be corrected for astigmatism by moving the stop**.

**Distortion is affected by all other aberration types** but it can be neglected as it only produce a shift in the metric of the image but does not impair image quality and can even eventually be corrected by software after image acquisition.

Only **spherical and Petzval can never be corrected by shifting the stop**. There are however other methods to handle them but this goes beyond of the scope of this post!

In a future post, I will show how it is possible to produce an acomatic (*S~2~=0*), anastigmat (*S~3~=0*), system using only a singlet lens by controlling its shape and stop position. All that made possible by the stop-shift equations! Can you already spot how I intend to achieve that?

I would like to give a big thanks to **Young**, **Samuel**, **Arif**, **Mehmet**, **James** , **Lilith**, **Vaclav**, **Hitesh**,** ****Jesse**, **Sivaraman**, **Jon**, **Themulticaster**, **Sebastian**, **Eric**, **Cory**, **Karel**, **Alex**, **Kewei**, and **Marcel** who have supported this post through [Patreon](https://www.patreon.com/thepulsar). 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!