#DevOptical Part 24: The Schmidt Corrector

We have already seen how to create a system with no coma and no astigmatism in [»] this post using [»] thinlens aberration and the [»] stop-shift theory and in [»] this post using the Seidel equations directly. The two systems studied had good off-axis performances but still suffered from spherical aberration (SA) which ultimately limited the overall MTF. One of the systems performed better than the other in that regard with a difference in SA proportional to the square of the refractive index of the glass being used.

More recently, we also discussed [»] aspheric lenses and how tuning the shape of the surface using a power series could cancel the spherical aberration of a system.

It would be tempting to try to apply some asphericity to our thinlens to also compensate SA directly but we need to recall from the [»] stop-shift theory that some spherical aberration is required to cancel coma by a proper placement of the STOP:

with S₁ the spherical aberration, S₂ the STOP-centered coma, Q the eccentricity factor due to a shift of the STOP and S₂^* the coma after the shift of the STOP.

If S₁ had been zero, there would be no position of the STOP (Q) that would allow to cancel the intrinsic coma. So, we cannot modify the thinlens directly and the only solution is to add another element. We will use what is called a Schmidt corrector.

A Schmidt corrector, or Schmidt corrector plate, is a plane window where one of the surfaces has been modified by a power series with no power (a₂=0). We have seen that an aspheric surface located at the STOP only adds spherical aberration so a natural position to use a Schmidt corrector is to place it at the STOP:

When the Schmidt corrector is placed at another position in the system, we must account for the effects of the stop-shift theory.

Since we are using a plane window for our corrector plate, we must also account for the effects of the window itself:

where u₀ is the slope of the marginal ray, u₀ is the slope of the chief ray, T the thickness of the window and n the glass refractive index.

The easiest case to process is when the marginal ray at the STOP equals zero (u₀=0) since the window does not add any aberration at all and we only have the asphericity. This corresponds to our planoconvex lens with the STOP placed at a distance P*(n-1)/n from the lens where P is the power of the lens. This is shown in Figure 1.

In this configuration, we need to solve a₄ such that

Note that I assumed that the window is made of the same glass as the lens, but this is by no means compulsory.

The MTF corresponding to the system of Figure 1 is given in Figure 2. We see that the system is close to diffraction limit up to 4°. The remaining aberration is Petzval curvature which produces a defocus proportional to field height. You may notice in Figure 2 that I optimized the focus position such that the field at mid-height is in-focus such as to limit the total defocus at 4°.

The second system, which is more generic, is a bit more complex to process since we have to account for the window this time. The layout is shown in Figure 3.

One possibility is to neglect the effect of the window by using one with very small thickness. A more formal solution is to alter the bending of the lens and the distance between the lens and the STOP to perfectly balance the system. At this stage, it becomes relatively complex to process through analytical computations and the best is to start by neglecting the effect of the window and let the optical design software optimizer do the modifications for us as we expect that the solution must not be that far away from the approximated solution.

The MTF of the system of Figure 3 is given in Figure 4. The performances are as good as in Figure 2 despite the initial system had more spherical aberration in the first place. You may see that I did not run the optimizer perfectly because there is now a touch of astigmatism at 4° (separation of the plain and dashed curves). In practice, if you were to build the system of Figure 3, you would not be able to see any difference in image quality compared to the one of Figure 1.

Using a system of only two elements (one singlet lens and one corrector plate) we were able to achieve S₁=S₂=S₃=0. We still suffer some Petzval curvature (S₄) that limits the off-axis performances, but the MTF is already pretty good up to 4° for a 20D lens. One downside however is that Schmidt corrector plates are expensive to manufacture, and we lose a part of the attractivity of the initial solution that was relying exclusively on planoconvex lenses which are standard elements that can be bought at any optical supplier shop. Nonetheless, this was an interesting way to start complexifying our systems by the introduction of a second element!

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I would also like to give a big thanks to Naif, Young, Samuel, James, Andrew, Lilith, Sebastian, Hitesh, Jesse, Alex, Aviv, Stephen, 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!