Solving the Corrector Plate Equation for Schmidt Systems

Unified Equation

As shown in the figure, the shapes of the primary and secondary mirrors are represented by \vec{R} and \vec{r} respectively. The optical path is \vec{e} \rightarrow (\vec{r}-\vec{R}) \rightarrow (\vec{K}-\vec{r}), where Q is the focus, \vec{K} is the focal vector, and |\vec{e}| = 1. For convenience, let \vec{D} = \vec{r} - \vec{R}, and let R, r, D be the magnitudes of \vec{R}, \vec{r}, \vec{D} respectively.

According to the law of reflection, the angle between \vec{D} and the tangent of \vec{R} is the same as the angle between \vec{e} and the tangent of \vec{R}. Therefore: \vec{e} \cdot d\vec{R} = \frac{\vec{D}}{D} \cdot d\vec{R} \tag{1} Similarly: \frac{\vec{D}}{D} \cdot d\vec{r} = \frac{\vec{K}-\vec{r}}{|\vec{K}-\vec{r}|} \cdot d\vec{r} = -\frac{\vec{K}-\vec{r}}{|\vec{K}-\vec{r}|} \cdot d(\vec{K}-\vec{r})

Using the identity \vec{r} \cdot d\vec{r} = r dr (obtained by differentiating both sides of \vec{r}^2 = r^2): \frac{\vec{D}}{D} \cdot d\vec{r} = -d|\vec{K}-\vec{r}| \tag{2} Subtracting (1) from (2) gives: -d|\vec{K}-\vec{r}| - \vec{e} \cdot d\vec{R} = \frac{\vec{D}}{D} \cdot d(\vec{r}-\vec{R}) = \frac{\vec{D}}{D} \cdot d\vec{D} = dD

Integrating once, we get: D + |\vec{K}-\vec{r}| + \vec{e} \cdot \vec{R} = C \tag{3} The other equation is provided by (1). In other words, (1) and (3) together determine \vec{r}. This is the general form of the corrector plate equation for non-parabolic telescopes. As seen in the figure, the focal length is f = C - R_-, where "R_-" is the magnitude of \vec{R} at the horizontal position. In practical applications, the equation needs to be simplified based on the specific form of the primary mirror. Below, we apply this logic to the case where the primary mirror is spherical (i.e., the Schmidt system). This is the core content of this article, the result of over a month of reflection from November to December last year...

Schmidt System

Establish a Cartesian coordinate system with the center of the circle at the origin, and set the following: \begin{aligned}\vec{R} &= (\cos\theta, \sin\theta), \quad \dot{\vec{R}} = (-\sin\theta, \cos\theta) \\ \vec{D} &= (D\cos\varphi, D\sin\varphi), \quad \vec{e} = (1, 0), \quad \vec{K} = (k, 0)\end{aligned} \vec{r} = (D\cos\varphi + \cos\theta, D\sin\varphi + \sin\theta); Initial condition: \theta = 0 \Rightarrow \varphi = \pi.

Substituting these into (1), we get: \sin\theta = \sin\theta \cos\varphi - \sin\varphi \cos\theta = \sin(\theta-\varphi) \Rightarrow -\pi-\theta = \theta-\varphi

Then we can write: \begin{aligned}\cos(\theta-\varphi) &= \cos(-\pi-\theta) = -\cos\theta = -\cos\left(\frac{\varphi-\pi}{2}\right) = -\sin\left(\frac{\varphi}{2}\right) \\ \sin\theta &= -\cos\left(\frac{\varphi}{2}\right)\end{aligned}

Substituting these into (3): \begin{aligned}D + \sqrt{D^2 + 1 + 2D\cos(\theta-\varphi) + k^2 - 2k(D\cos\varphi + \cos\theta)} + \cos\theta &= C \\ D + \sqrt{D^2 + 1 - 2D\sin\frac{\varphi}{2} + k^2 - 2k(D\cos\varphi + \sin\frac{\varphi}{2})} + \sin\frac{\varphi}{2} &= C \\ D^2 + 1 - 2D\sin\frac{\varphi}{2} + k^2 - 2k\left[D(1-2\sin^2\frac{\varphi}{2}) + \sin\frac{\varphi}{2}\right] &= \left(C - \sin\frac{\varphi}{2} - D\right)^2\end{aligned}

Simplifying gives: C^2 - 1 - k^2 + (2k - 2C) \sin \frac{\varphi}{2} + \sin^2 \frac{\varphi}{2} = 4k\left( \frac{C}{2k} - \frac{1}{2} + \sin^2\frac{\varphi}{2} - \frac{1}{k} \sin\frac{\varphi}{2} \right)D

At this stage, we can discuss two cases:

(1) When k = 0: C^2 - 1 + \sin^2 \frac{\varphi}{2} - 2C\sin\frac{\varphi}{2} = 4\left(- \sin \frac{\varphi}{2} + \frac{C}{2}\right)D

To make the form simpler, we take C=2, yielding 4D = 3 - \sin\frac{\varphi}{2}. If we substitute this result into \vec{r} = (D\cos\varphi + \sin\frac{\varphi}{2}, D\sin\varphi - \cos\frac{\varphi}{2}) and convert back to Cartesian coordinates, it is not difficult to find that this is a sixth-degree function. The telescope formed by it is shown in the figure above.

(2) When k \neq 0, we might try to make D a constant, which requires: \begin{aligned}C^2 - 1 - k^2 &= \frac{C}{2k} - \frac{1}{2} \\ C - k &= \frac{1}{2k}\end{aligned} (So that both sides can be reduced).

Surprisingly, the two equations above are actually self-consistent, meaning: C^2 - 1 - k^2 = \frac{C}{2k} - \frac{1}{2} \iff C - k = \frac{1}{2k}

Then we have D = \frac{1}{4k}, which is exactly the result we hoped for! This not only simplifies the equation form but also leaves the result non-unique (as k can be set freely).

According to \vec{r} = (D\cos\varphi + \sin\frac{\varphi}{2}, D\sin\varphi - \cos\frac{\varphi}{2}), let x = D\cos\varphi + \sin\frac{\varphi}{2} and y = D\sin\varphi - \cos\frac{\varphi}{2}. We have x^2 + y^2 = D^2 + 1 - 2D\sin\frac{\varphi}{2}.

x = D(1 - 2\sin^2 \frac{\varphi}{2}) + \sin\frac{\varphi}{2} = D\left[1 - 2\left(\frac{D^2 + 1 - x^2 - y^2}{2D}\right)^2\right] + \left(\frac{D^2 + 1 - x^2 - y^2}{2D}\right)

Expanding this yields: (x^2 + y^2)^2 - \left(1 + \frac{1}{8k^2}\right)(x^2 + y^2) + \left(\frac{1}{256k^4} - \frac{1}{16k^2}\right) + \frac{x}{2k} = 0

This is a quartic function (mathematical software can quickly expand it into a series form of x = f(y^2)). This type of curve is also called a "toric section" (toric section, just as a conic section is formed by the intersection of a cone and a plane, this is formed by the intersection of a plane and a torus). Its focal length is f = C - 1 = \frac{1}{2k} + k - 1.

When k=1, the structure is as follows:

This is the most common optical structure of the Schmidt system. The corrector plate equation is: (x^2 + y^2)^2 - \frac{9}{8}(x^2 + y^2) + \frac{x}{2} - \frac{15}{256} = 0

It is easy to see that the central part of the corrector plate is very close to a straight line. This indicates that when the spherical surface is not large, it can be used as a parabolic mirror with a focal length equal to half the radius. In fact, when x = 0.7499999 and y = 0.015—that is, with a precision of 0.0000001—a spherical mirror with an aperture (diameter) of 0.06 and a radius of 1 can be used as a parabolic mirror with a focal length of 0.5.

Analysis and Summary

This article uses vectors and basic calculus as tools to first derive the "general form of the corrector plate equation for non-parabolic telescopes." Based on this, a brief analysis of the Schmidt system was conducted, finally yielding results such as the corrector plate equation and the focal length formula for the Schmidt system. BoJone is an astronomy enthusiast and, even more so, a mathematics enthusiast. While writing this article, I did not consult current optical literature, so there may be discrepancies in terminology and notation compared to those works. The author purely analyzed this as a common problem of physics and mathematics to derive some small results and share them with everyone, hoping to serve as a modest contribution to stimulate further discussion.

References:

1. Pan Junhua, Design, Processing, and Testing of Optical Aspheric Surfaces.

2. Wikipedia "Toric section": https://en.wikipedia.org/wiki/Toric_section

3. Vector Applications: https://kexue.fm/search/vector/