• einstein1921
In summary: Now, the equation for a toroidal mirror is very similar to the equation for a lens:F(x,y,z) = A(x-h,y-v,z-w)where A is the mirror's aperture, h, v, and w are the distances from the mirror's center to the x-, y-, and z-axes, respectively.What this equation says is that, for a mirror, the distance between the image point and the sagittal plane (where the mirror is situated in space) is equal to the distance from the mirror's center to the focal point (where the light is focused). This is why a toroidal mirror is sometimes called a '

#### einstein1921

http://www.lasertechn.com/index.php/toroidal-mirrors
in the figure, what does S’s – for the sagittal plane means? the distance between the image point and the sagittal plane ??
I see in some papers,they use toroidal mirror to collimate the beam from a point source, how does this happen and why? some material say toroidal mirror have two focus point, where is it?
any thing help would be appreciated
best wishes!
Thank you!

S and S' are object and image vectors - notice the object and image are not on the same line?
The vectors have components along the saggital plane and the tangential plane.

Simon Bridge said:
S and S' are object and image vectors - notice the object and image are not on the same line?
The vectors have components along the saggital plane and the tangential plane.
thank you for your answer!how to use toroidal mirror to collimate the beam from a point source?

einstein1921 said:
thank you for your answer! how to use toroidal mirror to collimate the beam from a point source?
If you look at the equations - you can see that they look kinda like the lens-maker equation with a factor depending on the incident angle where you normally get the power. Presumably putting the source 1/P from the mirror will have a collimating effect - you'd get a rectangular beam.

OTOH: never tried - never seen it done this way.

einstein1921 said:
http://www.lasertechn.com/index.php/toroidal-mirrors
in the figure, what does S’s – for the sagittal plane means? the distance between the image point and the sagittal plane ??
I see in some papers,they use toroidal mirror to collimate the beam from a point source, how does this happen and why? some material say toroidal mirror have two focus point, where is it?
any thing help would be appreciated
best wishes!
Thank you!

This is a complicated topic, so let's go slowly.

First are some definitions: sagittal, meridonal, etc. In order to define these, we need basic notions of a rotationally symmetric optical system. If all rays from arbitrary object point P converge to a single unique image point P', the image is called 'stigmatic', and if not, the image is 'astigmatic'. Stigmatic images of a plane object OP, oriented normal to the optical axis, lie also on a surface OP', and a reference plane can be drawn that contains the optical axis, the object plane, and the image surface. This reference plane is defined as the 'tangential' or 'meridional' plane, and is the plane that is typically drawn in lens diagrams. All stigmatic rays lie within the meridional plane, but not all rays are stigmatic. These other rays, skew rays, are used to analyze astigmatic aberrations (stigmatic rays are used to analyze comatic aberrations). Typically, the skew rays lying in planes perpendicular to the meridional planes are chosen- those perpendicular planes are 'sagittal' planes and *are not constant throughout an optical system*- the sagittal plane changes after each surface refraction/reflection.

Your situation is even more complicated, because the optical surface is not rotationally symmetric. When a narrow beam is obliquely incident on a refracting/reflecting surface, astigmatism is introduced, and the image of a point source becomes a pair of focal lines, one sagttial pointing toward the lens axis and the other tangential, tangential to the lens axis, both lines perpendicular to the principal ray. The formulas on the page you cite are kinda-sorta like the Coddington equations that locate these focal lines.

How's that so far?

## What is a toroidal mirror?

A toroidal mirror is a type of curved mirror that is shaped like a donut or a ring. The surface of the mirror is created by rotating a parabola around a central axis.

## What is the purpose of a toroidal mirror?

Toroidal mirrors are used in various scientific instruments, such as telescopes and particle accelerators, to focus and reflect light or particles. They can also be used to correct aberrations in optical systems.

## How does a toroidal mirror differ from other types of mirrors?

The main difference between a toroidal mirror and other types of mirrors, such as spherical or parabolic mirrors, is its unique shape. Toroidal mirrors have a constant curvature along one direction and a varying curvature along the perpendicular direction.

## What are the advantages of using a toroidal mirror?

One advantage of using a toroidal mirror is that it can correct for certain types of aberrations more effectively than other types of mirrors. It also allows for a larger field of view and a shorter focal length, making it useful for certain applications.

## What are some real-world applications of toroidal mirrors?

Toroidal mirrors are commonly used in particle accelerators, where they are used to focus and steer particle beams. They are also used in some high-powered lasers and in certain types of telescopes, such as the Ritchey-Chrétien telescope.