## Cassegrain Optics

Here are the formulas, how to calculate the basic optical parameters of a cassegrain telescope. First we have to declare the sign conventions we will use:

### Sign Conventions

- Light entering the optical system travels from left to right.
- Distances from left to right are signed positive; those from right to left, negative.
- Curvatures with the convex side to the left are signed positive; otherwise they are negative.
- Intersection points above the optical axis are positive; those below the axis are negative.
- An angle between a ray and the optical axis is measured in the direction of the ray. When this angle is up from the axis, it is positive.
- The sign of the refractive index is the same as the sign of the direction in which light travels in the medium.
- In the case of reflection, therefore, the signs of the refractive indices are reversed.
- Surfaces are numbered in the sequence that they are hit by the rays.

### Basic parameters

All Cassegrain telescopes consist of a concave primary mirror with a small secondary mirror inside the focus of the primary; the secondary redirects starlight toward the primary. The image, in most cases, lies behind the primary. The convex secondary multiplies the focal length by a factor *M*. This factor *M* is termed secondary magnification, and is defined:

The focal length of the system is:

We can calculate the basic parameters seen in the picture with the formulas below:

Where <math>f_1</math> is the focal length of the primary, <math>f_2</math> is the focal length of the secondary (it is a negative value because of the conventions), and *d* is the separation. *b* is the back focus of the system, and *bfl* is the back focal length.

The focal surface has a radius of curvature:

Where <math>R_F</math> is the radius of curvature of the focal surface, <math>r_1</math> is the radius of curvature of the primary and <math>r_2</math> is the radius of curvature of the secondary.

The sagitta of the mirror is: (sagitta is the height of the mirror at a given radius.)

where *z* is the sagitta, *h* is the radius where we calculate the height of the mirror, *r* is the radius of curvature of the mirror, *SC* is the conic constant (explained later).

These equations do not deal with the field-of-view and detector size at the first glance. The relation between these is: <math>f\cdot\sin({\rm FOV}) = K</math> where FOV is the size of field-of-view (in degrees) while <math>K</math> is the image size (in millilmeters and/or in the same units in which <math>f</math> is given). The same equation stands for the sky (spatial) resolution and the pixel size, once FOV is replaced by the sky resolution and K is replaced by the pixel size.

### Conic constants

The conic constant, or Schwarzschild constant defines the shape of a conic section.

where *e* is the eccentricity of the conic section.

#### Calculating the conic constants

We will use the dimensionless quantities defined by Schwarzschild:

The **third-order Seidel Coefficients** for a two-mirror Cassegrain are:

**Spherical aberration:**

**<math>A_{cass} = 1 + SC_1 - \Bigg[SC_2 + \Bigg(\frac{M + 1}{M - 1}\Bigg)^2\Bigg]\frac{(M - 1)^3\cdot(1 - R)}{M^3}</math>**

**Coma:**

**Astigmatism:**

To simplify the formulas above, we will use additional quantities:

Since the spherical aberration is zero; <math>A_{cass} = 0</math>, we can calculate the different conic constants of the primary and secondary mirrors.

For the **classical Cassegrain**, the primary is parabolic <math>SC_1 = - 1</math>, so the secondary is:

For the **Ritchey-Chrétien**, both spherical aberration and coma are eliminated, so