# Cassegrain Optics

## 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:

$M = \frac{f}{f_{1}}$

The focal length of the system is:

$f = M \cdot f_1 = \frac{f_1 f_2}{f_1 + f_2 -d}$

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

$bfl = d + b = \frac{(f_1 - d)\cdot f_2}{f_1 + f_2 - d}$

$f = d + b + M \cdot d$

$f_1 = d + \frac{d + b}{M}$

$f_2 = \frac{-(d + b)}{M - 1}$

Where $f_1$ is the focal length of the primary, $f_2$ 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:

$\frac{1}{R_F} = \frac{2}{r_1} - \frac{2}{r_2}$

Where $R_F$ is the radius of curvature of the focal surface, $r_1$ is the radius of curvature of the primary and $r_2$ 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.)

$z = \frac{h^2}{r(1 + \sqrt{1-(h^2/r^2)(SC+1)})}$

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: $f\cdot\sin({\rm FOV}) = K$ where FOV is the size of field-of-view (in degrees) while $K$ is the image size (in millilmeters and/or in the same units in which $f$ 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.

$SC = -e^2$

where e is the eccentricity of the conic section.

#### Calculating the conic constants

We will use the dimensionless quantities defined by Schwarzschild:

$S = \frac{f_2}{f_1} = \frac{r_2}{r_1}$

$T = \frac{D_2}{D_1}$

$R = \frac{d}{f_1}$

$E = \frac{b}{f_1}$

$M = \frac{f}{f_1}$

$B = \frac{b}{f}$

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

Spherical aberration:

$A_{cass} = 1 + SC_1 - \Bigg[SC_2 + \Bigg(\frac{M + 1}{M - 1}\Bigg)^2\Bigg]\frac{(M - 1)^3\cdot(1 - R)}{M^3}$

Coma:

$B_{cass} = \frac{2}{M^2} + \Bigg[SC_2 + \Bigg(\frac{M + 1}{M - 1}\Bigg)^2\Bigg]\frac{(M - 1)^3\cdot R}{M^3}$

Astigmatism:

$C_{cass} = \frac{4(M - R)}{M^2(1 - R)} - \Bigg[SC_2 + \Bigg(\frac{M + 1}{M - 1}\Bigg)^2\Bigg]\frac{(M - 1)^3\cdot R^2}{M^3(1 - R)}$

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

$\alpha = \Bigg(\frac{M + 1}{M - 1}\Bigg)^2$

$\beta = \frac{(M - 1)^3\cdot(1 - R)}{M^3}$

$\gamma = \frac{(M - 1)^3\cdot R}{M^3}$

$\delta = \frac{2}{M^2}$

$\epsilon = \frac{4(M - R)}{M^2(1 - R)}$

$\vartheta = \frac{(M - 1)^3\cdot R^2}{M^3(1 - R)}$

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

For the classical Cassegrain, the primary is parabolic $SC_1 = - 1$, so the secondary is:

$SC_2 = - \alpha$

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

$A_{cass} = 0$ and $B_{cass} = 0$. The conic constants of the mirrors are:
$SC_1 = - \Bigg( 1 + \frac{\beta\cdot\delta}{\gamma}\Bigg)$
$SC_2 = - \Bigg(\alpha + \frac{\delta}{\gamma}\Bigg)$