# 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 f1 is the focal length of the primary, f2 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 RF is the radius of curvature of the focal surface, r1 is the radius of curvature of the primary and r2 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).

### Conic constants

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

SC = − e2

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; Acass = 0, we can calculate the different conic constants of the primary and secondary mirrors.

For the classical Cassegrain, the primary is parabolic SC1 = − 1, so the secondary is:

SC2 = − α

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

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