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:

<math>M = \frac{f}{f_{1}}</math>

The focal length of the system is:

<math>f = M \cdot f_1 = \frac{f_1 f_2}{f_1 + f_2 -d}</math>

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

<math>bfl = d + b = \frac{(f_1 - d)\cdot f_2}{f_1 + f_2 - d}</math>

<math>f = d + b + M \cdot d</math>

<math>f_1 = d + \frac{d + b}{M}</math>

<math>f_2 = \frac{-(d + b)}{M - 1}</math>

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:

<math>\frac{1}{R_F} = \frac{2}{r_1} - \frac{2}{r_2}</math>

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.)

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

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.

<math>SC = -e^2</math>

where e is the eccentricity of the conic section.

Calculating the conic constants

We will use the dimensionless quantities defined by Schwarzschild:

<math>S = \frac{f_2}{f_1} = \frac{r_2}{r_1}</math>

<math>T = \frac{D_2}{D_1}</math>

<math>R = \frac{d}{f_1}</math>

<math>E = \frac{b}{f_1}</math>

<math>M = \frac{f}{f_1}</math>

<math>B = \frac{b}{f}</math>

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:

<math>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}</math>

Astigmatism:

<math>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)}</math>

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

<math>\alpha = \Bigg(\frac{M + 1}{M - 1}\Bigg)^2</math>

<math>\beta = \frac{(M - 1)^3\cdot(1 - R)}{M^3}</math>

<math>\gamma = \frac{(M - 1)^3\cdot R}{M^3}</math>

<math>\delta = \frac{2}{M^2}</math>

<math>\epsilon = \frac{4(M - R)}{M^2(1 - R)}</math>

<math>\vartheta = \frac{(M - 1)^3\cdot R^2}{M^3(1 - R)}</math>

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:

<math>SC_2 = - \alpha</math>

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

<math>A_{cass} = 0</math> and <math>B_{cass} = 0</math>. The conic constants of the mirrors are:

<math>SC_1 = - \Bigg( 1 + \frac{\beta\cdot\delta}{\gamma}\Bigg)</math>

<math>SC_2 = - \Bigg(\alpha + \frac{\delta}{\gamma}\Bigg)</math>

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