CASSEGRAIN AND GREGORIAN TELESCOPE DESIGNS
In the book "Telescope Optics, Evaluation and Design", by Harrie
Rutten and Martin van Venrooij, the calculations for a Cassegrain
system are given in Chapter 21.2. This note is a derivation from
that source. The equation numbers assigned below match those
given in that chapter. The basic Cassegrain layout is shown in
Figure 1, below.
Figure 1: Cassegrain Layout
First, the definitions:
S = f2 / f1 = r2 / r1
T = D2 / D1
M = f / f1
B = b / f
P = d / f
where:
f1 and f2 are the primary and secondary
focal lengths
r1 and r2 are the
primary and secondary radii of curvature
D1 and D2 are the primary and
secondary diameters
f is the focal length of the system
d is the separation between the primary and secondary
b is distance of the final focal surface behind the primary
The equations to calculate the magnification are then:
M = ( 1 - T ) / ( T - B )
21.2.1
M = ( 1 - B - P ) / P
21.2.2
The auxilliary equations are:
P = ( 1 - B ) / ( M + 1 )
21.2.3
B = 1 - ( M + 1 ) * P
21.2.4
T = P + B
21.2.5
S = M * T / ( M - 1 )
21.2.6
f1 = f / M
f2 = S * f1
d = f * P
b = f * B
The Seidel coefficients are calculated from:
alpha = ( ( M + 1 ) / ( M - 1 ) )2
21.2.10
beta = ( ( M - 1 ) / M )3 * T
21.2.11
gamma = ( ( M - 1 ) / M )3 * ( 1 - T )
21.2.12
delta = 2 / M2
21.2.13
epsilon = 4 * ( 1 - P ) / ( M * T )
21.2.14
theta = ( M - 1 )3 * P2 / ( M * T )
21.2.15
With the Seidel coefficients themselves being:
Spherical Aberration:
Acass = 1 + SC1 - ( SC2 + alpha ) *
beta
21.2.7
Coma:
Bcass = delta + ( SC2 + alpha ) * gamma
21.2.8
Astigmatism:
Ccass = epsilon - ( SC2 + alpha ) * theta
21.2.9
The equations above are
consistent for Gregorian telescopes by letting the separation between
the primary and secondary
be larger than primary focal length, as shown in the
Gregorian layout of Figure 2, below. Due to the crossover, the
effect is that of making the secondary diameter appear negative (D2 < 0).
This effect is extended to the net focal length and secondary
magnification, making them appear negative, as well.
Figure 2: Gregorian Layout
The basic Cassegrain forms given in R&vV are still valid:
Classical (paraboloidal primary):
SC1 = -1
SC2 = -alpha
21.2.16
Dall-Kirkham (spherical secondary):
SC1 = alpha * beta - 1
21.2.17
SC2 = 0
Ritchey-Chretien (aplanatic):
SC1 = -( 1 + beta * ( delta / gamma ) )
21.2.18
SC2 = -( alpha + delta / gamma )
21.2.19
Pressman-Camichel (spherical primary):
SC1 = 0
SC2 = -( alpha - 1 / beta )
21.2.20
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