Equations Explained
1. Extension Factor (m)
m = Fsystem / Fprimary
How much the system focal length is extended by the secondary mirror.
2. Secondary to Primary Focus (A)
A = (F + b) / (m + 1)
Distance from the secondary mirror to the primary focus point.
3. System Focal Plane to Secondary (a)
a = m x A
Distance from the secondary mirror to the image plane.
4. Mirror Separation (B)
B = a - b
Physical separation between primary and secondary mirrors.
5. Radii of Curvature
R1 = 2 x F
R2 = (2 x A x m) / (m - 1)
Radii of curvature for the primary and secondary mirrors.
6. Conic Constants
K1 (Classical) = -1
K2 (Classical) = -[(4 x m) / (m - 1)²] - 1
K1 (Ritchey) = (-2 x (a / B)) / (m³) - 1
K2 (Ritchey) = ((-4 x m x (m - 1) - 2 x (m + (a / B))) / ((m - 1)³)) - 1
Describes the conic shape of the primary and secondary mirrors.
7. Coma Coefficient
coma (Classical) = 0.5
coma (Ritchey) = (m² /2 ) x ((1 / m²) - (1 + K1) x (B / (2 x B - R1)))
For the true Cassegrain Coma = 0.5, i.e. exactly the same as for a paraboloid of the same diameter .
8. Astigmatism Coefficient
astigmatism = m x [(2B - mR1) / ((2B - R1) x m²)]
astigmatism = m x [((2B - mR1) / (m²(2B-R1))) - (B / (2B-R1))² x (1 + K1)]
This coefficient quantifies how much astigmatism is introduced by the geometry of the mirror system.
9. Petzval Curvature
Petzval = m x [1 + (R1 x (m - 1)) / (m x (2B - R1))]
Petzval curvature represents the mean field curvature from both mirrors and affects how a flat image plane curves in space.
10. Field Curvature (COF)
Field Curvature (COF) = Petzval - 2 x astigmatism
The total field curvature, taking into account both Petzval and astigmatic effects. A positive curvature indicates a focal surface that is concave towards the incident light.
11. Coma and Astigmatism (Angular)
COMA = (3 / 8) x (D1 / Fsystem)² x coma x θ x 103132
ASTIG = (D1 / Fsystem) x θ² x astigmatism x (103132 / 4)
Conversion to arcseconds based on system geometry and field of view, θ (in radians).
12. Radius of Curvature of Field
RCOF = Fsystem / COF
How much the image plane curves due to field curvature aberration.
13. Secondary Mirror Axial Diameter
D2 = (D1 x A) / F
The physical diamter of the secondary mirror required to capture all the light reflected from the full aperature of the primary.
13. Secondary Mirror Optical Diameter
DOptical = θ x Fsystem
The diameter of the light cone from an off-axis object. Uses field angle θ (in radians).
14. Epsilon (Wavefront Error)
E1 = (b1 / 32) x (D1 / 2)⁴ / R1³
E2 = (b2 / 32) x (DOptical / 2)⁴ / R2³
The maximum surface deviation from the sphere is given by the above equation.
15. Edge Slope of Mirrors
θ1 = (b1 / 4) x (D1 / 2)³ / R1³
θ2 = (b2 / 4) x (DOptical / 2)³ / R2³
The slope at the edge relative to the sphere is given by the derivative of Eq. 14.