Cassegrain Formulas and Tips by Mike Lockwood

Copyright 2006-2007

A Cassegrain telescope is a bit more complicated to design and build than a Newtonian.  It requires careful placement of the optics in its setup, and often manipulation of the equations during design.  This discourages some, and annoyed me.  It always seemed that I was going through three different books or articles to locate the equations I was looking for.

Therefore, this web page was created to merge the important sources of Cassegrain formulas that I have come across.  I chose some common names for variables to make things easier to read, since every author seems to have different variables for the same quantities!  I also give a couple formulas I have derived that show how sensitive the Cassegrain design is to certain quantities.  These are fairly simple to derive, but I don't see them in the amateur publications that I have.  So, I put them here.

The first part of this page covers general Cassegrain formulae.  All the formulas given here are applicable to all these types  - classical, Ritchey-Chrétien, Dall-Kirkham.

The differences between the various types of Cassegrains are the figures of the primary and secondary mirrors, and they will be covered in the next section, followed by other useful facts and procedures that I have seen fit to include.

1)  Diagram, Variables, and Design Formulae

Cassegrain telescopes are built to reduce the physical length of the telescope tube.  By folding the light path and slowing down the convergence of the light cone, we shorten the tube to about the focal length of the primary mirror, while the telescope has a focal length of somewhere between 3 and 15 times the focal length of the primary!  This allows high magnification to be obtained from a compact instrument, and it can reduce some of the mechanical demands cause by having a long tube structure.  However, it requires a second optical surface to be made and positioned quite accurately, and the whole structure must maintain the alignment of the mirrors no matter where the telescope is pointed.  The secondary mirror also typically is of significant size, leading to a central obstruction of between 20% and 40%.

Cassegrain diagram, ala Lockwood
Figure 1:  A diagram of a Cassegrain telescope is shown above.  The distances between the mirrors and other quantities are illustrated.

The important quantities here are:
    the diameters of the primary and secondary mirrors, D1 and D2,
    the focal length of the primary f1,
    the primary-secondary separation, d,
    the system focal point to secondary distance, q,
    the primary mirror surface to focal plane distance, e, and
    the distance from the secondary surface to the focal point of the primary, p.

The purpose of a Cassegrain secondary mirror is to slow the convergence of the light cone from the primary (thereby increasing the system focal length), direct it to the focal plane, and possibly compensate for some of the image aberrations caused by the primary.  The secondary has a "magnification factor" that is referred to as M.  This factor tells by how much the secondary mirror magnifies (or multiplies) the focal length of the primary by.  The overall focal length of the instrument is designated as F.  With that introduction, here are the important equations for designing a Cassegrain telescope.  They can be manipulated as needed to solve for various quantities.  Most of these equations are from Texereau, another is from another source.
F  = f1*M             (System focal length)
M  = q / p            ("Magnification" of system)
p  = (f1+e)/ (M+1)    (secondary mirror to focal plane distance)
R2 = 2*p*M / (M -1)    (Radius of curvature of the secondary mirror)
D2 = D1*p / f1        (Diameter of secondary required to fully illuminate ONLY the center point of the field.)
s  = q / d            (Defined for convenience for later formulas)
Here is a handy equation - how to calculate the radius of curvature of the secondary if you know p and q.  This is required if you already have a primary that you wish to use in a Cassegrain.  After you choose a magnification, you know p and q, and all you need is the secondary's radius of curvature, R2.  Below is a manipulated form of the equation for R2 given by Richard Buchroeder from page ~135 of "Advanced Telescope Making Techniques, Vol. 1".  The first equation is solved for R2.  The second equation is the same relationship solved for q.
R2 = 2 / ((1/p) - (1/q))
 q = R2*p / (R2 - 2*p)      
The second equation is important because q determines where the focal plane will be.  If we observe that we have q in terms of both p and R2, we realize that we can find the sensitivity of the focal plane position as a function of the location of the secondary and as a function of the radius of curvature (R2) of the secondary.  This means we can figure out how much moving the secondary moves the focal plane, and how much an error in the secondary's curvature (caused by grinding in a curve that's slightly off) will affect the focal plane position.  To get these quantities, we simply take the derivative of the second equation with respect to p and R2.  Those formulae are given below.
dq / dp   =  R22 / (4*p2 - 4*p*R2 + R22)
dq / dR2  =  p/(R2 - 2*p) - R2*p/(R2 - 2*p)2
For example, for my 12.5" F/12.5 classical Cassegrain, R2 = 47.35", and p = 15.72".  Plugging these numbers into the equation yields  dq / dp = 8.86,  dq / dR2 = -1.95.  These results are unitless.

The second number means that an error of plus (or minus) one millimeter in R2 moves the focal plane 1.95mm closer to (or farther from) the secondary mirror.

The first number means that for every unit of length (say a millimeter) that the secondary is moved towards the primary mirror, the focal plane moves 8.86 millimeters farther behind the primary.  This derivative (rate of change) is only valid for the position of the secondary, and it will change if the secondary is moved.  (The p value should be recalculated if large moves are anticipated.)

Now, you might wonder why the focal plane moves so much more than we expect.  Common sense seems to indicate that, for the 12.5" F/12.5 Cass, with an F/4.2 primary and a secondary magnification of 12.5/4.2 = 3, moving the secondary 0.01" toward the primary should move the focal plane back 0.01" * 3 = 0.03".  But this is NOT true.  Instead, it moves 0.0886" back.  So where does the extra sensitivity come from?  Well, when you decrease the spacing between the secondary and primary, the focal length of the system actually gets longer (!), and this adds dramatically to the sensitivity to spacing changes.  This is why adjusting the spacing between an aspheric secondary and the primary changes the system correction (see tip a below).

The sensitivity to changes in these quantities increases as the magnification rises.  For a 14.25" F/21 Cassegrain, dq / dp = 31.2,  dq / dR2 = -10.51.  As we can see, both quantities rise quite significantly.  So, that planetary Cassegrain you're thinking about building is going to require you to design in only a small amount of secondary travel to put the focal plane exactly where you want it when you assemble it the first time.  Be aware that the change in length of the tube structure due to temperature will cause focus shift over the course of a night because that causes the distance between the primary and secondary mirrors to decrease.

2)  Mirror Prescription/Design Formulas

Here I'll give the formulas used to calculate the "prescriptions" of the primary and secondary mirrors for three types of Cassegrain telescopes - the Dall-Kirkham, the classical, and the Ritchey-Chrétien.  (No, no Pressmann-Camichels.  They're not very useful for astronomy.)  The prescription is the form of the aspheric shape of the mirror.  First, realize that in these designs, the mirrors require various amounts of aspherization (correction).

Now, you don't really need this next formula, but some of the more math-oriented ATMs might enjoy seeing it.  It was also obtained from "ATMT Vol 1.", and I will add the author's name and the page number soon.  A general formula for the shape of a cross section of a mirror's surface is given by:
z = (y2/R) / (1 + [1 - (1+b)(y/R)2]1/2)        (Don't worry - you don't need this to design your Cassegrain!)
The optical axis lies along the z-axis, and the y-axis is the transverse axis, so z represents the mirror's height as a function of y, the distance off the optical axis.  The center of the mirror is at y = 0, and its edge is at y = mirror radius.  R is the radius of curvature of the mirror.  The variable b represents the conic constant, which is an important quantity.

    For a sphere,         b =  0   (Dall-Kirkham secondary)
    For an ellipse,   0 > b > -1   (Dall-Kirkham primary)
    For a parabola,       b = -1   (Newtonian and classical Cassegrain primary)
    For a hyperbola,      b < -1   (Ritchey-Chrétien primary and secondary, classical Cassegrain secondary)

Basically, the parameter b determines the type of shape a concave or convex mirror has.  (If b > 0, we have an oblate spheroid, by the way.)  In order to design a Cassegrain, we need to calculate it according to the type of Cassegrain we wish to make.  Thankfully, the formulas are already worked out in Richard Buchroeder's article mentioned above - he has provided a convenient table, and I have merely substituted the variables names I have used here and made simple modifications to them.

"Flavor" of Cassegrain
 b for primary
 mirror
  b for secondary
 mirror

Dall-Kirkham
 s(M-1)(M+1)2
 ------------ - 1
   M3(M+s)
0  (sphere)

Classical Cassegrain
-1 (parabola)
        -4M
      ------ - 1
      (M-1)2

Ritchey-Chrétien
     -2s       
   ----- - 1
     M3
  -4M(M-1) - 2(M+s)
 ----------------- - 1
      (M-1)3
Table I:  Equations to determine the conic constant of primary and secondary mirrors in Cassegrain telescopes

So how to figure the mirrors?  Calculate b for the mirrors in your Cassegrain.  For example, in my 12.5" F/12.5 Cassegrain, b = -1 for the primary and b = -4.00 for the secondary.  The primary is a parabola, so we know how to test that.  Since b = -4.00 for the secondary, it is a hyperbola with 4.00 times the correction of a parabola, so if we are to make a concave test plate, we simply multiply all the ideal knife edge displacements by 4.00.  It's that simple!

Let's say for an R-C primary, b = -1.04167.  This means that for Foucault testing, we simply multiply the ideal Foucault knife edge positions for a parabola by 1.04167, and that gives us the ideal knife edge positions for the hyperbolic primary mirror (or concave test plate for the secondary mirror).  Many are surprised to find out that the primary mirror for an R-C is only 1.04167 times more corrected than a parabola.  The things that make an R-C difficult are the typically fast focal ratio of the primary and the significant correction that must be applied to the secondary mirror.

3)  Other Cassegrain Tips

I'm sure this section will get expanded in the future.

a)  Correction issues for Cassegrains with hyperboloidal secondary mirrors (classical, R-C):
The correction of a Cassegrain optical system with an aspheric secondary mirror (classical, R-C) is dependent on the primary-secondary distance.  This is because the secondary's conic constant, b is a function of how far it is from the primary mirror and the distance to the focal plane.  (This is why it is difficult to match a secondary mirror that is already finished to a primary mirror - the chances of the secondary having the right value of b are small.)  If the secondary has the right conic constant and it is placed at the proper design location, the system will be perfectly corrected for spherical aberration.  However, if you see spherical aberration, digest the following:

-If the secondary mirror is moved AWAY from the primary mirror, the correction of the system will INCREASE.
-If the secondary is moved TOWARD the primary mirror, the correction of the system will DECREASE.  (This assumes it is large enough to be moved and still intercept all of the light from the primary)
-If you are figuring a Cassegrain secondary using star testing, increasing the correction of the secondary will decrease the correction of the system.

So, if you see a bit of overcorrection in your classical or R-C,  move the secondary mirror closer to the primary and deal with the focus shift, or figure the secondary to have a bit more correction.  Why is this true?  Think about it this way - if we move the secondary farther from the primary, we use only a smaller, central portion of the secondary.  In effect, the hyperbola in the center of the secondary has less correction than that of the full mirror, and this causes the light in the outer zones of the mirror to converge slightly more slowly, which is the same as increasing the correction of the whole system.  So, the effect on a Cassegrain system of changing the correction of the secondary mirror is the OPPOSITE of changing the correction of the primary mirror.

b)  Correction issues for Dall-Kirkhams:
For Dall-Kirkhams, the position of the (spherical) secondary makes a difference, too, as evidenced by the presence of the variables s and M in the equation for the value of b for the primary mirror.  Those two variables are dependent on q, d and p, which vary with the location of the secondary mirror.  A quick numerical calculation using the mirror spacings for my 12.5" classical Cassegrain indicate that if it were a Dall-Kirkham, then moving the secondary toward the primary would require a slightly less corrected primary. So, moving the secondary toward the primary should slightly increase the correction of a Dall-Kirkham system in the system.  This is the opposite of what happens in the other Cassegrains with hyperboloidal secondaries.  I believe this system is less sensitive to movement of the secondary than a classical or R-C.  I still need to double-check these results, though.

c)  I collimate my Nasmyth-focus classical Cassegrain by inserting a laser collimator and using the following procedure:
  1. First, I adjust tilt of the focuser to aim the laser dot at the center of the tertiary mirror, adjusting the mechanical positions of the tertiary and focuser if necessary, and making sure that the focuser is square to the tube.  This adjustment should not need to be done very often.
  2. Second, I adjust the tertiary's tilt to send the laser dot to the exact center of the secondary mirror.  Before I install the secondary, I use a permanent marker to draw a ring exactly centered on the secondary's surface.  This makes the tertiary tilt adjustment (or focuser alignment, for those without a tertiary) much easier and more accurate.
  3. Third, I adjust the secondary's tilt to return the laser beam to the center of the tertiary and the center of the laser collimator.
  4. Fourth, and lastly, I remove the laser collimator and put in an eyepiece.  I find a bright star or planet in the eyepiece.  While looking at the star in the telescope, I adjust the primary's tilt until I remove all of the coma from the image at the center of the field of view.  This takes a little practice, but once you get the hang of it it is simple, quick, and very accurate.
This whole procedure can be accomplished in under 5 minutes once it is understood and practiced.  Most of the time is spent on the primary mirror.  There are other techniques (including use of holographic laser collimators, etc.) that can be used to align the primary.  It should be accepted that critical collimation of a Cassegrain telescope (especially if the primary mirror is fast) is a necessity in order for the telescope to function properly.  If you understand this before you build the tube structure, you will avoid many problems.

d)  Size of illuminated field:
In order for an area of the focal plane to be "fully illuminated", it must get light from all of the primary and secondary mirrors.  For a Cassegrain this means that the secondary mirror must be large enough to allow the entire primary mirror to be "seen" from the parts of the focal plane that we wish to fully illuminate.  For example, if we know the location of the focal plane and we put our eye there at its center, we should see the reflection of the primary mirror centered in the secondary mirror.  If only the very center of the field is fully illuminated (say the central 0.05"), then we should see the reflection of the edge of the primary mirror right at the edge of the secondary mirror.  If the fully illuminated field is larger, say 0.5" in diameter, then we should be able to move our eye off the optical axis by 0.25" and still see the entire primary mirror in the secondary mirror, though it will no longer be centered in the secondary.  (This will soon be illustrated below.)

More sections will probably be added in the future as I figure out what information is useful to people.

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