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# Exact formulation for Newtonian diagonals

3 replies to this topic

### #1 Mike I. Jones

Mike I. Jones

Aurora

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Posted 04 August 2020 - 05:18 PM

There was an earlier thread here that discussed the derivation of formulas for the Newtonian diagonal.  I decided to do that very thing from scratch, and validated it with the Zemax lens code.  The basis for this derivation is the 2D intersection of two vectors, in this case a ray from the upper side of the primary and semi-field, or the lower side of the primary and semi-field, with the 45ยบ line representing the diagonal face.  Earlier derivations (Ralph Dakin's in particular) have appeared in literature, but they were based on paraxial optics (flat primary rather than curved, and zero aberrations). The added twists I've added here are that

(1) The sagitta of the primary mirror is accounted for, and

(2) The third-order tangential coma height is used rather than just the paraxial image height.

For almost any practical Newtonian, (1) and (2) can be set to zero to give a good approximation.  However, since I personally hadn't previously seen an exact derivation that includes (1) and (2), I present it here.

The formulas look a bit messy, but program nicely in Excel (attached).  I validated this work using several different Zemax files and it does indeed give very precise results.  If I have made any typos, please bring them to my attention.

Mike

Derivation calcs.xlsx   12KB   9 downloads

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### #2 Mike I. Jones

Mike I. Jones

Aurora

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Posted 04 August 2020 - 05:34 PM

This is the Zemax validation for the example Newtonian used in the derivation.  The diagonal is correctly dimensioned and decentered

### #3 Mike I. Jones

Mike I. Jones

Aurora

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Posted 04 August 2020 - 05:39 PM

For you coder types like me out there, this is the Zemax macro I wrote that implements the formulation.  It can be re-expressed in Python, Visual Basic or whatever suits you.

#  Exact calculation of Newtonian diagonal dimensions and decentration
#  Mike Jones - Aug 4, 2020
#  Accounts for sagitta and tangential coma of paraboloidal primary

OUTPUT SCREEN

#  INPUTS:
dp = 200  #  Primary diameter
f  = 500  #  Primary focal length
l  = 150  #  Distance from optical axis to Newtonian focus
hp = 5    #  Paraxial field height = FOV / 2

print
print "INPUTS:"
format 8.3
print "Primary diameter = ", dp
print "Primary focal length = ", f
print "Axis to Newtonian focus = ", l
print "paraxial image height = ", hp

# CALCULATIONS:
r = dp / 2
fno = f / dp
s = r * r / (4 * f)
ht = hp * (1 + (3 / (8 * fno * fno) ) )
sa = (r - ht) / (f - s)
xa = (f - l - r - (s * sa)) / (1 - sa)
ya = sa * (xa - s) - r
sb = (ht - r) / (f - s)
xb = (f - l + r - (s * sb)) / (1 - sb)
yb = sb * (xb - s) + r
q = xa - f + l
ma = sqrt(q * q + ya * ya)
q = xb - f + l
mb = sqrt(q * q + yb * yb)
major_axis = ma + mb
minor_axis = major_axis * sqrt(0.5)
zemax_x_width = minor_axis / 2
zemax_y_width = major_axis / 2
decentration = sqrt(0.5) * (ya + yb)

print
print "DIAGONAL CALCULATIONS:"
format 16.8
print "Zemax X-width = ", zemax_x_width
print "Zemax Y-width = ", zemax_y_width
print "Major axis   = ", major_axis
print "Minor axis   = ", minor_axis
print "Zemax Y decentration = ", decentration

### #4 TOMDEY

TOMDEY

Fly Me to the Moon

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Posted 04 August 2020 - 06:02 PM

Hi Mike; quite amazing! Ralph originally introduced me to the Rochester Astronomy Club back in the 1970s, when we were both working at Bausch & Lomb. And yes, people have delightfully obsessed over more and more accurate derivations over the decades ~just because~. I later got involved in Imaging Satellites at Kodak... alignment and testing. Those comprise a lot of flat folds at all sorts of angles. We would always push for generous oversizing, just so the build would be easier, without having to be unnecessarily nominal. We also experienced one subtle nuance that occasionally matters a lot, and often overlooked: The optical and mechanical designers would define a minimum envelope that cleared all light rays. But the true wave nature of light interacts with the entire environment. So edges that get too close to the rays indeed begin to introduce noticeable Fresnel Diffraction effects... even though the rays clear everything. This is a big topic in Coronagraphs and Planet Finder satellites. We also noticed it in our onboard Knife Edge Foucault Wavefront Sensor.

So, I think that diffraction is the last vestige of exactitude... but with that peculiar feathered/ringing edge. In the limit... the position of my fingers on this keyboard effect (admittedly ever so slightly!) --- the data being collected at Kitt Peak. It's the kinda stuff that Feynman delighted it.    Tom

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