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What is the actual mathematical curve of a Schmidt corrector plate?

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#1 mirpis

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Posted 17 October 2021 - 10:23 PM

I've always seen it called an "asphere", but that doesn't tell me much.


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#2 Codbear

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Posted 17 October 2021 - 11:09 PM

I don't think there is a mathematical term for the given curves of an asphere.

 

It is interesting though that the name of the corrector plate lens is named after what it prevents as opposed to what it is.

 

For example, the four conical curves are a circle, ellipse, parabola and hyperbola, which are all descriptions of what the curves are.

 

However, aspheric describes what the lens prevents - spherical abberation...in the case of Schmidt Cassegrains, the spherical abberation generated by the spherical mirror.

 

If anybody finds another name for the aspheric, I would like to know as well.



#3 BGRE

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Posted 17 October 2021 - 11:42 PM

An asphere is merely a non spherical surface.

 

It may or may not be used to correct spherical aberration.

 

An example of the latter is a 2 lens laser beam shaper that converts from Gaussian to a flat top beam.

Such a lens combination changes the beam profile as well as minimising the output wavefront aberrations.

 

Another example is the Ritchey-Chretien telescope which uses 2 aspheric surfaces to correct both SA and coma.


Edited by BGRE, 18 October 2021 - 01:34 AM.

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#4 luxo II

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Posted 18 October 2021 - 01:50 AM

It is a quartic polynomial see equation 101.1

https://www.telescop...midt-camera.htm

 

Apologies to some others here, but I am still more than just good at maths, even at my age I still do that stuff, analytically.


Edited by luxo II, 18 October 2021 - 01:53 AM.

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#5 BGRE

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Posted 18 October 2021 - 02:53 AM

You mean a quartic radial polynomial?



#6 davidc135

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Posted 18 October 2021 - 02:55 AM

The actual mathematical curve-

 

You might think of the difference between a paraboloid and nearest sphere. To correct a sphere you could add material in a smooth curve that is deepest at the .71 radius (neutral) zone but which reduces to zero at both the mirror's centre and edge.

 

The Schmidt curve, by removing material accomplishes the same thing but being refractive must be of the opposite sign and, depending on the ref. index, is approx 4 times as deep. So, if imposed on a flat corrector surface, the neutral zone will again be at the 0.71r position.

 

If the Schmidt curve is put onto a surface of very slight convexity and then tested by contact interference with an optical flat the neutral zone will appear to have moved out further. Sometimes 0.866 radius NZ is chosen.

 

The approx formula for the Schmidt profile is Ar^2 + Br^4 + Cr^6 etc with A,B and C etc being coefficients that depend on the specs. It's sometimes called a fourth order paraboloid as, unless the instrument is very fast, knowing the first two terms is usually accurate enough. Hope I have that right.

 

David


Edited by davidc135, 18 October 2021 - 03:12 AM.


#7 luxo II

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Posted 18 October 2021 - 03:16 AM

This was discussed in the SciAm ATM books, long ago. From memory putting the maximum depth of the curve at 0.71r arises from seeking to have the edge thickness the same as the centre thickness. This however ignores the maximum gradient (steepness) of the curve at the edge, and the resulting spherochromatism. If the condition is that the maximum positive gradient and maximum negative gradient are equal, the result is minimal spherochromatism and raises when the maximum depth is at 0,866r. The edge thickness is thinner than the centre by a few microns.


Edited by luxo II, 18 October 2021 - 03:17 AM.


#8 MKV

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Posted 18 October 2021 - 08:17 AM

The name of the equation is biquadratic parabola formula.

 

schm corr delta.gif

 

(See ATM Book 3 p. 360)

 

where x is the aperture radius

r is the zonal radius, k is a NZ position factor (k = 1 for 0.707, and 1.5 for 0.866)

n is the refractive index of the schmidt corrector glass substrate

R is the raidus of curvature of the primary mirror.

 

As for the 0.707 neutral zone vs. 0.866 zone, this comes into play only for very fast systems 9such as Schmidt cameras ≤ f/2.  This is easily determined by raytracing.



#9 MKV

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Posted 18 October 2021 - 08:27 AM

You might think of the difference between a paraboloid and nearest sphere. To correct a sphere you could add material in a smooth curve that is deepest at the .71 radius (neutral) zone but which reduces to zero at both the mirror's centre and edge.

You're not adding, David. When you parabolize a spherical mirror you're removing. When shaping a Schmidt corrector you're also removing, when figuring a Cassegrain secondary, you're also removing -- selectively.

 

Conceptually, shaping the Schmidt corrector affects only the optical path length for different aperture zones, creating an exact opposite wavefront deformation to that created by the primary telescope/camera mirror, or the entire optical system.  



#10 Asbytec

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Posted 18 October 2021 - 08:50 AM

It's my understanding the 0.71r is neutral (by removing glass) on the Schmidt corrector because that is the zone of best focus of a spherical primary. Then the edge and center are left corrected accordingly. The thicker glass at center and the edge correct for the nearest reference parabola.

Edited by Asbytec, 18 October 2021 - 08:54 AM.


#11 davidc135

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Posted 18 October 2021 - 08:52 AM

You're not adding, David. When you parabolize a spherical mirror you're removing. When shaping a Schmidt corrector you're also removing, when figuring a Cassegrain secondary, you're also removing -- selectively.

 

Conceptually, shaping the Schmidt corrector affects only the optical path length for different aperture zones, creating an exact opposite wavefront deformation to that created by the primary telescope/camera mirror, or the entire optical system.  

Obviously, you're not adding when parabolising by polishing. I was trying to help visualise the simple relationship between the paraboloid (in my example) and Schmidt profiles as many people think of the latter as some strange and complex beast.

 

Mind you, corrections have sometimes been made by addition. Eg Astroscan and, I think, some Vixen products.

 

David


Edited by davidc135, 18 October 2021 - 09:02 AM.


#12 jimhoward999

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Posted 18 October 2021 - 02:08 PM

Generally a Schmidt Plate is modeled in optical design as having the form  z = A(r^2 - r^4)  where r is the normalized radial coordinate and A is a constant that sets how much SA correction that you want.  That is assuming you want the best fit sphere to be a flat. 

 

In real life because the Schmidt plate is made (at least by Celestron) by pulling a vacuum on a plate, polishing it flat and then releasing it the real shape is no doubt a bit different than a simple polynomial....but that should be close.



#13 MKV

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Posted 18 October 2021 - 02:24 PM

Generally a Schmidt Plate is modeled in optical design as having the form  z = A(r^2 - r^4)  where r is the normalized radial coordinate and A is a constant that sets how much SA correction that you want.  That is assuming you want the best fit sphere to be a flat. 

The exact 3rd order equation is given in #8.  



#14 freestar8n

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Posted 18 October 2021 - 04:24 PM

From Smith, Caragioli, Berry the surface is expressed as a radial polynomial with even powers:

 

z = Ar^2 + Br^4 + Cr^6 ...

 

You can get the exact terms from the prescription of a system.  They put the A term just as the normal radius of curvature of the surface, so the extra asphere terms are just B and C.

 

For 8" f/10 spherical sct the radius is -56118.28, with B = 6.431005 E-10, C = 3.113976E-16

 

From the size of those numbers it's clear the variation on the surface is very small.

 

In diagrams the Schmidt corrector is normally depicted as a somewhat wavy surface - and although that is greatly exaggerated, the overall shape is correct.  It mainly amounts to a convex paraboloid (the r^2 term) compensated by a higher order concave term to keep it overall flat.

 

I don't think there is a simple analytic form for where the terms come from since nowadays they would come from an overall optimization of the system.

 

I assume you can get more correction with more higher order terms but the benefit is minimal - and two terms is enough.

 

I guess some sct's have aspheres on both sides of the plate, so it would be about half as much on each side.

 

Frank



#15 davidc135

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Posted 18 October 2021 - 05:37 PM

From Smith, Caragioli, Berry the surface is expressed as a radial polynomial with even powers:

 

z = Ar^2 + Br^4 + Cr^6 ...

 

You can get the exact terms from the prescription of a system.  They put the A term just as the normal radius of curvature of the surface, so the extra asphere terms are just B and C.

 

For 8" f/10 spherical sct the radius is -56118.28, with B = 6.431005 E-10, C = 3.113976E-16

 

From the size of those numbers it's clear the variation on the surface is very small.

 

In diagrams the Schmidt corrector is normally depicted as a somewhat wavy surface - and although that is greatly exaggerated, the overall shape is correct.  It mainly amounts to a convex paraboloid (the r^2 term) compensated by a higher order concave term to keep it overall flat.

 

I don't think there is a simple analytic form for where the terms come from since nowadays they would come from an overall optimization of the system.

 

I assume you can get more correction with more higher order terms but the benefit is minimal - and two terms is enough.

 

I guess some sct's have aspheres on both sides of the plate, so it would be about half as much on each side.

 

Frank

How do the terms in the 8'' sct example translate into profile depth?  Does the A coefficient contribute as the sag for that paraxial sphere at a particular radius? I think the NZ is .866r in your example(?)

 

David

 

PS Mirpis, there is an online Schmidt calculator by Matt Considine which is fun to play about with.


Edited by davidc135, 18 October 2021 - 06:41 PM.


#16 freestar8n

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Posted 18 October 2021 - 06:09 PM

How do the terms in the 8'' sct example translate into profile depth at say .71r assuming that is the NZ?  Does the A coefficient contribute as the sag for that paraxial sphere at a particular radius?

I get around 32 waves max depth for this corrector.

 

David

The prescription shows the plate is flat in the front and has thickness of 4mm, so the polynomial is then a variation on top of that 4mm offset from the front of the plate.  Yes, instead of providing a value for A they say the surface is spherical with the specified radius - and the B, C terms get added to the displacement corresponding to the sphere.  That way if B and C are zero - it is a sphere.  So it is a bit confusing they write it as that polynomial - because the B and C terms will be different depending on departure from Ar^2 vs. departure from a sphere.  But it lets you approximate the plate in paraxial terms as a sphere with the given radius.

 

With regard to the "neutral zone" - they say "In theory, it is best if the neutral zone is place at 86.6% of the full radius of the corrector, since then the slopes on the corrector are smallest overall and so produce the least chromatic aberration."   i.e. you want to mix the sphere with the asphere in such a way that it is kept as flat as possible everywhere, with no local steep gradients.

 

Frank


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#17 davidc135

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Posted 19 October 2021 - 04:00 AM

Calculations for the 8'' sct would be for a modified 8'' F/2 (around F/2.25 equivalent Schmidt camera) but you still wouldn't need the C term.

 

David


Edited by davidc135, 19 October 2021 - 04:09 AM.


#18 MKV

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Posted 19 October 2021 - 08:48 AM

Calculations for the 8'' sct would be for a modified 8'' F/2 (around F/2.25 equivalent Schmidt camera) but you still wouldn't need the C term.

Correct. Based on residual third order spherical aberration, the equivalent Schmidt camera mirror RoC would be about 896 mm, which if f/2.4 for a 200 mm  aperture diameter. 



#19 MKV

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Posted 19 October 2021 - 09:05 AM

However, optimization favors f/2.3 Schmidt equivalent mirror RoC because of significant 5th order SA.




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