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#1 mark1234

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Posted 30 January 2013 - 08:47 AM

Maybe Mike or someone with experience of making a plate after tracing in OSLO EDU to get 4th and 6th order coefficients can jump in here and clarify things. When optimising for spherochromatism the only way to shift the neutral zone from the origin out to 0.707 of corrector radius is by making surfaces also weakly curved. But this information will not feature in calculating the deflection when polishing the plate with a vacuum pan - input to the formulae is via the A + B coefficients only.
Is this why vacuum pan method needs spherical tool and does it end up the same shape as OSLO ray trace model?
I have not seen this position of neutral zone mentioned in the calculation of sagitta of tool or deflection.
I am hovering with plate, plaster, tiles and pan. It is not an academic exercise - if plate is badly wrong it will be a huge waste of time.
I have made some practice plastic plates for a very short focus mirror, using basic Ingalls information and zonal grinding. The depth of curve finally arrived at was a lot deeper than calculated to get a good correction, I put this down to a much lower index for the acrylic than I estimated. 1.3 is a lot weaker than 1.5, getting towards half, twice as deep curve. They worked very well in the application for an exhibition installation which needed an extremely bright dot of light to scan objects. A Cree LED was imaged by a 150mm f0.8 spherical mirror very efficiently. Well done Schmidt!

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#2 mark1234

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Posted 30 January 2013 - 08:58 AM

sorry, intial post has attached wrong file.
used similar name for something else.
this attachment is the Schmidt Newt configuration only - no confusing additional optics.

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#3 DAVIDG

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Posted 30 January 2013 - 09:24 AM

Mark,
There is very good article in June 1972 Sky and Tel about calculating the plates. I know that both Gerry Logan and Bob Plaff have used these calculations to make some award winning schmidts. http://www.considine...pfaff/pfaff.htm
In OSLO EDU there is an example file for a Schmidt, besides aspheric coeff, the file also uses a conic of -1.

- Dave

#4 mark1234

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Posted 30 January 2013 - 11:33 AM

thanks Dave,
It's not the calculations or how to grind the plates I have a problem with - I'm familiar with the Bob Plaff and everything else I can find on the web. If I just ignore OSLO and use the formulae available, I feel kind of - well naked, because I've gotten used to ray-tracing in OSLO and have optics that work out ok.
Do you understand what I mean about the lateral colour curves starting at the origin (ie centre of the plate - which means it has a flat centre) in the optimisation in OSLO?
So is the -1 conic on the corrector plate going to still give correct 4th, 6th...order coefficients that I can safely input to formulae?
I don't have any recall of a parabolic conic being used as well. I don't have the S&T, but the maths are the same in all the different sources I've found.

#5 mark1234

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Posted 30 January 2013 - 11:56 AM

Dave,
The conic in the OSLO Schmidt demo file makes hardly any difference over a 10-0--10 range.
The Schmidt plate does have curvature. As set in the demo file it gives the neutral zone at the 0.86 of corrector radius height position. Which is no longer used for minimum chromatic abberations. Monochromatic design in this case as well.
Many thanks for suggesting I look that ref up.
Interesting, in that the 'flat' plate has a radius!
And it is integral to the prescription.
I need an OSLO pro here I think.
Let me know if something occurs to you about this, because as it stands it looks like I'm the first person who is trying to make a plate from an OSLO ray-trace as an ATM.

Mark

#6 mark1234

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Posted 30 January 2013 - 12:23 PM

to anyone looking at my original post -

this line implies I know very little about Schmidt plates

'I have not seen this position of neutral zone mentioned in the calculation of sagitta of tool or deflection.'

to clarify - in conjunction with OSLO derived coefficients!I am referring to the ability to shift the neutral zone around in OSLO using a long radius of curvature on the corrector, but then only end up with two coefficients. (to 6th order gives sufficient correction in my case).
Is the optical effect of the curve 'contained' within the coefficients by virtue of their values having to change to retain the minimum wavefront error?
I don't know how to confirm this using the available maths.
Stumped.

#7 Mike I. Jones

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Posted 30 January 2013 - 12:50 PM

and I don't know how to do it in OSLO optimization error functions. In ZEMAX it's simply a matter of changing the normalized pupil radius at the "pinch point", the zone in the pupil where all the colors are brought together for minimum chromatic aberration. That can be at the 70% zone for minimum material removal, at the roughly 85% zone for minimum chromatic aberration, or anywhere else from 0.0 (center) to 1.0 (edge). You have to experiment with the pinch point zone value, as it varies with aperture, field width and corrected spectral band. The optimization takes care of calculating the radius, 4th and 6th order coefficients.
Mike

#8 mark1234

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Posted 30 January 2013 - 01:11 PM

This is the way I have optimised where the neutral zone on the plate is in OSLO.
This example is what happens when there is no curve on the plate - the neutral zone is at 0 radius height or the centre of the plate.
Nobody in their right mind would attempt to figure a plate like this - the shape is a shallow dish (with constantly changing radius of curvature), with a steeply (but constantly changing radius!) margin.
The plot doesn't go through the origin because of the central obstruction (Newtonian mirror)

Mark

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#9 mark1234

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Posted 30 January 2013 - 02:45 PM

Many thanks Mike,
This is a screenshot with the curves on the sliders as well.
I would be great to pin down this curvature business if ZEMAX gives an output which includes a curve as well - what to do with it in the formulae...

You have to re-open the Optimise/Slider Wheel in order to increase the resolution on the plots - tedious, but then this incredibly powerful software is free. It would take more than a lifetime to calculate by sliderule alone.
It's interesting that Schmidt probably didn't use his vacuum technic to make the 14" camera. see here:
Astronomical Optics and Elasticity Theory By Gérard René Lemaitre
His old notes and papers had stuff about bending steel beams - no vacuum pan was found. Maybe he was human enough to give a cheeky and misleading demo of his secret 'method'. Doesn't sound like much of a salesman, content with Scnapps and a cigar. He only ever made the one plate as far as I can find out. Found some archive negs from the Hamburg Observatory - impressive even by todays CCD standards. Eventually he was persuaded to demonstrate it in California and they built a giant one on Mt Palomar.It's still his idea to use stress polishing which is at the heart of modern astronomical optics. They use a fancy dual vacuum technic and polish plates flat now as well. I won't be going down that route.

Mark

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#10 wh48gs

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Posted 31 January 2013 - 02:46 AM

When optimising for spherochromatism the only way to shift the neutral zone from the origin out to 0.707 of corrector radius is by making surfaces also weakly curved. But this information will not feature in calculating the deflection when polishing the plate with a vacuum pan - input to the formulae is via the A + B coefficients only.
Is this why vacuum pan method needs spherical tool and does it end up the same shape as OSLO ray trace model?
I have not seen this position of neutral zone mentioned in the calculation of sagitta of tool or deflection.



I'm not sure which formula you are referring to, but the radius factor is the first term in the Schmidt profile formula (Eq.101.1) - which combines corrector's radius of curvature (which is the radius of its central zone) with the aberration terms (or "coefficients" as they often call them). The radius term itself is directly dependent on the relative focus term, which itself determines the relative position of the neutral zone.

Tool profile in vacuum pan method is the inverse of corrector's profile, so it has to be calculated using those parameters as well.

Vla

#11 MKV

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Posted 31 January 2013 - 01:31 PM

Is the optical effect of the curve 'contained' within the coefficients by virtue of their values having to change to retain the minimum wavefront error?

I tink Vla pretty much covered it, but you may have come across a more commonly seen expansion series of the type

x = AY^2 + BY^4 + CY^6...

whose derivative is 0 = 2AY + 4BY^3 + 6CY^5 ...

The 2A (whose value will vary according to where the neutral zone is) is the vertex radius (Rv) term, or factor, or coefficient, such that Rv = 1/2A.

The value of A is defined as (z/r)^2*D^2/[8*(n-1)*R^3], where z is the zonal height, r is the diaameter of the corrector, D is its diameter, n is the refractive index of the glass used, and R s the radius of curvature of the mirror.

Thus, for D = 200 mm, r = 100, z = 86.6, n = 1.5168, ad R = 1200, your A = (0.866^2)*(200^2)/[8*0.5168*(1200^3)] = 4.2^-6, 2A = 8.4^-6, and and 1/2A ~ 119,000 inches.

Mladen

#12 mark1234

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Posted 31 January 2013 - 02:05 PM

Thank you Vla,
I hoped you would respond to this.
Your posts are very helpful.
I've gone through the equations on telescopeoptics, Ingalls and Pfaff again before reading your post.
I now have some agreement between all three on the 0.707 zone depth. But not quite confident in the process yet.
Also worked through the 300mm Schmidt example with these variations on the formulae as well. The wavelength used is not specified. I assumed about 587nm.
There is one outstanding issue to do with minus aspheric parameters in equation 101.1 - I optimised the 300mm example in OSLO and then used the 4th and 6th order parameters and vertex radius on corrector. The curvature is positive, the parameters are negative numbers, adding together gives a negative depth with too little neutral zone depth by 60% compared to the example result. Ignoring the negative sign on the 4th order parameter gives almost same result as in example. Maths is maths and one cannot ignore sign.
Ivan's Jodas result doesn't seem to fit at all and something doesn't correlate with his vertex radius and 1st parameter value - there is an applet to use on the web.
There is a lot of pro optics reference on the internet which highlights interactions between 4th and 6th order+ parameters and under-sampling of software routines. I've double checked the drawn shape of the profiles to make sure they are actually the expected curve, in any case OSLO slider optimisation seems to respond correctly to gross inputs.
I need some more correlation on using the values from OSLO.
Many thanks Vla


Mark

#13 MKV

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Posted 31 January 2013 - 02:57 PM

Mark, you can optimize without sliders. The example below is an 200 mm f/3 Schmidt camera whose corrector has a neutral zone at 86.6% of the corrector diameter. No sliders were used. Just OSLO's automatic optimization function. I only provided the correct vertex radius of curvature of ~ -119,000.

Mladen

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#14 mark1234

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Posted 31 January 2013 - 05:52 PM

Thanks Mladen,
I used the sliders while I tried different radius of vertex curvature - while just getting a feel for how to shift the neutral zone and seeing how all the parameters affected each other in an interactive way.
I'm hoping to leave the smooth curve from the vacuum pan method as is, without polishing in zones while attempting further corrections.
Otherwise I might as well just use a petal polisher on a turntable.
It has occured to me that it may be possible to pull a vacuum on a 6mm thick 480mm disc with a "profile" of aluminium foil layers on a flat - after all that is basically what the celestron patent describes using a master glass negative. I bet it's not that simple in practice though! The accurate matching of both optical centres with a two-sided corrector is not simple in practice either with any method.
Meanwhile I'm still trying to track down the cause of slightly different results as regards the depth of curve.

Mark

#15 MKV

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Posted 31 January 2013 - 09:20 PM

Mark, using sliders has its role and is quick and easy. But for if you want a corrector with 86.6% NZ then there is only one correct vertex radius. You set the Rv and then caluclate your deformation based on that.

If I were you I would not attempt to put the correction on both sides of the corrector precisely for the reason you mention -- unless you have some kinematic method of flipping the corrector and maintaining perfect axial alignemnt.

As for the vacuum method and using a template -- that's the way to go. You can make a template easily, then drill holes in it that will allow the corrector to be sucked against the template and supported evenly. Probably less of a chance of breaking during polishing when force much larger than grinding is applied.

But to make a single corrector -- I think that's way too much work. I'd rather hyperbolize the mirror and go with the Rosin design.

Mladen

#16 wh48gs

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Posted 31 January 2013 - 10:59 PM

Hi Mark,

It sounds right for the Schmidt profile on the front surface, light traveling left to right, the first (corrector radius) term should be numerically positive, and the aberration terms negative. For the rear surface, reversed.

There is no good reason for the (correctly typed in) formulae not to give correct values. If you look at FIG. 168 left, it is a graphic illustration of the parameters in the Schmidt profile equation. The first term is the corrector r.o.c. sagitta, and the aberration terms are effectively the (inverse) wavefront deformation "magnified" by the factor of light slowdown inside glass (so it is not d^4/4R3 s.a. coefficient for air, rather d^4/4(n'-n)R^3 for the first aberration term). The corrector radius term is merely used to flex this glass shape based on the aberrated wavefront, moving the neutral zone along the radius.

From this, it is obvious that needed depth of the corrector is directly related to the P-V wavefront error at chosen focus location, given by

W=xd^4/4R^3,

with x=[1+1.875N^2(-2+2N^2)]

being the relative error for the specific focus (i.e. neutral zone) location in units of the P-V error at paraxial focus, where "d" is the aperture radius, R the mirror r.o.c. and N the relative height of the neutral zone in units of aperture radius.

Maximum profile depth - the one at the neutral zone - is simply W/(n'-n). It gets more complicated with higher-order terms, but since the primary aberration term accounts for nearly all of it, it is a good checkup.

For instance, a 300mm f/3 Schmidt, so d=150, R=-1800, N=sqrt.0.5 (for the minimum spherochromatism and depth), Rc=-(n'-n)R^3/(Nd)^2 - taking the rear surface so the entry media (glass) index for BK7 and e-line is n=1.518 and the exit media index n'=1 - Rc=-269,000, and the first aspheric parameter (often called "coefficient")
A1=1/4(n'-n)R^3=8.28^-11.

These give the profile as z=(1/2Rc)(rd)^2+A1(rd)^4 - and taking the 0.707 relative pupil height of the neutral zone for the value of normalized in-pupil height "r" as the location of maximum depth - z=-0.021+0.0105=-0.0105mm.

The P-V wavefront error at best focus - which applies to this neutral zone position - is given by W=D/2048F^3 which, for D=300mm and F=3 gives W=0.0054mm. Hence, the needed corrector depth at 0.707 zone is z=0.0054/0.518=0.104mm, in good agreement with the formula.

After the primary spherical is corrected, the next higher order term is added to the profile curve. Since it is a 6th power function, it becomes significant only toward the edge. The effect of adding it is illustrated at the graph, grossly exaggerated, since it is usually much smaller than primary spherical. Effectively, it raises the edge, but since glass cannot be added, this means that the inner portion of corrector, from 0.7 zone to the center, need to be evenly deepened as much as it is the value of the second aberration term for r=1 (solid blue vs. dashed red). The deepest zone shifts just slightly below 0.707 radius, but that is insignificant. As with primary spherical, the actual P-V wavefront error is enlarged by a factor of 1/(n'-n) in the glass profile, which means that the paraxial focus error is by a factor (n'-n) smaller than the secondary aberration term itself (for r=1). However, since the amount of secondary spherical at its best focus is 2.6 times smaller than at the paraxial focus - which is the focus of correction for secondary spherical - leaving it uncorrected results in as much smaller secondary spherical, i.e. smaller by a factor of (n'-n)/2.6 than the secondary aberration term.

Vla

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#17 mark1234

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Posted 01 February 2013 - 07:06 PM

wh4hqs, thank you.
This is starting to make more sense. I'm going to re-read all the contributions so far and sleep on it.
Setting the Rv first for 0.707 or 0.868 zone has me puzzled just now. From where is this derived if OSLO cannot be left to optimise it?
When you point it out I'll realise it was obvious. Or freak out at a page of polynomial equations.

Mark

#18 mark1234

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Posted 01 February 2013 - 07:23 PM

Yes, just spotted the relevant line for Rc in your post wh48qs.
Now I can sleep on this.

Many thanks to everyone, the issue seems to be resolved.

Well, except for making sure I can actually measure centre depth of plate while under vacuum accurately enough etc.
It would be nice to use an off-axis portion of the aperture for visual use when seeing allowed. Which needs the best figure I can get on the corrector.

Mark

#19 MKV

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Posted 01 February 2013 - 09:35 PM

Setting the Rv first for 0.707 or 0.868 zone has me puzzled just now. From where is this derived if OSLO cannot be left to optimie it?

You choose it, Mark! See my post #5654333 above. Once you decide where your you want your "neutral zone" you calculate the vertex radius. I call it it Rv, Vla calls it Rc.

In my post I mentioned that all I had to do was choose the "NZ", calculate and enter the Rv and then let OSLO calculate deformation coefficients. No slider wheels were used to get that configuration. The "NZ" is implied by your choice of the vertex radius.

If you're not familiar with OSLO optimization techniques, you can read up on it in the online manual plus numeorus sites on the Internet.

Mladen

#20 mark1234

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Posted 02 February 2013 - 08:48 AM

Thanks Mladen,

and 1/2A ~ 119,000 inches.

I was assuming the ~ was 'when taking the reciprocal of the half A you end up with' - took a few goes to realise the inches was a typo. and the way you set it out is very clear and is as you intended vis. 'approximately'!

Mark

#21 MKV

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Posted 02 February 2013 - 10:59 AM

Mark I couldn't find the "approximate equal" sign (two wiggly lines), so I used tilde to to say the vertex radius (Rv) or 1/2A = (approximately) 119,000. The plate is essentially a zero-power lens so, rounding off the vertex radius to the nearest whole number is fine. It's a theoretical value that changes as soon as oyu move off center, so it;s only function is to determine where the neutral zone will be.

Oh, by the way, the 4th order and subsequent deformation coefficients (i.e. BY^4 + CY^6...) have nothing to do with the vertex radius or the neutral zone location but with 3rd and 5th order SA (i.e. 4BY^3 + 6CY^5...). The 4th order coefficient coefficient B is simply = 1/[4*(n-1)R^3)].

For systems f/3 and slower, as Vla already mentioned, the 5th order SA is usually so small that it can be ignored (i.e. it can be refocused), so only the third order SA term is used. The same probably holds true for even faster systems. refocuisnf cuts down the totla 3rd order SA by a factor of 4.

To illustrate the effects of using only the 3rd order vs. the whole series of coeffcients, this is best illustrated by raytrace. I am using a 200 mm f/3 Schmidt camera as the example below.

You're right about the inches rather than mm. Sorry about that.

Mladen

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#22 wh48gs

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Posted 02 February 2013 - 02:03 PM

However, since the amount of secondary spherical at its best focus is 2.6 times smaller than at the paraxial focus - which is the focus of correction for secondary spherical ...



Well, this is what the Schmidt profile relation implies, but it is not necessarily the best practical solution. Just like primary spherical, the secondary spherical can also be corrected by shaping up the profile to compensate for wavefront deviation at best focus location. I.e. instead of slowing the edge portion of aberrated wavefront by thickening the edge, it can be done by advancing the wavefront section around 0.7 zone. If so, then the secondary aberration term is not A2(dr)^6, but A2[d^6(r^6-r^2)]. Unlike the paraxial function, this one has negative values, directly adding to the depth, which requires much less figuring. In addition, as the graph shows, this term also adds less to the depth, by a ratio (r^6-r^2)/r^6, which for the zone of maximum depth r~0.7 in the numerator, and r=1 in the denominator, gives 0.375, i.e. 2.66 times smaller extra depth at the maximum. I don't know which way it is done by those making the plates, but this seems better.

Vla

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#23 wh48gs

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Posted 02 February 2013 - 02:25 PM

Mladen,

For systems f/3 and slower, as Vla already mentioned, the 5th order SA is usually so small that it can be ignored



It is easy to check out, so there's no need to guess. The value of secondary aberration coefficient A2 is the P-V wavefront error at paraxial focus magnified by 1/(n-1) factor, so the error is (n-1)A2. Best focus P-V error is smaller by a factor of 2.6, or W=(n-1)A2/2.6, ignoring the sign. Since A2=3d^6/(n-1)8R^5, that comes to the best focus P-V error W=d^6/7R^5 (ratio P-V/RMS is nearly identical as with primary spherical, nearly 3.4).

For a 200mm f/3 Schmidt (d=100, R=-1200), it comes to 1/10 wave P-V.

Vla

#24 mark1234

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Posted 02 February 2013 - 05:07 PM

Vla and Mladen, this is one area where ray tracing indulges with too much perfection on offer, but it is interesting to see how the shift in zone for best OPD chromatic error has been rationalised for example. It is likely to be easier to figure for one thing.
I set the plate up with the spherometer as a dummy run. I realised there is a gap in my comprehension of David Rowe's formulae - attached, hope he doesn't mind because I can't fathom why Poissons ratio is re-introduced "s" when the whole point is to pull the plate into a catenary with a calculated centre displacement and grind a shallow sphere into it - at that stage the plate could be made of anything as long as the original (say flat underside) returns to its previous (say flat) shape with the asphere on the ground side. Maybe not then? His spreadsheet doesn't work with my software - too old or something. It means I can't see what he's put in the calculation boxes either.

Mark

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#25 mark1234

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Posted 03 February 2013 - 06:25 AM

I guess the answer here is that it's the physics of plate deformation at work and that's how the equations have been reduced to get the needed values. I can't argue with Poisson and the variation in the actual value for my sample of glass is not going to make a significant difference to the deflection? s=0.21 seems the best guess based on the most recent information on the web for plate glass.
Thanks for all your help so far.

Mark






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