Calculating the sagitta of a lens

[dark star]

I am making a lens. It is 4.25 inches in diameter (107.95 mm).  It wil be convex on both sides. The radius of curvature of the first lens surface (R1) is 27.133 inches (689.1782 mm). The focal length is half the radius of curvature,13.5665 inches.

The radius of curvature of the second lens surface R2 (back of the same lens) is 302.4 inches (7680.96 mm). Focal length 151.20 inches.

I am having problems working out what the sagitta of R1 and R2 should be. Obiously the sagitta is positive as both surfaces are convex.

Putting these figures in to an on line sagitta calculator (https://www.bbastrodesigns.com/sagitta.html)

I get the sagitta of R1 to be 0.0833 inches (0.00948182 mm)

and  the sagitta of R2 to be 0.0075 inches (0.1905 mm)

However, using the formula for calculating sagitta below I get different numbers for sagitta. Quite possibly I have made a mistake,, as I am not great at maths.

Sagitta (s)  =  Mirror_Radius² / ( 4 × Focal_Length )  =  r² / 4F

Using this formula I work out that the sagitta of R1 should be 0.0416 inches (1.05664 mm)

R2 should be 0.0003733 inches (0.00948182 mm).

Is the calculation different because it is a lens?

I asked about this on Cloudy Nights webiste and was told that the sagitta for R1 should be 0.0737 inches and the sagitta of R2 should be 0.00661 inches, which is different to both sets of figures above! Can someone who is good at maths please tell me what I am doing wrong?

This is for making a Schupman telescope. It uses a singlet lens (as above) and  a corrector. Both the lens and corrector should be made of the same kind of glass. I am using BK7 glass. The blank for the corrector is also 4.25 inches. in diameter.  I am making the design where the objective lens is the same size as the corrector. In some designs it is smaller. There is also a Mangin mirror. This is a lens and mirror together, the front surface is a lens and the back surface is a silvered mirror, the light goes through the lens and is reflected back through the lens by the mirror. There is also a small spherical mirror. It is quite a complicated design, but if made correctly it will be apochromatic. In other words, you can look at the planets or at a star ant there will be no false colour, as there is in an achromatic refractor. It is much cheaper to make than the cost of buying a 4 inch apochromatic refractor. The version I am making will be f/12.

I will not go in to detail as to how the design works as this can be found on line.  Also, I am still in the process of reading more in order to fully understand this myself!

A few Schupmans have been made in America and Germany. I am not sure if any have been made in the UK.  Please let me know if anyone has made one in the UK.

I have made a 14 inch Dobsonian, including the mirror. I was making a 20 inch Dobonian but have now moved to a flat with 2 flights of stairs up to it. So this is no longer practical. I think that I am going to enjoy working on a small lens rather than a 20 inch mirror,

which was hard work!

I have an optical flat which I can use to test the lens and corrector  in double pass, using a Ronchi screen or a Foucault tester. I also have a Bath interferometer, but I am not sure if I will need to use this.

I have also made a small spherometer. I am in the process of making a wedge tester, to test the wedge of the lens and corrector, so that I can remove this if necessary.

I also have a 4 inch BK7 blank and a 4 inch flint blank. So if this project proves to be too challenging I will make an achromatic refrator instead. I may make one at some point anyway.

David

[dark star]

I have realised that I made a very simple mistake. The lens blank is 4.25 inches in diameter However,  the outer 0.25 inches will be hidden by the lens cell. Therefore on Cloudy Nights I said that the diameter of the lens is 4 inches, as this is the effective aperture. This accounts for the mistake. On Cloudy Nights the formual given was  Sagitta = D²/ (8 R) 

This appears to be the same  in effect as the other formula: Sagitta (s)  =  Mirror_Radius² / ( 4 × Focal_Length )  =  r² / 4F

Anyway, I now get the same results using both formula.

I now need to measure the distance between the ball feet on my spherometer and work out the correct numbers so that I can measure the sagitta of the lens. It is also possible to measure the sagitta of a mirror of a known focal length with the spherometer and use this to calibrate the spherometer. I have a couple of mirrors I could use, but I have not figured out yet how to work this out mathematically, there are formulas on line but they are quite complicated. The spherometer is 86 mm in diameter.

If anyone can explain how to do this I would be grateful. I have measured the sagitta of a  250 mm diameter, 1200 mm focal length Orion Optics UK mirror that I have. The reading on the spherometer is 0.32 mm.

David

 

 

[davidc135]

Great project! Hopefully we'll be able to follow your progress.

I think that you skipped a 2 in the denominator. I got .0832'' for R1.

Just to make things more complicated, looking up a 'sagitta of a chord' calculator, it gives the equation as the more precise:

S = R-sqrt(R^2-r^2) where R = radius of curvature giving S1 as .0857ins, a difference of 63 microns.

A lens shouldn't make any difference.

David

[davidc135]

I should scrub the Schupmann in my signature as after I'd made the Mangin I discovered that I'd made a basic mistake and so it's only half (or less) made.  David

[davidc135]

The three ball spherometer formula is: 

r = R^2/2H + H/2 +/-B (- for convex surfaces)

where r = surface radius of curvature

R=radial distance to centre of balls

B = radius of balls

H = measured sag

Calibrating the spherometer against a concave surface of known curvature in order to find R, I get:

R = sqrt(2Hr-H^2-2HB) if I have it right

and the first formula with B negative to calculate r of a convex surface.

David

[dark star]

On Cloudy nights I was given this calculation: (the diameter of the lens being 4 inches).

Sagitta = D²/ (8 R)   so R1 = 16 / (8 x 27.133) =  0.0737"     and for  R2  0.00661" 

For R1 I got the same number.

I am glad someone has at least tried making one in the UK.  It may be over ambitious, but if necessary I can always decide to make the achromatic refractor first, for practise with lenses. if I get stuck with the Schupman. It is supposed to be possible to make the lens and corrector quickly for this size of Schupman, but this was someone who knew exaclty what they were doing! And had made a lot of optics, lenses and mirrors.

[davidc135]

2 minutes ago, dark star said:

On Cloudy nights I was given this calculation: (the diameter of the lens being 4 inches).

Sagitta = D²/ (8 R)   so R1 = 16 / (8 x 27.133) =  0.0737"     and for  R2  0.00661" 

For R1 I got the same number.

I am glad someone has at least tried making one in the UK.  It may be over ambitious, but if necessary I can always decide to make the achromatic refractor first, for practise with lenses. if I get stuck with the Schupman. It is supposed to be possible to make the lens and corrector quickly for this size of Schupman, but this was someone who knew exaclty what they were doing! And had made a lot of optics, lenses and mirrors.

Ah yes, I was using 4.25'' aperture.  David

[dark star]

Could you please explain the formula for Calibrating the spherometer against a concave surface in simple terms?

What is sqrt? I only got as far as GSCE maths, and have forgotten most of it!

 

 

[davidc135]

I'm hardly in the best position, not having finished my 4.5'' version but I think the difficulty level is in the same league as an achromat. Definitely, you should be confident of success.

I messed up the maths on my Mangin but the optical fabrication was straightforward, especially as you have the flat etc.

David

[davidc135]

6 minutes ago, dark star said:

Could you please explain the formula for Calibrating the spherometer against a concave surface in simple terms?

What is sqrt? I only got as far as GSCE maths, and have forgotten most of it!

 

 

sqrt is square root.

[davidc135]

The formula for deriving R from a known concave surface is simply the earlier formula (in terms of r) rearranged.

What is half the diameter of the ball supports?

But you could do with the formula rearranged again to give the required Sag, or H. My Maths is poor as well, anyone?

David

[davidc135]

I've a feeling that for the Schupmann surfaces, just r^2/2R may be close enough especially if the spherometer feet balls are very small.  David

[dark star]

I will post more on this next weekend. When I will have time to measure the dimensions of the spherometer.  Also, I need to find my digital caliper in order to do this.

I have a small powered turntable, that I bought around 3 years ago, thinking it might come in usefull, and it has. It is just right for putting the 4 inch lens on, it rotates and I will grind  the lens by hand.

[dark star]

The balls on the spherometer are 6 mm  in diameter. They are ruby balls, which I got from Edmund Optics. As I read that metal balls can eventually go slightly flat where they contact the glass. Also, there is less chance of them scratching.

[davidc135]

Given your figures of radius of curvature of 2400mm and measured sag of 0.32mm on the 10'' mirror, I make the spherometer radius to be Ooops 39.19mm if just the simple equation:

R = sqrt(2Hr) is used. Adding either or both of the two extra terms only adds a tiny error. Is 2400mm exact?

David

Edited Sunday at 21:51 by davidc135

[dark star]

I don't actually know if the focal length is exact, I am just going on what the focal length is according to Orion Optics UK.  I will use my Foucault tester to measure the radius of curvature/ focal length using a tape measure.

I will also measure the distances between the balls on the spherometer and put this in to the formular So that I can compare the 2 figures.