Yet another SA200 configuration ...

[vlaiv]

I'm probably on a verge of spamming this section of SGL with my "ideas", but here is another one ...

Again topic is how to maximize SA200 effectiveness to achieve best possible resolution. 200 lines/mm is quite "potent" low (maybe to mid?) resolution grating, depending how it is utilized.

I'm aware that ultimately seeing will be limiting factor for any sort of slit-less spectrograph.

In converging beam we have two "forces fighting" to limit the resolution. One being beam width, or number of illuminated lines, other is the beam angle that results in coma - shallower the beam convergence, more coma is introduced that lowers the resolution.

So I asked my self, how to "widen" the beam and make the beam itself more steep. One obvious answer is to use collimating lenses, but I wanted to see if I can use any gear that I already have in my "astro box". Bonus would be if I did not have to purchase or fabricate any sort of adapters to put the whole assembly together.

Then idea came - how about following setup:

sensor - focal reducer - SA200 - barlow lens - scope

So I have all the bits and bobs needed for this setup:

All sorts of adapters (1.25" filter thread to T2 thread, various extenders in 1.25", T2 and 2") to put everything together, but with limited options for spacing. I have 0.5x 1.25" reducer (GSO variety), I have 2 nose piece 1.25" barlow elements - x2 GSO and x2.7 APM (coma corrected).

Barlow lens, being negative lens, will diverge incoming F/8 beam (8" RC scope) - making into F/16 or even F/21.6 (depending on barlow element used), and reducer element will "reduce" dispersion, so grating can be mounted optically further away from sensor - making beam width larger (at least this is my reasoning, not 100% sure).

Problem is of course, that I have no idea how to calculate distances, resulting dispersion, beam width on grating, and in general any of parameters

I don't even have the idea if such a setup will properly come to focus. I suspect it will, since RC has quite a large back focus, and it may even happen that focus position is not altered by much (barlow moves focus out, reducer brings it back in, might be that those two cancel out). There will be significant vignetting, but that is not important for point sources like stars (rest of the field will be "strange", but star close to optical axis should be ok even with 1.25" elements).

Any thoughts on this?

[andrew s]

You can also use an eyepiece instead of a reducer. I think Robin has a similar setup on his "three hills observatory" site.

Regards Andrew 

[vlaiv]

11 minutes ago, andrew s said:

You can also use an eyepiece instead of a reducer. I think Robin has a similar setup on his "three hills observatory" site.

Regards Andrew 

Yes, but beauty of this approach is that I have everything I need to assemble it

I did some more research and it turns out that I might be able to do a "fully collimated" beam with this setup. That would eliminate coma completely.

Barlow can produce parallel beam from converging beam if it is placed so that focal point of barlow is at same position as focal point of telescope.

On cheap x0.5 reducer - it turns out that these work the best on parallel beams - faster cones introduce SA and other aberrations. It looks like cheap achromatic reducers are objective lenses for 25mm binoculars and finders, according to this source:

https://www.cloudynights.com/topic/532924-are-gso-05x-focal-reducers-any-good/

I've also found that FL of reducer is around 101-103mm, and I guess that x2 barlow element has something like 70-80mm FL. So it is at least starting point in coming up with some sort of calculations for spacing and distances.

 

[andrew s]

Look forward to seeing your results

Regards Andrew 

You can put the Barlow in the scope and with your naked eye focus it on a distant object. This should with a relaxed eye give a parallel beam as it is a Galilean telescope.

[vlaiv]

Follow up based on some calculations.

This approach is not quite going to work in above configuration.

It turns out that dispersion is going to be too large, so in order to fit first order spectrum on sensor - 0 order star image will have to be quite offset to optical axis - and subject to much vignetting.

Also combination of barlow and reducer in any sensible configuration will still give magnification bigger than x1 - which will increase resolution and seeing blur will be spread over more pixels - thus additionally limiting spectrum resolution.

I calculated resulting focal length being over 2300mm (can't remember now exact value, but I've seen values in range 2300-2600mm when simulating setup). In any case, too much for my usual 3" FWHM star in long exposure - sampling being close to 0.3"/pixel - so 3" FWHM would translate into 10pixels star image, and with spectrum resolution of 2.6A/pixel - that equates to 26A blur or R230 (@6000A) so far away from expected R400-R600

Other option would be to use such setup on 80mm F/6 scope that has 480mm FL, and in that case I would end up with sensible base resolution where seeing would not impact spectrum too much, but that would limit light throughput quite a bit (more than x6 less light).

Also I figured that there is very simple way to reduce chromatic coma in prime focus, without using any additional glass - by use of aperture mask - again matter of trading off some light throughput.

In general I concluded that I don't fully understand cause and magnitude of chromatic coma from grating in converging beam. Spreadsheet used to calculate it, and also formula quoted on couple of websites assumes that chromatic coma is independent of the seeing. But as I understand it, these two should be additive in some sense (either pure addition or a form of convolution of respective PSFs). Also I was unable to find derivation of the expression. I tried to find the source of that formula and the furthest I got was:

Daniel J. Schroeder Astronomical Optics

There expression is listed without any derivation in section that deals with converging beam of spherical mirror. I'm not clear whether it is caused by spherical aberration from mirror or is it general property of grating in converging beam.

On Astrosurf section that talks about grating in converging beam Christian Buil offers some spot diagrams, which I believe (maybe wrongly so) are derived with geometrical optics approximation. This in turn would mean that chromatic blur is consequence of light rays in converging beam hitting grating at different angles thus being refracted at different angles to each other according to:

m * lambda = d * (sin a - sin b )

But if I try to derive delta_lambda from this equation (being error in wavelength due to refraction at different incident angle for base wavelength) - I get expression that depends on grating resolution (and gives very nebulous results - so it's obviously wrong) and original one does not. Here is quick derivation (m=1 for first order):

lambda = d * ( sin a - sin b )   // expression for outer most ray in converging beam incident at angle a

lambda + dlambda = d * ( sin b )  // at some other wavelength chief ray will be at 0 angle of incidence

We want to find dlambda when outer most ray of our base wavelength is equal to chief ray of "shifted" wavelength - this will give us blur extent (or half extent to be precise).

from these two equations:

dlambda = d * sin a

fr/2 = tan a = sin a / cos a = sin a  / sqrt(1 - sin^2 a)    // calculating sin a from focal ratio

fr/2 * sqrt(1 - sin^2 a) = sin a

fr^2 * (1 - sin^2 a) / 4 = sin^2 a

fr^2  - fr^2 * sin^2 a = 4 * sin^2 a

fr^2 = sin^2 a * (4 + fr^2)

sin a  = sqrt ( fr^2 / (4 + fr^2))

so dlambda = d * sqrt( fr^2 / (4 + fr^2))

Now let's calculate dlambda for SA200 at F/8, 500nm

it gives dlambda of ~312nm - which is obviously wrong