| article: The Performance of a Telescope - The Modulation Transfer Function MTF |
The Telescope: a low-pass Filter
As Harold Richard Suiter explained in his book about
Star Testing,
telescopes can be regarded to be low-pass filters.
In other words: The telescopes's finite aperture is cutting the high frequencies and hence is limiting the resolution.
Also some other filters are the
atmosphere
and other parts involved in the imaging process with a telescope.
Just like the bass and treble filters of an audio amplifier is changing the "flat" frequencies of music, a telescope will change the transfer of a "plane" wavefront.
The "bass" of an image is the low spatial frequency and the "treble" is the high spatial frequency.
High spatials are the finest details in an image, i.e the rilles on the Moon and close double stars.
Low spatials are the coarse detail like the evenly illuminated bottom of a crater of the Moon.
Obviously Rayleigh, Dawes, Sparrow and the term "diffraction limited" are only dealing with the highest spatial frequencies!
So, would it be reasonable to judge the performance of an audio system just by the highest frequency it can deliver?
The response of a telescope to a spatial frequency can be plotted as the MTF.
The term Modulation is also called the Contrast.
Giving a perfect seeing, no stray light and a linear optical system the MTF is describing very precisily the resulting contrast in the focal plane of an optical system.
The telescope is filtering (modulating) the 100% contrast of an object to something lower.
The higher the frequency the lower the contrast will be because as we have seen already because the telescope is a low-pass filter.
When the telescope reaches it's resolution limit the contrast transfer will be 0.
The resulting contrast of the object (i.e. planetary detail) together with the contrast lowered by the telescope at the spatial frequency of the detail is deciding whether we can image a detail in correct proportions or not.
This does *not* mean that a too small and hence unresolved detail cannot be imaged at all!
Otherwise the stars would never be seen on an stronomy image with amateur telescopes.
Unresolved details are just not shown "correctly", i.e. the Saturn's Cassini division might be imaged with smallest scopes but to wide and with too low of contrast.
Normalized MTF of unobstructed, perfect aperture (black) and a 30% obstructed aperture (red) [aberrator V2.11]
As can be seen both telescopes drop to 0 contrast at 100 % resolution.
It's interesting to see how the 30% of central obstruction is delivering better contrast at higher spatial frequencies compared to the unobstructed.
The old rule of thumb that obstruction is decreasing contrast is true only for 50% of maximum resolution and below hence for lower spatial frequencies.
If you cannot accept that fact just think of the obstruction beeing the treble filter of an audio amplifier - just an optical one.
However the perfect response of the amplifier clearly would be the linear one.
But for someone's taste the "emphasized" treble might be nicer.
If someone's task is to split double stars that kind of "treble boost" might just be right.
The MTF and the real world
If we want to compare the performance of real telescopes and their MTFs we need to know about
- how to calculate an MTF
- how to model the central obstruction and spider vanes
- what does 100% spatial frequency mean in arc seconds to de-normalize the MTF
How to calculate an MTF
As Hecht, Optics 2002 has shown, real life lenses (and objective mirrors) are actually working like Fourier Transformers.
He called them optical computers for real-time Fourier Transformation.
In other words: the Fourier Transform is describing very well the behaviour of a lens while image formation is going on.
This is one way to understand why telescopes are low-pass filters.
A Fourier low-pass filter would cut away the outer parts of the image after it was Fourier-transformed (hence in the frequency domain).
And the same is true for lenses (or mirrors) with finite diameters!
On the other hand an annular aperture (Newtonian secondary mirror) is cutting away the inner parts of the Fourier Transform and that is exactly what a high-pass Fourier Filter does.
You can take this as a model to understand the Newtonians lowered contrast at low spatial frequencies.
For calculating any MTF we could use some property of the Fourier Transform and Fourier Optics:
... (to be continued, sorry for the delay)
Unfortunately my own software is not ready yet.
Meanwhile I left the MTF-calculation to the freeware program Aberrator and copied the values for 0.95 to 0.05 % of contrast to a spread sheet.
The precision is limited but it can still give a first impression.
How to model the central obstruction and spider vanes
We have seen in the previous chapter that the central obstruction and the spider vanes can be modelled by blocking the Fourier Transform with exactly the shape of the secondary mirror support.
The precise relative thickness of the vanes compared to the central obstruction is important.
Also the objective diameter must be in proper relation.
Now if we take at least two pixels for the vanes of 0.5 mm (Nyquist Sampling Theorem) we end up with at least 4 pixels per mm.
The total size of the sampling plane for a 300 mm objective is then at least 1200 pixels.
(The resulting MTF-Plots are delayed, sorry.)
What does 100% spatial frequency mean
I find it reasonable to use the Sparrow's limit as the cut-off frquency of a telescope.
That's how I denormalized the graphs below.
I am still investigating this (all comments are very welcome).
Resulting MTF
As described above this result was estimated with Aberrator (not taking spieders into account).
The first plot is showing the absolute performance difference of a perfect 6 inch (150 mm) Refractor and Newtonian Reflector as well as a 8 inch (200 mm) Newtonian.
Absolute MTF of APO vs. Newtonian [aberrator V2.11]
The first arrow at about 5 seconds of arc is indicating that the APO should outperform both Newtonians for lower spatial frequencies.
The second arrow is indicating the the APO is outperforming the Newtonian of equal size even to relative fine details of up to 2.8 seconds of arc.
But for finest detail the 6 inch Newtonian is even better than the APO.
The overall performance of the 8 inch Newtonian is comparable or slightly better than the performance of the 6 inch APO.
I think this is quite reasonable and very often reported from practical tests.
Not commonly accepted is the result that the 8 inch Newtonain should clearly outperform the 6 inch APO in the domain of finest planetary detail.
The next plot is giving an overview of the absolute performance of various telescopes.
The 30% of obstruction for the Newtonian is what you have in a f/4 Newtonian with a fully illuminated field suitable for a typical astronomical CCD camera.
Absolute MTF of various telescopes - an estimation [aberrator V2.11]
Note the very comparable contrast at 10 seconds of arc and below for the tiny 2.4 inch (60 mm) APO compared to the 16 inch (400 mm) Newtonian.
In fact, with a pixelsize in that order on our CCD camera we can talk about wide angle photography.
That means that a very small telescope can produce astonishing images for the bigger objects in the sky.
Please have a look at the Andromeda galaxy taken with the Baby Tak to see an example (Don't misss the high resolution version!).
In the high detail domain the mirrors with huge diameters and a central obstruction are clearly dominating the scene.
Note that the contrast at a detail level of 1 second of arc can be doubled compared to the 6 inch APO!
Though a lot of amateurs are claiming this to be the other way round.
Have a look at the best planetary images and compare the telescopes.
These pictures are saying more than thousand words and are IMHO consistent with the MTF shown above.
Though the image gallery of planets of the astroscopic labs is currently by no means representing the state of the art, there is already a hint of what I meant with the above.
Best of Planetary Imaging Gallery
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