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Hi Jase,
I'd like to call time out and get back (for the time being) to the thread.
Sejanus must be thoroughly confused with all this
.
Question and Answer
The question posed was in the comparison of two types of telescope.
The answer to the question lies in the treatment of the telescope as an optical system - not just a primary mirror or lens.
In general:
To compare the telescopes optically we need to take into account the performance of each the optical element in the telescope then "combine" the performance of each of these elements such that we get a measure of the performance of the telescope as a whole. The overall performance of each 'scope is then compared.
The metric chosen here is the system Strehl ratio. As we have seen the Strehl ratio tells us the proportion of light reaching the Airy disk.
By definition:
SR = (Intensity at centre of PSF of the aberrated optic)
/(Intensity at centre of PSF of a diffraction limited optic)
PSF = Point spread function ie. the response of an optical system to a point image
cf. Impulse response in the time domain.
What happens next?
The psf of each element comprising the telescope must be known.
The PSF carries all the information about the optical element - to make life easy let us talk about one colour only.
I now try to describe in words the application of "Linear Systems Theory" to this telescope system.
(Why this theory may be applied to these systems is a long story that
starts with the fundamental question as to why we get diffraction in the
first place!!)
The light leaving the optical element is the light entering the
element modified by the psf of the element itself. This "modification" is
actually a mathematical operation called convolution (*). What may be said is that the function representing the incoming light (iin(x,y) is convolved with the psf(x,Y) to produce the output light function (io(x,y)) or
io(x,y) = iin(x,y)*psf(x,y)
For a system with a number of psf's in a line (eg. Telescope)
i(x,y) is the intensity profile. ii = input, io = output
io(x,y) = iin *(psf1*psf2*psf3)
Where psf1, psf2 etc are the individual psf's
and psft = psf1*psf2*psf3 ... psft = system psf
The equation: io(x,y) = iin(x,y)*psft(x,y) is very computationally intensive to implement - there must be an easier way (?) and there is!
We invoke an animal called the fourier transform - you may take the fourier transform of the above equation so:
if F{io(x,y)} = Io(wx,wy) ... notice change from i to I
note:x and y are spacial (eg mm) and wx and wy are frequency (cycles per mm)
In the time domain x and y would be seconds and wx and wy would be Hertz.
Io(wx,wy) = F{iin(x,y)*psft(x,y)}
or
Io(wx,wy) = Iin(x,y) . F{psft(x,y)} Where "." = multiply
note: We have changed the complex operation of convolution
to the simple operation of multiplication
Next, to simplify things by invoking the property of radial symmetry
we will remove one of the variables
So: Io(wx) = Iin(x) . F{psft(x)}
or write the above as:
Io = Iin . F{psft} ... Nice and simple!
Now remember that: psft = psf1*psf2*psf3
and
F{psf1*psf2*psf3} = F{psf1} . F{psf2} . F{psf3}
Now F{psf} is a complex function ie. It has amplitude and phase for
this exercise we will use the amplitude of F{psf} or |F{psf}|
In plain english:
|F{psf}| is the Modulus of the Fourier Transform
of the Optical Transfer Function and this is called the
Modulation Transfer Function (MTF)
Io = Iin . MTF1 . MTF2 . MTF3
or to extend the above to a 'scope with n components
Io = Iin.MTF1.MTF2.MTF3 ---- MTFn-1 . MTFn
MTF in optics is the same as frequency response in audio systems.
If we used the zero frequency term only then
I0o = Iin0o.S01 . S02 . S03 ----- S0 = Strehl ratio (SR)
I0o = Iin0o . SR1 . SR2 . Sr3 -----
So evaluating the component SR's in a telescope and multiplying them
gives us the overall SR which is a good measure on the performance of the scope.
The above sets the argument behind my original post. We have talked about the telescope as a system.
The next stage would be to discuss the points you raised but this is outside this thread.
It could be discussed as a separate thread but I am sure it would bore most people. The points would would address the tie between Strehl ratio and the phase front of the energy and how the optics affect this phase front. Once we understand this one could look at the atmosphere and the issue
of image de-convolution - notice many programmes have de-convolution algorithms and we have been talking about convolution - do you have a strange feeling that if we go down that path we will be going in the opposite
direction to the one we've been taking!!
Regards,
Jerry.
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