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West Coast Engineering

Posted: Tue Nov 04, 2003 8:16 am

Guest

Hi,

Well not really other than the fact that I'm working on this on my
time from home rather than needing to do it for my day job.

I form the diffraction OTF in the X and Y planes via auto-correlation
of the pupil function rather than via the FFT of the PSF since it is
faster when I only need responses for vertical and horizontal bars.

Anyone know how to extend this auto-correlation calculation to include
partial spatial coherence. That is, the very same source split very
slightly in space into two sources as in Young's double slit or more
correctly double pinhole experiment?

Please don't refer me to Born and Wolf. I've already read it with the
attendant headache it causes.

Sincerely,

West Coast Engineering

Phil Hobbs

Posted: Tue Nov 04, 2003 8:16 am

Guest

West Coast Engineering wrote:

Quote:

I form the diffraction OTF in the X and Y planes via auto-correlation
of the pupil function rather than via the FFT of the PSF since it is
faster when I only need responses for vertical and horizontal bars.

Anyone know how to extend this auto-correlation calculation to include
partial spatial coherence. That is, the very same source split very
slightly in space into two sources as in Young's double slit or more
correctly double pinhole experiment?

Dear West:
There's a terminological thicket here. It isn't deliberate, like
radiometry, colour science, linear programming, or workplace safety,
where the thicket is intended to hide the basic simplicity of the
subject (and therefore preserve job security for the devotees), but
rather a legacy of pervasive confusion on the topic of transfer functions.

The classical formulation of optical transfer functions is not a good
analogue to transfer functions as used in circuit theory, ordinary
differential equations, and so forth, although it looks like it.

Problems like yours usually come from starting somewhere too far from
the basic physics, i.e. time-varying electric and magnetic fields. (I
know I'm sounding worse than B&W, but I'm not really, don't worry.) The
coherent transfer functions, i.e. the ones that apply to _fields_ and
not _intensity_ (*ahem*, I mean of course Irradiance, almost forgot to
appease the Radiometry Gods!) are the ones to work with. They require
many fewer dubious assumptions and stay closer to the physics. If you
need the optical transfer function--the one that works on Irradiance and
assumes incoherent illumination--you can get it trivially at the end of
the calculation via a random phasor sum.

If you have some source field distribution S, and get a (coherent) PSF
P, then when you add a shifted copy of S, you get a phase-shifted copy
of P by the shift theorem of Fourier transforms,

FT(g(x-a)) = exp(-i2*pi*f*a)*FT(g(x)).

When you add them together, you get sinusoidal ripples in the coherent PSF,

FT(g(x)+m*g(x-a)) = (1+m*exp(-i2*pi*f*a))*FT(g(x)).

If you're going to need the optical transfer function eventually, then
the relevant illumination source for the coherent transfer function is a
point source, because at the end we're going to have to take the RMS sum
of all those point-source contributions to the image irradiance.

Each point source becomes a single Fourier ccmponent at the pupil of the
optical system. Similarly, in your case, each double point source
becomes two Fourier components, or one component with a sinusoidal
envelope, as above.

If you retrace your computation, following the physics in the above way,
the problem should become easier. The behaviour of fields is much more
intuitive than that of image irradiance, because the fields exist
throughout the optical system, whereas image irradiance doesn't.

There are other ways in which the OTF isn't really a transfer function,
the most important one being that you can't compute the OTF of two
systems in cascade by simply multiplying the individual OTFs.

For example, consider a 1:1 relay system consisting of two lenses of
focal length f, spaced 2f apart. With an object at -f from the first
lens, there will be a good image at the centre of the system and another
one at f past the second lens. If I choose my reference plane for the
individual OTFs to be the centre, everything works reasonably well. On
the other hand, if I choose it to be off centre, the image at that plane
will be out of focus, leading to an ugly OTF, falling off very rapidly
from zero spatial frequency. The second lens will also be defocused,
leading to another ugly OTF, so if I multiply them together, my
calculated OTF will be ugly squared. This is exactly the right answer,
*provided I put a diffuser at the reference plane*.

In real life, of course, an odd choice of reference plane doesn't affect
the system operation at all--the defocus of the first half is undone by
the defocus of the second, leading to a good image. The OTF gets this
wrong, but the CTF gets it right--the phase curvatures of the two CTFs
compensate correctly, and you get the right answer.

Lest anyone say that I'm just being silly, that nobody would set up a
calculation that way, let's go a bit deeper into the problem. A
symmetric optical system such as I've described has no odd-order
aberrations, because the second half's aberrations cancel out the first
half's. (The even orders add). Computing the overall OTF by
multiplying the two half-OTFs will get this wrong, because the phase
information is lost, so all the aberrations add in RMS instead of
directly. Odd order contributions will be overestimated, and even-order
ones underestimated. Yet this weird OTF thing is called "the transfer
function" and tossed about as though it had physical meaning.

That's the terminological thicket I'm talking about. It (and the
conceptual obscurity it creates) comes from the less-physical quantity
(the OTF) having seized the higher ground in early days when coherent
optical systems were a rarity.

Cheers,

Phil Hobbs

Helpful person

Posted: Tue Nov 04, 2003 3:21 pm

Guest

West Coast Engineering <westcoastengineering@westcoastengineering.com> wrote in message news:<q79fqv8gnmegv9j4qlk3qt87oca004k3lf@4ax.com>...

Quote:

It's many years since I studied the field of partial coherence.
However, from what I remember, image formation in partial coherent
light is not linear. In other words, a single frequency sine wave
object does not produce a single frequency sine wave image. If this
is correct, OTF will have no meaning.

AES/newspost

Posted: Tue Nov 04, 2003 7:46 pm

Guest

In article <vqfn7c5q4oocfe@corp.supernews.com>,
Phil Hobbs <pcdhSpamMeSenseless@us.ibm.com> wrote:

Quote:

West Coast Engineering wrote:

I form the diffraction OTF in the X and Y planes via auto-correlation
of the pupil function rather than via the FFT of the PSF since it is
faster when I only need responses for vertical and horizontal bars.

Anyone know how to extend this auto-correlation calculation to include
partial spatial coherence. That is, the very same source split very
slightly in space into two sources as in Young's double slit or more
correctly double pinhole experiment?

Dear West:
There's a terminological thicket here. It isn't deliberate, like
radiometry, colour science, linear programming, or workplace safety,
where the thicket is intended to hide the basic simplicity of the
subject (and therefore preserve job security for the devotees), but
rather a legacy of pervasive confusion on the topic of transfer functions.

The classical formulation of optical transfer functions is not a good
analogue to transfer functions as used in circuit theory, ordinary
differential equations, and so forth, although it looks like it.

Problems like yours usually come from starting somewhere too far from
the basic physics, i.e. time-varying electric and magnetic fields. (I
know I'm sounding worse than B&W, but I'm not really, don't worry.) The
coherent transfer functions, i.e. the ones that apply to _fields_ and
not _intensity_ (*ahem*, I mean of course Irradiance, almost forgot to
appease the Radiometry Gods!) are the ones to work with. They require
many fewer dubious assumptions and stay closer to the physics. If you
need the optical transfer function--the one that works on Irradiance and
assumes incoherent illumination--you can get it trivially at the end of
the calculation via a random phasor sum.

Agree totally with Phil Hobbs.

There seem to be two nonoverlapping approaches to situations like this:
coherence theory, and linear (mostly) system theory or basic physical
laws with random or probabilistic inputs.

From my experience (with stereotyping mode turned up to full blast
here), coherence theorists like to define coherence functions and
related theorems and language of ever increasing complexity; manipulate
these abstract constructs into ever more complicated forms; and rarely
and only when forced to do so actually apply them to meaningful
problems. (Maybe that final clause is unnecessarily nasty.)

Non-coherence theorists, as Phil says, prefer to start with the basic
coherent and most often linear equations or physical laws of the
situation; solve them as needed (in the process learning something about
the physics involved); and then if some of the inputs are actually
random or probabilistic variables, feed this into the solutions, taking
appropriate care whenever things are not linear or any kind of squared
or product terms appear.

I believe there are some coherence theorists who will quite seriously
argue that there are "incoherent" physical situations that are somehow
more than just ordinary physical situations with random or
probabilistically definable inputs -- that there are physical situations
that can fundamentally only be described using coherence theory as in
the second paragraph above, and not by using standard physical laws plus
standard probability and noise concepts as in the third paragraph. I've
never been able to understand this argument, and in fact keep worrying
that there's something I'm missing in not being able to understand it.

I think Phil may think similarly.

Phil Hobbs

Posted: Tue Nov 04, 2003 9:32 pm

Guest

AES/newspost wrote:

Quote:

I believe there are some coherence theorists who will quite seriously
argue that there are "incoherent" physical situations that are somehow
more than just ordinary physical situations with random or
probabilistically definable inputs -- that there are physical situations
that can fundamentally only be described using coherence theory as in
the second paragraph above, and not by using standard physical laws plus
standard probability and noise concepts as in the third paragraph. I've
never been able to understand this argument, and in fact keep worrying
that there's something I'm missing in not being able to understand it.

I think Phil may think similarly.

I'm a bit out of my depth in these waters, having dropped my statistical
optics class in grad school halfway through, but I think there are
situations where it isn't trivial to do a partially-coherent imaging
calculation by integrating over the source. One possible example is
atmospheric turbulence, in which the randomness comes in as fluctuations
in the optical system, rather than the source itself. I think that in
this situation it's probably easier to use the van Cittert-Zernike
theorem for the propagation of the complex degree of coherence than to
try averaging over an ensemble of optical systems. At least it
compartmentalizes the randomness a bit.

I have tried to teach myself statistical thinking several times over the
years, including sitting down one vacation and working about halfway
through Papoulis's statistical signal processing book, but it has never
become intuitive for me. It seems as though I'm just missing that
lobe. (I comfort myself with Rutherford's famous remark, "If your
experiment needs statistics, you ought to have done a better
experiment", but that's just sour grapes.)

Cheers,

Phil Hobbs

Posted: Wed Nov 05, 2003 12:55 pm

Guest

In article <siegman-C0F042.16462904112003@news.stanford.edu>, AES/newspost
<siegman@stanford.edu> wrote:

Quote:

From my experience (with stereotyping mode turned up to full blast
here), coherence theorists like to define coherence functions and
related theorems and language of ever increasing complexity; manipulate
these abstract constructs into ever more complicated forms; and rarely
and only when forced to do so actually apply them to meaningful
problems. (Maybe that final clause is unnecessarily nasty.)

I agree with the description "unnecessarily nasty" but I'm disappointed
that such comments would come from you.

Quote:

Non-coherence theorists, as Phil says, prefer to start with the basic
coherent and most often linear equations or physical laws of the
situation; solve them as needed (in the process learning something about
the physics involved);

I'm curious to find out what you mean by "solve them as needed".
Application of coherent equations in _partially_ coherent situation may
be, in the immortal words of Pauli, "not even wrong".

Quote:

I suspect you are missing their point and while I do not claim to
represent coherence theorists, I would like to set the record straight:
One _can_ apply a non coherent theory based treatment for either the fully
coherent or the fully incoherent limits and get some insight into a
problem. However, I think that most coherent theorists will tell you is
that the interesting physics is actually in between these two states and a
proper treatment may include theoretical constructs that IMHO are no more
abstract than anything else in physics.

A.G.

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