So now we have the fourth piece.
If the coherent impulse response little H,
the airy disk that side,
is related to the incoherent impulse response,
has the absolute magnitude of the little H squared.
Then all we have to do,
is for you to transform magnitude of H squared to find the incoherent transfer function.
That is, the incoherent impulse response should relate to the you
transform to the incoherent transfer function.
This has got a name, it's so common it's called the optical transfer function.
The question of the best name,
it would be better if we have a coherent and incoherent transfer functions.
But this is so often worked with incoherent light,
that the notation for this incoherent transfer function is simply the OTF.
And with just a little bit of fourier transform wizardry,
we can find that from things here already have.
So let's look here at two D,
there is a circular hole which you might see looking through a set of binoculars,
and one D there's just that wrecked function we had before.
That's our coherent transfer function, Fh,
it simply represents the people function light at low angles gets through,
and light at high angles high spatial frequencies hits them up and doesn't get through.
We know just a moment ago that the incoherent impulse response is,
the absolute magnitude squared of the coherent impulse response.
And this is the fourier transformer above little h in clear transfer function.
So little h is the coherent impulse response.
So there's a fourier relationship that tells us that,
if we know the fourier transform capital H of the coherent impulse response,
little H, how would we calculate
the incoherent transfer function
that's the fourier transform of the magnitude of H squared?
And it's simply the auto correlation which I've written as this circle pulls across.
So that's just a fourier transformer.
How taking the intensity absolute magnitude squared in the real domain,
corresponds to an auto correlation in the fourier domain.
Auto correlation is simply taking the function,
sliding it by itself and finding the amount of overlap.
So in the case of the wrecked,
we see as we slide one wrecked by another,
we get a rectangular area and of course we just get a triangle function out of that.
If the two rectangles are just touching,
we get no value if the two rectangles are completely overlapping,
or at max [inaudible] rectangle slide by each other again,
and we get back to zero.
And so that looks like a linear function.
And if this coherent transfer function had an absolute limit of f-zero,
the cutoff frequency, the incoherent transfer function goes out to twice that.
So we get more bandwidth through an incoherent system,
but at the expense that it's always got this sloppy characteristic.
It's not a nice flat topped pass-band,
but we always have this slope to
the transfer function of light through an incoherent system.
In two D, you have to think a little bit harder how to
slide two circles past each other through every possible offset,
and find their overlap.
But you get a function which pretty much looks the same.
It's got a little curvature to it,
but it's a circle,
it's twice as big,
it's zero at the edges and one at
the center and it's got the same slope sort of characteristic.
We're not going to calculate these functions in this class.
But now we have all four pieces.
We understand, hopefully the airy disk,
the impulse response of a coherent system,
you could take to be the intensity of that.
So you have no negative values.
That's the incoherent impulse response for moving some light through a system.
And you fourier transform for
those impulse responses to get the associated transfer functions.
In the case of the coherent system,
it's just pupil, it's just the light and the plane waves getting through.
In the case of the incoherent system,
you have this auto correlation,
and you end up with these triangle like
functions that go up to twice the total cutoff frequency.
And that's enough in this class.
We can now use those pieces to patch
up this problem we have with what happens at the focus.
