As a reminder, the reason we've gone through this set of calculations to
understand geometrical optical systems from the perspective of fourier optics,
where we use spatial frequencies and fourier transforms,
is because at the focus of our lenses,
geometrical optics would predict an infinitely small point spread function,
an infinitely small gathering of the energy there.
And we know that's not physical.
So, optic studio and all such geometrical optics design codes,
therefore have to include the math we've just gone through.
Because at the end of the day, you want to know how
the light is actually distributed to focus.
So, as an example, let's go into OpticStudio and see what that looks like.
I've set up the world's simplest system here.
This is a paraxial lens with a default 100 millimeter focal length,
and an object distance of 200 millimeters,
and an image distance of 200 millimeters.
So this is a nice magnification equals minus one simple paraxial system.
And as we would expect, if we zoom in on this image focus here,
it's infinitely small all the way down.
Because it's perfect paraxial lens,
and that's all geometrical optics can tell us.
But we can go up, move this over,
to p s f,
point spread function that's Optix jargon for impulse response.
And here's the FFT, fast Fourier transform,
that's the numerical version for Fourier transform point spread function.
And that's what I pulled up right here.
So, this shows us in microns,
what we would expect the array disk,
the diffraction limited blurred distribution
of intensity or formally we'll learn soon irradiance,
to look like at the focus.
How this is calculated is the math, we've just gone through.
Zemax finds the limiting aperture of the system,
which we're about to spend some more time looking at,
the pupil we've been talking about.
And it fourier transforms it to get to
the focal position and tells us then that's what the spot is expected to look like.
So let's a quick check here.
This is a 20 millimeter semi diameter and a 100 millimeter lens.
The numerical aperture of the cone of light going into,
and in this case out of this lens,
because it's symmetric is the radius of the pupil, or semidiameter,
in OpticStudio jargon 20 millimeters,
over the distance to the object.
A very common mistake students often make is to instead
take as the denominator of our numerical aperture expression,
the focal length of a lens, that's here.
Numerical aperture is a property of cones of light.
Lenses can operate at various numerical apertures,
but it's the light that we need to understand and use in these expressions.
So, the numerical aperture of this kind of light is 20 over 200 or 0.1.
To make life easy,
I have set the wavelength length over here to be point five microns.
So, you would expect the radius of the first null to be point six lambda, over NA.
Lambda is point five, NA is point one.
So lambda over NA is five.
Five times point six is three,
and three microns is just where we see that first null.
So indeed that works.
If for example I was to go up here and double the diameter of the lens.
Now this is getting to be a fairly fast lens.
Notice that my point spread function got narrower and it got narrower by a factor of two.
Now my first null is exactly at 1.5 microns.
So, that expresses this tradeoff we've seen as the cone of angles,
as the ray angle goes up,
the point spread function size goes down.
That's that Heisenberg tradeoff again.
Let me go back to my NA point one system.
So that's the impulse response of the system,
and notice this is our incoherent impulse response.
We have all positive values here.
Let me close this.
The next thing we'd like to look at,
is the optical transfer function.
So, this is the incoherent description
of what spatial frequencies are getting off of my object,
and are able to get over to my image.
And the fact that all of these don't get through the system with the same efficiency,
and that's the math we just went through.
So, this is under the next thing MTF,
modulation transfer function, that's the absolute magnitude.
Therefore, it's a real function of the potentially complex optical transfer function,
the incoherent transfer function of the system.
As we would expect, it has this triangle like
shape that we saw in the derivations, it's slightly curved.
And that has to do with circles overlapping with circles,
give you a little bit of a curve to it.
But if you want to think about rectangles overlapping with rectangles,
the one dimensional case and think of this as a triangle,
you're obviously pretty darn close.
It goes out to we see here,
a total maximum spatial frequency of 400 in units of inverse millimeters.
Let's see if that makes sense.
So, the cut off spatial frequency for
this system which would correspond to the largest Ray angle,
getting off of the object and through our people our the aperture stop here,
would be zero the cutoff frequency.
That would be numerical aperture over the wavelength.
Well the numerical aperture is point one,
the wavelength is point five,
point one over point five is point two inverse microns,
or 200 inverse millimeters.
And there's that 200 inverse millimeters sitting right in the middle of this plot,
because remember when we do that autocorrelation
to get from a coherent to the incoherent transfer function,
we double the width of the transfer function,
and so now it gets out to a total maximum frequency of 400 inverse millimeters.
So, all of the math we just did it seems to hold together,
and is supported by OpticStudio.
And these are the kind of functions you would look
at to understand the performance of your system.
So, one more here I kind of like.
And so let's come back to that.
This is a different version of the point spread function.
It's called the diffraction encircled energy.
And you see that next on our little tab bar up here.
This takes the point spread function.
Let me bring that back up.
And integrates it radially.
And this is another way of looking at the response of the system.
So, it's just the accumulated integral started in radius zero, and integrating out.
It's therefore got zero on access.
This is like taking a pinhole and just making it bigger and bigger and bigger.
So, you get no energy through the pinhole when the radius of the pinhole is zero.
You get one all the energy through when the radius is infinite.
And you see that for example this integral flattens out at about three microns,
which is just where my point spread function has a null.
So that kind of makes sense that the interval gets flat there for a while.
This view is kind of nice when you want to think about putting a pixel
out to integrate up the energy that I'm getting through this imaging system,
and I want to know how much energy gets onto the pixel.
And so for example,
this tells us that if we put a pixel that was
just the diameter of my point spread function,
between its first nulls,
we would get about 85% of the total energy onto that pixel.
And that's kind of nice to know.
If we went out to the second null,
which is out here to at around six microns,
we get about 90%.
So not a big gain for doubling the size of our pixel.
One of the reasons this particular view is nice and we'll see this later on,
is when we put imperfect real multi paraxial lenses in this system.
The point spread function will always grow,
because we can't do any better than our perfect paraxial lens here.
So we'll get bigger point spread functions.
And this encircle energy plot will move out.
It will take us a bigger pixel to get the same amount of energy,
because the system has gotten blurry.
And so I find this a very useful view of deciding when my system is good enough,
because I'm often interested in integrating
the energy onto something like a pinhole or a pixel.
So, the point is,
we have now seen that OpticStudio uses the math that we just went through.
It uses these fourier transform concepts,
to get from the infinitely small point spread function over here,
to the more realistic point spread function we have to have through diffraction.
It's all built in.
And to some extent maybe you didn't need to
know all of the concepts we just went through,
but you need to understand where these functions come from.
And the fact that really what's going on is OpticStudio is doing a lot of
calculations to find the fields in the pupil,
and then fourier transform from the pupil to the focus.
And that so then tells you what goes on the focus.
So, you need to understand that's the math,
that's the machinery going on underneath though.
Amy SullivanResearch Associate
Robert McLeodProfessor
