0:15

And to potentially help understand the Lagrange invariant a little more,

maybe, introduce a concept called phase space.

It's just another way of sort of visualizing the spacial and

angular content of a bundle of light, of a set of rays, moving through a system.

And this is really wrapping up the concepts that we are teaching in this

module.

That using the marginal and chief ray allows you to

understand how the ensemble of rays moves the system.

Because the marginal and chief ray bound the angular and

spacial extent of the object and image.

They can change character through the system, but at the object and

image plains, the marginal tells you the angular extent.

The chief tells you the field, or the spatial extent of a system.

1:30

I have drawn in the marginal ray and the chief ray.

Of course, the chief ray goes through the center of the aperture stop.

And it happens to be parallel to the axis out here in object space,

because it's telecentric in object space.

I'm imagining I have some sort of source here, let's say this is a CCD.

Sorry, LCD, liquid crystal display.

And I have 1,000 pixels that are 5 microns on a side.

I'm inside the 5 millimeter total field of view here in object space.

So I can go ahead and calculate my Lagrange invariant.

Of course, in object and image spaces,

one of these terms is always 0, so that's convenient.

And that's because, remember bar represents chief ray, the height of

the marginal ray, y here, is always 0 at object and image spaces.

So this second term goes away.

So from the first term of the angle of my marginal ray happens to be one-tenth.

And that's set by where my aperture stop is and its diameter.

And the height of the chief ray is half of the 2.5 mm field of view.

2:46

So let's check and see if that makes sense,

because I get a Lagrangian variant of 0.25 millimeters in the units I've used here.

We said the number of spots the system could carry is

the wavelength over 2 divided into the Lagrange invariant.

So wavelength over 2 would be 0.25 microns.

This is 250 microns, so that's 1,000.

Hey, this system is set up exactly to carry 1,000 spots,

which is convenient because we have 1,000 pixels.

3:22

Let's see what that means in terms of its setup to exactly carry

a number of spots equal to the number of pixels.

And a way to think about that is to think about what does the point spread function

look like at the object plane here?

That would be thinking about this in the Fourier domain.

So I'm going to take a single 5 micron pixel here, and

I'm going to Fourier transform that little rectangular function.

4:04

So this system, if I radiate out little square pixels here.

When I get to the aperture stop, and

notice this is that 4D transform plane right behind the lens.

I would see each of the spectra,

the spatial frequency spectra of those pixels laid out in space.

And the edge of the aperture stop would be right on the first null of that sine x

over x function.

And that's not a bad design principle,

because that's really where the information is.

The extra bits of this Fourier transfer carry the shape of the pixel.

But really this blob in the middle between the first nulls

carries the information that there is a pixel there.

So what this Fourier space is, or this phase space, is I plot it here at

the plane of the object, where the light is in terms of position.

There is no light, then there's a lot of light, then there's none.

And this light is within plus or minus 2.5 millimeters, the size of the field.

I've imagined just the clarity, I've turned one pixel right in the middle off.

So I have pixels that are on, and

then the white line represents one pixel that is off.

And then the rest of the pixels are on and

that's just sort of a convenient little marker.

And notice that my marginal ray bounds the angular, or

spatial frequency extent of the object.

But it's at the object plane 0 position, yep that's right there.

While the chief ray bounds the positional extent of this bundle of information.

But it's at 0 angle, yep that's right there.

So this phase space is a way of simultaneously looking at the spatial and

angular content of my beam of light.

And we see here in this nice telecentric case that we have this nice little box.

The area of that box, and notice that's a unitless quantity,

from here down to here turns out to be 2000.

6:03

It turns out that the Fourier transform is symmetric for real objects.

And so the positive and negative side bands carries the same information.

And so really, the information content of this system is the upper half.

And that's 1,000 which is exactly the number of spots we just calculated.

So this system is designed to carry in its phase space the field and

angular extent of the system, 1000 spots.

That's another way of looking at my Lagrange Invariant.

Well, now let's just advance through the system and see what happens.

6:39

So I've used the transfer equation to move all of this light,

and to replot how it would look right before the lens.

So first let's do the Lagrange invariant.

First of all, the angle of the chief ray, u bar here, is still 0.

So the second term in my Lagrange invariant is still 0,

the, Angle of the marginal ray, and the height of the chief ray haven't change.

So the first term didn't change at all, so I still have the same H.

So good, my invariant is indeed invariant.

It turns out what's happened to my phase space here,

you can see the marginal ray is still at the same spatial frequency, that is angle.

That's what I see here, but it's moved over into a positive position.

7:30

Yep, the chief array hasn't done anything at all, stayed right where it was.

So that's an indication of what's happens in the phase space,

is everything just skews.

That rectangle becomes a parallelogram.

And it turns out if you remember your basic geometry,

the area of that parallelogram is the same as the area of the associated rectangle.

The area didn't change.

I still have the same information content in my phase space.

All right, let's go through the lens.

7:59

Now I've applied the refraction equation to both my rays and

to all of this space up here.

Because every space here represents a ray.

And I've just drawn two rays, the chief and the marginal.

Okay let's calculate my Lagrange invariant,

that's a little harder now, a tiny bit harder.

My marginal ray angle has become negative, there it is, minus one-twentieth.

But the height of my chief ray didn't change,

so that's still there at the edge of the field, yep.

The chief ray angle has become negative, and

it's parallel to the marginal ray, so that's also minus one-twentieth.

And the height of the marginal ray I calculated.

I put that in, and

when I calculated that, look at that, Lagrange in variant is still the same.

8:49

I'm still carrying the same number of spots.

Over here in my phase space,

I've just done another skewing operation, it turns out.

And when you skew parallelograms and rectangles, their area doesn't change.

So once again, the area that I'm using in this phase space, the set of angles and

positions occupied by my rays, are all quite different now.

But if I sketch out the volume in phase space that they're occupying,

it's still the same.

That's another way of thinking about you can't squeeze the bundle of light.

If I make the angles go up, the positions go down.

This is a plot of that angle in position space.

9:32

Let's stop at the aperture shot just to see what happens,

maybe it's interesting space.

Well, let's see, now because by the definition of

the chief ray, the chief ray height is 0.

The first term in the Lagrange invariant expression goes to 0, but

the second term shows up.

So now I'm going to use the angle of my chief ray,

still minus one-twentieth as it was after the lens.

But the height of my marginal ray there 5, so note what's happened now.

I still get the same Lagrange invariant, is the marginal ray and

chief ray have switched their function.

The chief ray now is bounding the angular extent of my bundle of rays.

While the marginal ray is bounding the positional extent of my bundle of rays.

That's why you need two terms in a Lagrange invariant,

they can swap back and forth.

And in the object plane, and in this case the aperture plane,

they have absolutely opposite character, but they switch what they do.

Intermediate planes, it's mixed, but

this term describes how that mixture never really changes.

And I still have the same Lagrangian variant and the same number of spots.

10:49

And we can kind of see that something

magical has happened over here in the phase space.

Because now the chief ray is at a position of 0.

Remember before,

here at the object it had a position at the edge of the field at an angle of 0.

It was bounding over here and the marginal ray is now switched,

it is now at, has a finite position.

11:17

And so it's bounding the edge of the bundle of rays in position.

So again you can kind of see how that these have switched back and forth.

Before they were lined up up and down, now they are lined up side to side.

And once again that area is completely skewed.

Notice that if we just shove the detector here, in position,

what we see is the Fourier transform of these beams.

If we slice across this, and integrate up all the rays coming in at

different angles, what we'd see is that sine x over x function.

That's very cool,

that tells us we get to use a lens to take a Fourier transform of our pixels.

And that's another concept from Fourier optics.

11:57

And finally, let's go to the image plane.

Notice that my little pixel that I turned off just for

clarity here back at the center of the object.

If I now put a camera here,

I would see all of the light from that pixel at the same position.

Hey I'm back in focus, so

all of my light from individual pixels is stacked up again.

And it's only at individual positions out here in the image plane.

Once again, the marginal ray height has gone to 0.

So the second term in the Lagrange invariant has disappeared.

That has to be the case, because that's the definition of the marginal ray.

My field has doubled in height.

This is a magnification -2 system, because now y bar is 5 instead of 2.5.

But of course the angle of my marginal ray,

remember it was positive one-tenth, has now gone to minus one-twentieth.

So the angle has come down, the height has gone up, and

I have the same Lagrange invariant.

And then once again because when you skew things the system doesn't change.

And I'd get the same number of spots represented by the area as well.

So I hope this example has been useful to sort of understand this Lagrange

invariant, and its relationship to the number of spots.

That in this nice simple telecentric example, the marginal and

chief rays described here, angle and position, but here the opposite character.

But if you're careful about that, and understand the Lagrange invariant.

The number of spots, the amount of information, the object size,

angle, product cannot be changed, as you move through the system.