Let's use these concepts to show how simply you can

constrain the design of a very typical imaging system.

I called this a projector

and that could be a slide projector,

it could be the conference room projector

that you might have seen

hanging on a ceiling in a conference room,

it could be a lithographic projector sending

an image off of the mask down onto a chip.

The general idea if we want to start with a lamp,

we would gather some light off of

the lamp within an imaging system or an optical system,

typically called a condenser,

so that we illuminate an object that could

be the chip in

your conference room projector that

then is imaged somewhere else,

we'll call that the screen.

The question is, how do

we begin to design that problem and how do we use

these radiometric concepts to

immediately constrain and put some bounds on the problem?

I've drawn our traditional two rays here and I've

used the red and blue colors that we typically use.

But there's an important little distinction

here is that they actually flip

their character going through

the system and this is super common,

microscopes use this same idea.

So here's my lamp. Let's say and I will tilt up

a paraxial marginal ray up off of that lamp,

and I'll be limited by

my numerical aperture that I

captured from the lamp by the condenser.

Of course whatever aperture that I stopped,

that i having this condensing system,

will appear as an entrance pupil,

perhaps just the edge of the condenser lens itself.

So my chief ray for

the condensing system would be

centered in the entrance pupil of the condenser.

So I've drawn that,

and that chief ray is going to determine

the edge of my field at the lamp.

The thing that's odd is I typically will put my,

I've called it slide here,

but there was a slide projector or this would

be the chip of your conference room projector.

This is my object.

Note here, I have switched the two rays.

I have my chief ray for

the condenser at the middle of my object.

The reason for that is,

is if I did it the other way around,

if I imaged the lamp to

the object then I would see all of the humps and bumps

and filaments or whatever shape I had in my lamp in

my object and eventually I'd see that projected out

of the screen and that's not what I would like.

So instead I flipped the nature of these two rays.

Now, every point on

the object is illuminated by every point on the lamp.

So really what's sprayed out here across

the object is the angular spectrum of the lamp.

You can see the blue ray here.

That's the angular content of the lamp and

that is what is spread out in space across the object.

So that's a common way to get a bumpy lamp like

a wire filament to look very uniform here at my object.

Then, my object now the red ray is my marginal ray.

I tilt it up until I hit some aperture stop in my system,

perhaps the aperture of

the projection lens that can

be viewed as an entrance pupil here.

Notice my little dotted lines as I'd find the image of

my lamp in the entrance pupil of my projection lens and

the blue ray now for

the projection lens is

my chief ray and so it determines

the edge of field for my object.

So, they're the same two rays

it's just they flipped their character going through.

All right. So how do we design this system?

It turns out that radiometry

is fundamentally terribly simple.

You probably have a specification

that you need a certain amount of power

on the screen that could be because it's

a camera and it needs to get a certain amount of photons,

it's a photosensitive material it needs

a certain amount of power or you're sitting in

a conference room in the eyeballs that are

sitting out in that conference room

need a certain amount of power.

It's got a certain area A,

and so you divide those two and that gives

you the irradiance that you need on

that screen and it makes sense

that we're talking about irradiance

on the screen because that's watts per unit area.

Well, now we can jump all the way back

to the fact that the best we can do,

and of course this is probably not actually the case,

you probably need to put efficiency factors in,

but the best we can do is,

the total power we got on the screen was

exactly the total power we

managed to accept from the lamp.

Once we know that, we can say well,

the lamp has a certain radiance spec.

I can look that up or know that from the lamp

and that's going to give me a Lagrange invariant H

that I have to have for my condenser and the rest of

my system in order to get

the total power from the lamp to the screen.

As soon as I have Lagrange invariant,

I can pick anywhere in that system I'd

like maybe the object and say well,

if it's got a certain size H

and that's going to give me

a numerical aperture and I'm done.

That then defines

the Lagrange invariant through the whole system.

Now, the height or

numerical aperture at every point

on the system that product had better

be at least that if I'm going to have

Lagrange invariant big enough to

carry this power all the way to the screen.

There's some discussion here

of which those would you like to make slightly larger.

In general, let's say the conference room projector.

You're going to find that,

that chip whether it's

a liquid crystal or Deformable Mirror Device,

DMD, that's going to be your limiting aperture.

It's going to have a certain size

and you can't make that bigger because

that's the chip and it's going

to have a certain acceptance angle.

Again, you can't make that bigger,

that's going to be on the physics of whatever limits,

whatever how about chip works.

So that's going to determine

a Lagrange invariant right there.

That tells you that

your condenser had better have that Lagrange invariant,

maybe a little bit more,

but certainly not less because if

your condenser had a lower Lagrange invariant,

then this limiting aperture,

you're imaging chip or your display chip,

then you'd not be

optimally gathering light up off the lamp.

On the other hand, if you made your condenser have

a much bigger Lagrange invariant

than that of your limiting object

here, your display chip,

you would just either be putting

light outside the display chip in

area or bigger in

angle than the chip could work

on and you'd be throwing it away.

You'd be wasting it.

There's no reason to make

the condenser Lagrange invariant

significantly bigger than your display chip.

The projection lens probably needs to have

a Lagrange invariant a little bit

bigger than the limiting aperture,

let's say you're imaging chip.

If it's smaller, you're going to vignette

or you're going to throw away

power and that would be silly.

So that's a very typical thing,

your theaters that use digital chips now,

that's the calculation all the way through.

It's you have this enormous bulb

back in that projection chamber,

you have a very expensive projection lens

but the thing that determines how the whole system works.

The total optical power you can get onto

the screen is really determined by

the Lagrange invariant it which is set by

the smallest Lagrange invariant in your system,

the display chip and then everything else in the system

you simply have to design to have at least that.

The point is, radiometry isn't super hard if you use

these concepts of course the details

are more difficult, but fundamentally,

the question we asked at the beginning of this module,

how do I understand,

how much light I can get off of

that light bulb and out of an optical system?

This is the math for it.

It's really just understanding

these units that we've defined carefully and

the super important concept of

conservation of radiance and how

it relates to the Lagrange invariant.