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Up to now our first order ray tracing has involved

the distance of the object and the image from the lens.

So if you were tracing a multiple lens system,

you would find that as rays move through the system,

you'd continually have to find where the image of lens one would be,

then relate to that to where lens two was.

And you'd find it's actually quite awkward.

So we now want to move towards a more sophisticated view of ray tracing where

we simply deal with the distances between elements.

We're really only concerned with where is the object and where is

the image at the very first element and the very last element in the system.

And in the intermediate spaces, we simply want to direct the ray,

what's its height, what's its angle, as it moves through the system.

And notice that height and angle at least if we're in a single plane,

which we'll deal with here.

We'll assume that we have rotationally asymmetric system so

we can stay in a single plane.

Knowing two variables that specify a ray, which are its height and angle.

So we now want to move to a system where we track those two variables for

a ray as it moves all the way through a system.

So to derive those quantities,

let's start with our single lens imaging system with you're familiar with.

We'll define for the ray a paraxial angle before and after the lens.

We can relate those angles to the height of the ray at the lens,

y, and the object and image distances through simple triangles.

We're quite used to that.

If we then write the thin lens equation but replace the object and

image distances now with these triangle relations up here,

such that we end up with angles and

the ray height of the lens, we get a new equation.

And this equation actually is, in essence, gone backwards.

Because here we have nu equals nu after the lens equals nu

before the lens, that's just our paraxial cells law.

And we find that the modification when you refract through a lens is that the ray

angle is changed proportional to the height that the ray hits the lens and

proportional to the power of the lens.

So we'll call this a refraction equation.

It gets rays across lenses.

The height, y, stays the same, but the angle changes from u to u prime.

Then it's very, very simple geometry now to simply

move the ray to some new surface a distance, d, away.

Again, we're going to move to d, the distance between surfaces,

instead of t, the distance to object or images.

So just a little bit of first order trigonometry since that the ray

height changes like the ray angle times the distance the ray travels.

Notice I've reminded you here of the sign convention, that if we're going

from the surface k forward, that's d prime moving forward, a positive number.

But if we were to look at surface k+1 and ask how far backwards do we have to go,

that would be d, no prime, because I'll always have primes after the surfaces and

no primes before, and that would be a minus quantity.

So these two values are equal, though of opposite sign.

So with these two things, these two equations, we can refract

changing the angle at of lens, keeping the height of the ray the same.

Then we can keep the angle of the ray the same as we move between optics and

change the height.

And we'll call this the transfer equation because it transfers us between surfaces.

Now most of the time, we have rays traveling through vacuum or

air, and so we'll take the refractive index as one.

But if we're within a lens, imagine we're treating two surfaces, the front and

the back of the lens, then we need to keep track of the refractive index.

And that actually can sometimes be awkward to keep track of all of the refractive

indices.

And you can, and those equations we just showed you,

have a refractive index in them.

But as an alternative approach,

one can actually scale the refractive index out of the equations.

I actually like to do this in my homework but it's up to you.

So let's start with Snell's law, n sine theta = n prime sine theta prime.

Go to our paraxial approximation.

Notice the variable u here,

we'll use to remind ourselves that we're in a paraxial regime.

So u is the paraxial version of theta.

So let's define, since we see nu here seems to be an interesting quantity,

let's define u-hat.

A scaled angle to be the refractive index times the paraxial ray angle.

Second, let's look at our Gaussian thin lens equation.

And here we notice that wherever we see a distance,

it's divided by the local refractive index.

And that suggests that maybe we should scale distance by

the local refractive index.

So we multiply the ray angle by n.

We divide distance by n.

If you make those two substitutions and rewrite the last two equations,

the refractive index drops out.

And we now get a ray angle changes only by refractive power of the lens.

The u-hat going to u-hat prime, here,

it doesn't change because we've embedded Snell's law,

the dependance on refractive index, into the definition of the scale view.

And here the refractive index that we multiplied into the n and

divided out of the d actually cancel out, and so this equation is written unchanged.

So when I'm working, I actually like to work in the scaled coordinate systems.

I don't pay attention to refractive index and I work with these variables.

And then when I'm done I come back and simply scale all distances and

angles by their refractive index.

However, if you don't like that, just work with the previous equations

that have the refractive index included in them.