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This can be a bit confusing, so let's do another example.

And this time, we'll add a little bit, which is translating from curved

surfaces to the paraxial representation, and then tracing through the system.

So we'll do a kind of real world example here.

What we'd like to know is, if we have an object in front of a real thick lens now,

how do we now do the single lens imaging problem?

And of course, we don't have single thin paraxial lens, so

we can't apply our Gaussian or Newtonian, the lens equation.

So, we pretty much do the same thing we did before.

First, we label every surface.

The object should always be 0.

It doesn't really matter but it's good to have habits and be able to count things.

So let's always make the object 0.

And I'm going to label the 1, 2, 3 surfaces of this lens.

And then finally, 4 is the image plane.

So, note that I have some refractive indices now.

And so I'm going to have to remember to use the versions of my refraction

equation that have the refractive indices in them.

So the first thing I'm going to do is I add a couple of rows to my

table here with the curvature and the index of every surface.

So for example, surface 0 has no curvature because it's just the object and

it's living in index 1.

And again, to keep track of whether I mean before or after the surface,

I put a prime, so I mean after surface 0 indeed has index 1.

Surface number 1 has a curvature of 0.02,

because its radius is 50.

Remember, curvature is the inverse of radius.

And we don't have units on this, but

it doesn't really matter as long as we keep the same units all the way through.

So it can be meters if you like these as your big lenses.

And the index after surface 1 is 1.5.

The next radius, notice,

is negative because the center of curvature is to the left of the element.

And so we have a -50, so the curvature is -0.02.

And this refractive index after surface 2 is 1.6.

The last surface is plane 0 now.

So that would mean the radius is infinite or the curvature is 0 and

the index after surface 3 and prime 3 is 1.

So the reason we tabulate all those is so now we can go ahead and

calculate the equivalent paraxial system.

So we replace all of that glass, all of those curved surfaces with,

once again, thin optics that are separated by the former distances.

And we'll calculate the equivalent paraxial power.

because what we want to do is do a paraxial analysis of that non-paraxial or

real world lens.

So now, we just apply our equation for the power of a single surface,

the curvature times the difference in the refractive indices.

And we have the refractive indices nicely laid out on either side of

the curvature here.

So that makes it very easy to calculate the power of the three surfaces

to be positive 0.01, -0.002, and finally, the surface is plane 0, so it's 0.

The rays will bend at surface 3 because of course the refractive

index changes from 1.6 to 1.

But not because the surface has power, only because of Snell's law.

So now we have the powers, we can go ahead and fill in the distances.

Remember, distances here always mean to the intersection of the curved

surface with the optical axis, with the vertex of the surface.

That's what we mean in our paraxial description by distance.

So we had that on the original diagram, 300, 10, and 2, in whatever units.

And what we'd like to find is this last one, which I put a box around,

which is what is d3 prime, the distance after surface 3 to surface 4?

Well now, we're just back to the problem we had before.

And at this point, we can go forward.

So let's just launch a ray from, we can launch it from anywhere.

But it seems like the easy thing to do would be to launch it from the axis and

with an arbitrary ray angle.

It shouldn't matter, this is a linear description.

And I've chosen 0.01 just because it keeps the numbers kind of near unity.

But you could've chosen pi if you'd like, or whatever you'd like.

So we launch this ray, we have to choose or

specify two values about the ray, its height and its angle.

And once we do that,

then it's just mechanical, we can go through the whole system.

So we use the refraction and transfer equations whenever we're

calculating a ray height, we'll use the transfer equation.

And then once we have the ray height on this surface,

we have to get across the lens by refracting through it.

Remember to use the two refractive indices here.

And note at surface 3 here, right here,

the angle changed from after surface 2 to after surface 3,

even though the surface has no power.

And that's because right here, that's essentially Snell's Law.

So finally, we'll specify that the image height we're interested in,

the ray height is 0.

That means it's the image because we know we should come back to the axis.

And we finally write the last transfer equation and solve for

the missing distance.

And we do that, we come up with the distance we are looking for.

So this is how we do ray tracing,

paraxial ray tracing of real systems.