Chapter 27: Geometrical Optics – Reflection & Refraction

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome back to the Deep Dive.

Today we're opening up a really fundamental physics toolkit, the kind engineers use constantly.

Yeah, we're getting into geometrical optics.

Exactly, and it's all about this, well, practical, but powerful way of thinking about light retracing.

It's the physics behind, you know, your phone camera, big telescopes, everything.

That's right.

And this deep dive, we're aiming to follow Feynman's path here, starting simple, building up.

Our mission is really foundational.

Foundational how?

We need to really distill the physics, how light behaves going from, say, just one surface then to full lens system.

And crucially, figure out where this whole geometrical approach starts to, well, break down.

When you need something more.

Precisely, when you have to jump to wave theory.

But first, we'll get those core equations optical engineers need and really understand why they work.

Okay, so where do we start?

Not with lenses, right?

You said foundational.

No, we start with the bedrock principle, the physical law that honestly governs everything else in this field.

Fermat's principle of least time, simple idea, profound consequences.

These time.

So light is what, lazy?

Ha!

Well, efficient, maybe.

Between any two points, light always picks the path that takes the minimum travel time.

That single constraint dictates reflection, refraction,

focusing.

It all stems from that.

And our source material gives a little formula for visualizing that, right?

The delay difference.

It shows how the extra time delay, let's call it delta, for a ray that's a bit off the central axis.

Well, it depends on how far off it is.

Like the height off the axis?

Exactly.

The delay delta is approximately h squared, divided by twice the distance along the surface.

So delta, approx.

h22 level.

Okay, so the delay grows faster, the further off axis you go.

Right.

And that insight is key.

If you could design a lens shape, maybe a curve, that makes the total travel time the same for all paths from an object point to an image point.

Then all the light arrives at the same instant.

Bingo.

You get a perfect focus.

So geometrical optics, at its heart, it's really about engineering shapes and materials to equalize those travel times.

All right.

Principle established.

Let's build something.

Simplest case.

A single curved surface.

Imagine light going from, say, air into glass.

Okay.

Air has a refractive index of basically one.

Glass is higher.

Let's call it n.

Now, to make this manageable, we have to use a crucial simplification.

Ah, the fine print.

You could call it that.

It's the paraxial approximation.

It's like an engineering cheat code.

What does it mean?

We assume we're only looking at rays traveling very close to the central axis.

Rays where that height saw it, we talked about, is really, really small.

Why do that?

Because it lets us treat the complex curves almost like straight lines.

We use as the first, simplest terms in the math.

It makes the whole system solvable, much easier to handle, but it's a limitation we need to remember.

Got it.

So with that approximation, what happens at that single air to glass boundary?

If we make the travel times equal for these paraxial rays?

You derive the absolute core equation for imaging with a single surface.

It connects everything.

Lay it on us.

Okay.

It looks a bit dense, but conceptually, it ties together the material, the shape, and the distances.

For air to glass, index n, it's n divided by s prime.

S prime being the image distance.

Right.

Plus one divided by s.

The object distance.

Correct.

And that sum equals n minus one, all divided by r, the radius of curvature of the surface.

So, frocken's 1r.

Wow.

Okay.

What's the practical takeaway there?

It shows you the focusing power of that single surface depends on two things.

How much it curves, that's the r, and the change in refractive index, the n.

It's the interplay between geometry and material.

And this lets us define focal lengths for just one surface.

Exactly.

Two key ones.

If light comes in parallel, like from infinitely far away.

So, object distance s is infinity.

Right.

Then one is zero.

The equation tells you where the image forms, and that distance is prime is what we call a secondary focal length.

Five dollars.

Okay.

And the other one?

If you want light leaving the surface to be parallel, so the image is formed at infinity.

Then s prime is infinity, and ns prime is zero.

Precisely.

The equation then tells you where the object as must have been.

That's the primary focal length.

Five dollars.

And what's neat is the relationship.

Five dollars is always n times the secondary focal length is longer by a factor of the refractive index.

Interesting.

Okay.

Before we combine services,

the dreaded sign conventions, why are these so necessary?

They always feel a bit arbitrary.

They feel arbitrary, but they're absolutely non -negotiable.

If you want one equation to work for all situations, convex, concave, real image, virtual image.

Without strict rules, you'd need a different formula for every setup.

So what are the rules we need to stick to?

Rule one.

Object distance as is positive if the object is to the left of the surface where the light usually comes from.

Makes sense.

Rule two.

Image distances prime is positive if the image forms to the right of the surface.

And if the math gives a negative, that's prime.

That's your signal.

It means the image is on the wrong side back on the left where the light came from.

It's a virtual image.

Like in a magnifying glass, you can see it looking through, but you can't project it on a screen.

Okay.

And the radius are?

Four dollars is positive if the center of curvature is to the right of the surface.

Think of a surface bulging towards the left, like a typical converging lens surface.

Its center is to the right, so R is positive.

This convention ensures converging surfaces have positive power in the equations.

Sticking to those rules lets one formula cover everything.

Exactly.

It's the universal calculator for optical designers.

Right.

That single surface formula is like the engine.

Now let's put two together.

How do we build the thin lenses we see everywhere?

We just apply that engine, that single surface formula, twice in a row.

A lens is just two surfaces.

So you calculate the image formed by the first surface.

Let's say it forms its side dollars.

That image then acts as the object for the second surface.

Ah, the output of the first becomes the input for the second.

Precisely.

So 222 is related to five dollars.

You plug that into the formula again for the second surface.

Now it gets a bit messy algebraically, but if you make a key simplification - Well, I guess the lens has to be thin.

You got it.

Assume the lens is thin and it's sitting in air on both sides, then all that complexity boils down beautifully.

You get the famous thin lens equation.

Which is much cleaner.

Oh, incredibly elegant.

It just becomes one over s plus one over its prime equals one over s.

Crack one, strut one, strut.

Simple.

All the details of the glass and the curves are now packed into that one number, f.

Exactly.

The focal length f.

And where does f come from?

It comes from that two surface calculation we just talked about.

That gives us the lens maker's equation.

Which tells you how to make a lens with a desired f.

Correct.

It shows how the power of the lens, which is one dollar, depends on the glass index n and the two radii of curvature.

It's one dollars and two dollars.

The formula is one dollars and one one times the quantity.

Read on one and that directly links the physical construction to the focal length.

It does.

And it confirms that useful rule of thumb.

If you place an object at a distance of two dollars from a thin lens, the image forms at two dollars on the other side.

Right.

And it's inverted.

Same size.

Perfect for 1 .1 copying.

Very practical.

Okay, so that covers objects on the axis.

What if the object has some height?

How big is the image?

That's magnification, right?

Yes.

The magnification, usually written as Miley, the ratio of image height to object height.

We figured that out using geometry, basically drawing rays.

The easiest one is the ray that goes straight through the center of a thin lens.

It doesn't bend at all.

Simple triangles, then.

Pretty much.

Similar triangles give you the magnification ratio, but here's where it gets really quite neat.

If you take that magnification ratio and substitute it back into the thin lens formula, one dollars plus one zedler's one F4, and do a bit of algebra.

You get something even simpler.

You get the Newtonian form.

It's beautiful.

Instead of measuring distances and us's prime from the lens center, you measure from the focal points.

Let's call these distances x and x prime.

So x is how far the object is past the front focal point, and x prime is how far the image is past the back focal point.

Exactly.

And the relationship becomes incredibly simple.

x times x prime equals f squared.

Six dollars, get x equals f22.

That's it.

That's it.

A perfect parabolic relationship derived purely from geometry under the paraxial assumption.

It means the whole focusing behavior is captured by f.

No f, no x, you instantly know x prime.

Remarkable simplification.

And just quickly, does this still work for complex systems, like a big telescope objective or a microscope with many lenses?

Can we still use these ideas?

Fundamentally, yes.

Any complex system of lenses, no matter how many elements, can be mathematically reduced.

You find its overall principal planes and calculate a single effective focal length.

Then, for ray tracing purposes, that whole complicated instrument behaves just like a single thin lens with that effective f.

Okay, so we've built this really elegant mathematical picture using that key assumption, the paraxial approximation, rays near the center.

But reality isn't always paraxial.

What happens when that assumption breaks down, when the lens is wide?

Right.

That's where we hit the aberrations.

These are the ways real lenses deviate from that simple ideal model.

The first big one is spherical aberration.

Is that where rays hitting the edge of the lens don't focus at the same spot as rays through the middle?

Precisely.

Remember that delay formula, delta -approx -h22 -zebel.

It's an approximation that works well for small h.

When h gets large for rays, far from the axis, hitting a wide open lens, that approximation isn't good enough, the actual focus point for those edge rays shifts inwards, closer to the lens.

So instead of a sharp point focus, you get a smear.

Exactly, a blurred spot.

It tells you that the perfect shape for focusing all rays isn't actually a sphere, it's a parabola.

But making perfect parabolas is hard and expensive.

So engineers use combinations of spherical lenses to try and cancel out the error.

That's the usual approach, yes.

Carefully designed combinations can minimize spherical aberration.

Okay, what's the other main problem with our simple model?

The second major one isn't about the shape, it's about the material itself.

It's chromatic aberration.

Chromatic, meaning color.

Yes.

The problem is that the refractive index n of glass isn't actually constant, it changes slightly depending on the wavelength, the color of the light passing through it.

Ah, so blue light bends differently than red light.

Typically, yes.

Blue light bends a bit more than red light in most common glasses.

Now remember the lensmaker's equation, one dollar f is equals n one times.

The focal length f depends directly on n.

So if n is different for red and blue.

Then the focal length f is different for red and blue.

The lens focuses red light and blue light at slightly different distances.

Meaning if you focus the red perfectly, the blue is a bit blurry and vice versa.

You got it.

That's why cheap telescopes or binoculars sometimes show color fringes, maybe reddish edges on bright objects, bluish edges on dark ones.

That's chromatic aberration made visible.

And the fix for that.

You have to use multiple lenses made of different types of glass with different color dispersion properties.

These are called achromatic doublets, designed to bring, say, the red and blue focal points together.

Okay, so we can fight spherical aberration with shape, fight chromatic aberration with material combinations.

But eventually, even with a perfect lens, we hit a wall, right?

We can't just magnify forever and see infinitely small details.

Correct.

There's an absolute fundamental limit.

And this is where the whole idea of ray tracing geometrical optics finally has to give way completely to the wave nature of light.

This is about resolving power, being able to tell two close together things apart.

Exactly.

The limit comes from diffraction.

Because light acts like a wave, it spreads out slightly when it passes through an aperture, like the lens opening.

So even light from a perfect point source doesn't focus back to a perfect point, but to a tiny diffraction pattern.

Right.

And if you have two point sources very close together, their diffraction patterns overlap.

If they overlap too much, you just can't distinguish them anymore.

Is there a rule for when you can distinguish them?

A wave -based rule?

There is.

It connects time and size again, interestingly.

To resolve two points, the difference in the time it takes light to travel from those two points to opposite edges of the lens.

That time difference, $2 .21, must be greater than the period of the light wave itself.

The period being one over the frequency, $1 .01.

Yes.

If the time difference is less than one oscillation period, the waves arrive too much in sync and the sources blur together.

You can't resolve them.

And that condition, T111, leads directly to the practical limit on resolution.

It does.

It translates into a limit on the smallest angle you can resolve.

That angle turns out to be proportional to the wavelength of light, divided by the diameter of the lens, roughly theta -approx lambda.

So smaller wavelength helps, but bigger lens diameter really helps.

Absolutely.

That's why astronomers build enormous telescopes.

A bigger diameter D means a smaller, minimum resolvable angle theta, letting them see finer details in distant galaxies.

You can make the perfect lens according to geometrical optics, but diffraction sets the ultimate limit.

So let's recap this deep dive.

We started from that fundamental idea, Format's Principle of Least Time.

That gave us the math for a single surface.

Which we then used twice to get the workhorse thin lens equation, $1 plus one equals one upper.

Right.

And also that elegant Newtonian form, 6 ton up XC is X stock two two when measuring from the focal points.

But we had to acknowledge this is all based on the paraxial approximation.

Exactly.

And when that breaks down, you get spherical aberration from geometry and chromatic aberration, because the material properties depend on color.

And even if you fix those, you eventually hit the ultimate physical limit,

set by diffraction, the wave nature of light, which determines the best possible resolution.

That's the essence of it.

Geometrical Optics provides the essential toolkit for designing almost any optical instrument.

And it's a fantastic toolkit built on that clever approximation.

It works amazingly well.

But thinking about that resolution limit,

the fact that you simply can't see things that are too close together, not because of lens flaws, but because light itself is a wave.

Well, it forces you to remember the deeper physics.

That light isn't just rays on paper.

It's a propagating electromagnetic field, something to ponder next time you're focusing a camera.

Even a perfect design is still bound by the fundamental wiggle of a wave.