Chapter 33: Reflection & Transmission from Surfaces

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Welcome to the Deep Dive.

Today we're looking at something you see constantly light hitting, say, glass or water.

And we're asking, why does some of it bounce back?

Why does some go through?

It feels intuitive maybe, but it's not arbitrary at all.

It's actually dictated precisely by the fundamental laws of electromagnetism, by Maxwell's equations right where the light hits the surface.

So this isn't just about geometry?

Not entirely.

We're diving into a chapter from the Feynman Lectures on Physics, Volume 2, called Reflection from Surfaces.

Our goal here is to unpack the physics of how light, as an electromagnetic wave, deals with a boundary between two different materials.

We want to get the essence without getting lost in, you know, all the heavy math on the page.

Okay, so what's the plan?

Where do we start?

We'll start with the rules you probably remember, like Snell's Law, then only to talk about how we actually describe light as a wave, mathematically.

Then comes the really crucial part, the boundary conditions.

What must happen right at the surface?

And from there, we'll see how those conditions actually explain things like shiny metals and total internal reflection.

Sounds like a solid path.

So let's kick off with those familiar rules.

We all learn that the angle the light bounces off, the angle of reflection,

is exactly the same as the angle it came in at, the angle of incidence.

Yep.

Thedar, thetai.

That's locked in.

And for the light that gets through the transmitted light, its angle is given by Snell's Law.

Ah yes.

Hen, dollar synthetai, n2 synthetide.

That connects the angles to the materials themselves through their indices of refraction at model one and two or two.

Exactly.

But like you hinted, those laws tell us where the light goes, not how much light goes where.

That's where it gets, well, much more interesting from a physics perspective.

Because the amount reflected isn't fixed.

Right.

The intensity, the fraction of light that reflects, it depends very strongly on the angle, yes, but also critically on the light's polarization.

Polarization, meaning the direction the electric field part of the light wave is oscillating.

Precisely.

Is the electric field oscillating parallel to the surface it's hitting, or perpendicular to it, or somewhere in between?

The formulas for the reflection amount are completely different for those cases.

Okay, so that's why polarized sunglasses work, right?

They cut out glare, which is often light reflected off horizontal circles, meaning it's strongly polarized horizontally.

Exactly.

They block that specific orientation.

And Feynman adds this really neat point here.

Reflection isn't just about the bulk properties, the N dollar and 10 dollar two deep inside the materials.

What else is there?

He emphasizes it's often a surface property, sometimes just a very thin layer, maybe even just molecules thick right at the interface can dominate how the light reflects.

Like those anti -reflection coatings on camera lenses, they're super thin.

Perfect example, you change that tiny layer and the reflection behavior changes dramatically, even if the air and the main glass are the same.

It's all about that initial interaction zone.

Okay, so to really understand how this interaction forces these outcomes, we need a way to describe the light wave itself, right?

Not just rays, but the wave.

Yes.

And the standard tool for that is the sinusoidal plane wave.

Mathematically, we often use complex exponentials that sort of EEIEI omega t math beer form.

Right, I've seen that.

Looks a bit intimidating, but what's the key physical information packed in there?

Well, omega is the frequency, which determines the color.

But the really crucial bit for this discussion is the vector math BFK, the wave vector.

What does math BFK tell us?

Two main things.

First, its direction is the direction the wave is traveling.

Second, its magnitude, its length dollars is related to the wavelength, and more importantly, to the wave speed in the medium.

Remember, the phase velocity is NC.

Well, Taylor is the negative NC tall.

So math BFK directly involves the materials index of refraction.

Oh, okay.

So the wave vector itself changes when light enters a different medium because the main changes.

Exactly.

And this wave description also makes dealing with Maxwell's equations a bit simpler for these plane waves.

Operations like divergence or curl can sometimes be replaced by just multiplying by parts of I math BFK.

Mathematical shortcut.

Sort of.

And it makes the link between the electric field math BFE and the magnetic field math BF very clear.

For these waves, math BFE is always related to math BFE times math BFB.

Which means math BFB and math BFB are always perpendicular to each other, and both are perpendicular to the direction the wave is going, math BFKB.

Correct.

They're locked together in this transverse structure.

Okay, so we have our incident wave described like this coming in, and we expect a reflected wave going out and a transmitted wave going through, each with its own math BFB after and its own amplitude, e dollars of all.

How do we figure out the amplitudes of the transmitted waves?

That's the core problem, isn't it?

That is the core problem.

And the answer comes from demanding that Maxwell's equations hold up right at the boundary between the two materials.

This brings us to the boundary conditions.

What exactly are these conditions demanding?

Can you give us a feel for it?

Imagine zooming right in on that surface.

Maxwell's equations basically tell us that the electromagnetic fields can't just abruptly jump or break as you cross from medium 1 to medium 2.

Certain parts of the fields have to be, well, continuous.

They have to match up.

Like ensuring a smooth transition, no sudden gaps or spikes.

Precisely.

Specifically, the components of the electric field that run parallel to the surface must be the same on both sides of the boundary.

If they weren't, you'd have crazy infinite fields, which isn't physical.

Okay, parallel E field components match.

And the same applies to the magnetic field components parallel to the surface.

They also have to match across the boundary.

These matching rules are the constraints.

They are what link the incoming wave to the outgoing waves.

So we have three waves, incident, reflected, transmitted.

Each has its own E and B fields.

And we're saying the sum of the parallel components on side one must equal the parallel component on side two.

Exactly right.

You apply those continuity rules mathematically to the wave formulas we just discussed.

And when you do that, some really important consequences just fall out.

Consequences like what?

First, think about time.

For these equations involving the fields matching up to hold true at all moments in time, there's only one way.

The frequency, omega, must be the same for all three waves.

Incident, reflected, and transmitted.

Light doesn't change color when it reflects or refracts.

Okay, that makes sense.

It does make sense.

But here, it's a direct mathematical requirement of the boundary conditions holding over time.

The second consequence comes from demanding the conditions hold at all points along the surface in space.

What does that force?

It forces the components of the wave vectors, the math bfk vectors, that run parallel to the surface to be equal for all three waves.

Let's call this the way direction if the surface is the xy plane.

So incident y reflected, y k transmitted, y away.

Wait a minute.

Just demanding that the parallel parts of math b a k match up, what does that give us?

Yeah, it gives us everything.

Let's take the first equality.

Can we go natural?

Would we wa?

We increase other incident and we wares reflected, we wa.

If you draw the vectors and remember that the magnitude dollar is the same in medium one for both incident and reflected waves, this equality geometrically forces the angle of reflection to equal the angle of incidence.

Thetar equals the tie.

Wow.

Okay, so the law of reflection isn't just an observation.

It's required by field continuity.

Required.

Now, take the second equality.

Two incident, we fxsk to wa, y edaw.

Remember that the magnitude terrors is in the two media.

Two equals is omega ncai.

When you write out what tessie means in terms of the angles, say tie and the tar and include the indices of refraction under dollar and two two, this equation mathematically transforms directly into Snell's law.

One in dollars and two since the tar.

That's actually incredible.

The two fundamental laws of geometric optics just pop out because the waves have to connect smoothly across the boundary in space.

Isn't that something?

It shows these laws are deeply rooted in electromagnetism.

Once you have these angle laws fixed by the spatial continuity, you then use the actual field amplitude matching conditions, the parallel E and parallel B rules to solve for the remaining unknowns,

the amplitudes of the reflected and transmitted waves relative to the incident one.

And those solutions are the Fresnel equations we hear about.

Those are the Fresnel equations.

They give you the exact reflection and transmission coefficients for light polarized parallel or perpendicular to the plane of incidence for any angle.

Okay, so the boundary conditions are the engine driving all of this.

Let's use that understanding now.

Why are metals like silver or aluminum so shiny?

Why do they reflect almost all the light that hits them?

Good question.

It comes down to the index of refraction.

Noller again.

For materials like glass or water, another is just a real number.

For metals, though, not as a complex number.

It has a real part and an imaginary part.

An imaginary part.

What does that to physically?

The imaginary part is related to absorption, how strongly the material absorbs light energy.

Metals are typically very good conductors, meaning electric fields inside them drive currents, which dissipates energy.

So they tend to have a large imaginary part to their index $1.

So they're good absorbers, but they reflect really well.

That sounds like a contradiction.

It does, doesn't it?

But here's the twist.

When you plug a complex index $1 with a large imaginary part into those Fresnel equations derived from the boundary conditions, the math shows that the reflection coefficient gets very, very close to one, meaning almost 100 % reflection.

How does that work?

Why does strong absorption lead to strong reflection?

Think about the boundary condition for the parallel electric field.

Inside the metal, the field dies out extremely quickly because of the absorption and the conductivity.

To maintain continuity right at the surface, the reflected waves electric field has to be almost equal and opposite to the incident waves field just outside the metal.

The only way to satisfy the boundary condition in the face of such rapid internal damping is to have a huge reflection.

So the metal's eagerness to kill the field inside forces it to reject the field almost entirely at the surface.

That's a great way to think about it.

The absorption property fundamentally drives the high reflectivity through the boundary conditions.

Mind blown.

Okay, what about the other case?

Total internal reflection.

This happens when light goes from a denser medium, like water, to a less dense one, like air, right?

Correct.

From higher N -dollar to lower two N -dollar dollars.

As you increase the angle of incidence, you eventually reach a point where Snell's law, N02 into Sanatak, would require Nth to be greater than one for the transmitted angle of effect.

Which is impossible.

San can't be greater than one.

Exactly.

The angle where Synthetak would hit one is called the critical angle, Sanatak.

It's defined by Synthetak increases N to N1 point.

If your angle of incidence Theta is greater than this critical angle, no real angle, Fanta, can satisfy Snell's law.

And experimentally, we see that all the light reflects back into the first medium.

Total internal reflection.

Right.

All the energy is reflected.

But what happened to the transmitted wave in our equations?

Does it just vanish?

Good point.

If the math requires continuity, there should still be something on the other side, shouldn't there?

There is.

When Synthetak gets larger than two hundred and dollar one, the term under a square root in the calculation for the transmitted wave vector component perpendicular to the surface, Klee transmitted, becomes negative.

Square root of a negative number.

That means it becomes imaginary.

Precisely.

The wave vector component pointing into the second medium becomes purely imaginary.

Now, what does an imaginary kilodollar mean for the wave nu?

Omega, T, K, Shu, well, and just ABBA, which times a null.

A null or times a null is a minus one.

So instead of oscillating in the Z direction, it decays.

Got it.

The wave doesn't propagate into the second medium carrying energy away.

Instead, its amplitude decays exponentially as you move away from the surface.

It becomes an evanescent wave.

So there is an electromagnetic field in the second medium, but it's stuck to the surface and dies off really quickly.

Exactly.

It penetrates maybe a few wavelengths deep, but that's it.

No energy flows across in the long run.

Feynman describes a beautiful experiment to show this isn't just math.

How can you see it if it doesn't carry energy away?

You take two prisms, shine light into one past the critical angle so you get total internal reflection.

Normally, no light comes out the other side, but now bring the surface of the second prism really close to the first one, maybe separated by an air gap thinner than a wavelength of the light.

That exists in the tiny air gap can then leak or tunnel into the second prism.

And suddenly light does come out of the second prism.

You frustrated the total internal reflection by giving the evanescent field somewhere to go.

So the field really was there in the gap, just waiting.

That proves the boundary conditions still hold even when reflection is total.

It's a fantastic demonstration.

The field had to be there to maintain continuity, even if it couldn't propagate far on its own.

This has been quite the journey.

We started with simple observations about light bouncing off stuff, and we've ended up seeing that the angle rules Snell's law, why metals shine, why fiber optics work.

It all comes down to Maxwell's equations demanding smooth continuity for the electric and magnetic fields right at the surface.

It really does.

All those phenomena are just necessary consequences.

The boundary conditions are the linchpin that connects the incident wave to whatever reflected and transmitted waves are needed to keep the fields consistent across that interface.

The specific properties of the materials, like nuller being real or complex, just determine what those reflected and transmitted waves look like.

The big takeaway seems to be the power and universality of these boundary conditions.

They're not just for light.

They appear everywhere in physics, don't they?

Absolutely.

Wherever you have different regions or materials meeting, ensuring some kind of continuity across the boundary, whether it's fields, temperature, fluid flow, often dictates the entire behavior of the system.

It makes you wonder just how many physical laws are at their heart, simply statements about how things must connect smoothly at interfaces.

What aspect of that connection stands out most to you?