Fresnel Equations Explained: Reflection, Polarization, and Brewster's Angle

Why glass reflects 4% of light, its angle dependence, and why polarized sunglasses work.

The Question Every Optics Student Asks First

You're standing at the edge of a lake on a sunny afternoon. The water is glaring so bright it hurts your eyes. You put on polarized sunglasses, and the glare vanishes. Suddenly you can see straight to the bottom.

What just happened? Why is the reflected light polarized? And what determines how much light a surface reflects in the first place?

The answers come from a set of equations that Augustin-Jean Fresnel worked out in 1823, two centuries ago and still among the most useful results in all of optics. They're also the foundation of every thin-film coating simulation, including ours. If you read our tutorial on anti-reflection coatings, you already saw the punchline: bare glass reflects about 4.2% of light at normal incidence. This tutorial explains where that number comes from, and what happens when light hits a surface at an angle.

Light at a Boundary: What Can Happen

When light hits a flat interface between two transparent materials (say, air and glass), three things happen:

1. Some light reflects back.
2. Some light transmits through and refracts (bends).
3. If the material absorbs, some energy is lost, but we'll stick with transparent materials for now.

The fraction that reflects is the reflectance ($R$), and the fraction that transmits is the transmittance ($T$). Energy conservation says:

$R+T=1$ (for non-absorbing materials)

So if you know $R$, you know $T$. The question is: what determines $R$?

Two things:

• The refractive indices of the two materials ($n_1$ and $n_2$)
• The angle of incidence (${\theta }_i$)

And here's the twist: the answer depends on the polarization of the light.

Two Polarizations, Two Different Answers

Light is a wave where the electric field oscillates perpendicular to the direction the light is traveling. When light hits a surface, the geometry of the reflection defines two natural polarization directions:

• s-polarization (from the German senkrecht, perpendicular): the electric field oscillates parallel to the surface, perpendicular to the plane of incidence.
• p-polarization (parallel): the electric field oscillates in the plane of incidence.

Here's a way to picture it. Imagine a beam of light hitting a tabletop. The plane of incidence is the vertical plane containing the incoming and reflected beams. s-polarization oscillates side-to-side (parallel to the table surface). p-polarization oscillates up-and-down (in that vertical plane).

At normal incidence (straight on), there's no "plane of incidence" and the two polarizations are equivalent. But as soon as the light comes in at an angle, s and p behave differently, sometimes dramatically so.

The Fresnel Equations

For light going from material 1 (index $n_1$) into material 2 (index $n_2$), with angle of incidence ${\theta }_i$ and angle of refraction ${\theta }_t$ related by Snell's law ($n_1\sin {\theta }_i=n_2\sin {\theta }_t$):

Amplitude reflection coefficients:

Intensity reflectances:

For unpolarized light (like sunlight), the reflectance is the average:

These are the Fresnel equations. Let's see what they tell us, starting with the simplest case.

A note on notation: $r_s$ and $r_p$ are amplitude coefficients. They can be positive or negative, where a negative sign means the reflected wave picks up a 180° phase flip. Squaring them gives the intensity reflectances, which are what you actually measure with a power meter.

Normal Incidence: The Simplest Case

At normal incidence (${\theta }_i=0$), Snell's law gives ${\theta }_t=0$, and the two polarizations become identical:

For an air–glass interface with BK7 ($n$ = 1.5185 at 550 nm):

That's where the "4% rule" for glass comes from. A single surface reflects about 4% of light. A window with two surfaces (front and back) transmits about (1 − 0.0424)² = 91.7%, and that's before you even account for absorption.

For water ($n\approx 1.333$): $R$ = 2.0%. For diamond ($n\approx 2.417$): $R$ = 17.2%.

Higher refractive index means more reflection. This is why diamonds sparkle, and why water barely reflects when you look straight down at it.

What Happens as the Angle Increases

Normal incidence is just one point on the curve. The real story unfolds when you tilt the beam.

Here's what happens for light hitting BK7 glass ($n$ = 1.5185) as the angle goes from 0° to 90°:

Three things jump out:

1. Both polarizations start at 4.24% and end near 100%. Every surface becomes a near-perfect mirror at grazing incidence. This is why you can see reflections in a road surface at a shallow angle, even though asphalt isn't remotely shiny.
2. s-polarization always reflects more than p-polarization (for external reflection off a denser medium). The gap widens as the angle increases.
3. At 56.6°, something remarkable happens: $R_p$ drops to exactly zero. This is Brewster's angle.

Brewster's Angle: Where p-Polarization Vanishes

At Brewster's angle, the p-polarized reflectance drops to exactly zero. Not approximately zero. Not "very small." Exactly zero.

The condition is simple:

• For air/BK7: ${\theta }_B=\arctan (1.5185)={\mathbf{56.63}}^{\circ }$
• For air/water: ${\theta }_B=\arctan (1.333)={\mathbf{53.12}}^{\circ }$
• For air/diamond: ${\theta }_B=\arctan (2.417)={\mathbf{67.52}}^{\circ }$

There's an elegant geometric explanation. At Brewster's angle, the reflected and refracted rays are exactly perpendicular to each other (${\theta }_i+{\theta }_t={90}^{\circ }$). The p-polarized reflected wave would need to oscillate along its own direction of travel, but light is a transverse wave, so it can't vibrate in the direction it's moving. The reflection simply doesn't happen.

The result: at Brewster's angle, all reflected light is purely s-polarized, regardless of what polarization the incoming beam had. The surface acts as a perfect polarizing filter for the reflected light.

Why Polarized Sunglasses Work

Now we can answer the lake question from the opening.

Sunlight is unpolarized: it contains equal amounts of s and p components. When it reflects off a water surface at a steep angle, more s-polarization reflects than p-polarization. At a typical viewing angle near 50–60° (which happens to be close to Brewster's angle for water at 53°), the reflected glare is almost entirely s-polarized.

Polarized sunglasses contain a filter oriented to block s-polarized light, the horizontally oscillating component. They let p-polarized light through, which is mostly the light coming from objects under the water, not the glare bouncing off the surface.

That's why you can suddenly see fish through the water when you put on polarized sunglasses. You're not reducing the total light. You're selectively blocking the reflected glare while keeping the transmitted light that carries the image.

The same principle shows up everywhere:

• Photography: Polarizing filters reduce glare from windows and water surfaces. Photographers rotate the filter to find the angle of maximum glare suppression, which is strongest near Brewster's angle.
• Laser windows: Brewster-angle windows inside laser cavities introduce zero loss for p-polarized light, forcing the laser to oscillate in a single polarization. Gas lasers (like He-Ne) use tilted glass windows at exactly Brewster's angle for this reason.
• Fiber optics: Light reflected from fiber end faces is partially polarized, which matters for polarization-sensitive measurements.

The Connection to Thin-Film Coatings

The Fresnel equations describe reflection from a single interface. But a thin-film coating stack is just many interfaces in sequence. The transfer matrix method (TMM) that powers our thin-film simulator works by applying the Fresnel equations at every interface and tracking how the multiply-reflected waves interfere with each other.

The normal-incidence formula $R=((n_1-n_2)\mathrm{/}(n_1+n_2){)}^2$ is exactly what motivated the anti-reflection coating in our previous tutorial. That 4.24% reflectance is the problem. The coating is the solution. The Fresnel equations are the diagnosis.

At oblique incidence, s and p polarization reflect differently, which is why coating specs always include the angle of incidence, not just the wavelength. To go deeper into how polarization states propagate through optical systems, see our tutorial on Jones matrices and polarization states. To see how Fresnel reflections combine in multilayer coatings to create mirrors with R > 99.99%, see how high-reflector mirrors work.

References

1. E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4: "The Propagation of Light." The standard undergraduate treatment of Fresnel equations.
2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Sec. 1.5: "Reflection and refraction of a plane wave." The rigorous derivation from Maxwell's boundary conditions.
3. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 6: "Polarization Optics." Clear treatment of s/p polarization and Brewster's angle.
4. H.A. Macleod, Thin-Film Optical Filters, 5th ed. (CRC Press, 2017), Ch. 2: "Basic Theory." How Fresnel equations generalize to multilayer stacks via the transfer matrix method.