S and P Polarization: 5 Easy Facts & Fresnel Equations Guide

S and p polarization describe two perpendicular orientations of light's electric field relative to the plane of incidence. S-polarization (from the German senkrecht, meaning perpendicular) has the electric field pointing perpendicular to the plane of incidence. P-polarization has the electric field lying parallel to that plane. This distinction matters because the two polarisations reflect and transmit very differently at any surface — which is why you can cancel glare with polarised sunglasses, why some camera filters work better than others, and why Brewster's angle exists.

Here is what we cover: the plane of incidence and why it matters, the definition of s and p polarisation, the Fresnel equations that govern reflection and transmission, Brewster's angle where p-polarised light vanishes, and real-world applications from sunglasses to laser optics.

What is the plane of incidence?

The plane of incidence is the imaginary flat surface that contains three rays: the incoming (incident) ray, the reflected ray, and the transmitted (refracted) ray. It is always perpendicular to the interface between the two media.

Picture a light beam hitting a flat water surface at an angle. The incident ray comes downward, the reflected ray bounces upward, and the refracted ray continues into the water. All three lie in the same plane — that is the plane of incidence. If you draw it on paper, the plane of incidence is the page itself.

The plane of incidence is the reference for defining s and p polarisation. Any light wave can be described as a combination of two components: one perpendicular to this plane (s) and one parallel to it (p).

S-polarisation — perpendicular to the plane of incidence

S-polarisation has the electric field vector pointing perpendicular to the plane of incidence. If the plane of incidence is the page, the s-polarised electric field points straight out of the page — like an arrow coming toward you.

Picture a flag on a pole. The flag waving side to side is like s-polarisation — the motion is perpendicular to your line of sight. Or picture Venetian blinds: the slats are vertical, and light that vibrates parallel to the slats passes through, while light perpendicular to them is blocked.

The s stands for senkrecht, the German word for perpendicular. In engineering contexts, s-polarisation is also called TE (Transverse Electric) polarisation because the electric field is transverse to the plane of incidence.

S-polarised light reflects more strongly than p-polarised light at most angles. This is why glare from water, glass, and roads — which is reflected light — is predominantly s-polarised. Polarised sunglasses have their transmission axis oriented vertically, which blocks the horizontally polarised s-component of glare.

P-polarisation — parallel to the plane of incidence

P-polarisation has the electric field vector lying within the plane of incidence. If the plane of incidence is the page, the p-polarised electric field oscillates up and down within the page.

Picture a skipping rope being shaken up and down. The wave travels along the rope, but the vibration stays in the vertical plane. That vertical plane aligned with your view is like p-polarisation — the motion is parallel to the plane you are looking at.

The p stands for parallel — the electric field is parallel to the plane of incidence. In engineering contexts, p-polarisation is also called TM (Transverse Magnetic) polarisation because the magnetic field is transverse to the plane of incidence.

P-polarised light is the component that disappears at Brewster's angle. When unpolarised light hits a surface at exactly the right angle, the p-polarised component is completely transmitted — none of it reflects. The reflected light is then 100% s-polarised.

Fresnel equations — how s and p behave at surfaces

The Fresnel equations are the mathematical rules that govern how much of each polarisation reflects and transmits at an interface. They depend on the angle of incidence and the refractive indices of the two media.

For s-polarisation, the reflection coefficient rₛ and transmission coefficient tₛ are:

rₛ = (n₁ cos θᵢ − n₂ cos θₜ) / (n₁ cos θᵢ + n₂ cos θₜ)

tₛ = (2 n₁ cos θᵢ) / (n₁ cos θᵢ + n₂ cos θₜ)

For p-polarisation, the reflection coefficient rₚ and transmission coefficient tₚ are:

rₚ = (n₂ cos θᵢ − n₁ cos θₜ) / (n₂ cos θᵢ + n₁ cos θₜ)

tₚ = (2 n₁ cos θᵢ) / (n₂ cos θᵢ + n₁ cos θₜ)

Where n₁ and n₂ are the refractive indices, θᵢ is the angle of incidence, and θₜ is the angle of transmission (from Snell's law).

The key insight: these coefficients are different for s and p. At normal incidence (θᵢ = 0°), they are the same. But as the angle increases, they diverge. The s-coefficient rises steadily, while the p-coefficient drops — reaching zero at Brewster's angle.

The reflected intensity is the square of the reflection coefficient: Rₛ = |rₛ|² and Rₚ = |rₚ|².

For a more detailed look at the physics of polarisers and reflection, see our guide on how polarisation works.

Brewster's angle — where p-polarisation disappears

Brewster's angle is the specific angle of incidence where p-polarised light reflects with zero intensity. Only s-polarised light reflects. The reflected beam is therefore 100% linearly polarised.

Brewster's angle θ_B is given by:

θ_B = arctan(n₂ / n₁)

For light in air (n₁ = 1.00) reflecting off water (n₂ = 1.33), Brewster's angle is about 53°. For crown glass (n₂ = 1.52), it is about 57°.

The physical reason: at Brewster's angle, the reflected and transmitted rays are perpendicular to each other. The electric field of p-polarised light in the transmitted beam oscillates parallel to the reflected direction. Since an oscillating dipole does not radiate along its own axis, no p-polarised light is reflected.

This is why polarised sunglasses are most effective when the sun is at around 37° above the horizon (90° − 53° for water). At that position, the glare from lakes and roads is maximally polarised, and a vertical polariser blocks it almost completely.

For a practical comparison of different polariser types, including how circular and linear polarisers handle reflected light, see our guide on types of polarisation.

Phase changes on reflection for s and p

There is an important difference in phase behaviour between s and p polarisation upon reflection.

When light reflects off a higher-index medium (external reflection, n₁ < n₂):

• S-polarisation undergoes a 180° phase shift for all angles up to grazing incidence.
• P-polarisation undergoes a 180° phase shift only for angles smaller than Brewster's angle. At Brewster's angle, the phase jumps abruptly — there is no reflected p-wave to have a phase.

When light reflects off a lower-index medium (internal reflection, n₁ > n₂):

• S-polarisation undergoes no phase shift for angles below the critical angle. Past the critical angle (total internal reflection), the phase shift varies continuously with angle.
• P-polarisation similarly has no phase shift below the critical angle and a varying phase shift above it.

These phase shifts are critical in designing thin-film coatings, waveplates, and interference-based optical devices.

Real-world applications of s and p polarisation

The difference between s and p polarisation is exploited in many practical applications.

Polarised sunglasses. The glare from horizontal surfaces is s-polarised. Sunglasses with vertical transmission axes block this s-component while transmitting p-polarised light, reducing glare without darkening the entire scene.

Brewster windows. In gas lasers, the laser tube often has windows mounted at Brewster's angle. These windows transmit p-polarised light with zero reflection loss, so the laser output is naturally p-polarised.

Anti-reflection coatings. Thin-film coatings exploit the phase differences between s and p reflections to cancel reflected light at specific wavelengths and angles.

Ellipsometry. By measuring the change in polarisation (the ratio of s and p reflection coefficients) when light reflects off a surface, ellipsometers determine film thickness and refractive index with nanometre precision.

Photography. Camera polarising filters work by selectively blocking s-polarised light from reflections. Rotating the filter adjusts which plane of polarisation is blocked, allowing photographers to control glare and enhance colour saturation.

Key takeaways

• S and p polarisation are defined relative to the plane of incidence — the plane containing the incident, reflected, and transmitted rays.
• S-polarisation (senkrecht/perpendicular) has the electric field perpendicular to the plane of incidence. Also called TE polarisation.
• P-polarisation (parallel) has the electric field parallel to the plane of incidence. Also called TM polarisation.
• The Fresnel equations give different reflection and transmission coefficients for s and p.
• At Brewster's angle, p-polarised reflection drops to zero — the reflected light is 100% s-polarised.
• S-polarised light reflects more strongly than p-polarised at most angles, which is why glare from water and roads is predominantly s-polarised.
• Applications include polarised sunglasses, Brewster windows in lasers, anti-reflection coatings, ellipsometry, and photography.

External resources

• Wikipedia: Fresnel Equations — comprehensive technical reference on the Fresnel coefficients for s and p polarisation
• HyperPhysics: Fresnel's Equations — interactive guide with reflection/transmission curves and worked examples
• RP Photonics: Fresnel Equations — detailed reference covering s and p polarisation, Brewster's angle, and applications