Optics, electromagnetic theory Definition

Brewster's angle

Builds on Fresnel equations Refractive index Polarization of light

Definition

Let a flat interface separate two lossless, isotropic, non-magnetic media with refractive indices $n_1 > 0$ (incident side) and $n_2 > 0$ , with $n_1 \neq n_2$ . Brewster's angle is the angle of incidence $\theta_B = \arctan(n_2/n_1)$ . It is the unique angle in $(0, \pi/2)$ at which the $p$ -reflection coefficient $r_p$ of the Fresnel equations vanishes.

Remarks

Check: $r_p = 0$ means $n_2 \cos\theta_i = n_1 \cos\theta_t$ ; together with $n_1 \sin\theta_i = n_2 \sin\theta_t$ this gives $\sin\theta_i \cos\theta_i = \sin\theta_t \cos\theta_t$ with $\theta_i \neq \theta_t$ , so $\theta_i + \theta_t = \pi/2$ and $\tan\theta_i = n_2/n_1$ . Example: from air ( $n_1 = 1$ ) to glass ( $n_2 = 1.5$ ), $\theta_B = \arctan 1.5 \approx 56.3^\circ$ . From glass to air, $\theta_B = \arctan(1/1.5) \approx 33.7^\circ$ , below the critical angle $41.8^\circ$ . Non-example: for $s$ -polarization, $r_s$ never vanishes when $n_1 \neq n_2$ , since $n_1 \cos\theta_i = n_2 \cos\theta_t$ and Snell's law would force $n_1 = n_2$ . So no Brewster angle exists for $s$ -polarized light at a non-magnetic interface.

Reading path

To understand this page from the ground up, read these 75 pages in order. The foundations come first.

Calculus of variations concept
Constitutive relations of electromagnetism concept
Electric displacement field concept
Euclidean space object
Euler–Lagrange equation concept
Function concept
Function composition concept
Geodesic concept
Infimum concept
Magnetization concept
Maxwell's equations concept
Normal vector concept
Plane wave concept
Polarization of light concept
Refractive index concept
Supremum concept
Fréchet derivative definition
Total derivative concept
Fermat's principle statement
Function of several variables concept
Vector-valued function concept
Jordan measure definition
Differentiable function of several variables concept
Iterated integral definition
Limit of a function of several variables concept
Partial derivative concept
Parametrised curve concept
Snell's law statement
Gradient concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Differentiable on an open set definition
Double integral definition
Jacobian matrix definition
Chain rule in several variables statement
Gradient definition
Higher partial derivatives definition
Fubini's theorem statement
Iterated derivative definition
Triple integral definition
Chain rule for a composition with a curve statement
C^k function characterised by continuous derivatives statement
Differential form definition
Parametrised curve definition
Vector field definition
Wedge product of forms definition
Closed curve definition
Rectifiable curve definition
Parametrised surface definition
Conservative vector field definition
Divergence definition
Curl definition
Tangent vector of a curve definition
Line integral of a scalar field definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Regular curve definition
Green's theorem statement
One-form definition
Surface area definition
Arc length parametrisation definition
Divergence theorem on a rectangle in the plane statement
Surface integral definition
Exterior derivative definition
Line integral as an integral over the parameter interval statement
Orientable surface definition
Gradient theorem statement
Flux through a surface definition
Divergence theorem statement
Generalised Stokes theorem statement
Stokes' theorem statement
Interface conditions for electromagnetic fields statement
Fresnel equations statement
Brewster's angle definition

Tag	`X3BVPWA03E`
URI	`m2://X3BVPWA03E`
Kind	definition
Writing	W0 stub
MSC	78A40, 78A10
Defines	Brewster's angle
Uses	Fresnel equations, Refractive index, Polarization of light
Changed	3 November 2026

Brewster's angle

Definition

Remarks

Reading path

Further reading

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