p-polarization

p-polarisation (from German parallel) refers to an electromagnetic wave striking an interface with its electric field vector in the plane of incidence — the plane containing the incoming k-vector and the surface normal. In waveguide and antenna literature the same state is called TM (transverse magnetic) polarisation because the magnetic field is entirely transverse to the plane of incidence.

The distinctive feature of p-polarisation at a dielectric interface is that the Fresnel reflection coefficient r_p = (n₂cosθ₁ − n₁cosθ₂)/(n₂cosθ₁ + n₁cosθ₂) passes through zero at Brewster's angle θ_B = arctan(n₂/n₁). At this angle the p-polarised wave is fully transmitted with no reflection. This vanishing of r_p — combined with the non-vanishing r_s — means that unpolarised light reflected at Brewster's angle is purely s-polarised, a principle Brewster exploited in his 1815 systematic measurements. Brewster windows in laser cavities use this property to eliminate reflective losses for one polarisation: inserting the window at θ_B ensures the p-polarised gain-medium output passes without loss, while the s-polarised component is discriminated against enough to force oscillation on the clean p-polarised mode.