  A general theory for the intrinsic (Lorentzian) linewidth of
photonic--crystal surface--emitting lasers (PCSELs) is presented. The effect of
spontaneous emission is modeled by a classical Langevin force entering the
equation for the slowly varying waves. The solution of the coupled--wave
equations, describing the propagation of four basic waves within the plane of
the photonic crystal, is expanded in terms of the solutions of the associated
spectral problem, i.e. the laser modes. Expressions are given for photon
number, rate of spontaneous emission into the laser mode, Petermann factor and
effective Henry factor entering the general formula for the linewidth. The
theoretical framework is applied to the calculation of the linewidth--power
product of air--hole and all--semiconductor PCSELs. For output powers in the
Watt range, intrinsic linewidths in the kHz range are obtained in agreement
with recent experimental results.
