  We present a perturbative method for calculating phonon properties of an
insulator in the presence of a finite electric field. The starting point is a
variational total-energy functional with a field-coupling term that represents
the effect of the electric field. This total-energy functional is expanded in
small atomic displacements within the framework of density-functional
perturbation theory. The linear response of field-polarized Bloch functions to
atomic displacements is obtained by minimizing the second-order derivatives of
the total-energy functional. In the general case of nonzero phonon wavevector,
there is a subtle interplay between the couplings between neighboring k-points
introduced by the presence of the electric field in the reference state, and
further-neighbor k-point couplings determined by the wavevector of the phonon
perturbation. As a result, terms arise in the perturbation expansion that take
the form of four-sided loops in k-space. We implement the method in the {\tt
ABINIT} code and perform illustrative calculations of the field-dependent
phonon frequencies for III-V semiconductors.
