  We shall give a priori conditions on the illuminations $\phi_i$ such that the
solutions to the Helmholtz equation $-div(a \nabla u^i)-k q u^i=0$ in \Omega,
$u^i=\phi_i$ on $\partial\Omega$, and their gradients satisfy certain non-zero
and linear independence properties inside the domain \Omega, provided that a
finite number of frequencies k are chosen in a fixed range. These conditions
are independent of the coefficients, in contrast to the illuminations
classically constructed by means of complex geometric optics solutions. This
theory finds applications in several hybrid problems, where unknown parameters
have to be imaged from internal power density measurements. As an example, we
discuss the microwave imaging by ultrasound deformation technique, for which we
prove new reconstruction formulae.
