Lack of Critical Phase Points and Exponentially Faint Illumination

The Stationary Phase Principle (SPP) states that in the computation of oscillatory integrals, the contributions of non-stationary points of the phase are smaller than any power n of 1/k, for k. Unfortunately, SPP says nothing about the possible growth in the constants in the estimates with respect to the powers n. A quantitative estimate of oscillatory integrals with amplitude and phase in the Gevrey classes of functions shows that these contributions are asymptotically negligible, like exp(–ak^b), a,b > 0. An example in Optics is given.