The stationary phase or differential method for the determination of a light beam shift by total reflexion

Abstract

We consider a light beam the electric field of which depends only on two dimensions, that case being suited to a linear source. The shift by total reflexion of such a beam depends on the direction, relative to the mean incident plane, of the plane P containing the propagation vectors of the plane waves into which the incident wave may be decomposed. The present work computes this influence, and explains some apparent inconsistancies in the application of the stationary phase method; we confirm the known results in the cases of a plane P coincident with the incidence plane or perpendicular to it. We show first that this method is equivalent to the search of the maximum energy or flux density, and that it implies a symmetric incident beam, with a peak the smoother that one is closer to the limit of total reflexion. In its first form (stationary phase), the method is only applicable when the field components parallel or perpendicular to the plane P, undergo the same phase variation by reflexion. This condition ascertains two eigendirections of the field, and the incident wave may be decomposed into two waves corresponding to these directions, which exhibit orthogonal elliptic polarizations. In the case of only one reflexion, the corresponding maxima of the energy are not separated from each other; the resulting shift may be also computed directly without getting through the eigenwaves, using the second form of the method (maximum of energy). In the same way, in the case of many successive reflexions with polygonal or isosceles prism devices, one may compute two eigenwaves, and if the two maxima are not separated from each other, the resulting shift. In the case of only one reflexion, the method may be extended to a beam, the intensity of which is axially symmetric. Finally, the results of the computation are compared with those of Imbert's experience using an equilateral prism.