Longitudinal shift of a microwave beam at total internal reflection

Energy-flux pattern in the goos-Hanchen effect

The (two-dimensional) wave beam fields and the associated energy flux in the Goos-Hanchen effect are studied analytically and numerically. In particular, the time-averaged Poynting vector and its flux lines around the interface are calculated for both states of wave polarization, irrespective of a positive shift (the usual one) or a negative shift of the reflected beam. For a given transverse field profile of the incident beam, the flux lines associated with the evanescent waves for the two cases of shift are shown to have the same shape yet to take opposite directions; they are parabolas if the profile is Gaussian. The flux lines in the first medium are shown to connect to those in the second medium on the two sides of the interface. In the case of positive shift, the whole flux pattern expectedly shows the supply of energy from the incident beam to the evanescent wave on one side and the return of energy on the other side to the reflected beam. In the case of negative shift, on the other hand, the flux lines nearby the interface form loops, in addition to the expected incoming-outgoing flux pattern in the remaining region.