Effective Fresnel number and the focal shifts of focused partially coherent beams

We define the effective Fresnel number of the cylindrical lens illuminated by a plane wave or Schell-model beams. On the basis of the concept of the effective Fresnel number, the focusing properties of the cylindrical lens illuminated by the Schell-model beam are investigated in a simple way. It is shown that the relative focal shift can be evaluated by an analytical formulation, which is expressed as a function of the effective Fresnel number. To evaluate our approach, we make the comparison between the results obtained by our method and the numerical calculation based on the diffraction integral. The results indicate that we can simply and exactly evaluate the focal shifts with our method.