Vectorial diffraction methods provide accurate analysis of subwavelength diffractive optical elements, but they
usually consume a lot of computing time and computer memory. On the other hand, scalar diffraction methods
have simpler physical models and cost much less computer resources at the expense of a large numerical error.
Therefore, an approximate method is highly required, which can substantially relieve the computational burden
and possess a high accuracy. We firstly propose such an approximate vectorial method, i.e., the improved first
Rayleigh-Sommerfeld method (IRSM1), and apply it to analysis of two-dimensional (2-D) cylindrical microlenses
and metallic cylindrical focusing micromirrors. The IRSM1 is clearly proved to be much more accurate than
the scalar diffraction method. In addition, compared with the vectorial boundary element method, the IRSM1
consumes much less computing time and computer memory. Besides of the above, the IRSM1 is expected to
be used in optical designs of subwavelength diffractive optical elements by incorporating with the simulated
annealing method or the Yang-Gu algorithm.