The S-matrix propagation algorithm, presented in detail in Chapter III, completely eliminates the numerical problems linked with the integration process and matrix inversion in the classical differential method. However, as explained in the first two chapters, the Fourier series of the field for highly conducting gratings used in TM polarization remained slowly converging. This problem remained a mystery for quite long time. Of course, one could try to convince himself of the basic difference in nature between TE and TM polarization problems in saying that in TE polarization, we only represent into Fourier series a continuous function Ez(x, y), while in TM polarization, on top of Hz(x,y), we also represent into Fourier series a discontinuous function Ex whose coefficients En(y) only decrease as 1/n, instead of 1/n2. Thus the desire of working with continuous functions. Today, it is clear that the difficulties of the differential method for TM case lie in the correct presentation of the Fourier series of different products of discontinuous functions, when truncated to a finite number of Fourier components.