## ABSTRACT

This method was widely popularized by Moharam and Gaylord at the beginning of the 1980s. They called it “rigorous coupled-wave analysis” [I.20]. The qualifier “rigorous” was used to distinguish it from the previously developed “coupled-wave theory” [VI. 1] and [VI.2] and to point out the fact that it does not introduce any hypothesis or simplification in Maxwell equations. But its denomination is somehow confusing in the sense that the previously developed differential theory is as rigorous as it is; moreover they are both “coupled-wave” theories. Li has called it “coupled-wave method” [I.44], but recently proposed to call it “Fourier modal method” [VI.3] or “modal method by Fourier expansion” [VI.4]. But the qualifier “modal” may introduce confusion with the classical modal theory [I.16] and [I.17]. Thus, in this book, we will use the denomination of “rigorous coupled-wave theory” (RCW) despite its inconveniences, because it is the most widely used denomination. Indeed, the first presentation of the RCW theory can be found in the early work of Peng et al. [I.18], who refered to the work of Lewis and Hessel [VI.5]; an independent work was also published in a Russian proceeding in 1977 [VI.6]. However, the method was not widely used before the beginning of the 1980s. This is due to the fact that it is restricted to particular situations in which matrix M in Eq. (V.22) is independent of y as it happens for lamellar (or rectangular groove) gratings. This may occur in slanted volume holographic gratings after a convenient choice of the coordinates [VI.6], too. This type of profile was not common before the advent of binary optics and the ion-etching technique. Both Peng, Tamir and Bertoni on one side [I.18], and Moharam and Gaylord on the other side [VI.7] assumed that it was possible to extend its validity to arbitrary profiles via a staircase approximation, in view of having a general approach to analyze any kind of gratings. However, this hypothesis, which may seem quite natural, was never seriously checked. Until recently, the convergence of the staircase approximation was not analyzed in detail, and turns out to fail in TM polarization, as will be shown in Section VI.5.1.2. For surface relief gratings, the RCW method is then limited to lamellar profiles; it is then really performant. It replaces the numerical integration of Eq. (V.22) by the search of eigenvalues and eigenvectors of matrix M. However, it does not avoid the numerical contamination by growing exponential functions, since the 114eigenvalues are computed with a finite accuracy. In order to avoid the inversion of the global T matrix of a deep grating, the S-matrix algorithm has to be used too. Moreover, since the RCW method uses the same Fourier basis as the differential method, it suffered for a long time from the slow convergence of the Fourier series of the field when analyzing metallic gratings lighted with TM polarized waves. How this point was resolved will be explained in Section VI.2.