The modified integral method and real electromagnetic properties of echelles

The deep research of all types of echelle gratings, working from low (8) up to very high (1431) diffraction orders with use of the rigorous modified integral method of the analysis is presented. The modified integral approach allows one, with the help of the standard program (PCGrate (TM) 2000X) and a rather small PC, to simulate one of the most hard-to-converge diffraction efficiency problems, what the behaviour of echelle is. In comparison with detailed paper of E. Loewen et al. "Echelles: scalar, electromagnetic, and real-groove properties" the significant difference was found in calculation values for some examples in TM polarization. The difference between the compared theoretical data with the same refractive index for 316 gr/mm r-2 echelle at 632.8 nm in the 9 order and for TM polarization is up to 25% of absolute efficiency. The difference between calculated curve and measured data for the same grating and polarization at 441.6 nm in 12 and 13 orders is small (one-two percents) in opposite to the data of E. Loewen et al., where the difference is many times more because of weak convergence of their method. The appreciable difference also exists for the medium and high orders. The presented results for the given refractive indices (basically, taken from the book E. Palik) have the best coincidence to experiment in all cases. Numerical research of two largest monolithic echelles, made on the project SOFIA also is included. The new record of rigorous calculations for r-10 EXES echelle, working in 1431 order was achieved at 10.6 mum. Because of the very small a wavelength-to-period ratios (similar to0.001) it is necessary to increase truncation parameter for such a case up to such value, that in result the matrices with the order about three thousands turn out.