## Abstract

For lack of alternatives, echelle-grating diffraction behavior has in the past been modeled on scalar theory, despite observations that indicate significant deviations. To resolve this difficulty a detailed experimental, theoretical, and numerical study is performed for several echelles that work at low (8–13), medium (35–55), high (84–140), and very-high (to 660) diffraction orders. Noticeable deviations from the scalar model were detected both experimentally and numerically, on the basis of electromagnetic theory: (1) the shift of the observed blaze position was shown to decrease with the wavelength-to-period ratio, and it tends to zero more rapidly than the decrease of the maximum width, so that the TE- and TM-plane responses tend to merge into each other; (2) cut-off effects (Rayleigh anomalies) were found to play a significant role for high groove angles, where passing-off orders are close to the blaze order. A possibility for evaluation of the blaze angle from angular, rather than from spectral, measurements is discussed. Several reasons for the differences between real and ideal echelles (material-index deviations, profile deformations, and groove-angle errors) are analyzed, and their effects on the performance of echelles is studied.

© 1995 Optical Society of America

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### Equations (14)

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(1)
$$\frac{1}{{M}^{2}}{I}_{g}\left(p\right)\equiv {I}_{f}{B}_{f}={\left[\frac{\text{sin}\left(M\frac{kdp}{2}\right)}{M\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{kdp}{2}\right)}\right]}^{2}{\left[\frac{\text{sin}\left(\frac{ksp}{2}\right)}{\frac{ksp}{2}}\right]}^{2},$$
(2)
$$p\equiv \text{sin}\phantom{\rule{0.2em}{0ex}}{\theta}_{d}+\text{sin}{\theta}_{i}=\frac{N}{M}\frac{\lambda}{d}.$$
(3)
$$2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{B}=-N\frac{\lambda}{d},$$
(4)
$$\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta}_{N}=\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta}_{i}+N\frac{\lambda}{d},$$
(5)
$${\lambda}_{{B}_{N}}\propto \frac{1}{N}.$$
(6)
$${\lambda}_{{B}_{N}}-{\lambda}_{{B}_{N+1}}\propto \frac{1}{N}-\frac{1}{N+1}\approx \frac{1}{{N}^{2}}.$$
(7)
$${\Delta}_{{\text{max}}_{N}}\propto \frac{1}{{N}^{2+\delta}}.$$
(8)
$$\frac{{{\lambda}_{{B}_{N}}}^{\text{TE}}-{{\lambda}_{{B}_{N}}}^{\text{TM}}}{{{\lambda}_{{B}_{N}}}^{\text{TM}}-{{\lambda}_{{B}_{N+1}}}^{\text{TM}}}=\frac{1}{{N}^{\delta}}\underset{\lambda \to 0}{\to}\{\begin{array}{ll}\infty \hfill & \delta <0.\hfill \\ 0\hfill & \delta >0\hfill \end{array}$$
(9)
$$2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta}_{i}=-n\frac{\lambda}{d}.$$
(10)
$${\theta}_{N}={\varphi}_{B}-{\Delta}_{\theta},$$
(11)
$${\theta}_{i}={\varphi}_{B}+{\Delta}_{\theta}.$$
(12)
$$\text{cos}\phantom{\rule{0.2em}{0ex}}{\Delta}_{\theta}=\frac{{\lambda}_{\Delta}}{{\lambda}_{B}}.$$
(13)
$$\text{sin}\left({\theta}_{{N}_{C}}\right)=N\frac{\lambda}{d}-1,$$
(14)
$$\text{sin}\left[{\theta}_{{\left(-1\right)}_{C}}\right]=1-\frac{\lambda}{d},$$