Variation of the blaze peak efficiency with $\theta (= \alpha - \theta_B)$ and $\gamma$

We calculated the blaze peak efficiencies for R2.00 and R2.75 echelles. Figures 10 and 11 show the variation of the blaze peak efficiency with $\theta  (= \alpha - \theta_B)$ for R2.00 and R2.75 echelles, respectively. The blaze peak efficiency decreases with $\cos \alpha / \cos \beta$ with increasing $\theta$. These results confirm the results of Schroeder (1981) and prediction of Bottema (1981). The decrease of the blaze peak efficiency with $\theta$ is explained as follows. The effective width of a groove gets smaller with increasing $\theta$, so that the blaze function gets broader. Thus the interference maxima for $\Delta m = \pm 1$ are located inside the main diffraction envelope as shown in Figure 9. The integrated intensities for $\Delta m = \pm 1$ and $\pm 2$ are shown in Figures 12 and 13 for $\theta =$ 0, 2 - 6. As seen in Figure 12 and 13, the integrated intensities for the $\Delta m = \pm 1$ orders increase significantly with increasing $\theta$. As a result, the blaze peak efficiency decreases with increasing $\theta$ (Figures 10 and 11).

Figures 14 and 15 show the variation of the blaze peak efficiency with $\gamma$. The efficiency varies only a few percent with $\gamma$.

In Figures 10 and 11, several dips exist on the curves with their location changing with the spectral order. In Figures 14 and 15, the blaze peak efficiencies vary with spectral orders. These are related to Wood's anomallies, which are discussed in the following subsection.

Figure 10: Variation of the blaze peak efficiency with $\theta$ ($=$ $\alpha$ $-$ $\theta _{B}$) for an R2.00 echelle.

Figure 11: Variation of the blaze peak efficiency with $\theta$ ($=$ $\alpha$ $-$ $\theta _{B}$) for an R2.75 echelle. $\bar{m} = {m}$.

Figure 12: $I_m/I_0$ for the R2.00 echelle when $I_0 = 1$. $I_m$ is the integrated intensity of the sub-order $\Delta m$: $N^2 I(\delta) \Delta \beta_m$. $I_0$ is the integrated intensity at the main peak( $\Delta m = 0$): $N^2 I(0) \Delta \beta_0$. $+$ (red): $\Delta m = -2$, $\times$ (green): $\Delta m = -1$, $\Box$ (blue): $\Delta m = +1$, $\odot$ (pink): $\Delta m = +2$.

Figure 13: $I_m/I_0$ for the R2.75 echelle. The marks have the same meanings as for Figure 12.

Figure 14: Variation of the blaze peak efficiency with out-of-plane angle $\gamma$ for an R2.00 echelle.

Figure 15: Variation of the blaze efficiency with out-of-plane angle $\gamma$ for an R2.75 echelle.