Variation of the blaze peak efficienty with spectral orders and Wood's anomaly

The variation of efficiency with spectral order is shown in Figures 16 and 17, in which several dips are seen on the blaze peak efficiency curves. The dips get deeper and wider with decreasing spectral order.

These dips are related to Wood's anomallies discovered by Wood in 1902 as a phenomenon that the intensity distribution of the diffracted light varies abruptly. Description about these anomallies is seen in Kitchin (1995):

``.... These are sudden brightenings of the spectrum with a sharp onset and a slightly less sharp decline to longer wavelengths. The anomallies arise from that energy that would go into spectral orders behind the grating, if these were possible, being redistributed back into the actually visible orders. ....''

The first sentence in the above quotation exactly corresponds to the shape of the dips. In order to confirm the fact that the dips are caused by Wood's anomallies, locations of the anomallies are calculated. The spectral orders at which the anomallies occur can be calculated by the following equations,

$$m_{+} = \frac{\sigma \cos \gamma}{\lambda_B} ( \sin \alpha + 1) ,$$ (38)

$$m_{-} = \frac{\sigma \cos \gamma}{\lambda_B} (\sin \alpha + \sin(\theta_B - 90)),$$ (39)

where $m_{+}$ is the order number at $\beta = 90 ^{\circ}$, $m_{-}$ is the order number at $\beta = \theta_B - 90 ^{\circ}$, $\lambda_B$ is the blaze wavelength at $m$, and $\gamma$ is the out-of-plane angle. Because $\lambda_B = (\sigma \cos \gamma / m) \cdot 2 \sin \theta_B \cos \theta$,

$$m - m_{+} = m \left( 1 - \frac{\sin \alpha + 1}{2 \sin \theta_B \cos \theta} \right),$$ (40)

$$m - m_{-} = m \left( 1 - \frac{\sin \alpha + \sin (\theta_B -90)}{2 \sin \theta_B \cos \theta} \right).$$ (41)

The locations are shown in Figures 18 and 19 with arrows. By comparison, it is confirmed that the dips occur at the expected positions where Wood's anomallies occur. The anomallies are strong for lower $\Delta m$ because the interference maxima for lower $\Delta m$ exist in the higher intensity region of the diffraction envelope. The biggest dips and brightenings at the 18th order for an R2.00 echelle and at the 30th order for an R2.75 echelle are related to the anomallies that the interference maximum for the spectral order $\Delta m = -1$ disappears beyond the dispersion limit ( $\theta_B - 90^{\circ} < \beta < 90^{\circ}$) with decreasing $\bar{m}$ (see Figures 12 and 13) and the intensity is re-distributed into remnants of the interference maxima within the dispersion limit.

The locations of the anomallies are shifted toward lower orders with increasing $\theta$ as seen in Figures 16 and 17. The blaze peak efficiency in the dips is less than the value of ( $\cos \alpha / \cos \beta$).

Figure 16: Variation of the blaze peak efficiency with order number for an R2.00 echelle.

Figure 17: Variation of the blaze peak efficiency with order number for an R2.75 echelle.

Figure 18: Locations of Wood's anomallies for an R2.00 echelle. Arrows indicate their positions.

Figure 19: Locations of Wood's anomallies for an R2.75 echelle. Arrows indicate their positions.