Blaze Function and the Groove Shadowing Effect

The blaze function determines a diffraction envelope and is given by

where is the phase difference between the center and edge of a single groove of effective width . The value of is given as,

where is the incident angle, is the diffraction angle, and is the blaze angle of the grating. The grating interference function is given by

where is the phase difference between the centers of adjacent grooves and is the number of grooves lighted up by incident beam on the grating. The value of is given as follows,

where is the groove spacing.

For calculating the integrated intensity of spectral images, we use the peak intensity and effective image width of interference maxima. The peak values of the interference maxima are . The effective image width of the th order interference maximum, , can be considered as a half of the separation of the first minima. The value of is,

where is the diffraction angle of the th order interference maximum. The integrated intensity of the th order inerference maximum is given by

(a) Case for

The effective groove width varies with the incident angle (Figure 8) and can be written as

where is . The blaze function will thus be

(b) Case for

In this case, the diffracted beam is vignetted by neighboring grooves (Figure 8) and only the fraction contributes to the image (Bottema, 1981). The effective groove width is determined by the diffracted beam and can be written in the form similar to Eq. 28.

The blaze function will be

The peak values of interference maxima are also smaller than those for the case . As a result, the integrated intensity will be

where .

The effective blaze function () can be written as follows,

Figure 9 shows the EBF when 0 and 4 degrees. When , the amplitude of the EBF is a factor of smaller than that of the EBF for , and the effective width of a single groove gets narrower with increasing . As a result, the main envelope of the EBF for gets broader for lower amplitudes and the side-lobes are negligible.

The blaze function is affected by polarization. The polarization effect is related to the ratio of wavelength to groove spacing, , and the blaze function. The scalar theory can be used as a good approximation when is less than 0.2 (Loewen et al., 1977). For the Littrow case, the ratio can be re-written as follows

Thus the polarization effect is negligible for high orders. For examples, the ratio is less than 0.2 when for an R2.00 echelle ( ) and an R2.75 echelle ( ).

2003-05-29