On the Resolving Power of an Echelle Grating in the Presence of a Corrugated Wavefront of Incidence

The resolving power of a grating is determined by the ability of the system to separate the in-tensity maxima produced by two infinitesimally close wavelengths. The design parameters of a grating address the two issues- (a) improvement of the resolving power and (b) enhancement of dispersive property[ll. An echelle grating can ideally satisfy these criteria giving a resolving power- lo6- 10'. This high resolution property presupposes that the wave front of incidence is perfectly coherent. The question asked here is- what happens to the resolving power of the grat-i?g if the phase coherence in the incidence beam be lost beyond a distance p, = A/2x6,, where, 8, is the 'seeing ' angle. When incident on the grating however, the coherence is compressed to a distance r, by collimation, where r, = (dlD)p,, where D is the diameter of the telescope and 'd ' being the length of the grating. We consider the scattering process from the echelle grating to be equivalent to that from a "periodic rough surfaceV[2] and in terms of the standard definitions therein, we have the scattering wave vectors, v, = k(sin 81- sin 82 cos 03); vz =-k(cos el + ~ 0 ~ 8 ~ ) (1) with k = 2x12; and R being the wavelength of light with the groove being in x direction and scattering in y direction is absent. By addition of the complex amplitude of all the rays diffracted by the grating with N groove of width '2a ' each and blazing angle a, we find the intensity to follow, where, $,,(x, y) is the random phase at the point of incidence (2na t x) on the nth groove of the grating. Considerin