Fraunhofer diffraction by plane grating: Plane grating consists of multiple parallel, equidistant slits on an opaque screen. It is an extension of Young’s double slit, where both interference and diffraction play their part. To find the intensity distribution by a grating first consider interference of the waves originating from N sources, having equal amplitude, a. The  phase will change by an equal amount, d from one slit to the next. Therefore  the  resultant complex amplitude, Ae^iq , is the sum of the series,

Ae^iq =a (1 + e^id + e^i2d + e^i3d + ...... + e^i(N-1)d )

=a(1 - e^iNd) / (1 - e^Id )

By multiplying with the complex conjugate

A² = a² sin² (Nd /2) / sin² (d /2) = a² sin² N g/ sin² g

where   g  = d /2 = (1/2) kd sin q , with d as the distance between the centers of adjacent slits. The  factor  a² represents the intensity due to diffraction from a single slit.  Therefore the intensity distribution due to an ideal grating is

I= [A_o² sin² b/ b ²] [sin² N g/ sin² g]

Principal maxima: we note that,

 Lim _g ® mp sin g [sin N g/ sin g] = ± N

where m is an integer positive or negative. Therefore g = mp , is the condition of maxima. This gives,

d sin q = m l [grating equation]

m gives the order of the diffraction pattern. m=0 is the zeroth order; m=1 the first order and so on. The maximum value of m is obtained for q =90^o. We observe from the grating equation that maxima for different wavelengths will be at different angle q . The principal maxima are called the spectral lines. since the lines are narrow they will be seen separate. As one goes from lower to higher order the separation will be larger. The grating can thus used for obtaining spectrum which is its chief application.

The  chromatic  resolving  power, that is  the power of the grating to resolve two nearby wavelengths l and l + D l is given by,

l / D l= m N

where m is the order of spectrum and N is the number of slits.