Diffraction Gratings

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Dear all
I've been pondering this problem for a while but am daunted by the maths I suspect is required.

Consider the irradiance from a diffraction grating:

I=I0(sinB/B)^2 . (sin Na/sin a)^2

where a=0.5kdsin(theta)
b=0.5kwsin(theta)

I0= max irradiance
N= No of slits
k= wavevenumber
d= distance between adjascent slits
theta= angle of diffraction
w= width of anyone slit


My question is how to derive a similar equation for a grating in which the slit separtion is not uniform but is instead, a function of y (the distance along the length of the grating).

Any help would be grately appreciated.

Many thanks
 

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