Deriving Irradiance of Diffraction Gratings with Non-Uniform Slit Separation

[billyboy]

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

 

[Valerie Park]

The equation for the irradiance from a diffraction grating with non-uniform slit separation can be derived by replacing the sin Na/sin a term with an expression that takes into account the varying slit separation. The resulting equation is as follows:I=I0(sinB/B)^2 . (sin N(y)/sin a(y))^2where a(y)=0.5kd(y)sin(theta) b(y)=0.5kw(y)sin(theta)I0= max irradianceN(y)= No of slits at position yk= wavevenumberd(y)= distance between adjacent slits at position ytheta= angle of diffractionw(y)= width of anyone slit at position yBy substituting the appropriate terms into the equation, you can calculate the irradiance at any given point along the length of the grating.