Transmission function of diffraction grating aperture

[FiberOptix]

Homework Statement

The aperture has vertical opaque strips of width, a, separated by open spaces of width 2a. The width of the overall aperture is L = 19a.

a) Write a transmission function for this aperture.

b) What would the field distribution be, at a plane in the Fraunhofer zone, if the aperture was
irradiated by a plane wave directed along the normal of the aperture? Draw a sketch of the
irradiance pattern along the X and Y axes of the far field plane.

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Despite searching through some notes and several textbooks I still cannot find a clear expression for a transmission function that includes the specifics of this question. If someone can help or make some useful references that would be very helpful, thank you.

 

[Redbelly98]

This looks like a pretty specialized, so I doubt you'll find a final formula for it.

They are probably expecting you to figure out how the phase contributions from all the openings combine to produce a pattern. Not sure off the top of my head how to do that, but do you have a derivation for the single-slit you can refer to? That would probably illustrate how to proceed for this case.