The transmittance function for a clear aperture of width w is
For a circular aperture of diameter d (so d = 2x2 at a given z2 )
get standard Bessel type function for transform of circular aperture (Airy Disk).
If monochromatic light falls normally upon a grating whose amplitude transmittance varies sinusoidally, three beams will exit. To show this, define the transmittance function T as
where p is the pitch (equal to the wavelength) of the grating, R is the transmittance variation, and Q is the average transmittance, express the cosine in exponential form, and collect the phase factor in front of the integral into C to form the new constant C', get
where the first integral is just the diffraction by a clear aperture with an extra factor Q, which is a measure of the overall transmittance of the grating, attenuating the beam. The second integral is the same as a clear aperture with a spatial frequency shift f = 1/p in the exponential. This spatial frequency modification imposes a coordinate shift in the transform plane. For a small aperture, the transform is the sinc function centered at the spatial frequency coordinate in the transform plane. The value of f can be found by
A short and relatively simple book about Fourier Transforms and their properties is:
Brigham, E. Oran, Fast Fourier Transform, Prentice Hall, 1974.
Send Mail: matthysd@vms.csd.mu.edu
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