Diffraction Through an Aperture

2 Computer Generated Holography 1.2.1 Diffraction Through an Aperture 

In order to understand how nano technology can be used in computer generated holography, we need understand the basics of diffraction theory. This is covered in much better detail in several fundamental texts [12, 13], so we will only summarise the key aspects in this chapter. Let us assume we have an arbitrary aperture (hole) function, A x, y) in the plane S as shown in Fig. 1.1, with coordinates [x, y]. The light passing through this aperture will be diffracted at its edges and the exact form of this pattern can be calculated. We want to calculate the field distribution at an arbitrary position away at the point P, which is a distance R from the aperture. 

we need to change coordinates to the plane containing the point P, which are defined as [a, / ] to give the full expression for each wavelet in terms of x and 

Such an expression can only be solved directly for a few specific aperture functions. To account for an arbitrary aperture, we must approximate, simplify and restrict the regions in which we evaluate the diffracted pattern. If the point P is reasonably coaxial (close to the z axis, relative to the distance R) and the aperture A(x, y) is small compared to the distance R, then the lower section of (1.2) for dE can be assumed to be almost constant and that for all intents and purposes, r = R. The similar expression in the exponential term in the top line of (1.2) is not so simple. It cannot be considered constant as small variations are amplified through the exponential. To simplify this section we must consider only the far field or Fraunhofer region where. 

In this case, the final term in the exponential can be considered negligible. To further simplify, we use the binomial expansion, and keep the first two terms only to further simplify the exponential expression. Hence the simplified version of the field dE, can be expressed as. 

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