Fundamentals of Diffraction and Image Formation

Effects of a diffracting aperture on the angular spectrum of a perturbation

We place an aperture Σ in the plane xOy (at z=0 ). We note Ui(x,y,0) the incident field just before Σ and Ut(x,y,0) the transmitted field just after Σ (see figure II-3).



   

    Image II-3 -Diffracting aperture located at the plane xOy
Image II-3 -Diffracting aperture located at the plane xOy [zoom...]Info

We define the transmittance t(x,y) of the aperture Σ as the ratio between the complex amplitudes taken immediately after and before Σ.

In the case of figure II-3, Σ being an aperture:

General case, valid for any incident wave

Considering the definition of t: Ut(x,y,0)=Ui(x,y,0)t(x,y), the convolution theorem implies:

II-13 Info

with

Particular case of a planar incident wave

In this case, the result is simpler:

By applying (II-13):

Therefore, we directly obtain the angular spectrum of the transmitted wave by calculating the FT of the aperture transmittance.

Introducing a diffracting aperture which spatially limits the incident wave results in broadening the perturbation angular spectrum. Indeed, the smaller is the aperture and the wider are both its FT and the angular spectrum of the transmitted wave. A similar effect occurs in the temporal domain, which is characterized by the relation: . The shorter the pulse duration and the wider the frequency spectrum.

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