Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMSYyk ′k0θx−YFigure 13.2 Diffraction grating of width 2Y with light of wavelength 2π/kbeing diffracted through an angle θ.The factor exp[ik ′ · (r 0 − yj)] represents the phase change undergone by the lightin travelling from the point yj on the screen to the point r 0 , and the denominatorrepresents the reduction in amplitude with distance. (Recall that the system isinfinite in the z-direction and so the ‘spreading’ is effectively in two dimensionsonly.)If the medium is the same on both sides of the screen then k ′ = k cos θ i+k sin θ j,and if r 0 ≫ Y then expression (13.8) can be approximated by∫A(r 0 )= exp(ik′ · r 0 ) ∞f(y) exp(−iky sin θ) dy. (13.9)r 0 −∞We have used that f(y) =0for|y| >Y to extend the integral to infinite limits.The intensity in the direction θ is then given byI(θ) =|A| 2 = 2πr2 |˜f(q)| 2 , (13.10)0where q = k sin θ.◮Evaluate I(θ) for an aperture consisting of two long slits each of width 2b whose centresare separated by a distance 2a, a>b; the slits are illuminated by light of wavelength λ.The aperture function is plotted in figure 13.3. We first need to find ˜f(q):∫1 −a+b˜f(q) = √ e −iqx dx + 1 ∫ a+b√ e −iqx dx2π 2π−a−b] −a+ba−b] a+b= √ 1 [− e−iqx+ 1 [√ − e−iqx2π iq−a−b 2π iqa−b= −1iq √ [e −iq(−a+b) − e −iq(−a−b) + e −iq(a+b) − e −iq(a−b)] .2π438