CALC It is possible to calculate the intensity in the single-slit Fraunhofer diffrac...

CALC It is possible to calculate the intensity in the single-slit Fraunhofer diffraction pattern without using the phasor method of Section 36.3. Let y’ represent the position of a point within the slit of width a in Fig. 36.5a, with y’ = 0 at the center of the slit so that the slit extends from y’ = -a/2 to y’ = a/2. We imagine dividing the slit up into infinitesimal strips of width dy_, each of which acts as a source of secondary wavelets. (a) The amplitude of the total wave at the point O on the distant screen in Fig. 36.5a is E₀. Explain why the amplitude of the wavelet from each infinitesimal strip within the slit is E₀(dy’/a), so that the electric field of the wavelet a distance x from the infinitesimal strip is (b) Explain why the wavelet from each strip as detected at point P in Fig. 36.5a can be expressed as

where D is the distance from the center of the slit to point P and k = 2π/λ. (c) By integrating the contributions dE from all parts of the slit, show that the total wave detected at point P is

(The trigonometric identities in Appendix B will be useful.) Show that at θ = 0^o, corresponding to point O in Fig. 36.5a, the wave is E = E₀ sin(kD – ωt) and has amplitude E₀, as stated in part (a). (d) Use the result of part (c) to show that if the intensity at point O is I0 , then the intensity at a point P is given by Eq. (36.7).