Chapter 36, Problem 36.66CP

Intensity Pattern of N Slits. (a) Consider an arrangement of N slits with a distance d between adjacent slits. The slits emit coherently and in phase at wavelength λ. Show that at a time t, the electric Held at a distant point P is

$\begin{array}{l}{E}_P(t)=E_0\cos (kR-\omega t)+E_0\cos (k_R-\omega t+\varphi )\\ {\text{ + E}}_{\text{0}}\cos (kR-\omega t+2\varphi )+\mathrm{...}\\ {\text{ +E}}_{\text{0}}\cos (kR-\omega t+(N-1)\varphi )\end{array}$

where E₀ the amplitude at P of the electric field due to an individual slit, ϕ = (2π/sinθ)/λ, θ is the angle of the rays reaching P (as measured from the perpendicular bisector of the slit arrangement), and R is the distance from P to the most distant slit. In this problem, assume that R is much larger than d. (b) To carry out the sum in part (a), it is convenient to use the complex-number relationship = e^iz = cosz + i sin z, where i = $\sqrt{-1}$. In this expression, cosz is the real port of the complex number e^iz, and sinz is its imaginary port. Show that the electric field E_P(t) is equal to the real part of the complex quantity

${\displaystyle \underset{n=0}{\overset{N-1}{\sum }}}{E}_0{e}^{{}^i(kR-\omega t+n\varphi )}$

(c) Using the properties of the exponential function that = e^Ae^B = e^(A+B) and (e^A)ⁿshow = e^n,A, that the sum in part (b) can be written as

$E_0(\frac{e^{iN\varphi }-1}{e^{i\varphi }-1})e^{{}^i(kR-\omega t)}$

= $E_0(\frac{e^{iN\varphi /2}-e^{-iN\varphi /2}}{e^{{}^{i\varphi }/2}-e^{-iN\varphi /2}})e^{{}^i[kR-\omega t+[N-1]\varphi /2]}$

Then, using the relationship e^iz = cosz + isinz, show that the (real) electric field at point P is

$E_P(t)=[E_0\frac{\sin (N\varphi /2)}{\sin (\varphi /2)}]\cos [kR-\omega t+(N-1)\varphi /2]$

The quantity in the first square brackets in this expression is the amplitude of the electric field at P. (d) Use the result for the electric-field amplitude in part (c) to show that the intensity at an angle θ is

$I=I_0{[\frac{\sin (N\varphi /2)}{\sin (\varphi /2)}]}^2$

where I₀ is the maximum intensity for an individual slit, (e) Cheek the result in part (d) for the case N = 2. It will help to recall that sin2A = 2 sin A cosA. Explain why your result differs from Eq. (35.10), the expression for the intensity in two-source interference, by a factor of 4. (Hint: Is I₀, defined in the same way in both expressions?)