Multi-Slit Diffraction

Suppose that the opaque screen in our interference/diffraction apparatus contains $N$ identical, equally spaced, parallel slits of finite width. Let the slit spacing be $d$, and the slit width $\delta$, where $\delta < d$. It follows that the aperture function for the screen is written

$$F(x) = \frac{1}{N}\sum_{n=1,N} F_2(x-x_n),$$ (10.59)

where

$$x_n = [n-(N+1)/2]\,d,$$ (10.60)

and

$$F_2(x)=\left\{ \begin{array}{ccc} 1/\delta&\mbox{\hspace{0.5c... ...elta/2\\ [0.5ex] 0 &&\vert x\vert> \delta/2 \end{array}\right..$$ (10.61)

We recognize $F_2(x)$ as the aperture function for a single slit, of finite width $\delta$, that is centered on $\theta =0$. [See Equation (10.54).]

Assuming normal incidence (i.e., $\theta_0=0$), the interference/diffraction function, which is the Fourier transform of the aperture function, takes the form [see Equation (10.52)]

$$\bar{F}(\theta) = \int_{-\infty}^{\infty} F(x)\,\cos(k\,\sin\theta\,x)\,dx.$$ (10.62)

Hence,

$$\bar{F}(\theta) = \frac{1}{N}\sum_{n=1,N}\int_{-\infty}^\infty F_2(x-x_n)\,\cos(k\,\sin\theta\,x)\,dx$$

$$=\frac{1}{N}\sum_{n=1,N} \left[\cos(k\,\sin\theta\,x_n)\,\int_{-\infty}^\infty F_2(x')\,\cos(k\,\sin\theta\,x')\,dx'\right.$$

$$~~~~\left. - \sin(k\,\sin\theta\,x_n)\,\int_{-\infty}^\infty F_2(x')\,\sin(k\,\sin\theta\,x')\,dx'\right]$$

$$=\left[\frac{1}{N}\sum_{n=1,N} \cos(k\,\sin\theta\,x_n)\right] \int_{-\infty}^\infty F_2(x')\,\cos(k\,\sin\theta\,x')\,dx',$$ (10.63)

where $x'=x-x_n$. Here, we have made use of the result $\int_{-\infty}^\infty F_2(x')\,\sin(\alpha\,x')\,dx'=0$, for any $\alpha$, which follows because $F_2(x')$ is even in $x'$, whereas $\sin(\alpha\,x')$ is odd. We have also employed the trigonometric identity $\cos(x-y)\equiv \cos x\,\cos y-\sin x\,\sin y$. (See Appendix B.) The previous expression reduces to

$$\bar{F}(\theta) = \bar{F}_1(\theta)\,\bar{F}_2(\theta).$$ (10.64)

Here [cf., Equation (10.38)],

$$\bar{F}_1(\theta)= \int_{-\infty}^\infty F_1(x)\,\cos(k\,\sin\the... ...ac{\sin[\pi\,N\,(d/\lambda)\,\sin\theta]}{\sin [\pi\,(d/\lambda)\,\sin\theta]},$$ (10.65)

is the interference/diffraction function for $N$ identical parallel slits of negligible width that are equally spaced a distance $d$ apart, and

$$F_1(x) = \frac{1}{N} \sum_{n=1,N} \delta(x-x_n),$$ (10.66)

is the corresponding aperture function. Furthermore [cf., Equation (10.55)],

$$\bar{F}_2(\theta) = \int_{-\infty}^\infty F_2(x)\,\cos(k\,\sin\theta\,x)\,dx = {\rm sinc}\left[\pi\,\frac{\delta}{\lambda}\,\sin\theta\right],$$ (10.67)

is the interference/diffraction function for a single slit of width $\delta$.

Figure 10.12: Multi-slit far-field interference pattern calculated for $N=10$, $d/\lambda = 10$, and $\delta /\lambda =2$, assuming normal incidence.

We conclude, from the preceding analysis, that the interference/diffraction function for $N$ identical, equally spaced, parallel slits of finite width is the product of the interference/diffraction function for $N$ identical, equally spaced, parallel slits of negligible width, $\bar{F}_1(\theta)$, and the interference/diffraction function for a single slit of finite width, $\bar{F}_2(\theta)$. We have already encountered both of these functions. The former function (see Figure 10.8, which shows $[\bar{F}_1(\theta)]^{\,2}$) consists of a series of sharp maxima of equal amplitude located at [see Equation (10.39)]

$$\theta_j = \sin^{-1}\left(j\,\frac{\lambda}{d}\right),$$ (10.68)

where $j$ is an integer. The latter function (see Figure 10.10, which shows $[\bar{F}_2(\theta-\theta_0)]^{\,2}$) is of order unity for $\vert\theta\vert\lesssim \sin^{-1}(\lambda/\delta)$, and much less than unity for $\vert\theta\vert\gtrsim \sin^{-1}(\lambda/\delta)$. It follows that the interference/diffraction pattern associated with $N$ identical, equally spaced, parallel slits of finite width, which is given by

$${\cal I}(\theta) \propto \left[\bar{F}_1(\theta)\,\bar{F}_2(\thet... ...pto\left[\bar{F}_1(\theta)\right]^{\,2}\, \left[\bar{F}_2(\theta)\right]^{\,2},$$ (10.69)

is similar to that for $N$ identical, equally spaced, parallel slits of negligible width, $[\bar{F}_1(\theta)]^{\,2}$, except that the intensities of the various maxima in the pattern are modulated by $[\bar{F}_2(\theta)]^{\,2}$. Hence, those maxima lying in the angular range $\vert\theta\vert< \sin^{-1}(\lambda/\delta)$ are of similar intensity, whereas those lying in the range $\vert\theta\vert> \sin^{-1}(\lambda/\delta)$ are of negligible intensity. This is illustrated in Figure 10.12, which shows the multi-slit interference/diffraction pattern calculated for $N=10$, $d/\lambda = 10$, and $\delta /\lambda =2$. As expected, the maxima lying in the angular range $\vert\theta\vert< \sin^{-1}(0.5) = \pi/6$ have relatively large intensities, whereas those lying in the range $\vert\theta\vert> \pi/6$ have negligibly small intensities.