Multi-Slit Interference

Suppose that the interference apparatus pictured in Figure 10.1 is modified such that $N$ identical slits of width $\delta\ll \lambda$, running parallel to the $y$-axis, are cut in the opaque screen that occupies the plane $z=0$. Let the slits be located at $x=x_n$, for $n=1,N$. For the sake of simplicity, the arrangement of slits is assumed to be symmetric with respect to the plane $x=0$. In other words, if there is a slit at $x=x_n$ then there is also a slit at $x=-x_n$. The distance between the $n$th slit and a point on the projection screen that is an angular distance $\theta$ from the plane $x=0$ is [cf., Equation (10.3)]

$$\rho_n = R\left[1-\frac{x_n}{R}\,\sin\theta + {\cal O}\left(\frac{x_n^{\,2}}{R^{\,2}}\right)\right].$$ (10.31)

Thus, making use of the far-field orderings (10.10), where $d$ now represents the typical spacing between neighboring slits, and assuming normally incident collimated light, Equation (10.5) generalizes to

$$\psi(\theta,t)\propto \sum_{n=1,N}\cos(\omega\,t - k\,R -\phi+k\,x_n\,\sin\theta),$$ (10.32)

which can also be written

$$\psi(\theta,t)\propto \cos(\omega\,t-k\,R-\phi)\sum_{n=1,N} \cos(k\,x_n\,\sin\theta) \sin(\omega\,t-k\,R-\phi)\sum_{n=1,N}\sin(k\,x_n\,\sin\theta),$$ (10.33)

or

$$\psi(\theta,t)\propto \cos(\omega\,t-k\,R-\phi)\sum_{n=1,N} \cos(k\,x_n\,\sin\theta).$$ (10.34)

Here, we have made use of the fact that arrangement of slits is symmetric with respect to the plane $x=0$ [which implies that $\sum_{n=1,N} \sin(k\,x_n\,\sin\theta)= 0$]. We have also employed the trigonometrical identity $\cos(x-y) \equiv \cos x\,\cos y+\sin x\,\sin y$. (See Appendix B.) It follows that the intensity of the interference pattern appearing on the projection screen is specified by

$${\cal I}(\theta) \propto \langle \psi^{\,2}(\theta,t)\rangle \pro... ...um_{n=1,N} \cos\left(2\pi\,\frac{x_n}{\lambda}\,\sin\theta\right)\right]^{\,2},$$ (10.35)

because $\langle\cos^2(\omega\,t-k\,R-\phi)\rangle=1/2$. The previous expression is a generalization of Equation (10.12).

Figure 10.8: Multi-slit far-field interference pattern calculated for $N=10$ and $d/\lambda = 5$ with normal incidence and narrow slits.

Suppose that the slits are evenly spaced a distance $d$ apart, so that

$$x_n = [n-(N+1)/2]\,d$$ (10.36)

for $n=1,N$. It follows that

$${\cal I}(\theta) \propto \left[\sum_{n=1,N} \cos\left(2\pi\,[n-(N+1)/2]\,\frac{d}{\lambda}\,\sin\theta\right)\right]^{\,2},$$ (10.37)

which can be summed to give

$${\cal I}(\theta)\propto \frac{\sin^2[\pi\,N\,(d/\lambda)\,\sin\theta]}{\sin^2[\pi\,(d/\lambda)\,\sin\theta)]}.$$ (10.38)

(See Exercise 1.)

The multi-slit interference function, (10.38), exhibits strong maxima in situations in which its numerator and denominator are simultaneously zero; that is, when

$$\sin\theta=j\,\frac{\lambda}{d},$$ (10.39)

where $j$ is an integer. In this situation, application of l'Hopital's rule yields ${\cal I} = N^{\,2}$. The heights of these so-called principal maxima in the interference function are very large, being proportional to $N^{\,2}$, because there is constructive interference of the light from all $N$ slits. This occurs because the distances between neighboring slits and the point on the projection screen at which a given maximum is located differ by an integer number of wavelengths; that is, $\rho_n-\rho_{n-1}=d\,\sin\theta=j\,\lambda$. All of the principal maxima have the same height.

The multi-slit interference function (10.38) is zero when its numerator is zero, but its denominator non-zero; that is, when

$$\sin\theta=\frac{l}{N}\,\frac{\lambda}{d},$$ (10.40)

where $l$ is an integer that is not an integer multiple of $N$. It follows that there are $N-1$ zeros between neighboring principal maxima. It can also be demonstrated that there are $N-2$ secondary maxima between neighboring principal maxima. However, these maxima are much lower in height, by a factor of order $N^{\,2}$, than the primary maxima.

Figure 10.8 shows the typical far-field interference pattern produced by a system of ten identical, equally spaced, parallel slits, assuming normal incidence and narrow slits, when the slit spacing, $d$, greatly exceeds the wavelength, $\lambda$, of the light (which, as we saw in Section 10.2, is the most interesting case). It can be seen that the pattern consists of a series of bright fringes of equal intensity, separated by much wider (relatively) dark fringes. The bright fringes correspond to the principal maxima discussed previously. As is the case for two-slit interference, the innermost (i.e., low $j$, small $\theta$) principal maxima are approximately equally spaced, with a characteristic angular spacing ${\mit\Delta}\theta\simeq \lambda/d$. [This result follows from Equation (10.39), and the small-angle approximation $\sin\theta\simeq \theta$.] However, the typical angular width of a principal maximum (i.e., the angular distance between the maximum and the closest zeroes on either side of it) is $\delta\theta \simeq (1/N)\,(\lambda/d)$. [This result follows from Equation (10.40), and the small-angle approximation.] The ratio of the angular width of a principal maximum to the angular spacing between successive maxima is thus

$$\frac{\delta\theta}{{\mit\Delta}\theta}\simeq \frac{1}{N}.$$ (10.41)

Hence, we conclude that, as the number of slits increases, the bright fringes in a multi-slit interference pattern become progressively sharper.

The most common practical application of multi-slit interference is the transmission diffraction grating. Such a device consists of $N$ identical, equally spaced, parallel scratches on one side of a thin uniform transparent glass, or plastic, film. When the film is illuminated, the scratches strongly scatter the incident light, and effectively constitute $N$ identical, equally spaced, parallel line sources. Hence, the grating generates the type of $N$-slit interference pattern discussed previously, with one major difference. Namely, the central ($j=0$) principal maximum has contributions not only from the scratches, but also from all the transparent material between the scratches. Thus, the central principal maximum is considerably brighter than the other ($j\neq 0$) principal maxima.

Diffraction gratings are often employed in spectroscopes, which are instruments used to decompose light that is made up of a mixture of different wavelengths into its constituent wavelengths. As a simple example, suppose that a spectroscope contains an $N$-line diffraction grating that is illuminated, at normal incidence, by a mixture of light of wavelength $\lambda$, and light of wavelength $\lambda+{\mit\Delta}\lambda$, where ${\mit\Delta}\lambda\ll \lambda$. As always, the overall interference pattern (i.e., the overall wavefunction at the projection screen) produced by the grating is a linear superposition of the pattern generated by the light of wavelength $\lambda$, and the pattern generated by the light of wavelength $\lambda+{\mit\Delta}\lambda$. Consider the $j$th-order principal maximum associated with the wavelength $\lambda$ interference pattern, which is located at $\theta_j$, where $\sin\theta_j = j\,(\lambda/d)$. [See Equation (10.39).] Here, $d$ is the spacing between neighboring lines on the diffraction grating, which is assumed to be greater than $\lambda$. (Incidentally, the width of the lines is assumed to be much less than $\lambda$.) The maximum in question has a finite angular width. We can determine this width by locating the zeros in the interference pattern on either side of the maximum. Let the zeros be located at $\theta_j\pm \delta\theta_j$. The maximum itself corresponds to $\pi\,N\,(d/\lambda)\,\sin\theta_j = \pi\,N\,j$. Hence, the zeros correspond to $\pi\,N\,(d/\lambda)\,\sin(\theta_j\pm\delta\theta_j)= \pi\,(N\,j\pm 1)$ (i.e., they correspond to the first zeros of the function $\sin[\pi\,N\,(d/\lambda)\,\sin\theta]$ on either side of the zero at $\theta_j$.) [See Equation (10.38).] Taylor expanding to first order in $\delta\theta_j$, we obtain

$$\delta\theta_j = \frac{\tan\theta_j}{N\,j}.$$ (10.42)

Hence, the maximum in question effectively extends from $\theta_j-\delta\theta_j$ to $\theta_j+\delta\theta_j$. Consider the $j$th-order principal maximum associated with the wavelength $\lambda+{\mit\Delta}\lambda$ interference pattern, which is located at $\theta_j+{\mit\Delta}\theta_j$, where $\sin(\theta_j+{\mit\Delta}\theta_j)= j\,(\lambda+{\mit\Delta}\lambda)/d$. [See Equation (10.39).] Taylor expanding to first order in ${\mit\Delta}\theta_j$, we obtain

$${\mit\Delta}\theta_j = \tan\theta_j\,\frac{{\mit\Delta}\lambda}{\lambda}.$$ (10.43)

In order for the spectroscope to resolve the incident light into its two constituent wavelengths, at the $j$th spectral order, the angular spacing, ${\mit\Delta}\theta_j$, between the $j$th-order maxima associated with these two wavelengths must be greater than the angular widths, $\delta\theta_j$, of the maxima themselves. If this is the case then the overall $j$th-order maximum will consist of two closely spaced maxima, or “spectral lines” (centered at $\theta_j$ and $\theta_j+{\mit\Delta}\theta_j$). On the other hand, if this is not the case then the two maxima will merge to form a single maximum, and it will consequently not be possible to tell that the incident light consists of a mixture of two different wavelengths. Thus, the condition for the spectroscope to be able to resolve the spectral lines at the $j$th spectral order is ${\mit\Delta}\theta_j>\delta\theta_j$, or

$$\frac{{\mit\Delta}\lambda}{\lambda} > \frac{1}{N\,j}.$$ (10.44)

We conclude that the resolving power of a diffraction grating spectroscope increases as the number of illuminated lines (i.e., $N$) increases, and also as the spectral order (i.e., $j$) increases. Incidentally, there is no resolving power at the lowest (i.e., $j=0$) spectral order, because the corresponding principal maximum is located at $\theta =0$ irrespective of the wavelength of the incident light. Moreover, there is a limit to how large $j$ can become (i.e., a given diffraction grating, illuminated by light of a given wavelength, has a finite number of principal maxima). This follows because $\sin\theta_j$ cannot exceed unity, so, according to Equation (10.39), $j$ cannot exceed $d/\lambda$.