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Multi-Slit Interference

Suppose that the interference apparatus pictured in Figure 62 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 (995)]

$\displaystyle \rho_n = R\left[1-\frac{x_n}{R}\,\sin\theta + {\cal O}\left(\frac{x_n^{\,2}}{R^{\,2}}\right)\right].$ (1023)

Thus, making use of the far-field orderings (1002), where $ d$ now represents the typical spacing between neighboring slits, and assuming normally incident collimated light, Equation (997) generalizes to

$\displaystyle \psi(\theta,t)\propto \sum_{n=1,N}\cos(\omega t - k R -\phi+k x_n \sin\theta),$ (1024)

which can also be written

$\displaystyle \psi(\theta,t)$ $\displaystyle \propto \cos(\omega t-k R-\phi)\sum_{n=1,N} \cos(k x_n \sin\theta)$    
  $\displaystyle    - \sin(\omega t-k R-\phi)\sum_{n=1,N}\sin(k x_n \sin\theta),$ (1025)

or

$\displaystyle \psi(\theta,t)\propto \cos(\omega t-k R-\phi)\sum_{n=1,N} \cos(k x_n \sin\theta).$ (1026)

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

$\displaystyle {\cal I}(\theta) \propto \langle \psi^2(\theta,t)\rangle \propto ...
...um_{n=1,N} \cos\left(2\pi \frac{x_n}{\lambda} \sin\theta\right)\right]^{ 2},$ (1027)

because $ \langle\cos^2(\omega\,t-k\,R-\phi)\rangle=1/2$ . The previous expression is a generalization of Equation (1004).

Figure 69: Multi-slit far-field interference pattern calculated for $ N=10$ and $ d/\lambda = 5$ with normal incidence and narrow slits.
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Suppose that the slits are evenly spaced a distance $ d$ apart, so that

$\displaystyle x_n = [n-(N+1)/2]\,d$ (1028)

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

$\displaystyle {\cal I}(\theta) \propto \left[\sum_{n=1,N} \cos\left(2\pi [n-(N+1)/2] \frac{d}{\lambda} \sin\theta\right)\right]^{ 2},$ (1029)

which can be summed to give

$\displaystyle {\cal I}(\theta)\propto \frac{\sin^2[\pi N (d/\lambda) \sin\theta]}{\sin^2[\pi (d/\lambda) \sin\theta)]}.$ (1030)

(See Exercise 1.)

The multi-slit interference function, (1030), exhibits strong maxima in situations in which its numerator and denominator are simultaneously zero: that is, when

$\displaystyle \sin\theta=j\,\frac{\lambda}{d},$ (1031)

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 (1030) is zero when its numerator is zero, but its denominator non-zero: that is, when

$\displaystyle \sin\theta=\frac{l}{N}\,\frac{\lambda}{d},$ (1032)

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 69 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 11.2, 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 $ \Delta\theta\simeq \lambda/d$ . [This result follows from Equation (1031), 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 (1032), and the small angle approximation.] The ratio of the angular width of a principal maximum to the angular spacing between successive maxima is thus

$\displaystyle \frac{\delta\theta}{\Delta\theta}\simeq \frac{1}{N}.$ (1033)

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+\Delta\lambda$ , where $ \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+\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 (1031).] 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 (1030).] Taylor expanding to first order in $ \delta\theta_j$ , we obtain

$\displaystyle \delta\theta_j = \frac{\tan\theta_j}{N j}.$ (1034)

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+\Delta\lambda$ interference pattern, which is located at $ \theta_j+\Delta\theta_j$ , where $ \sin(\theta_j+\Delta\theta_j)= j (\lambda+\Delta\lambda)/d$ . [See Equation (1031).] Taylor expanding to first order in $ \Delta\theta_j$ , we obtain

$\displaystyle \Delta\theta_j = \tan\theta_j\,\frac{\Delta\lambda}{\lambda}.$ (1035)

In order for the spectroscope to resolve the incident light into its two constituent wavelengths, at the $ j$ th spectral order, the angular spacing, $ \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+\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 $ \Delta\theta_j>\delta\theta_j$ , or

$\displaystyle \frac{\Delta\lambda}{\lambda} > \frac{1}{N\,j}.$ (1036)

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 (1031), $ j$ cannot exceed $ d/\lambda$ .


next up previous
Next: Thin Film Interference Up: Wave Optics Previous: Coherence
Richard Fitzpatrick 2013-04-08