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# Multi-Slit Interference

Suppose that the interference apparatus pictured in Figure 62 is modified such that identical slits of width , running parallel to the -axis, are cut in the opaque screen that occupies the plane . Let the slits be located at , for . For the sake of simplicity, the arrangement of slits is assumed to be symmetric with respect to the plane . In other words, if there is a slit at then there is also a slit at . The distance between the th slit and a point on the projection screen that is an angular distance from the plane is [cf., Equation (995)]

 (1023)

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

 (1024)

which can also be written

 (1025)

or

 (1026)

Here, we have made use of the fact that arrangement of slits is symmetric with respect to the plane [which implies that ]. We have also employed the trigonometrical identity . (See Appendix B.) It follows that the intensity of the interference pattern appearing on the projection screen is specified by

 (1027)

because . The previous expression is a generalization of Equation (1004).

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

 (1028)

for . It follows that

 (1029)

which can be summed to give

 (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

 (1031)

where is an integer. In this situation, application of l'Hopital's rule yields . The heights of these so-called principal maxima in the interference function are very large, being proportional to , because there is constructive interference of the light from all 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, . 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

 (1032)

where is an integer that is not an integer multiple of . It follows that there are zeros between neighboring principal maxima. It can also be demonstrated that there are secondary maxima between neighboring principal maxima. However, these maxima are much lower in height, by a factor of order , 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, , greatly exceeds the wavelength, , 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 , small ) principal maxima are approximately equally spaced, with a characteristic angular spacing . [This result follows from Equation (1031), and the small angle approximation .] 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 . [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

 (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 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 identical, equally spaced, parallel line sources. Hence, the grating generates the type of -slit interference pattern discussed previously, with one major difference. Namely, the central ( ) 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 ( ) 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 -line diffraction grating that is illuminated, at normal incidence, by a mixture of light of wavelength , and light of wavelength , where . 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 , and the pattern generated by the light of wavelength . Consider the th-order principal maximum associated with the wavelength interference pattern, which is located at , where . [See Equation (1031).] Here, is the spacing between neighboring lines on the diffraction grating, which is assumed to be greater than . (Incidentally, the width of the lines is assumed to be much less than .) 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 . The maximum itself corresponds to . Hence, the zeros correspond to (i.e., they correspond to the first zeros of the function on either side of the zero at .) [See Equation (1030).] Taylor expanding to first order in , we obtain

 (1034)

Hence, the maximum in question effectively extends from to . Consider the th-order principal maximum associated with the wavelength interference pattern, which is located at , where . [See Equation (1031).] Taylor expanding to first order in , we obtain

 (1035)

In order for the spectroscope to resolve the incident light into its two constituent wavelengths, at the th spectral order, the angular spacing, , between the th-order maxima associated with these two wavelengths must be greater than the angular widths, , of the maxima themselves. If this is the case then the overall th-order maximum will consist of two closely spaced maxima, or spectral lines'' (centered at and ). 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 th spectral order is , or

 (1036)

We conclude that the resolving power of a diffraction grating spectroscope increases as the number of illuminated lines (i.e., ) increases, and also as the spectral order (i.e., ) increases. Incidentally, there is no resolving power at the lowest (i.e., ) spectral order, because the corresponding principal maximum is located at irrespective of the wavelength of the incident light. Moreover, there is a limit to how large 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 cannot exceed unity, so, according to Equation (1031), cannot exceed .

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