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One-Dimensional Fourier Optics

We have already considered the interference of monochromatic light produced when a plane wave is incident on an opaque screen, coincident with the plane $ z=0$ , in which a number of narrow (i.e., $ \delta\ll \lambda$ , where $ \delta$ is the slit width) slits, running parallel to the $ y$ -axis, have been cut. Let us now generalize our analysis to take slits of finite width (i.e., $ \delta\gtrsim \lambda$ ) into account. In order to achieve this goal, it is convenient to define the so-called aperture function, $ F(x)$ , of the screen. This function takes the value zero if the screen is opaque at position $ x$ , and some constant positive value if it is transparent, and is normalized such that $ \int_{-\infty}^\infty F(x) dx = 1$ . Thus, for the case of a screen with $ N$ identical slits of negligible width, located at $ x=x_n$ , for $ n=1,N$ , the appropriate aperture function is

$\displaystyle F(x)=\frac{1}{N}\sum_{n=1,N} \delta(x-x_n),$ (1041)

where $ \delta(x)$ is a Dirac delta function.

The wavefunction at the projection screen, generated by the previously mentioned arrangement of slits, when the opaque screen is illuminated by a plane wave of phase angle $ \phi$ , wavenumber $ k$ , and angular frequency $ \omega $ , whose direction of propagation subtends an angle $ \theta_0$ with the $ z$ -axis, is (see the analysis in Sections 11.2 and 11.4)

$\displaystyle \psi(\theta,t)\propto \cos(\omega t-k R-\phi)\sum_{n=1,N} \cos[k x_n (\sin\theta-\sin\theta_0)].$ (1042)

Here, for the sake of simplicity, we have assumed that the arrangement of slits is symmetric with respect to the plane $ x=0$ , so that $ \sum_{n=1,N} \sin(\alpha x_n) =0$ for any $ \alpha$ . Using the well-known properties of the delta function [see Equation (723)], Equation (1042) can also be written

$\displaystyle \psi(\theta,t) \propto \cos(\omega\,t-k\,R-\phi)\,\bar{F}(\theta),$ (1043)

where

$\displaystyle \bar{F}(\theta) = \int_{-\infty}^\infty F(x) \cos[k (\sin\theta-\sin\theta_0) x] dx.$ (1044)

In the following, we shall assume that Equation (1043) is a general result, and is valid even when the slits in the opaque screen are of finite width (i.e., $ \delta\gtrsim \lambda$ ). This assumption is equivalent to the assumption that each unblocked section of the screen emits a cylindrical wave in the forward direction that is in phase with the plane wave which illuminates it from behind. The latter assumption is known as Huygen's principle. [Huygen's principle can be justified using advanced electromagnetic theory (Jackson 1975), but such a proof lies well beyond the scope of this book.] The interference/diffraction function, $ \bar{F}(\theta)$ , is the Fourier transform of the aperture function, $ F(x)$ . This is an extremely powerful result. It implies that we can calculate the far-field interference/diffraction pattern associated with any arrangement of parallel slits, of arbitrary width, by Fourier transforming the associated aperture function. Once we have determined the interference/diffraction function, $ \bar{F}(\theta)$ , the intensity of the interference/diffraction pattern appearing on the projection screen is readily obtained from

$\displaystyle {\cal I}(\theta) \propto \left[\bar{F}(\theta)\right]^{\,2}.$ (1045)


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