Two-Slit Interference

Consider a monochromatic plane light wave, propagating in the $z$-direction, through a transparent dielectric medium of refractive index unity (e.g., a vacuum). (Such a wave could be produced by a uniform line source, running parallel to the $y$-axis (say), that is located at $z=-\infty$.) Let the associated wavefunction take the form

$$\psi(z,t) = \psi_0\,\cos(\omega\,t-k\,z-\phi).$$ (10.1)

Here, $\psi(z,t)$ represents the electric component of the wave, $\psi_0>0$ the wave amplitude, $\phi$ the phase angle, $k>0$ the wavenumber, $\omega=k\,c$ the angular frequency, and $c$ the velocity of light in vacuum. Let the wave be normally incident on an opaque screen that is coincident with the plane $z=0$. See Figure 10.1. Suppose that there are two identical slits of width $\delta$ cut in the screen. Let the slits run parallel to the $y$-axis, and be located at $x=d/2$ and $x=-d/2$, where $d>\delta$ is the slit spacing. Suppose that the light that passes through the two slits travels to a cylindrical projection screen of radius $R$ whose axis coincides with the line $x=z=0$. In the following, it is assumed that there is no variation of wave quantities in the $y$-direction.

Figure 10.1: Two-slit interference at normal incidence.

Provided the two slits are much narrower than the wavelength, $\lambda=2\pi/k$, of the light (i.e., $\delta\ll \lambda$), we expect any radiation that passes through them to be strongly diffracted. (See Section 10.7.) Diffraction is a fundamental wave phenomenon that causes waves to bend around small (compared to the wavelength) obstacles, and spread out from narrow (compared to the wavelength) openings, while maintaining the same wavelength and frequency. The laws of geometric optics do not take diffraction into account, and are, therefore, restricted to situations in which light interacts with objects whose physical dimensions greatly exceed its wavelength. The assumption of strong diffraction suggests that each slit acts like a uniform line source that emits light isotropically in the forward direction (i.e., toward the region $z>0$), but does not emit light in the backward direction (i.e., toward the region $z<0$). It is possible to demonstrate that this is, in fact, the case (with certain provisos; see Section 10.10), using electromagnetic theory (Jackson 1975), but such a demonstration lies beyond the scope of this course. As discussed in Section 7.4, we would expect a uniform line source to emit a cylindrical wave. It follows that each slit emits a half-cylindrical light wave in the forward direction. See Figure 10.1. Moreover, these waves are emitted with equal amplitude and phase, because the incident plane wave that illuminates the slits has the same amplitude (i.e., $\psi_0$) and phase (i.e., $\omega\,t-\phi$) at both slits, and the slits are identical. Finally, we expect the cylindrical waves emitted by the two slits to interfere with one another (see Section 6.4) in such a manner as to generate a characteristic pattern on the projection screen. Let us determine the nature of this pattern.

Consider the wave amplitude at a point on the projection screen that lies an angular distance $\theta$ from the plane $x=0$. See Figure 10.1. The wavefunction at this particular point is written

$$\psi(\theta,t)\propto\frac{ \cos(\omega\,t-k\,\rho_1-\phi)}{\rho_... ...ho_2-\phi)}{\rho_2^{\,1/2}} + {\cal O}\left(\frac{1}{k\,\rho_2^{\,3/2}}\right),$$ (10.2)

assuming that $k\,\rho_1,\,k\,\rho_2\gg 1$. In other words, the overall wavefunction in the region $z>0$ is the superposition of cylindrical waves [see Equation (7.11)] of equal amplitude (i.e., $\rho^{-1/2}$) and phase (i.e., $\omega\,t-k\,\rho-\phi$) emanating from each slit. Here, $\rho=(x^{\,2}+z^{\,2})^{1/2}$. Moreover, $\rho_1$ and $\rho_2$ are the distances that the waves emitted by the first and second slits (located at $x=d/2$ and $x=-d/2$, respectively) have travelled by the time they reach the point on the projection screen in question.

Standard trigonometry (i.e., the law of cosines) reveals that

$$\rho_1 = R\left(1-\frac{d}{R}\,\sin\theta + \frac{1}{4}\,\frac{d^... ...,\frac{d}{R}\,\sin\theta + {\cal O}\left(\frac{d^{\,2}}{R^{\,2}}\right)\right].$$ (10.3)

Likewise,

$$\rho_2 = R\left[1+\frac{1}{2}\,\frac{d}{R}\,\sin\theta + {\cal O}\!\left(\frac{d^{\,2}}{R^{\,2}}\right)\right].$$ (10.4)

Hence, Equation (10.2) yields

$$\psi(\theta,t)\propto\cos(\omega\,t-k\,\rho_1-\phi)+ \cos(\omega\... ...2-\phi)+ {\cal O}\left(\frac{1}{k\,R}\right)+ {\cal O}\left(\frac{d}{R}\right),$$ (10.5)

which, making use of the trigonometric identity $\cos x + \cos y\equiv 2\,\cos[(x+y)/2]\,\cos[(x-y)/2]$ (see Appendix B), gives

$$\psi(\theta,t)\propto\cos\left[\omega\,t-\frac{1}{2}\,k\,(\rho_1+... ...\right]+ {\cal O}\left(\frac{1}{k\,R}\right)+ {\cal O}\left(\frac{d}{R}\right),$$ (10.6)

or

$$\psi(\theta,t)\propto\cos\left[\omega\,t-k\,R-\phi+ {\cal O}\left... ...\right]+ {\cal O}\left(\frac{1}{k\,R}\right)+ {\cal O}\left(\frac{d}{R}\right).$$ (10.7)

Finally, assuming that

$$\frac{k\,d^{\,2}}{R},\,\frac{1}{k\,R},\, \frac{d}{R}\ll 1,$$ (10.8)

the previous expression reduces to

$$\psi(\theta,t)\propto \cos(\omega\,t-k\,R-\phi)\,\cos\left(\frac{1}{2}\,k\,d\,\sin\theta\right).$$ (10.9)

Figure 10.2: Two-slit far-field interference pattern calculated for $d/\lambda = 5$ with normal incidence and narrow slits.

The orderings (10.8), which can also be written in the form

$$R\gg d,\,\lambda,\,\frac{d^{\,2}}{\lambda},$$ (10.10)

are satisfied provided the projection screen is located sufficiently far away from the slits. Consequently, the type of interference described in this section is known as far-field interference. One characteristic feature of far-field interference is that the amplitudes of the cylindrical waves emitted by the two slits are approximately equal to one another when they reach a given point on the projection screen (i.e., $\vert\rho_1-\rho_2\vert/\rho_1\ll 1$), whereas the phases are, in general, significantly different (i.e., $k\,\vert\rho_1-\rho_2\vert\gtrsim \pi$). In other words, the interference pattern generated on the projection screen is entirely a consequence of the phase difference between the cylindrical waves emitted by the two slits when they reach the screen. This phase difference is produced by the slight difference in the distance between the slits and a given point on the projection screen. (The phase difference becomes significant as soon as the path difference becomes comparable with the wavelength of the light.)

Figure 10.3: Two-slit far-field interference pattern calculated for $d/\lambda = 1$ with normal incidence and narrow slits.

The mean energy flux, or intensity, of the light striking the projection screen at angular position $\theta$ is

$${\cal I}(\theta) \propto \langle \psi^{\,2}(\theta,t)\rangle \pro... ...k\,d\,\sin\theta\right)\propto\cos^2\left(\frac{1}{2}\,k\,d\,\sin\theta\right),$$ (10.11)

where $\langle\cdots\rangle$ denotes an average over a wave period. [The previous expression follows from the standard result ${\cal I} \propto E^{\,2}/Z_0$, for an electromagnetic wave, where $E$ is the electric component of the wave, and $Z_0$ the impedance of free space. (See Section 6.8.) Recall, also, that $\psi\propto E$.] Here, use has been made of the easily established result $\langle\cos^2(\omega\,t-k\,R-\phi)\rangle=1/2$. The very high oscillation frequency of light waves (i.e., $f\sim 10^{14}\,{\rm Hz}$) ensures that experiments typically detect (e.g., by means of a photographic film, or a photo-multiplier tube) the intensity of light, rather than the rapidly oscillating amplitude of its electric component. For the case of two-slit, far-field interference, assuming normal incidence and narrow slits, the intensity of the characteristic interference pattern appearing on the projection screen is specified by

$${\cal I}(\theta) \propto \cos^2\left(\pi\,\frac{d}{\lambda}\,\sin\theta\right).$$ (10.12)

Figure 10.4: Two-slit far-field interference pattern calculated for $d/\lambda = 0.1$ with normal incidence and narrow slits.

Figure 10.2 shows the intensity of the typical two-slit, far-field interference pattern produced when the slit spacing, $d$, greatly exceeds the wavelength, $\lambda$, of the light. It can be seen that the pattern consists of multiple bright and dark fringes. A bright fringe is generated whenever the cylindrical waves emitted by the two slits interfere constructively at a given point on the projection screen. This occurs when the distances between the two slits and the point in question differ by an integer number of wavelengths; that is,

$$\rho_2-\rho_1 = d\,\sin\theta = j\,\lambda,$$ (10.13)

where $j$ is an integer. (This ensures that the effective phase difference of the two waves is zero.) Likewise, a dark fringe is generated whenever the cylindrical waves emitted by the two slits interfere destructively at a given point on the projection screen. This occurs when the distances between the two slits and the point in question differ by a half-integer number of wavelengths; that is,

$$\rho_2-\rho_1=d\,\sin\theta= (j+1/2)\,\lambda.$$ (10.14)

(This ensures that the effective phase difference between the two waves is $\pi$ radians.) We conclude that the innermost (i.e., low $j$, small $\theta$) bright fringes are approximately equally spaced, with a characteristic angular width ${\mit\Delta}\theta\simeq \lambda/d$. This result, which follows from Equation (10.13), and the small-angle approximation $\sin\theta\simeq \theta$, can be used experimentally to determine the wavelength of a monochromatic light source from a two-slit interference apparatus. (See Exercise 5.)

Figure 10.3 shows the intensity of the interference pattern generated when the slit spacing is equal to the wavelength of the light. It can be seen that the width of the central (i.e., $j=0$, $\theta =0$) bright fringe has expanded to such an extent that the fringe occupies almost half of the projection screen, leaving room for just two dark fringes on either side of it.

Finally, Figure 10.4 shows the intensity of the interference pattern generated when the slit spacing is much less than the wavelength of the light. It can be seen that the width of the central bright fringe has expanded to such an extent that the band occupies the whole projection screen, and there are no dark fringes. Indeed, ${\cal I}(\theta)$ becomes constant in the limit that $d/\lambda\ll 1$, in which case the interference pattern entirely disappears.

Figures 10.2–10.3 imply that the two-slit, far-field interference apparatus shown in Figure 10.1 only generates an interesting interference pattern when the slit spacing, $d$, is greater than the wavelength, $\lambda$, of the light.

Figure 10.5: Two-slit interference at oblique incidence.

Suppose that the plane wave that illuminates the interference apparatus is not normally incident on the slits, but instead propagates at an angle $\theta_0$ to the $z$-axis, as shown in Figure 10.5. In this case, the incident wavefunction (10.1) becomes

$$\psi(x,z,t) = \psi_0\,\cos(\omega\,t-k\,x\,\sin\theta_0- k\,z\,\cos\theta_0-\phi).$$ (10.15)

Thus, the phase of the light incident on the first slit (located at $x=d/2$, $z=0$) is $\omega\,t- (1/2)\,k\,d\,\sin\theta_0-\phi$, whereas the phase of the light incident on the second slit (located at $x=-d/2$, $z=0$) is $\omega\,t+ (1/2)\,k\,d\,\sin\theta_0-\phi$. Assuming that the cylindrical waves emitted by each slit have the same phase (at the slits) as the plane wave that illuminates them, Equation (10.2) generalizes to

$$\psi(\theta,t)\propto\frac{\cos(\omega\,t-k\,\rho_1-\phi_1)}{\rho_1^{\,1/2}} +\frac{\cos(\omega\,t-k\,\rho_2-\phi_2)}{\rho_2^{\,1/2}},$$ (10.16)

where $\phi_1= (1/2)\,k\,d\,\sin\theta_0+\phi$ and $\phi_2=-(1/2)\,k\,d\,\sin\theta_0+\phi$. Hence, making use of the far-field orderings (10.10), and a standard trigonometric identity, we obtain

$$\psi(\theta,t) \propto \cos\left[\omega\,t-k\,R-\frac{1}{2}\,(\phi_1+\phi_2)\right]\cos \left[\frac{1}{2}\,k\,d\,\sin\theta-\frac{1}{2}\,(\phi_1-\phi_2)\right]$$

$$\propto \cos(\omega\,t-k\,R-\phi)\,\cos \left[\frac{1}{2}\,k\,d\,(\sin\theta-\sin\theta_0)\right].$$ (10.17)

For the sake of simplicity, let us consider the limit $d\gg \lambda$, in which the innermost (i.e., low $j$) interference fringes are located at small $\theta$. (The projection screen is approximately planar in this limit, as indicated in Figure 10.5, because a sufficiently small section of a cylindrical surface looks like a plane.) Assuming that $\theta_0$ is also small, the previous expression reduces to

$$\psi(\theta,t)\propto \cos(\omega-k\,R-\phi)\,\cos \left[\frac{1}{2}\,k\,d\,(\theta-\theta_0)\right],$$ (10.18)

and Equation (10.12) becomes

$${\cal I}(\theta) \propto\cos^2\left[\pi\,\frac{d}{\lambda}\,(\theta-\theta_0)\right].$$ (10.19)

Thus, the bright fringes in the interference pattern are located at

$$\theta = \theta_0 + j\,\frac{\lambda}{d},$$ (10.20)

where $j$ is an integer. We conclude that if the slits in a two-slit interference apparatus, such as that shown in Figure 10.5, are illuminated by an obliquely incident plane wave then the consequent phase difference between the cylindrical waves emitted by each slit produces an angular shift in the interference pattern appearing on the projection screen. To be more exact, the angular shift is equal to the angle of incidence, $\theta_0$, of the plane wave, so that the central ($j=0$) bright fringe in the interference pattern is located at $\theta=\theta_0$. See Figure 10.5. This is equivalent to saying that the position of the central bright fringe can be determined via the rules of geometric optics. (This conclusion holds even when $\theta_0$ is not small.)