Coherence

A practical monochromatic light source consists of a collection of similar atoms that are continually excited by collisions, and then spontaneously decay back to their electronic ground states, in the process emitting photons of characteristic angular frequency $\omega={\mit\Delta}{\cal E}/\hbar$, where ${\mit\Delta}{\cal E}$ is the difference in energy between the excited state and the ground state, and $\hbar = 1.055\times 10^{-34}\,{\rm J\,s}^{-1}$ is Planck's constant divided by $2\pi$ (Hecht and Zajac 1974). An excited electronic state of an atom has a characteristic lifetime, $\tau$, which can be calculated from quantum mechanics, and is typically $10^{\,-8}\,{\rm s}$ (ibid.). It follows that when an atom in an excited state decays back to its ground state it emits a burst of electromagnetic radiation of duration $\tau$ and angular frequency $\omega$. However, according to the bandwidth theorem (see Section 8.5), a sinusoidal wave of finite duration $\tau$ has the finite bandwidth

$${\mit\Delta}\omega\sim \frac{2\pi}{\tau}.$$ (10.21)

In other words, if the emitted wave is Fourier transformed in time then it will be found to consist of a linear superposition of sinusoidal waves of infinite duration whose frequencies lie in the approximate range $\omega-{\mit\Delta}\omega/2$ to $\omega+{\mit\Delta}\omega/2$. We conclude that there is no such thing as a truly monochromatic light source. In reality, all such sources have small, but finite, bandwidths that are inversely proportional to the lifetimes, $\tau$, of the associated excited atomic states.

How do we take the finite bandwidth of a practical “monochromatic” light source into account in our analysis? In fact, all we need to do is to assume that the phase angle, $\phi$, appearing in Equations (10.1) and (10.15), is only constant on timescales much less that the lifetime, $\tau$, of the associated excited atomic state, and is subject to abrupt random changes on timescales much greater than $\tau$. We can understand this phenomenon as being due to the fact that the radiation emitted by a single atom has a fixed phase angle, $\phi$, but only lasts a finite time period, $\tau$, combined with the fact that there is generally no correlation between the phase angles of the radiation emitted by different atoms. Alternatively, we can account for the variation in the phase angle in terms of the finite bandwidth of the light source. To be more exact, because the light emitted by the source consists of a superposition of sinusoidal waves of frequencies extending over the range $\omega-{\mit\Delta}\omega/2$ to $\omega+{\mit\Delta}\omega/2$, even if all the component waves start off in phase, the phases will be completely scrambled after a time period $2\pi/{\mit\Delta}\omega = \tau$ has elapsed. In effect, what we are saying is that a practical monochromatic light source is temporally coherent on timescales much less than its characteristic coherence time, $\tau$ (which, for visible light, is typically of order $10^{\,-8}$ seconds), and temporally incoherent on timescales much greater than $\tau$. Incidentally, two waves are said to be coherent if their phase difference is constant in time, and incoherent if their phase difference varies significantly in time. In this case, the two waves in question are the same wave observed at two different times.

What effect does the temporal incoherence of a practical monochromatic light source on timescales greater than $\tau\sim 10^{\,-8}$ seconds have on the two-slit interference patterns discussed in the previous section? Consider the case of oblique incidence. According to Equation (10.16), the phase angles, $\phi_1= (1/2)\,k\,d\,\sin\theta_0+\phi$, and $\phi_2=-(1/2)\,k\,d\,\sin\theta_0+\phi$, of the cylindrical waves emitted by each slit are subject to abrupt random changes on timescales much greater than $\tau$, because the phase angle, $\phi$, of the plane wave that illuminates the two slits is subject to identical changes. Nevertheless, the relative phase angle, $\phi_1-\phi_2= k\,d\,\sin\theta_0$, between the two cylindrical waves remains constant. Moreover, according to Equation (10.17), the interference pattern appearing on the projection screen is produced by the phase difference $(1/2)\,k\,d\,\sin\theta-(1/2)\,(\phi_1-\phi_2)$ between the two cylindrical waves at a given point on the screen, and this phase difference only depends on the relative phase angle. Indeed, the intensity of the interference pattern is ${\cal I}(\theta)\propto \cos^2[(1/2)\,k\,d\,\sin\theta-(1/2)\,(\phi_1-\phi_2)]$. Hence, the fact that the relative phase angle, $\phi_1-\phi_2$, between the two cylindrical waves emitted by the slits remains constant on timescales much longer than the characteristic coherence time, $\tau$, of the light source implies that the interference pattern generated in a conventional two-slit interference apparatus is unaffected by the temporal incoherence of the source. Strictly speaking, however, the preceding conclusion is only accurate when the spatial extent of the light source is negligible. Let us now broaden our discussion to take spatially extended light sources into account.

Figure 10.6: Two-slit interference with two line sources.

Up until now, we have assumed that our two-slit interference apparatus is illuminated by a single plane wave, such as might be generated by a line source located at infinity. Let us now consider a more realistic situation in which the light source is located a finite distance from the slits, and also has a finite spatial extent. Figure 10.6 shows the simplest possible case. Here, the slits are illuminated by two identical line sources, $A$ and $B$, that are a distance $D$ apart, and a perpendicular distance $L$ from the opaque screen containing the slits. Assuming that $L\gg D,\,d$, the light incident on the slits from source $A$ is effectively a plane wave whose direction of propagation subtends an angle $\theta_0/2\simeq D/(2\,L)$ with the $z$-axis. Likewise, the light incident on the slits from source $B$ is a plane wave whose direction of propagation subtends an angle $-\theta_0/2$ with the $z$-axis. Moreover, the net interference pattern (i.e., wavefunction) appearing on the projection screen is the linear superposition of the patterns generated by each source taken individually (because light propagation is ultimately governed by a linear wave equation with superposable solutions; see Section 7.3.). Let us determine whether these patterns reinforce, or interfere with, one another.

The light emitted by source $A$ has a phase angle, $\phi_A$, that is constant on timescales much less than the characteristic coherence time of the source, $\tau$, but is subject to abrupt random changes on timescale much longer than $\tau$. Likewise, the light emitted by source $B$ has a phase angle, $\phi_B$, that is constant on timescales much less than $\tau$, and varies significantly on timescales much greater than $\tau$. In general, there is no correlation between $\phi_A$ and $\phi_B$. In other words, our composite light source, consisting of the two line sources $A$ and $B$, is both temporally and spatially incoherent on timescales much longer than $\tau$.

Again working in the limit $d\gg \lambda$, with $\theta,\,\theta_0\ll 1$, Equation (10.18) yields the following expression for the wavefunction at the projection screen:

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

$$~~~~+ \cos(\omega\,t-k\,R-\phi_B)\,\cos\left[\frac{1}{2}\,k\,d\,(\theta+\theta_0/2)\right].$$ (10.22)

Hence, the intensity of the interference pattern is

$${\cal I}(\theta)\propto \langle \psi^{\,2}(\theta,t)\rangle \propto \langle\,\cos^2 (\omega\,t-k\,R-\phi_A)\rangle\cos^2\left[\frac{1}{2}\,k\,d\,(\theta-\theta_0/2)\right]$$

$$~~~~+2\,\langle \cos (\omega\,t-k\,R-\phi_A)\,\cos (\omega\,t-k\,R-\phi_B)\rangle$$

$$\times \cos\left[\frac{1}{2}\,k\,d\,(\theta-\theta_0/2)\right]\,\cos\left[\frac{1}{2}\,k\,d\,(\theta+\theta_0/2)\right]$$

$$~~~~+\langle\,\cos^2 (\omega\,t-k\,R-\phi_B)\rangle\cos^2\left[\frac{1}{2}\,k\,d\,(\theta+\theta_0/2)\right].$$ (10.23)

However, $\langle \cos^2 (\omega\,t-k\,R-\phi_A)\rangle= \langle \cos^2 (\omega\,t-k\,R-\phi_B)\rangle=1/2$, and $\langle \cos (\omega\,t-k\,R-\phi_A)\,\cos (\omega\,t-k\,R-\phi_B)\rangle=0$, because the phase angles $\phi_A$ and $\phi_B$ are uncorrelated. Hence, the previous expression reduces to

$${\cal I}(\theta) \propto\cos^2\left[\frac{1}{2}\,k\,d\,(\theta-\theta_0/2)\right]+ \cos^2\left[\frac{1}{2}\,k\,d\,(\theta+\theta_0/2)\right]$$

$$= 1 + \cos\left(2\pi\,\frac{d}{\lambda}\,\theta\right)\,\cos\left(\pi\,\frac{d}{\lambda}\,\theta_0\right),$$ (10.24)

where use has been made of the trigonometric identities $\cos^2\theta\equiv (1+\cos 2\theta)/2$, and $\cos x + \cos y \equiv 2\,\cos[(x+y)/2]\,\cos((x-y)/2]$. (See Appendix B.) If $\theta_0=\lambda/(2\,d)$ then $\cos[\pi\,(d/\lambda)\,\theta_0]=0$ and ${\cal I}(\theta)\propto 1$. In this case, the bright fringes of the interference pattern generated by source $A$ exactly overlay the dark fringes of the pattern generated by source $B$, and vice versa, and the net interference pattern is completely washed out. On the other hand, if $\theta_0\ll \lambda/d$ then $\cos[\pi\,(d/\lambda)\,\theta_0]\simeq 1$ and ${\cal I}(\theta)\propto 1+\cos[2\pi\,(d/\lambda)\,\theta]=2\, \cos^2[\pi\,(d/\lambda)\,\theta]$. In this case, the two interference patterns reinforce one another, and the net interference pattern is the same as that generated by a light source of negligible spatial extent.

Suppose that our light source consists of a regularly spaced array of very many identical incoherent line sources, filling the region between the sources $A$ and $B$ in Figure 10.6. In other words, suppose that our light source is a uniform incoherent source of angular extent $\theta_0$. As is readily demonstrated, the associated interference pattern is obtained by averaging expression (10.24) over all $\theta_0$ values in the range 0 to $\theta_0$; that is, by operating on this expression with $\theta_0^{\,-1}\int_0^{\theta_0}\cdots\,d\theta_0$. In this manner, we obtain

$${\cal I}(\theta) \propto 1 + \cos\left(2\pi\,\frac{d}{\lambda}\,\theta\right)\,{\rm sinc}\left(\pi\,\frac{d}{\lambda}\,\theta_0\right),$$ (10.25)

where ${\rm sinc}(x)\equiv \sin x/x$. We can conveniently parameterize the visibility of the interference pattern, appearing on the projection screen, in terms of the quantity

$$V = \frac{{\cal I}_{\rm max} - {\cal I}_{\rm min}}{{\cal I}_{\rm max} + {\cal I}_{\rm min}},$$ (10.26)

where the maximum and minimum values of the intensity are taken with respect to variation in $\theta$ (rather than $\theta_0$). Thus, $V=1$ corresponds to a sharply defined pattern, and $V=0$ to a pattern that is completely washed out. It follows from Equation (10.25) that

$$V = \left\vert{\rm sinc}\left(\pi\,\frac{d}{\lambda}\,\theta_0\right)\right\vert.$$ (10.27)

The predicted visibility, $V$, of a two-slit interference pattern generated by an extended incoherent light source is plotted as a function of the angular extent, $\theta_0$, of the source in Figure 10.7. It can be seen that the pattern is highly visible (i.e., $V\sim 1$) when $\theta_0\ll \lambda/d$, but becomes washed out (i.e., $V\sim 0$) when $\theta_0\gtrsim \lambda/d$.

Figure 10.7: Visibility of a two-slit far-field interference pattern generated by an extended incoherent light source.

We conclude that a spatially extended incoherent light source only generates a visible interference pattern in a conventional two-slit interference apparatus when the angular extent of the source is sufficiently small that

$$\theta_0\ll \frac{\lambda}{d}.$$ (10.28)

Equivalently, if the source is of linear extent $D$, and located a distance $L$ from the slits, then the source only generates a visible interference pattern when it is sufficiently far away from the slits that

$$L\gg \frac{d\,D}{\lambda}.$$ (10.29)

This follows because $\theta_0\simeq D/L$.

The whole of the preceding discussion is premised on the assumption that an extended light source is both temporally and spatially incoherent on timescales much longer than a typical atomic coherence time, which is about $10^{\,-8}$ seconds. This is generally the case. However, there is one type of light source—namely, a laser—for which this is not necessarily the case. In a laser (in single-mode operation), excited atoms are stimulated in such a manner that they emit radiation that is both temporally and spatially coherent on timescales much longer than the relevant atomic coherence time.

Let us consider the two-slit far-field interference pattern generated by an extended coherent light source of angular extent $\theta_0$. In this case, as is readily demonstrated (see Exercise 2), Equation (10.25) is replaced by

$${\cal I}(\theta)\propto \cos^2\left(\pi\,\frac{d}{\lambda}\,\theta\right){\rm sinc}^2\left(\frac{\pi}{2}\,\frac{d}{\lambda}\,\theta_0\right).$$ (10.30)

It follows, from Equation (10.26), that the visibility of the interference pattern is unity; that is, the pattern is sharply defined, irrespective of the angular extent of the light source. (However, the overall brightness of the pattern is considerably reduced when $\theta_0\gtrsim \lambda/d$.) It follows that lasers generally produce much clearer interference patterns than conventional incoherent light sources.