5.3: Interference of Monochromatic Fields of the Same Frequency

Let us first recall the basic concepts of interference. What causes interference is the fact that light is a wave, which means that it not only has an amplitude but also a phase. Suppose for example we evaluate a time-harmonic field in two points $$\mathcal{U}_{1}(t)=\cos (\omega t), \quad \mathcal{U}_{2}(t)=\cos (\omega t+\varphi) . \nonumber$$

Here $\varphi$ denotes the phase difference between the fields at the two points. If $\varphi=0$, or $\varphi$ is a multiple of $2 \pi$, the fields are in phase, and when they are added they interfere constructively $$\mathcal{U}_{1}(t)+\mathcal{U}_{2}(t)=\cos (\omega t)+\cos (\omega t+2 m \pi)=2 \cos (\omega t) . \nonumber$$

However, when $\varphi=\pi$, or more generally $\varphi=\pi+2 m \pi$, for some integer $m$, then the waves are out of phase, and when they are superimposed, they interfere destructively.

$$\begin{aligned} \mathcal{U}_{1}(t)+\mathcal{U}_{2}(t) &=\cos (\omega t)+\cos (\omega t+\pi+2 m \pi) \\ &=\cos (\omega t)-\cos (\omega t) \\ &=0 . \end{aligned} \nonumber$$

We can sum the two fields for arbitrary $\varphi$ more conveniently using complex notation: $$\mathcal{U}_{1}(t)=\operatorname{Re}\left[e^{-i \omega t}\right], \quad \mathcal{U}_{2}(t)=\operatorname{Re}\left[e^{-i \omega t} e^{-i \varphi}\right] . \nonumber$$

Adding gives $$\begin{aligned} \mathcal{U}_{1}(t)+\mathcal{U}_{2}(t) &=\operatorname{Re}\left[e^{-i \omega t}\left(1+e^{-i \varphi}\right)\right] \\ &=\operatorname{Re}\left[e^{-i \omega t} e^{-i \varphi / 2}\left(e^{i \varphi / 2}+e^{-i \varphi / 2}\right)\right] \\ &=\operatorname{Re}\left[e^{-i \omega t} e^{-i \varphi / 2} 2 \cos (\varphi / 2)\right] \\ &=2 \cos (\varphi / 2) \cos (\omega t+\varphi / 2) \end{aligned} \nonumber$$ For $\varphi=2 m \pi$ and $\varphi=\pi+2 m \pi$ we retrieve the results obtained before. It is important to note that what we see or detect physically (say, the ’brightness’ of light) does not correspond to the quantities $\mathcal{U}_{1}, \mathcal{U}_{2}$. After all, $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ can attain negative values, while there is no such thing as ’negative brightness’. What $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ describe are the fields, which may be positive or negative. The ’brightness’ or the irradiance or intensity is given by taking an average over a long time of $\mathcal{U}(t)^{2}$ (see (1.7.8), we shall omit the factor $\sqrt{\epsilon / \mu_{0}}$ ). As explained in Chapter 2 , we see and measure only the long time-average of $\mathcal{U}(t)^{2}$, because at optical frequencies $\mathcal{U}(t)^{2}$

fluctuates very rapidly. We recall the definition of the time average over an interval of length $T$ at a specific time $t$ given in (1.8.4) in Chapter 2: $$\langle f(t)\rangle=\frac{1}{T} \int_{t}^{t+T} f\left(t^{\prime}\right) \mathrm{d} t^{\prime}, \nonumber$$ where $T$ is a time interval that is the response time of a typical detector, which is $10^{-9} \mathrm{~s}$ for a very fast detector, but this is still extremely long compared to the period of visible light which is of the order of $10^{-14} \mathrm{~s}$. For a time-harmonic function, the long-time average is equal to the average over one period of the field and hence it is independent of the time $t$ at which it is taken. Indeed for ($\PageIndex{5}$) we get $$\begin{aligned} I &=\left\langle\left(\mathcal{U}_{1}(t)+\mathcal{U}_{2}(t)\right)^{2}\right\rangle \\ &=4 \cos ^{2}(\varphi / 2)\left\langle\cos ^{2}(\omega t+\varphi / 2)\right\rangle \\ &=1+\cos (\varphi) \end{aligned} \nonumber$$ where $T \omega>>1$. It is important to note that one can use complex notation to obtain the factor $1+\cos (\varphi)$ more easily. Let us write $$\mathcal{U}_{1}(t)=\operatorname{Re}\left\{U_{1} e^{-i \omega t}\right\}, \quad \mathcal{U}_{2}(t)=\operatorname{Re}\left\{U_{2} e^{-i \omega t}\right\}, \nonumber$$ where $$U_{1}=1, \quad U_{2}=e^{-i \varphi} . \nonumber$$

Then we find $$\begin{aligned} \left|U_{1}+U_{2}\right|^{2} &=\left|1+e^{-i \varphi}\right|^{2} \\ &=\left(1+e^{i \varphi}\right)\left(1+e^{-i \varphi}\right) \\ &=1+1+e^{-i \varphi}+e^{i \varphi} \\ &=2+2 \cos (\varphi), \end{aligned} \nonumber$$ hence $$I=\frac{1}{2}\left|U_{1}+U_{2}\right|^{2} . \nonumber$$ To see why this works, recall Eq. ( 1.8.5) and choose $A=B=U_{1}+U_{2}$.

Remark. To shorten the formulae, we will omit in this chapter the factor $1 / 2$ in front of the time-averaged intensity.

Hence we define $I_{1}=\left|U_{1}\right|^{2}$ and $I_{2}=\left|U_{2}\right|^{2}$, and we then find for the time-averaged intensity of the sum of $U_{1}$ and $U_{2}$ : $$\begin{aligned} I &=\left|U_{1}+U_{2}\right|^{2}=\left(U_{1}+U_{2}\right)^{*}\left(U_{1}+U_{2}\right) \\ &=\left|U_{1}\right|^{2}+\left|U_{2}\right|^{2}+U_{1}^{*} U_{2}+U_{1} U_{2}^{*} \\ &=I_{1}+I_{2}+2 \operatorname{Re}\left[U_{1}^{*} U_{2}\right]. \end{aligned} \nonumber$$

Here, $2 \operatorname{Re}\left[U_{1}^{*} U_{2}\right]$ is known as the interference term. In the famous double-slit experiment (which we will discuss in a later section), we can interpret the terms as follows: let us say $U_{1}$ is the field that comes from slit 1 , and $U_{2}$ comes from slit 2 . If only slit 1 is open, we measure on the screen intensity $I_{1}$, and if only slit 2 is open, we measure $I_{2}$. If both slits are open, we would not measure $I_{1}+I_{2}$, but we would observe fringes due to the interference term $2 \operatorname{Re}\left[U_{1}^{*} U_{2}\right]$.

More generally, the intensity of a sum of multiple time-harmonic fields $U_{j}$ all having the same frequency is given by the coherent sum $$I=\left|\sum_{j} U_{j}\right|^{2} \nonumber$$

However, we will see in the next section that sometimes the fields are unable to interfere. In that case all the interference terms of the coherent sum vanish, and the intensity is given by the incoherent sum $$I=\sum_{j}\left|U_{j}\right|^{2} . \nonumber$$