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PHY.K02UF Molecular and Solid State Physics

Intensity of the scattered waves

When plane waves scatter off two point scatterers in two dimensions, the amplitude of the scattered waves is,

$$F_1\frac{\cos (k|\vec{r}-\vec{r}_1|-\omega t +kx_1)}{\sqrt{|\vec{r}-\vec{r}_1|}}+F_2\frac{\cos (k|\vec{r}-\vec{r}_2|-\omega t +kx_2)}{\sqrt{|\vec{r}-\vec{r}_2|}}.$$

Where $|\vec{r}-\vec{r}_1|$ is the distance from source 1 and $|\vec{r}-\vec{r}_2|$ is the distance from source 2. The wavenumber $k$ is related to the wavelength of the incident plane waves $k=\frac{2\pi}{\lambda}$, and the angular frequency $\omega$ is related to the period $T$ of the oscillations, $\omega=\frac{2\pi}{T}$. This expression is simply the sum of the waves radiated by the two scatterers.

If you focus on any one point in the interference pattern, it executes simple harmonic motion at angular frequency $\omega$. To calculate the intensity of the scattered waves, it is convenient to use a complex representation of harmonic motion. This is illustrated in the simulation below.

 Your browser does not support the canvas element. $$$\large e^{i\omega t} = \cos\omega t + i \sin\omega t$$$

The red ball represents the position of the complex number $e^{i\omega t}$ as it moves through the complex plane with real numbers plotted horizontally and imaginary number plotted vertically. The black circle is the unit circle. The blue ball represents the position of $\cos \omega t$ and the green ball represents the position of $i\sin \omega t$. The simulation on the left is a graphical representation of Euler's formula on the right.

Oscillations that can be described by $\sin \omega t$ or $\cos \omega t$ are called harmonic oscillations. From the simulation it is clear that there is a relationship between circular motion and harmonic oscillations. If you look at the motion of the red ball from above, it moves in a circle but if you look at the motion of the red ball from the side, it executes harmonic motion.

The relationship between circular motion and harmonic oscillations is described easily using complex numbers. Sometimes when we observe a harmonic oscillation it is convenient to imagine that we are looking at circular motion from the side. We can't measure the component of the motion in the imaginary direction, we just imagine it.

Written in complex form, the amplitude of the scattered waves is,

$$\frac{F_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1|-\omega t +kx_1)}+\frac{F_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| -\omega t+kx_2)}.$$

This expression can be factored into a time-independent part $\mathcal{A}(\vec{r})$ and a time-dependent factor $e^{-i\omega t}$,

$$\left( \frac{F_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1| +kx_1)}+\frac{F_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| +kx_2)} \right) e^{-i\omega t} = \mathcal{A}(\vec{r})e^{-i\omega t}.$$

The time-independent part is a complex number and the magnitude of this complex number $|\mathcal{A}(\vec{r})|$ is the amplitude of the oscillations at point $\vec{r}$. The intensity $I$ is proportional to the amplitude squared,

$$I\propto \mathcal{A}^*\mathcal{A}.$$