Angular Momentum and the Photon Picture

In the photon picture of light we envision photons as carrying energy and momentum. We have already discussed how photons can have linear momentum in the amount of $LaTeX: \hbar$ $\hbar$ k. The linear momentum is proportional to the magnitude of the k-vector which in tern depends on the inverse of the wavelength.

We can also expect light to carry angular momentum, however the reasoning is a bit different than how we derived the linear momentum. The power delivered to a system from a stream of photons is $LaTeX: \text{$\Delta \mathcal{E}$/$\Delta $t}$ $\text{$\Delta \mathcal{E}$/$\Delta $t}$ and the power generated by a torque $LaTeX: \Gamma$ $\Gamma$ acting on the system is $LaTeX: \omega \Gamma$ $\omega \Gamma$ . When only angular momentum is present we have

$LaTeX: \frac{\Delta \mathcal{E}}{\text{$\Delta $t}}=\omega \Gamma$ $\frac{\Delta \mathcal{E}}{\text{$\Delta $t}}=\omega \Gamma$

the torque is the time rate of change of angular momentum so that

$LaTeX: \frac{\Delta \mathcal{E}}{\text{$\Delta $t}}=\omega \frac{\text{$\Delta $L}}{\text{$\Delta $t}}$ $\frac{\Delta \mathcal{E}}{\text{$\Delta $t}}=\omega \frac{\text{$\Delta $L}}{\text{$\Delta $t}}$

over the same time interval

$LaTeX: \Delta \mathcal{E}=\omega \text{$\Delta $L}$ $\Delta \mathcal{E}=\omega \text{$\Delta $L}$

$LaTeX: \text{$\Delta $L}=\frac{\Delta \mathcal{E}}{\omega }$ $\text{$\Delta $L}=\frac{\Delta \mathcal{E}}{\omega }$

for an atomic system the change in energy upon absorption of a photon is quantized and equal to $LaTeX: \Delta \mathcal{E}=\hbar \omega$ $\Delta \mathcal{E}=\hbar \omega$

$LaTeX: \text{$\Delta $L}=\hbar$ $\text{$\Delta $L}=\hbar$

conversely, when an atomic system releases a photon the change in energy is $LaTeX: \Delta \mathcal{E}=-\hbar \omega$ $\Delta \mathcal{E}=-\hbar \omega$ leading to the general relationship for circularly polarized light

$LaTeX: \text{$\Delta $L}=\pm \hbar$ $\text{$\Delta $L}=\pm \hbar$ .

This means that circularly polarized light carries angular momentum in the amount of $LaTeX: \pm \hbar$ $\pm \hbar$ . One particularly useful applications of the fact that $LaTeX: \mathcal{R}$ $\mathcal{R}$ -state and $LaTeX: \mathcal{L}$ $\mathcal{L}$ -state light carry angular moment of $LaTeX: \pm \hbar$ $\pm \hbar$ is a phenomenon called magnetic circular dichroism (MCD Links to an external site.).

We will discuss dichroism in more detail in an upcoming section but MCD is the tendency for a material with a well defined magnetic moment to preferentially absorb $LaTeX: \mathcal{R}$ $\mathcal{R}$ -state OR $LaTeX: \mathcal{L}$ $\mathcal{L}$ -state light.

With a system capable of producing either $LaTeX: \mathcal{R}$ $\mathcal{R}$ -state or $LaTeX: \mathcal{L}$ $\mathcal{L}$ -state light, the magnetic properties of a material can be imaged. The following figure and text explain the principal.

XMCD

[From: C. Donnelly, V. Scagnoli, M. Guizar-Sicairos, M. Holler, F. Wilhelm, F. Guillou, A. Rogalev, C. Detlefs, A. Menzel, J. Raabe, and L. J. Heyderman, "High-resolution hard x-ray magnetic imaging with dichroic ptychography," Phys. Rev. B 94, 1-9 (2016).]

Figure caption:

(a) The dichroic ptychography setup. A diamond phase plate converts linearly polarized light into CL or CR light, which is then focused close to the sample plane. A piezoelectric stage is used to scan the sample across the beam. (b) The absorption part of reconstructed images taken with CL and CR polarized light at the Gd L3 edge with a photon energy of 7.2445 keV, which contain both electron density and magnetic contributions. The difference of these images removes the electron density contrast of Pt reference structures on the sample to give a purely magnetic image, i.e., with XMCD contrast. Scale bars represent 1 μm.