Pound-Drever-Hall Laser Frequency Stabilization

This article is distilled from Am. J. Phys. 69(1), January 2001, page 79.

The goal is to make a laser to output light at frequency $\omega$ with very little frequency variation.

To do this you take a local oscillator at frequency $\Omega$ and using a Pockels cell, do a phase modulation of the laser. That is, you convert the laser field $E_0 e^{i\omega t}$ to $E_0 e^{i\omega t + i\beta \sin(\Omega t)}$ . This essentially adds two fourier components at frequencies $\omega\pm\Omega$ . These fourier components are called “sidebands”.

$E_0 e^{i\omega t + \beta \sin(\Omega t)} = E_0[J_0(\beta) e^{i\omega t} + J_1(\beta)e^{i(\omega+\Omega)t}-J_1(\beta)e^{i(\omega-\Omega)t}]$

Here, the $J_0$ and the $J_1$ are Bessel functions.

Now you throw this modulated laser light onto a high-finesse Fabry-Perot cavity (with free spectral range $\text{FSR}$ , line-width $\text{LW}$ and mirror reflectivity $r \simeq 1$ ) and look at the light reflected from the Fabry-Perot. This multiplies a fourier component at $\omega$ by the complex number

$F(\omega) = r \dfrac{e^{i \omega/\text{FSR}}-1}{1-r^2 e^{i\omega/\text{FSR}}}$

We can calculate the power in the reflected light. There is a term proportional to $\cos(\Omega t)$ / $\sin(\Omega t)$ whose coefficient is proportional to the real/imaginary part of $F(\omega)F^*(\omega+\Omega)-F^*(\omega)F(\omega-\Omega)$ .

Consider the case of interest where $\omega$ is almost on resonance: $\omega = 2\pi N \times \text{FSR} + \delta \omega$ , and the sidebands are almost completely reflected. Then, we can approximate $F(\omega\pm\Omega)$ by $-1$ , and $F(\omega)$ by $i \delta \omega/\pi \text{LW}$ (since $\text{FSR}/\text{LW} \simeq \pi/(1-r^2)$ ). This means that the $\cos(\Omega t)$ term in the reflected power is almost zero and we are left with the $\sin(\Omega t)$ term with a coefficient proportional to (the negative of) the frequency offset.

Now by mixing the local oscillator signal (in the correct phase) and passing through a low-pass filter, we can isolate the $\sin(\Omega t)$ term and feed it to the laser, which supplies a negative feedback to the laser. This negative feedback accomplishes the frequency stabilization.

About Raghu Mahajan

Postdoctoral research fellow in theoretical physics, studying quantum gravity.

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