PDH method - How can we lock laser frequency? Theory Part

PDH method for laser locking: basic idea

Motivation

Pound-Drever-Hall (PDH) method: first used in microwave frequency stabilization, then expanded to optical field in LIGO project.

A naive idea to lock the laser is to lock the laser on the transmission signal of FP cavity.

Transmitted intensity profile for a cavity with two identical mirrors. The blue curve represents the transmission profile of a cavity without losses. Mirror losses are considered for the orange curve.

However, for photon in a high finesse cavity, the lifetime is pretty long ($\tau = \mathcal{F}L / \pi c$, ~10$\mu \mathrm{s}$), which means the transmission signal have a delay on laser frequency.

This is fatal, as any noise whose frequency is higher than $1/10\mu\mathrm{s} \sim 100\mathrm{kHz}$ cannot be cancelled.

The solution is to utilize the reflection signal.

Cavity

The cavity mirrors reflect and transmit at the red edge.

The reflected field amplitude

$$\begin{aligned} E_r &= E_{inc} r - \left( t^2 r e^{\mathrm{i}2\Delta\varphi} E_{inc} + t^2 r^3 e^{\mathrm{i}4\Delta\varphi} E_{inc} + \cdots \right) \\ &= E_{inc}\left[ r - t^2 r e^{\mathrm{i}2\Delta\varphi} \sum_{n=0}^{\infty} \left( r^2 e^{\mathrm{i}2\Delta\varphi} \right)^n \right] \\ &= \frac{r(1-e^{\mathrm{i}2\Delta\varphi})}{1-r^2e^{\mathrm{i}2\Delta\varphi}} E_{inc} \\ &= F(\omega) E_{inc} \end{aligned}$$

where the minus sign in the second term comes from half-wave loss (the light reflects from dense medium) and $\Delta\varphi = kL-\zeta(L) = \frac{\omega}{c}L-\zeta(L)$ is the single-trip accumulated phase (from one mirror to the other) for $\text{TEM}_{00}$ mode. Here mirror losses are not considered.

The reflected intensity

$$I_r = I_{inc} \frac{4 r^2 \sin^2 (\Delta\varphi)}{t^4 + 4 r^2 \sin^2 (\Delta\varphi)} = I_{inc} \frac{\sin^2 \left(\pi \frac{\nu}{\nu_{FSR}}\right)}{\left(\frac{\pi}{2\mathcal{F}}\right)^2 + \sin^2 \left(\pi \frac{\nu}{\nu_{FSR}}\right)}$$

where $\nu_{FSR} = \frac{c}{2L}$ is the free spectral range only dependent on the cavity length $L$, $\mathcal{F} = \frac{\pi r}{1-r^2}$ is the finesse of the cavity only dependent on the reflection coefficient $r$.

From the intensity expression, we can get the FWHM (assuming $\mathcal{F} \gg 1$ which is usually satisfied)

$$\Delta \nu_{FWHM} = \frac{\nu_{FSR}}{\mathcal{F}}$$

Insight: I

If we want to lock a laser on the edge of cavity signal, the signal should satisfy the following requirements:

1. The signal should change quickly when laser frequency changes.
2. The signal should be odd about the cavity resonance, to give information of the direction of laser drift.
3. The signal should be robust against perturbations.

The phase of $F(\omega)$ is a good signal for locking. However, we cannot directly measure phase.

Nevertheless, the imaginary part of $F(\omega)$ has similar properties as its phase, and PDH method provides a way to extract $\Im F(\omega)$.

PDH method

A EOM gives the laser a phase modulation (PM):

$$\begin{aligned} E_0 e^{-\mathrm{i} \omega_0 t - \mathrm{i} \Delta \varphi \sin (\Omega t)} &= E_0 J_0(\Delta \varphi) e^{-\mathrm{i} \omega_0 t} \\ &+ E_0 \sum_{n=1}^{\infty} J_n(\Delta \varphi) \left[ e^{-\mathrm{i} (\omega_0+n\Omega) t} + (-1)^n e^{-\mathrm{i} (\omega_0-n\Omega) t} \right] \end{aligned}$$

where $\Delta \varphi$ is the PM depth and $\Omega$ is the PM frequency. $J_n$ is the Bessel function, we can ignore $|n|>1$ terms if $\Delta \varphi \ll 1$.

If PM frequency $\Omega \gg \Delta\nu_{FWHM}$, two sidebands are reflected from the cavity unaffected. The reflected light will be

$$\frac{E_r}{E_{inc}} = F(\omega) J_0(\Delta \varphi) + F(\omega+\Omega) J_1(\Delta \varphi) e^{-\mathrm{i} \Omega t} - F(\omega-\Omega) J_1(\Delta \varphi) e^{\mathrm{i} \Omega t}$$

Hence the reflected intensity

$$\begin{aligned} \frac{I_r}{I_{inc}} =& \left( \frac{E_r}{E_{inc}} \right)^2 \\ =& |F(\omega)|^2 J_0^2(\Delta \varphi) + |F(\omega+\Omega)|^2 J_1^2(\Delta \varphi) + |F(\omega-\Omega)|^2 J_1^2(\Delta \varphi) \\ & (\text{DC terms}) \\ +& 2 J_0(\Delta \varphi)J_1(\Delta \varphi) \Re[F(\omega)F^*(\omega+\Omega) - F(\omega)F^*(\omega-\Omega)] \cos(\Omega t) \\ -& 2 J_0(\Delta \varphi)J_1(\Delta \varphi) \Im[F(\omega)F^*(\omega+\Omega) + F(\omega)F^*(\omega-\Omega)] \sin(\Omega t) \\ & (\text{$\Omega$ oscillating terms}) \\ +& \mathcal{O}(2\Omega) \end{aligned}$$

Insight: II

Note that when $\Omega \gg \Delta\nu_{FWHM}$, $F(\omega-\Omega) = F(\omega+\Omega) = 1$, then the imaginary part of $F(\omega)F^*(\omega+\Omega) + F(\omega)F^*(\omega-\Omega)$ is exactly $2 \Im F(\omega)$, which is the ideal signal we want. The real part term is trivial. We only need to extract the $\sin(\Omega t)$ term with a mixer.

The idea of PDH method is discussed above. The error signal is the reflected light instead of transmitted light. When laser frequency drifts, reflected light hence the error signal changes immediately. The instant laser phase interferes with the phase of the time-averaging frequency.

Mixer

A mixer forms the product of its inputs. Feed the RF signal driving the EOM and the PD signal into a mixer:

$$(\mathcal{R}\cos(\Omega t) + \mathcal{I}\sin(\Omega t)) \times \sin(\Omega t + \theta) = \frac{1}{2}\mathcal{R}\sin\theta + \frac{1}{2}\mathcal{I}\cos\theta + \mathcal{O}(2\Omega)$$

By choosing a proper $\theta$, we can compensate experimental delays and filter out unwanted signals.

Order of magnitude

$$\begin{aligned} \mathcal{F} &\sim 10^6 \\ L &\sim 10 \text{cm} \\ \nu_{FSR} &\sim 1 \text{GHz} \\ \Delta \nu_{FWHM} &\sim 1 \text{kHz} \\ \Delta \nu_{laser} &\sim (0.1\% - 0.01\%) \Delta \nu_{FWHM} \sim 1 \text{Hz} \end{aligned}$$

Reference

https://zhuanlan.zhihu.com/p/83987752

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