Errors in PDH offset locking due to spurious spectral features

TL;DR

This work identifies and quantifies a systematic PDH offset-locking error arising from the interaction between residual offset-sidebands and misaligned higher-order cavity modes, which shifts the laser lock point. It develops a simple analytic model incorporating misalignment-induced modes at offsets $\nu_h$ and $2\nu_h$, with a spurious shift $\Delta\xi_{spur}$ determined by the spur voltage and the PDH slope. The study compares ordinary dual-sideband offsets with serrodyne-modulated, spectrally-pure offsets, showing potential lock shifts up to $\delta\nu_c/2$ for DSB but suppressed to about $\delta\nu_c/40$ with serrodyne; these predictions are validated experimentally using an unbalanced Mach-Zehnder optical-frequency analyzer. The results provide practical guidance to reduce PDH offset-lock errors, improving precision in optical spectroscopy, clocks, and quantum information experiments by adopting spectrally-pure offset strategies.

Abstract

The Pound-Drever-Hall (PDH) technique is widely used to stabilize the frequency of lasers. Here we report on a routinely underestimated source of error in PDH offset-locking: a shift in the lock point due to the unintended interaction between residual optical sidebands and higher-order spatial modes in misaligned Fabry-Perot cavities. Significant frequency deviations-up to 50% of the cavity linewidth-can arise when the optical offset is obtained from a sinusoidally driven EOM. We measure this deviation experimentally, find agreement with a simple model, and show how a spectrally-pure frequency offset can reduce the deviation by an order of magnitude. Our findings draw attention to a systematic effect of importance to precision optical spectroscopy, optical clocks, and quantum information science.

Figures (4)

• Figure 1: This figure illustrates the frequency deviation in a PDH-locked laser due to the interplay between imperfect cavity mode matching and unwanted spectral features in the offset laser spectrum. (a) Illustration of PD signal for light transmitted by a misaligned cavity (Eq.$\text{ }$\ref{['eq:misaligned-fpi-spectrum']}). The horizontal axis is the incident-laser detuning relative to a $\text{TEM}_{00}$ cavity mode at $\nu_{00}$. Here we assume a cavity with $d=100\,\mathrm{mm}$, $R_{1}=500\,\mathrm{mm}$, $R_{2}=\infty$, $\delta\nu_{c}=113.4\,\mathrm{kHz}$, $C_{1}=30$, $C_{2}=15$ and $C_{k}=\infty$ for $k>2$. The orange (blue) lines show the transmitted PD spectrum with (without) PDH phase modulation at $\Omega/2\pi=25\,\mathrm{MHz}$ and depth $\beta=1.082\text{ rad}$ where the PDH error signal slope $k_{e}$ (V/Hz) is maximized. (b) Illustration of the mechanism underlying a shift in the PDH locking point. The left-hand (right-hand) side shows a zoom-in around $\Delta\xi=0$ ($\Delta\xi=\nu_{h}$); the upper orange trace is the PD signal and the lower blue trace is the resulting PDH error signal with slope $k_{e}$ ($k_{e}^{\prime}$). Ideally, the laser is locked to the left-hand feature so that $V_{\text{err}}=0$ and the right-hand feature is absent. The dashed horizontal line marks the spurious PDH error signal due to laser light near the higher-order mode at $\nu_{h}$ which gives rise to an offset $V_{\mathrm{spur}}$ in the PDH error signal and a corresponding spurious lock point shift $\Delta\xi_{\mathrm{spur}}=V_{\mathrm{spur}}/k_{e}$. Note that $\delta\nu_{c}$, $\Omega/2\pi$, and the contrasts for this figure are exaggerated to articulate certain effects. Physically realistic parameters are used in the calculation in Figure \ref{['fig:lock-shift-calc-plot']}
• Figure 2: Here we plot the shift in a PDH-locked laser due to interaction of the $\mathrm{HG}_{01}$, $\mathrm{HG}_{10}$ and $\mathrm{LG}_{10}$ cavity modes with unwanted spectral features of the EOM-offset laser light. Here we assume a cavity with $d=100\,\mathrm{mm}$, $R_{1}=500\,\mathrm{mm}$, $R_{2}=\infty$, $\delta\nu_{c}=113.4\,\mathrm{kHz}$, $\Omega/2\pi=25\,\mathrm{MHz}$, $C_{1}=30$, $C_{2}=15$ and $C_{k}=\infty$ for $k>2$. $f_{m}$ is DSB or serrodyne EOM modulation frequency. $f_{m}=\nu_{0,00}-\xi$ for an offset lock to a $\text{TEM}_{0,00}$ mode. The blue trace corresponds to single-tone drive of the EOM with depth $\beta\approx1.84\text{ rad}$ and PDH locking the $\text{TEM}_{0,00}$ cavity mode to the upper (+1) optical sideband. The calculation includes the $0,\pm1,\pm2$ spectral features emitted by the EOM. The orange trace corresponds to serrodyne drive of the EOM and PDH locking the $\text{TEM}_{0,00}$ cavity mode to the $+1$ feature. The calculation includes the $0,\pm1,+2$ and $+3$ serrodyne features (SF). The relative peak heights follow those observed experimentally: relative to the $+1$ feature, $0,+2$ are 13 dB lower and $-1,+3$ are $16\text{ dB}$ lower. Here we distinguish cavity modes in different FSR ranges by adding a new index $q$ (e.g., $\text{TEM}_{q,00}$) because spurious features will interact with modes spanning multiple FSR. The vertical axis shows the relative lock point shift (see Fig.$\text{ }$\ref{['fig:misaligned-spectrum']}(b)). As an example, $\Delta\xi_{\mathrm{spur}}$ around $f_{m}=\nu_{h}$ arises due to the interplay between SFs and cavity modes indicated in the red dashed boxes. (b) Lock point shift with symmetric (red trace) and asymmetric (yellow and green) SFs. The red trace, which is the same one in plot (a), has equal power in -1 and +2 features, as well as in 0 and +3 features. In this case $\Delta\xi_{\mathrm{spur}}$ is zero at multiples of $\nu_{\mathrm{FSR}}/2$. The other two traces are more practical examples where SFs are asymmetric.
• Figure 3: Diagram of our OFA. Red lines represent free-space beam propagation, blue lines represent propagation in non-PM fiber. The long arm of the unbalanced MZI consists of a spool of SMF-28 fiber encapsulated in spray foam to reduce its sensitivity to vibration. Light exiting the MZI was incident on a balanced photodiode pair (Thorlabs PDB450A) and digitized with an oscilloscope.
• Figure 4: Plot of the measured and predicted frequency deviations of a PDH offset locked laser due to spurious optical frequencies interacting with various modes of the cavity. These are the five most prominent features in the center of Fig. \ref{['fig:lock-shift-calc-plot']}(a). We did not synchronously record the MZI balanced photodiode output with the frequency sweep, thus there is an ambiguity in the sign of the frequency deviations extracted from the trace data. We selectively inverted the extracted frequency deviations prior to averaging to obtain the plots shown here. The dark blue line is the averaged extracted frequency excursion from the data, the shaded light blue region is the corresponding standard deviation of the measured excursions at each sample point. The orange curve depicts theoretical predictions of the induced frequency excursion for a DSB modulation scheme. Error in the theory curve is too small to see in the plot above.