1. Introduction

A common goal in precision optics is to employ feedback [1] to stabilize (lock) the frequency of a continuous-wave (CW) laser to that of an external system, such as a Fabry-Perot optical cavity [2] or atomic transition [3,4]. This can be used to stabilize the laser itself, or to continuously monitor the dynamics of the external system. For example, by locking a laser to a sufficiently stable cavity, it is possible to produce extraordinarily coherent light [5], which can then be applied to the precision control and spectroscopy of trapped atoms [6], high-accuracy optical clocks [7, 8], or interferometric detection of passing gravitational waves [9]. Alternatively, a locked laser can be used to probe atomic or molecular absorption within the cavity [10], or to resolve the quantum zero-point motion of an embedded mechanical element [11, 12] (see also [13]). Within the latter context (optomechanics [14]), a stable, low-power lock is particularly important for systems combining lightweight, highly compliant, ultrahigh-Q mechanical system [15, 16] with a high-finesse cavity, wherein the radiation pressure from an average cavity occupancy of one photon should in principle profoundly alter the mechanical trajectory [15]. Indeed, our failed attempts to lock the combined system of [15] (using traditional methods) is precisely what motivated the present work.

As reviewed in Sec. 2 below, all feedback schemes aim to simultaneously achieve the largest possible closed-loop gain (the degree to which noise can be suppressed) and sufficient dynamic range (headroom) to compensate all fluctuations. The closed-loop gain is ultimately limited by the speed with which corrections can be applied (the bandwidth), which itself is fundamentally limited by the delay of the signal propagating through the loop [1]. In many situations, however, the achievable gain is practically limited by other system nonidealities. For example, one means of tuning a laser’s emission frequency is to mechanically stretch an internal optical path, and the bandwidth is then practically limited by the structure’s mechanical resonances. For this reason, low-noise, mechanically tuned lasers (e.g. commercial Nd:YAG lasers) are typically limited to control bandwidths of ~100 kHz. Faster feedback can be achieved by controlling the laser’s pump, and commercial diode lasers (e.g.) routinely achieve sufficient pump modulation bandwidth that feedback is limited by other loop nonidealities; as such, using the pump to stabilize against an external cavity is often employed as a first stage to reduce their comparatively large noise [17]. The control bandwidth of the combined system, however, is then limited by the external cavity’s mechanical resonances. Cavity mirror actuation has improved in recent years, achieving 180 kHz with short-travel piezo actuation [18] and now up to ~700 kHz with the incorporation of photothermal tuning [19].

An alternative, laser-independent technique is to shift the light’s frequency after emission. For visible wavelengths, this is usually accomplished with an acousto-optical modulator (AOM), achieving ~200 kHz closed-loop bandwidth [5] with ~MHz-scale headroom on its own, and up to 2 MHz bandwidth when combined with an electro-optical modulator (EOM) to correct the high-frequency noise in parallel [20]. At near-infrared (telecom) wavelengths, low-cost fiber modulators are more commonly employed: using serrodyne techniques, wherein a voltage-controlled oscillator (VCO), nonlinear transmission line (NLTL), and EOM generate a saw-tooth phase that effectively shifts the carrier frequency [21, 22], or single-sideband modulation (SSM), wherein a VCO and Mach-Zehnder interferometer shift a small portion of the carrier [23], it is routine to achieve several-MHz feedback bandwidth and well over 100 MHz headroom.

A second common goal in precision optics is to perform heterodyne readout [24], wherein a weak “signal” beam is overlapped with a strong local oscillator (LO) beam detuned by an electronically measurable frequency. Landing on a photodiode, the beating between these two beams produces an amplified electronic signal with a spectrum shifted to the LO detuning, thereby providing simultaneous access to the signal’s amplitude and phase quadratures (or, equivalently, to its double-sided spectrum). In addition to spectroscopy, polarimetry, laser radar (and lidar), microscopy, and other applications [24], heterodyne readout enables a continuous, self-calibrating measurement of mechanical temperatures in the quantum regime of motion [11, 12].

Here we present and characterize a simple, low-power, high-bandwidth, post-emission laser locking technique with built-in heterodyne readout. This approach employs a high-speed VCO and a single EOM to control the signal beam frequency, and can be implemented with any laser. In contrast to serrodyne systems, it does not require a precise NLTL-generated saw-tooth waveform (or the extra EOM bandwidth to handle it), and, similar to SSM, shifts only a fraction of the laser light. In contrast to both, the carrier is exploited as an optical LO for heterodyne readout, and no alignment or relative path stabilization is required. Using the test ports of our chosen electronics, we directly measure the frequency-dependence of the closed-loop gain, demonstrating a delay-limited feedback bandwidth of 3.5 MHz (one integrator) and excellent agreement with a simple model based on ideal components. From this we propose a modified setup that should realistically achieve a gain of 4 × 10⁷ at 1 kHz (6.6 MHz bandwidth, two integrators). The headroom allowed by these components exceeds 500 MHz, limited only by a 10 V ceiling on the VCO programming voltage imposed by our amplifier (~1 GHz should be possible with this VCO/EOM combination, at the expense of added amplitude noise). Section 2 briefly reviews requisite concepts in laser feedback. Section 3 then introduces the “Pound-Drever-Hall” method for generating an error signal [2, 25, 26] (including a derivation of its dynamical response) and a simple electronics modification enabling heterodyne readout. We then present the technical details of our “proof-of-concept” system in Sec. 4, characterize its performance in Sec. 5, and conclude in Sec. 6.

2. A brief review of laser feedback

All laser frequency stabilization schemes rely on (i) generation of an “error” signal proportional to the frequency difference (detuning) δ between the laser and an external system, and (ii) processing and routing of this signal to a port that adjusts δ to compensate [1]. Figure 1(a) shows a conceptual diagram of a feedback loop for locking a laser to a cavity resonance. Cavity vibrations and laser frequency noise together introduce a nominal detuning δ_n (note these noise sources appear on equal footing, and only the difference δ will be stabilized) that is converted to an electronic signal by a photodiode “D”, amplified and filtered by assorted electronics “−A”, and sent to a “feedback” port “F” to adjust δ. This correction is added to the original noise (e.g., by tuning the laser frequency, cavity length, or both), resulting in a relationship for the net detuning δ = δ_n − CDAFδ, where C, D, −A, and F are complex, frequency-dependent gains (transfer functions) for the cavity, diode, electronics, and feedback port. Solving for δ yields

 

Fig. 1 Feedback stabilization. (a) Generic control loop for stabilizing a laser’s detuning δ from the resonance of an optical cavity. Noise δ_n enters, is converted to an optical signal by the cavity (transfer function C), collected by a diode (D), manipulated by electronics (−A) and sent to a “feedback” port (F). (b) Practical implementation using Pound-Drever-Hall readout. Straight red lines represent optical paths, straight black lines represent electrical paths, and dashed gray lines show potential feedback paths. The laser is phase-modulated, lands on a beam splitter (BS) and interacts with the cavity, which converts phase to amplitude modulation. This is recorded with a photodiode and mixed (demodulated) with a local oscillator. Inset shows the resulting steady-state voltage V_Y (δ), with a red dot indicating a stable lock point. The manipulated signal can be fed back to (i) the cavity length or (ii) the laser frequency. Feeding back to (iii) the oscillator frequency only adjusts the sidebands.

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This immediately highlights the central concerns for stabilization. First, it is desirable to make the “closed-loop gain” G ≡ CDAF as large as possible, to maximally cancel the noise. For |G| ≫ 1, the overall phase ϕ_G does not matter, but if G approaches −1 at a some “bad” frequency, then the noise at that frequency is amplified. This places unavoidable limits on G for the following reasons: (i) any delay t_d in the signal path introduces a phase factor $e^{-i{\omega }_n{t}_d}$ (where ω_n is the frequency of the noise we wish to cancel), forcing ϕ_G = −π at finite frequencies, regardless of what electronics are chosen for A, (ii) stability concerns impose that the magnitude of the gain at the lowest of these frequencies ω_−π should be less than 1, and (iii) causality places an upper bound $\left|G\right|<{\omega }_{-\pi }^2/{\omega }_n^2$ on how much the gain can increase below this frequency [1]. Since most noise occurs at low frequencies, it is therefore desirable to make ω_−π large, and to engineer a feedback circuit such that G increases as rapidly as possible with decreasing frequency.

A readout of δ (the error signal) can be obtained by several methods. A high-finesse optical cavity of length L, input mirror field transmission coefficient t₁ and power ringdown time τ has an overall field reflection coefficient (see Appendix A)

The reflected power (∝ |r|²) therefore follows a Lorentzian line shape. On resonance (δ=0), the reflected power cannot on its own be used for feedback since it does not provide information about the sign of δ. One can generate a bipolar error signal by tuning the laser away from resonance [27], but this technique couples laser power fluctuations to detuning errors. However, the phase of r(δ) does vary linearly on resonance, and can be extracted via phase modulation [2], heterodyne [24, 28], and homodyne [29] schemes, wherein the mode of interest interferes with one or more reference beams having different frequency or phase. Other techniques employ a second cavity mode as a reference, for example a mode of different polarization [30] or a higher order spatial mode [31, 32]. The powerful “Pound-Drever-Hall” technique [2] is discussed in the following section.

3. Modified Pound-Drever-Hall readout and dynamical response

A ubiquitous method for on-resonance laser stabilization is the “Pound-Drever-Hall” (PDH) technique [2, 25], a diagram of which is drawn in Fig. 1(b). Stated briefly, this technique effectively amounts to dithering the laser frequency with an electro-optical modulator (EOM) and measuring the induced modulation in the reflected power to infer the slope of |r(δ)|² (or Im[r]) [26]. The resulting error signal (inset blue curve, near red dot) can then be manipulated with electronics (−A) and fed back to either (i) the cavity length or (ii) the laser frequency, as described above. Feeding back to a voltage-controlled oscillator (VCO, iii) will not adjust the carrier frequency (or δ) in this configuration, but can be used to lock a sideband as discussed in Sec. 4. An elegant, pedagogical derivation of the steady-state error signal (V_Y in Fig. 1(b)) for this system can be found in [26]. This accurately captures the system’s ability to convert low-frequency detuning noise into an error signal, but breaks down when the detuning δ(t) contains frequencies comparable to the cavity’s linewidth 1/τ.

A straightforward means of deriving the dynamic response [33] is to propagate a small laser “noise” component through an EOM, cavity, diode, and demodulation (mixer) circuit in Fig. 1(b) to extract a combined transfer function, as follows (see Appendix B for more details). Suppose there exists a laser frequency noise component at ω_n that is the real part of $\tilde{\mathrm{\Omega }}(t)={\mathrm{\Omega }}_n{e}^{i{\omega }_{n}t}$, where Ω_n is a constant amplitude. This corresponds to phase modulation ϕ(t) = ϕ_n sin(ω_nt), where ϕ_n = Ω_n/ω_n. If this light is fed through a phase modulator (EOM) driven by voltage V_osc = V_e sin(ω_et), the field landing on the cavity is

This motivates the use of a “proportional-integral” (PI) amplifier for the conditioning electronics (A in Fig. 1). A PI amplifier has a transfer function

The total system behaves like an integrator over all frequencies, with increasing gain at low frequencies. The overall phase is far from −π, preventing the system’s total delay factor $e^{-i{\omega }_n{t}_d}$ from forcing the closed-loop gain below 1 at a low frequency. This phase margin furthermore provides “wiggle room” for loop nonidealities such as indirectly driven resonances that can cause a temporary excursion in phase (see, e.g., [18]). However, even if the bandwidth of the feedback port F is effectively infinite and / or we have precisely compensated for all of its artifacts, the ultimate gain is limited by the signal delay t_d – in this case from the output of the EOM to the cavity, back to the diode, through the electronics, and through the feedback port – which forces the closed-loop gain to be less than 1 at frequency ω_−π < π/4t_d for this choice of electronics.

It is also possible to lock the first-order sidebands (ω_l ± ω_e) to the cavity. Following the same analysis for the case of either sideband resonant with the cavity produces a transfer function

Finally, similarly propagating an amplitude noise component through this system (i.e., setting $E_l\to (1+Re[\widetilde{\mathit{ϵ}}])E_l$, where $\widetilde{\mathit{ϵ}}(t)={\mathit{ϵ}}_n{e}^{i{\omega }_{n}t}$ with constant ϵ_n ≪ 1) has no impact on ${\tilde{V}}_Y$ or ${\tilde{V}}_{Y,\pm }$. However, introducing a relative π/2 phase shift between the mixer’s LO and signal ports provides access to a similar readout of the laser’s amplitude noise ${\tilde{V}}_X$ with an overall transfer function

 

Fig. 2 Sideband locking with heterodyne readout. Many parts are from Thorlabs (TL), Newport (NP) and Minicircuits (MC). (a) A VCO (MC ZX95-1600W-S+) signal is split (MC ZX-10-2-20-S+) and amplified (MC ZX60-4016E-S+) feeding an EOM (TL LN65-10-P-A-A-BNL-Kr with shortened output fiber) and the LO (“L”) ports of two mixers (MC ZFM-5X-S) for quadrature readout. Laser (Koheras Adjustik E15) feeds a 14.2-dBm (26.3 mW) carrier through the EOM, producing a 10.8 dBm carrier and -2.2 dBm (5%) sidebands, set by a variable attenuator (MC ZX73-2500-S+, ~15 dB) leading to the EOM. Once collimated (TL F260APC-1550), the beam passes through a beam splitter (BS, TL BS018 50:50), mode-matching lenses (−5 cm and 10 cm focal length, see (b)), and steering mirrors (M) before landing on a cavity comprising a flat (NP 10CM00SR.70F) and curved (NP 10CV00SR.70F) supermirror, the second of whose position is swept by a piezo mirror mount (TL K1PZ). The transmitted beam is focused on a photodiode (PD, TL PDA10CF), while the reflected beam is rerouted by the BS, passes through an isolator (TL IO-2.5-1550-VLP), and is focused upon a 2-GHz photodiode (PD, Femto HSA-X-S-2G-IN). Low-frequencies signals (< 20 MHz) are eliminated with a high-pass (MC SHP-20+), before amplification (MC ZX60-P105LN+) and splitting by a π/2 splitter (MC ZX10Q-2-13-S+). The phase-shifted signals are fed to the mixers’ RF (R) ports and demodulated to the IF (I) ports. The “phase” quadrature (V_Y) is fed to a PI amplifier (−A, NP LB1005) for feedback to the VCO. Inset shows the steady-state voltage of the “amplitude” quadrature V_X (δ). (b) Photograph of optics. (c) Simultaneously acquired V_X and V_Y, for three different VCO controls: 0 V (lightest), 0.8 V, and 1.8 V (darkest).

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4. Apparatus for sideband locking with heterodyne readout

Figure 2(a) shows our test setup for locking a first order sideband (at ω_l ± ω_e) to a Fabry-Perot cavity resonance. Sidebands are created with a fiber EOM driven at ω_e by a VCO with 90 MHz modulation bandwidth and 0.65–1.75 GHz tuning range. Light from the EOM passes through a beam splitter (BS) and mode-matching optics (shown in Fig. 2(b)), reflects from the cavity, and is collected by a high-bandwidth photodiode. The resulting signal is filtered and amplified before passing through a power splitter that produces a phase shift of 0 and π/2 at its outputs. These two signals are separately mixed with that of the VCO to produce V_X and V_Y. The VCO output is split prior to the EOM, delayed, and used as the electronic LO for both mixers. In order to maintain a fixed phase between the mixers’ LO and signal ports over the full range of VCO frequencies ω_e, the delay between the two signal paths must match. Any difference Δt_d produces a relative phase ω_eΔt_d that must remain small compared to π/2 at the highest VCO frequency. Here this imposes that Δt_d ≪ π/2ω_e ~ 1 ns, corresponding to a free-space path difference ≪ 30 cm, which is mostly compensated for with a combination of cables and extension adapters (Fig. 2(a)), with mm-scale fine-tuning of the photodiode’s position. The higher precision required for larger-ω_e systems can be easily implemented with the diode optics mounted on a translation stage.

Figure 2(b) shows a photograph of the optical path; the electronics are mounted on a nearby platform. The detuning δ between the laser and cavity can be widely adjusted with long-travel piezos in the second mirror mount (“Piezo M”). Figure 2(c) shows a diagnostic measurement of V_Y (δ) and V_X (δ) recorded during cavity length sweeps for a few values of ω_e. Each sweep was performed “quickly” (16 ms over the full range) to reduce run-to-run variations from the ambient vibrations of the test cavity. The insensitivity of the quadrature readout to ω_e indicates the delay is matched (see Appendix C for a larger range). The cavity has a power ringdown time τ = 1.2 ± 0.1 µs (finesse 4700±400), and so these fast sweeps produce a transient response [34] resulting in a measured V_Y (top plot of (c)) that is consistently not symmetric about V_Y = 0, and a measured V_X (bottom plot of (c)) that deviates from a simple peak. This artifact can be highly misleading when tuning the relative delay, and so rather than trying to symmetrize V_Y, we recommend slowly modulating ω_e while quickly sweeping the cavity, and adjusting Δt_d to produce a signal shape that does not vary with ω_e.

The error signal V_Y is then fed through a tunable PI amplifier having the transfer function of Eq. (5), with ω_PI = 110 kHz and g = 105 = 40 dB (measured) before finally being fed back to the VCO. Due to the sidebands’ opposed frequency response, one sideband is always stabilized by this feedback and the other is always destabilized; here we (arbitrarily) lock the upper sideband (verified by the negative value of V_X). Despite the open-air design and flagrant disregard for vibration isolation, this system readily locks and remains so indefinitely (it is impervious to chair scoots, door slams, claps, and shrieks, but fails if the table surface is tapped with a wrench).

5. Performance

Once locked, we increase the feedback gain G₀ until the system rings (at ~3 MHz for this implementation), indicating that the gain at ω_−π ~ 3 MHz has exceeded unity. We then reduce G₀ until the remaining noise in V_Y is minimized. The most sensitive estimate of V_Y is achieved by referring the PI amplifier’s output back to its input using its known (measured) transfer function; together with an independent measurement of the error signal slope on resonance 2π × ∂_δV_Y = 388 ± 40 mV/MHz, we estimate that the stabilized RMS detuning noise δ_RMS/2π is below 70 Hz (0.0005 cavity resonance full-widths). This is a factor of 3000 lower than the pre-stabilized value of 240 kHz (1.6 fullwidths, corresponding to 0.3 nm RMS cavity length noise), as estimated directly from the PI output and the VCO specifications (52 MHz/V). Figure 3(a) shows the power spectral densities of these two inferred detuning signals. The square root of their ratio provides a basic estimate of the closed-loop gain magnitude |G(ω_n)| ~ 1000 at ω_n/2π = 1 kHz. We note that this estimate of the pre-stabilized noise is made while the system is locked – the cavity’s inherent mechanical noise and narrow linewidth together preclude an open-loop estimate – and so this data mainly serves as a consistency check for the closed-loop gain measurement below and our assumptions about the other system components.

 

Fig. 3 (a) Detuning noise power spectral density (PSD) before and after lock, recorded while locked. The pre-feedback noise (red) is inferred from the proportional-integral (PI) amplifier output and the VCO conversion factor 52 MHz/V, while the post-feedback noise (blue) is inferred from the PI output referred back to its input and the independently measured slope of the error function (388 ± 40 mV/MHz) at the lock point. (b) Measured (blue) and modeled (red) closed-loop transfer function. The model includes the cavity (green, ring-down time τ = 1.1 µs), PI amplifier (yellow, ω_PI =110 kHz, and g=105), and a delay (brown, 70 ns, 52 ns from the PI amplifier). Transfer functions of other components are assumed to be “flat” on this scale. The gray dashed line shows a closed-loop gain that could be achieved with optimizations: replacement of the PI amplifier and further delay reductions to 10 ns and two PI filters, one with ω_PI/2π = 70 kHz, 1/g = 0), and the other with ω_PI/2π = 15 MHz and g = 10⁵.

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To directly measure G(ω_n), we inject a small amount of “noise” into the locked system and observe how it is suppressed. The PI amplifier provides a second (inverted) input, and an isolated monitor of the in-loop error signal. Using a lock-in amplifier, we apply an oscillatory signal V_n of frequency ω_n to this input and record both quadratures of the error signal V_Y at ω_n (correcting for the transfer functions between the input and error monitor, as well as the lock-in and its measurement cables). Using the same analysis of Fig. 1(a) with CDAF → G, δ_n → V_n, and δ → V_Y, we solve for the closed-loop gain G = V_n/V_Y − 1, which is plotted in Fig. 3(b) (blue). Importantly, the observed gain smoothly decreases with ω_n (approximately as 1/ω_n), and the phase crosses −π at ω_n/2π = 3.5 MHz, where |G| < 1, consistent with the observed ringing frequency. The measurement noise increases at low frequencies due to the reduced signal at high gain. It is worth pointing out that, despite the addition of sidebands to the VCO output (at ω_e ± ω_n), the measured transfer function through the EOM, cavity, diode, and mixer is identical to that of laser frequency noise (this can be seen by tracking these extra sidebands through a calculation similar to that of Appendix B).

The red line in Figs. 3(b) and 3(c) represents a simple model for G(ω_n) comprising the product of (i) the PI transfer function (Eq. (5)) with measured ω_PI =110 kHz and g=105, (ii) the cavity transfer function (Eq. (7)) with τ=1.1 µs (i.e. one standard deviation below the measured value), (iii) a closed-loop delay t_d = 70 ns, and (iv) an overall scaling factor chosen to match the measured G(ω_n). The yellow and green curves show the modeled PI and cavity transfer functions alone for reference, and the brown curve shows the phase contribution from the delay. The employed value of t_d is consistent with the signal travel time of the loop, independently estimated to be approximately 68 ns from the signal path of the lower VCO loop in Fig. 2(a): a combined cable and component length of 127″ traversed at 2/3 the speed of light (16 ns) plus the measured internal delay of the PI amplifier (52 ns). The agreement between the model and measurement suggests that the chosen components exhibit no important nonidealities up to ~10 MHz, and that the other components (the EOM, optics, diode, filters, mixers, amplifiers, attenuators, splitters, and connectors) can be assumed to have a flat response, adding a combined delay on the order of nanoseconds at most.

The phase plot of Fig. 3(b) highlights that the achieved bandwidth is limited primarily by the delay. Without it, the phase would remain above −π/2 to a significantly higher frequency, allowing for larger G₀. The PI amplifier accounts for 75% of the delay, implying the greatest gains can be made by replacing it with a faster (albeit less flexible) integrated circuit. Modern amplifiers routinely achieve sub-nanosecond delays, and the requisite PI filters can be realized with passives (capacitors and resistors). It is also straightforward to reduce the optical and electronic lengths: using compact mode-matching optics and shorter cables alone can reduce the delay to ~10 ns. Furthermore, replacing the existing PI filter with two – one having ω_PI/2π = 70 kHz and 1/g → 0 and the other having ω_PI/2π = 15 MHz and g = 10⁵ – for example, would produce a bandwidth of 6.6 MHz and (more importantly) a near-causality limited gain |G(2π × 1 kHz) ~ 4 × 10⁷ (Fig. 3(b), dashed line). This optimization will be the subject of future work.

To estimate the headroom, we change the cavity length L while locked and monitor the output voltage of the PI amplifier; the system remains locked over the full ~100 MHz tuning range presented in Fig. 2(b), in this case limited by the cavity’s small free spectral range: the lower sideband of an adjacent mode eventually becomes degenerate with the locked sideband, spoiling the error signal. Performing the same test on a 5-cm cavity, we find a headroom of 550 MHz, limited instead by the maximum output voltage of the PI amplifier (10 V), which covers only half the tuning range of the VCO. A headroom exceeding 1 GHz is in principle possible with these components; however, while more headroom is certainly useful for tracking large fluctuations, the frequency-dependencies of the VCO output, EOM, and other electronics will eventually couple these fluctuations to the amplitude of the optical signal and LO beams (see Appendix C). Further engineering effort is therefore best spent reducing the system’s inherent noise.

6. Conclusion

We have demonstrated a simple technique for locking a first order laser sideband to an optical cavity with a delay-limited feedback bandwidth of 3.5 MHz with a single integrator, and a headroom exceeding 500 MHz. We directly measured the closed-loop gain, finding excellent agreement with a model based on ideal components, and suggest simple modifications for realizing a gain exceeding 10⁷ at 1 kHz. Finally, we note that, by implementing an appropriately weighted sum of V_X and V_Y (or otherwise shifting the relative phase of the mixers’ electronic LO and signal ports), it should be possible to create an amplitude-insensitive locking point – a zero crossing in the resulting error signal – at arbitrary detuning.

Appendix A: Reflection from an asymmetric cavity

Consider a cavity of length L with mirrors that are not identical, having field transmission coefficients it₁ and it₂ and reflection coefficients −r₁ and −r₂ (where t_i and r_i are real positive and |t_i|²+|r_i|² ≤ 1). If the first mirror is driven by a field of amplitude E_in and angular frequency ω_l, the steady-state reflected field – a sum of the prompt reflection and all subsequent paths leaking from the input mirror – can be calculated from the resulting geometric series:

(9)$$\begin{array}{l}\frac{E_r}{E_{in}}=-r_1+t_1^2{r}_2{e}^{-2i{\omega }_{l}L/c}+t_1^2{r}_2{e}^{-2i{\omega }_{l}L/c}\left(r_1{r}_2{e}^{-2i{\omega }_{l}L/c}\right)\\ +t_1^2{r}_2{e}^{-2i{\omega }_{l}L/c}{\left(r_1{r}_2{e}^{-2i{\omega }_{l}L/c}\right)}^2+\dots \end{array}$$

(10)$$=-r_1+\frac{t_1^2{r}_2{e}^{-2i{\omega }_{l}L/c}}{1-r_1{r}_2{e}^{-2i{\omega }_{l}L/c}}$$

In the high-finesse limit, this can be expanded in terms of the small transmission t_i ≪ 1 and loss ${\rho }_i\equiv \sqrt{1-r_i^2}\ll 1$, such that the overall reflection coefficient

(11)$$r(\delta )\equiv \frac{E_r}{E_{in}}\approx \frac{t_1^2\tau c/L}{1+i2\tau \delta }-1,$$

where $\tau =\frac{2L/c}{{\rho }_1^2+{\rho }_2^2}$ is the power ringdown time and δ = ω_l − ω_r is the detuning from the m^th resonance frequency ω_r = mc/2L (δ is also assumed small compared to the mode spacing c/2L).

Appendix B: Transfer function from laser noise to error signal

B.1 Shorthand notation

To save space and avoid typos, we adopt the following shorthand notation.

(12)$$C_a,S_a\equiv \cos ({\omega }_{a}t),\sin ({\omega }_{a}t)$$

(13)$$C_{a\pm b},S_{a\pm b}\equiv \cos ({\omega }_{a}t\pm {\omega }_{b}t),\sin ({\omega }_{a}t\pm {\omega }_{b}t)$$

(14)$$C_{a\pm b\psi },S_{a\pm b\psi }\equiv \cos ({\omega }_{a}t\pm ({\omega }_{b}t+\psi )),\sin ({\omega }_{a}t\pm ({\omega }_{b}t+\psi ))$$

(15)$$C_{a\pm 2b\psi },S_{a\pm 2b\psi }\equiv \cos ({\omega }_{a}t\pm ({\omega }_{b}t+\psi )),\sin ({\omega }_{a}t\pm (2{\omega }_{b}t+\psi ))$$

The diode and mixer steps discussed below involve nonlinear operations, and so (as a matter of algebraic taste) we save complex notation for the end of the calculation.

B.2 Propagating laser noise through the readout system

We proceed by adding a small “noise” term to the field from a laser, and then propagate this through the electro-optical modulator (EOM), cavity, photodiode, and mixers to determine the readout’s overall transfer function. Suppose the laser has a “noise” component at frequency ω_n, with amplitude and frequency modulation

(16)$$\mathit{ϵ}(t)={\mathit{ϵ}}_n\cos ({\omega }_{n}t)$$

(17)$$\mathrm{\Omega }(t)={\mathrm{\Omega }}_n\cos ({\omega }_{n}t)$$

for constants ϵ_n and Ω_n. In this case, the field (nominally of the noiseless form $\sqrt{P_l}\cos ({\omega }_{l}t)$ for laser frequency ω_l and power P_l) is $E(t)=\sqrt{P_l}(1+{\mathit{ϵ}}_n{C}_n)\cos ({\omega }_{l}t+{\varphi }_n{S}_n)$, where ϕ_n ≡ Ω_n/ω_n is the phase modulation amplitude. If this passes through an electro-optical modulator (EOM) driven by a voltage-controlled oscillator (VCO) output of the form V_VCO(t) = V_e sin(ω_et) (for constant V_e and frequency ω_e), E is further modified to

(18)$$E(t)=\sqrt{P_l}(1+{\mathit{ϵ}}_n{C}_n+{\mathit{ϵ}}_e{S}_e)\cos ({\omega }_{l}t+{\varphi }_n{S}_n+{\varphi }_e{S}_e),$$

where ϕ_e = η_EOMV_VCO is the phase excursion for a “flat” EOM conversion efficiency η_EOM (units of rad/V), and we have included a small amount of “accidental” amplitude modulation ϵ_e due to imperfections in the EOM optics for generality. If we assume the modulations are small (ϵ_n, ϵ_e, ϕ_n, ϕ_e ≪ 1), the field can be expanded to second order and rewritten in terms of a carrier at ω_l plus 12 sidebands:

(19)$$\begin{array}{l}\frac{E(t)}{\sqrt{P_l}}\approx \left(1-\frac{1}{4}{\varphi }_e^2-\frac{1}{4}{\varphi }_n^2\right)C_l\\ +\frac{1}{2}{\mathit{ϵ}}_n(C_{l+n}+C_{l-n})+\frac{1}{2}{\varphi }_n(C_{l+n}-C_{l-n})\\ +\frac{1}{2}{\mathit{ϵ}}_e(S_{l+e}-S_{l-e})+\frac{1}{2}{\varphi }_e(C_{l+e}-C_{l-e})\\ +\frac{1}{8}{\varphi }_n^2(C_{l+2n}+C_{l-2n})+\frac{1}{4}{\varphi }_n{\mathit{ϵ}}_n(C_{l+2n}-C_{l-2n})\\ +\frac{1}{8}{\varphi }_e^2(C_{l+2e}+C_{l-2e})+\frac{1}{4}{\varphi }_e{\mathit{ϵ}}_e(-2S_l+S_{l+2e}+S_{l-2e})\\ +\frac{1}{4}{\varphi }_e{\varphi }_n(C_{l+e+n}-C_{l+e-n}-C_{l-e+n}+C_{l-e-n})\\ +\frac{1}{4}{\varphi }_n{\mathit{ϵ}}_e(S_{l+e+n}-S_{l+e-n}-S_{l-e+n}+S_{l-e-n})\\ +\frac{1}{4}{\varphi }_e{\mathit{ϵ}}_n(C_{l+e+n}+C_{l+e-n}-C_{l-e+n}-C_{l-e-n}).\end{array}$$

Now suppose these beams land on a cavity with either a resonance “A” tuned to the carrier at ω_l or a resonance “B” tuned to the upper sideband at ω_l +ω_e (we track both options through the calculation, allowing the choice of scenarios afterward). In the limit of “large” VCO frequency (ω_e ≫ ω_n, 1/τ), only the terms at ω_l, ω_l ± ω_n, and ω_l ± 2ω_n interact with resonance A and only the terms at ω_l + ω_e and ω_l + ω_e ± ω_n interact with resonance B, so the field reflected from the cavity becomes

(20)$$\begin{array}{l}\frac{-E(t)}{r_1\sqrt{P_l}}=\left(1-\frac{1}{4}{\varphi }_e^2-\frac{1}{4}{\varphi }_n^2\right){\rho }_{A0}{C}_l\\ +\frac{1}{2}{\mathit{ϵ}}_n\left({\rho }_{An}{C}_{l+n{\psi }_A}+{\rho }_{An}{C}_{l-n{\psi }_A}\right)+\frac{1}{2}{\varphi }_n\left({\rho }_{An}{C}_{l+n{\psi }_A}-{\rho }_{An}{C}_{l-n{\psi }_A}\right)\\ +\frac{1}{2}{\mathit{ϵ}}_e\left({\rho }_{B0}{S}_{l+e}-\alpha S_{l-e}\right)+\frac{1}{2}{\varphi }_e\left({\rho }_{B0}{C}_{l+e}-\alpha C_{l-e}\right)\\ +\frac{1}{8}{\varphi }_n^2\left({\rho }_{A2n}{C}_{l+2n{\varphi }_A}+{\rho }_{A2n}{C}_{l-2n{\varphi }_A}\right)\\ +\frac{1}{4}{\varphi }_n{\mathit{ϵ}}_n\left({\rho }_{A2n}{C}_{l+2n{\varphi }_A}-{\rho }_{A2n}{C}_{l-2n{\varphi }_A}\right)\\ +\frac{1}{8}{\varphi }_e^2\left(C_{l+2e}+C_{l-2e}\right)+\frac{1}{4}{\varphi }_e{\mathit{ϵ}}_e\left(-2S_l+S_{l+2e}+S_{l-2e}\right)\\ +\frac{1}{4}{\varphi }_e{\varphi }_n\left({\rho }_{Bn}{C}_{l+e+n{\psi }_B}-{\rho }_{Bn}{C}_{l+e-n{\psi }_B}-\alpha C_{l-e+n}+\alpha C_{l-e-n}\right)\\ +\frac{1}{4}{\varphi }_n{\mathit{ϵ}}_e\left({\rho }_{Bn}{S}_{l+e+n{\psi }_B}-{\rho }_{Bn}{S}_{l+e-n{\psi }_B}-\alpha S_{l-e+n}+\alpha S_{l-e-n}\right)\\ +\frac{1}{4}{\varphi }_e{\mathit{ϵ}}_n\left({\rho }_{Bn}{C}_{l+e+n{\psi }_B}+{\rho }_{Bn}{C}_{l+e-n{\psi }_B}-\alpha C_{l-e+n}-\alpha C_{l-e-n}\right).\end{array}$$

where we define

(21)$${\rho }_N(\delta )\equiv 1-\frac{t_1^2\tau c/L}{1+i2\tau \delta }$$

(22)$${\rho }_{N0}\equiv {\rho }_N(0)$$

(23)$${\rho }_{Nn}\equiv |{\rho }_N({\omega }_n)|$$

(24)$${\psi }_N\equiv \arg \left[\rho \left({\omega }_n\right)\right]=-\arg \left[\rho \left(-{\omega }_n\right)\right]$$

(25)$${\rho }_{N2n}\equiv \left|{\rho }_N\left(2{\omega }_n\right)\right|$$

(26)$${\varphi }_N\equiv \arg \left[\rho \left(2{\omega }_n\right)\right]=-\arg \left[\rho \left(-2{\omega }_n\right)\right]$$

for cavity resonance N ∈ (A, B), and we have attached a prefactor α = 1 to the lower sidebands to track their effect on the demodulated signal (see below).

When all of this light lands on a photodiode, the resulting signal comprises all frequencies generated by the quantity E²(t) that are within the diode’s electronic bandwidth (a few GHz in our case), which includes many frequency components. Anticipating the subsequent demodulation at ω_e and our eventual interest in the noise term at ω_n, we can ignore the vast majority, however. Assuming a constant photodiode conversion efficiency η_PD (units of V/W) and again keeping only terms up to second order in ϵ_n, ϵ_e, ϕ_n, ϕ_e, the photodiode’s output V_PD(t) is then given by

(27)$$\begin{array}{l}\frac{4V_{PD}(t)}{{\eta }_{PD}{r}_1^2{P}_l}\approx \dots +{\rho }_{A0}{\varphi }_e{\varphi }_n\left({\rho }_{Bn}{C}_{e+n{\psi }_B}-{\rho }_{Bn}{C}_{e-n{\psi }_B}+\alpha C_{e+n}-\alpha C_{e-n}\right)\\ +{\rho }_{A0}{\mathit{ϵ}}_e{\varphi }_n\left({\rho }_{Bn}{S}_{e+n{\psi }_B}-{\rho }_{Bn}{S}_{e-n{\psi }_B}+\alpha S_{e-n}-\alpha S_{e+n}\right)\\ +{\rho }_{A0}{\varphi }_e{\mathit{ϵ}}_n\left({\rho }_{Bn}{C}_{e+n{\psi }_B}+{\rho }_{Bn}{C}_{e-n{\psi }_B}-\alpha C_{e-n}-\alpha C_{e+n}\right)\\ +{\rho }_{An}{\mathit{ϵ}}_e{\mathit{ϵ}}_n\left({\rho }_{B0}+\alpha \right)\left(S_{e-n{\psi }_A}+S_{e+n{\psi }_A}\right)\\ +{\rho }_{An}{\varphi }_e{\mathit{ϵ}}_n\left({\rho }_{B0}-\alpha \right)\left(C_{e-n{\psi }_A}+C_{e+n{\psi }_A}\right)\\ +{\rho }_{An}{\mathit{ϵ}}_e{\varphi }_n\left({\rho }_{B0}-\alpha \right)\left(S_{e-n{\psi }_A}-S_{e+n{\psi }_A}\right)\\ +{\rho }_{An}{\varphi }_e{\varphi }_n\left({\rho }_{B0}+\alpha \right)\left(C_{e-n{\psi }_A}-C_{e+n{\psi }_A}\right)\end{array}$$

At this point, we can demodulate this signal with the VCO output ∝ sin(ω_et) to produce the “classic” Pound-Drever-Hall (PDH) error signal quadrature V_Y, and we can simultaneously demodulate with a phase-shifted VCO output ∝ cos(ω_et) to produce the other quadrature V_X. Assuming a “flat” mixer efficiency η_M, (i.e. V_Y = η_MV_PDS_e and V_X = η_MV_PDC_e) and keeping only terms at ω_n, the two quadratures simplify to

(28)$$\begin{array}{l}\frac{4V_Y(t)}{{\eta }_M{\eta }_{PD}{r}_1^2{P}_l}\approx -{\varphi }_e{\varphi }_n{\rho }_{A0}\left({\rho }_{Bn}{S}_{n\psi B}+\alpha S_n\right)+{\varphi }_e{\varphi }_n\left({\rho }_{B0}+\alpha \right){\rho }_{An}{S}_{n{\psi }_A}\\ +{\mathit{ϵ}}_e{\mathit{ϵ}}_n\left({\rho }_{B0}+\alpha \right){\rho }_{An}{C}_{n{\psi }_A}\end{array}$$

(29)$$\begin{array}{l}\frac{4V_X(t)}{{\eta }_M{\eta }_{PD}{r}_1^2{P}_l}\approx {\mathit{ϵ}}_e{\varphi }_n{\rho }_{A0}\left({\rho }_{Bn}-\alpha \right)S_{n\psi B}+{\varphi }_e{\mathit{ϵ}}_n{\rho }_{A0}\left({\rho }_{Bn}{C}_{n\psi B}-\alpha C_n\right)\\ +{\varphi }_e{\mathit{ϵ}}_n{\rho }_{An}\left({\rho }_{B0}-\alpha \right)C_{n{\psi }_A}-{\mathit{ϵ}}_e{\varphi }_n{\rho }_{An}\left({\rho }_{B0}-\alpha \right)S_{n{\psi }_A}.\end{array}$$

These expressions can be applied to many situations, and we now consider those relevant to the main text.

B.3 Case 1: Frequency noise only, ideal EOM, cavity resonant with carrier

If we consider only frequency noise, an ideal EOM, and the cavity resonant with the carrier, ϵ_e → 0, ϵ_n → 0, ρ_Bn → 1, ρ_B0 → 1 and ψ_B → 0, so the quadratures (Eqs. (28)–(29)) become

(30)$$V_Y(t)\to \sqrt{\frac{P_l{P}_e}{8}}{\eta }_M{\eta }_{PD}{r}_1^2{\varphi }_n\left(1+\alpha \right)\left({\rho }_{An}{S}_{n{\psi }_A}-{\rho }_{A0}{S}_n\right)$$

(31)$$V_X(t)\to 0.$$

where we have used Eq. (19) to rewrite the power in each EOM-driven sideband $\sqrt{P_e}\approx \sqrt{\frac{1}{2}{P}_l}{\varphi }_e$. The “amplitude” quadrature V_X is zero to lowest order, which is sensible considering the laser is exactly on resonance and we are only modulating its frequency. With both sidebands present (α = 1), the “phase” quadrature V_Y also provides the dynamic response of the classic PDH error signal

$$V_Y=\sqrt{\frac{P_l{P}_e}{2}}{\eta }_M{\eta }_{PD}{r}_1^2{\varphi }_n\left({\rho }_{An}{S}_{n{\psi }_A}-{\rho }_{A0}{S}_n\right).$$

If the lower sideband is not present, the signal is simply a factor of 2 smaller; the “classic” PDH error signal receives a contribution from both sidebands in proportion to the frequency noise.

Now that all the nonlinear operations (photodiode, mixer) are complete, we have a linear mapping from the original noise Ω(t) to the error signal V_Y. We can convert this into a complex transfer function by noting that Ω is the real part of $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_n{e}^{i{\omega }_{n}t}$, and that V_Y is the real part of

$${\tilde{V}}_Y\equiv -i\sqrt{\frac{P_l{P}_e}{2}}{\eta }_M{\eta }_{PD}{r}_1^2{\varphi }_n\left({\rho }_A\left({\omega }_n\right)e^{i{\omega }_{n}t}-{\rho }_A(0)e^{i{\omega }_{n}t}\right).$$

The complex transfer function is then the ratio

$$\frac{{\tilde{V}}_Y}{\tilde{\mathrm{\Omega }}}=\frac{\sqrt{2P_l{P}_e}{\eta }_M{\eta }_{PD}{r}_1^2{t}_1^{2}c{\tau }^2/L}{1+i2\tau {\omega }_n}=\frac{2{\varphi }_e{E}_l^2\beta {\tau }^2}{1+i2\tau {\omega }_n}.$$

where we have used the definitions of ϕ_n = Ω_n/ω_n and ρ_A (Eq. (21)) and defined the constant $\beta ={\eta }_M{\eta }_{PD}{r}_1^2{t}_1^{2}c/2L$. The cavity acts as a low-pass filter.

B.4 Case 2: Frequency noise only, ideal EOM, cavity resonant with upper sideband

If the cavity is instead resonant with the upper sideband, ρ_An → 1, ρ_A0 → 1 and ψ_A → 0, and the quadratures (Eqs. (28)–(29)) become

(32)$$V_Y(t)\to -\sqrt{\frac{P_l{P}_e}{8}}{\eta }_M{\eta }_{PD}{r}_1^2{\varphi }_n\left({\rho }_{Bn}{S}_{n{\psi }_B}-{\rho }_{B0}{S}_n\right)$$

(33)$$V_X(t)\to 0.$$

Following the same analysis as Case 1,

$$\begin{array}{l}\frac{{\tilde{V}}_Y}{\tilde{\mathrm{\Omega }}}=-\frac{\sqrt{P_l{P}_e}}{2}\frac{{\eta }_M{\eta }_{PD}c{\tau }^2{r}_1^2{t}_1^2/L}{1+i2\tau {\omega }_n}=-\frac{{\varphi }_e{E}_l^2\beta {\tau }^2}{1+i2\tau {\omega }_n}.\\ \end{array}$$

The error signal behaves exactly the same, but is half as large and inverted. The lower sideband does not play a role.

B.5 Case 3: Amplitude noise only, ideal EOM, cavity resonant with carrier

If we now consider only amplitude noise, an ideal EOM, and the cavity resonant with the carrier, ϵ_e, ϕ_n → 0, ρ_Bn → 1, ρ_B0 → 1, ψ_B → 0, and the quadratures (Eqs. (28)–(29)) become

(34)$$V_Y(t)\to 0$$

(35)$$V_X(t)\to \sqrt{\frac{P_l{P}_e}{8}}{\eta }_M{\eta }_{PD}{r}_1^2{\mathit{ϵ}}_n(1-\alpha )\left({\rho }_{A0}{C}_n+{\rho }_{An}{C}_{n{\psi }_A}\right)$$

The “phase” quadrature V_Y is zero, as expected, regardless of the presence of the lower sideband; this is consistent with the notion that changing the laser’s amplitude will not affect location of the error signal’s zero-crossing. Following the analysis of the previous cases,

$$\frac{{\tilde{V}}_X}{\widetilde{\mathit{ϵ}}}=\sqrt{\frac{P_l{P}_e}{2}}{\eta }_M{\eta }_{PD}{r}_1^2(1-\alpha )\left(1-\frac{c\tau t_1^2}{L}\left(\frac{1+i\tau {\omega }_n}{1+i2\tau {\omega }_n}\right)\right).$$

This signal is only nonzero if the lower sideband is either missing or is otherwise not identical to the upper sideband (i.e. α ≠ 1 and/or the EOM is not ideal).

B.6 Case 4: Amplitude noise only, ideal EOM, cavity resonant with upper sideband

If the cavity is instead resonant with the upper sideband, ϵ_e, ϕ_n → 0, ρ_An → 1, ρ_A0 → 1, ψ_A → 0, and

(36)$$V_Y(t)\to 0$$

(37)$$V_X(t)\to \sqrt{\frac{P_l{P}_e}{8}}{\eta }_M{\eta }_{PD}{r}_1^2{\mathit{ϵ}}_n\left({\rho }_{Bn}{C}_{n{\psi }_B}+{\rho }_{B0}{C}_n-2\alpha C_n\right)$$

meaning

$$\frac{{\tilde{V}}_X}{\widetilde{\mathit{ϵ}}}=\sqrt{\frac{P_l{P}_e}{2}}{\eta }_M{\eta }_{PD}{r}_1^2\left(1-\alpha -\frac{c\tau t_1^2}{L}\left(\frac{1+i\tau {\omega }_n}{1+i2\tau {\omega }_n}\right)\right)$$

with an overall offset determined by the lower sideband α. The presence of the lower sideband (α = 1) produces a PI-like behavior (compare to Eq. (5)).

$$\frac{{\tilde{V}}_X}{\widetilde{\mathit{ϵ}}}=-{\varphi }_e{E}_l^2\beta \tau \left(\frac{1+i\tau {\omega }_n}{1+i2\tau {\omega }_n}\right).$$

Appendix C: Error signal for wider range of VCO outputs

Figure 4 shows the two quadrature amplitudes of the photodiode signal, taken with a shorter cavity of L = 5 cm and for a wider range of VCO tune voltages than that of Fig. 2(c) in the main text. Over this range, the non-ideal response of the system components is visible. Note in particular the peaks of V_X (δ), systematically vary with ω_e by a factor of ~2, arising from a combination of VCO and EOM nonidealities (they are not “flat”). Large frequency noise will therefore produce amplitude noise even if it is perfectly tracked by a sideband, highlighting the fact that, while a large headroom is desirable to remain locked in the presence of large fluctuations, it is always preferable to engineer a stable, vibration-isolated cavity.

 

Fig. 4 Quadrature amplitudes of the photodiode signal for 11 different VCO voltages.

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Funding

NSERC (418459-12); FRQNT (NC-172619); Alfred P. Sloan Foundation (BR2013-088); CFI (228130); CRC (235060); INTRIQ; Centre for the Physics of Materials at McGill; T.M. acknowledges support by a Swiss National Foundation Early Postdoc Mobility Fellowship.

Acknowledgments

We thank Erika Janitz, Maximilian Ruf, Alexandre Bourassa, Simon Bernard, Abeer Barasheed, and Vincent Dumont for helpful discussions.

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