1. Introduction and motivation

Cavity-enhanced sensing is a powerful tool for measuring weak optical absorption signals, with applications ranging from molecular spectroscopy to biomolecular sensors [1]. By confining light within a cavity of finesse $\mathcal {F}$, the effective optical depth of its interaction with any intra-cavity material is enhanced by $\sim 2\mathcal {F}/\pi$ [2], making it possible to detect very weak changes in absorption with a compact apparatus. A wide variety of application-specific absorption sensing methods exist, usually based on detection of transmitted light [3–6] or measurement of the cavity lifetime [7]. Reflection-mode techniques like CEAMLAS [8,9], which can be immune to classical noise when the cavity is critically coupled, also provide access to single-port operation. Other sensing modalities employ heterodyne detection to suppress technical noise, such as NICE-OHMS [10,11], which allows for shot-noise-limited readout of narrow absorption features.

To continuously exploit the $2\mathcal {F}/\pi$ enhancement, the probe light must remain locked to the cavity resonance. As presented here, one simple solution is to employ a recently developed Pound-Drever-Hall [12] (PDH) based sideband-locking scheme that readily achieves MHz (delay-limited) feedback bandwidth [13] while providing continuous readout of intra-cavity absorption. This technique provides several advantages. First, the apparatus is comparatively simple, requiring a single laser, a single phase modulator, and low-cost RF electronics. Second, the feedback is all electronic, so neither the laser nor cavity require fast tuning capabilities, and the resulting high feedback bandwidth allows for a commensurately robust lock [14]. Third (in contrast to NICE-OHMS, e.g.), the technique is suited to sensing wide absorption features. Fourth (in contrast to CEAMLAS, e.g.), the light leaving the cavity is heterodyne-amplified by a large off-resonant carrier, making it ideal for situations requiring a weak probe beam. Finally, it requires only single-port access, enabling endoscopic applications.

Using a 5-cm-long, finesse-9,000 Fabry-Perot cavity as a testbed, we measure a sensitivity of $10^{-10}\;\textrm{cm}^{-1}/\sqrt {\textrm {Hz}}$ for frequencies between 0.03-1 MHz and a minimum sensitivity of ${6.6\times 10^{-11} \;\textrm{cm}^{-1}/\sqrt {\textrm {Hz}}}$ at 100 kHz. This proof-of-concept measurement is performed with a total of 3.5 mW landing on the cavity, with a phase-modulated sideband coupling 160 $\mathrm{\mu}$W into the cavity (520 mW circulating), and 38 $\mathrm{\mu}$W (from 93 mW circulating) collected by our photodiode after losses from diagnostic beam splitters. Importantly, without optimizing our choice of laser or modulation electronics – here chosen to optimize the robustness of the lock – the minimum sensitivity is within a factor of $\sim$10 of the shot noise limit, demonstrating a straightforward means of achieving low-noise transient readout with MHz bandwidth.

2. Measurement scheme and fundamental limits

To summarize the technique, Fig. 1 shows the employed optical circuit [13], which locks a laser sideband to a cavity resonance while monitoring intra-cavity losses. First, continuous-wave laser light at frequency $\omega _l$ is phase modulated by an electro-optical modulator (EOM) driven by a voltage-controlled oscillator (VCO) at frequency $\Omega$, thereby creating sidebands at frequencies ${\omega _l \pm \Omega }$. When the laser or a sideband is near a cavity resonance, phase modulation is converted to amplitude modulation in the reflected light, which is detected and demodulated with a delay-matched copy of the VCO output. The resulting “phase quadrature” $V_Y$ varies linearly with detuning, providing an error signal that is fed back to the VCO. In contrast, demodulation of a $\pi /2$-shifted copy of the signal yields the “amplitude quadrature” $V_X$; this value provides information about intra-cavity losses when a sideband is on resonance with the cavity, as discussed below. Note this scheme exploits the carrier as an inherently path-matched heterodyne local oscillator, which boosts the signals (both $V_X$ and $V_Y$) from the resonant sideband above the detector noise.

 

Fig. 1. Basic measurement scheme. Laser light at frequency $\omega _l$ is phase-modulated at frequency $\Omega$ by an electro-optical modulator (EOM) driven by a voltage-controlled oscillator (VCO), generating sidebands at ${\omega _l\pm \Omega }$. These beams are reflected from an optical cavity, generating amplitude modulation that is detected by a “fast” photodiode (PD). The PD signal and a $\pi /2$-shifted copy are simultaneously demodulated by a delay-matched copy of the original VCO signal, producing the steady-state, detuning-dependent voltages $V_X$ (“amplitude quadrature”, upper inset) and $V_Y$ (“phase quadrature”, lower inset). At the indicated sideband lock point, $V_Y$ provides an error signal to be fed back to the VCO via amplifiers and filters (transfer function $-A$), while $V_X$ records cavity losses.

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To predict the scheme’s fundamental sensitivity limit, consider a cavity of length $L$ with front and back mirrors of amplitude reflectivities $-r_1$ and $-r_2$, respectively (assuming real-valued $r_i$ for simplicity) and a “baseline” power loss fraction $2\delta$ per round trip (from just the mirrors). We model any medium inside the cavity as providing a power absorption coefficient per unit length $\alpha$. In the limit of high reflectivity, low mirror loss $\delta$, and low absorption $\alpha$, the cavity finesse is

For an incident phase-modulated laser beam (electric field ${ \propto e^{i\omega _l t + i\beta \sin {\Omega t}} }$, where $\beta$ is the modulation depth), a sideband that is exactly on resonance will experience an amplitude reflection coefficient

Importantly, $V_X$ depends on $r_{\textrm {res}}$, which itself depends on the cavity finesse (Eq. (2)) and can therefore provide a continuous record of the absorption coefficient $\alpha$ (Eq. (1)). The sensitivity of such a measurement depends on the variation of $P_X$ with $\alpha$ ($|dP_X/d\alpha |$) as well as how much noise is present. When detection is limited by the shot noise $\langle S_P\rangle = \sqrt {2 \bar {P} e/\eta }$ on the total average power $\bar {P}$ reaching the detector (where $e$ is the electron charge and $\eta$ is the detector responsivity), the sensitivity becomes

The black line in Fig. 2 shows how $\langle S_\alpha \rangle$ varies with the mode mismatch ${1-\epsilon }$ for an ideal “single-port” cavity (${r_2\rightarrow 1}$) and weak modulation (${\rho _{\mathrm {c}}\rightarrow 1}$), provided we collect all the reflected light. When perfectly coupled ($1-\epsilon =0$), the sensitivity is identical to that of the usual transmission techniques employing “symmetric” cavities (${r_1=r_2}$) of the same finesse and circulating power. For a single-port cavity, a factor of $\sqrt {2}$ improvement could nominally be achieved by locking a carrier (having power equal to that of our sideband) on resonance and directly monitoring the reflected power without heterodyne amplification, though this would require lower noise detectors for a given cavity power and fast tuning of either the cavity length or laser frequency. Alternatively, one could achieve the same $\sqrt {2}$ improvement by sending higher power into the other (high-reflectivity) mirror, and directly measuring the transmitted light; this would require access to both ports and (again) lower noise detectors, but one could then employ sideband locking to eliminate the need for cavity or laser tuning. Note that (perhaps counter-intuitively) our scheme does work even with critically coupled cavities, despite the absence of any reflected power on resonance; however, as with transmission measurements, this imposes another penalty of $\sqrt {2}$ in sensitivity since half the cavity light is lost. Imperfect cavity coupling (${\epsilon <1}$) adds additional uncoupled power to the reflection photodiode, thereby increasing the shot noise. Nevertheless, this penalty is a slowly increasing function in the range of “typical” values (dashed line). For reference, the red line shows the shot noise limit including losses for the commercially available mirrors discussed below.

 

Fig. 2. Shot-noise-limited sensitivity relative to that of an ideal transmission measurement through a symmetric cavity at the same circulating power. The black curve shows the relative sensitivity $1/\sqrt {\epsilon \rho _{\mathrm {c}}}$ versus uncoupled power fraction $1-\epsilon$ for an ideal cavity with $r_2,\ \rho _{\mathrm {c}}\rightarrow 1$ and $\rho _{\mathrm {sb}} \to 0$. The red curve corresponds to our chosen (commercially available) high-finesse mirrors (${r_1=0.999796}$, ${r_2=0.9999974}$), loss parameter ($\delta =143$ ppm) and modulation strength (${\rho _{\mathrm {c}}=0.8831}$, ${\rho _{\mathrm {sb}}=0.0590}$). The gray dashed line shows the coupling level for our system, aligned with free-space optics.

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3. Experimental setup

We experimentally test the technique using a 5-cm-long, finesse $\sim$ 9,000 cavity probed by a 1550 nm continuous-wave laser beam, phase-modulated at $\Omega /2\pi \sim 1$ GHz. Before the optical cavity, half the input beam is picked off by a non-polarizing beamsplitter to monitor $P_\mathrm {in}$, thereby allowing us to account for power drift and eliminate some classical laser noise (see Sec. 4.). Light reflected from the cavity is detected via the other port of this beamsplitter, where we measure both high- and low-frequency reflection signals; lacking a fast photodiode capable of detecting down to DC, we split the reflected light again so that half of the power lands on a DC-coupled ($\sim 150$ MHz bandwidth) “diagnostic” photodiode and the other half lands on an AC-coupled $\sim 2$ GHz bandwidth “signal” photodiode. Another DC-coupled detector is used to monitor the transmitted signal.

To estimate the cavity parameters, we simultaneously fit data from six measurements, three of which are shown in Fig. 3 (Appendix A for more details). The six independent parameters are: (1) the empty cavity power lifetime $\tau$, (2) the coupled power fraction $\epsilon$, (3) the back mirror reflectivity $r_2$, (4-5) the carrier and sideband power fractions $\rho _{\mathrm {c}}$ and $\rho _{\mathrm {sb}}$, and (6) the conversion gain factor $G$. We first measure the swept-cavity ringdown time [15] on the DC-coupled photodetector by rapidly sweeping the cavity length via the piezo-mounted back mirror (Fig. 3(a)). Next, we slowly sweep the cavity length to monitor quasi-steady-state reflected and transmitted power fractions $R$ (Fig. 3(b)) and $T$ without phase modulation. Finally, we turn on phase modulation and measure the reflected signal and $V_X$ (Fig. 3(c)) near the lower sideband and carrier frequencies. Fitting the dataset simultaneously (black curves) yields ${\tau = 495.9\pm 0.2}$ ns, ${\epsilon = 0.749\pm 0.005}$, ${r_2=0.9999974\pm 0.0000001}$, ${\rho _{\mathrm {c}}=0.8831\pm 0.0007}$, ${\rho _{\mathrm {sb}}=0.0590\pm 0.0004}$ and ${G=-80\pm 3}$ V/W for the empty cavity. The remaining parameters, ${t_1^{2}=0.00041 \pm 0.00001}$, and ${L=5.2 \pm 0.1}$ cm are measured directly.

 

Fig. 3. Empty cavity characterization with simultaneous fits. We extract the cavity lifetime $\tau$, demodulation gain $G$, power coupling $\epsilon$, back mirror reflectivity $r_2$, and carrier (sideband) power fraction $\rho _{\mathrm {c}}$ ($\rho _{\mathrm {sb}}$) by simultaneously fitting six cavity length sweep measurements, examples of which are shown: (a) swept ring-down spectroscopy; (b) cavity reflection $R$ without phase modulation; (c) demodulated PDH beat signal $V_X$ near the lower sideband (transmission $T$ and modulated reflection of carrier and sidebands not shown). Measurements are normalized by off-resonant values in (a) and (b), and $GP_\mathrm {in}$ in (c).

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4. Absorption sensitivity

Once locked to the (lower) sideband, the absorption sensitivity can be estimated from the noise power spectral density (PSD) of $V_X$ (Fig. 4(a), red) scaled by $dV_X/d\alpha$, similar to Eq. (4). In principle, we could predict $dV_X/d\alpha$ from the measured cavity parameters, but since

 

Fig. 4. Estimation of absorption sensitivity. (a) Voltage noise power spectral densities (PSDs). Red: noise of $V_X$ with the lower sideband locked to the cavity. Teal: noise while unlocked and off resonance. Yellow: shot noise limit (SNL) and electronics noise (laser off). Black: SNL for the collected light ($625\ \mathrm{\mu}$W). The vertical dotted line marks our $V_X$ measurement bandwith ($1/(4\pi \tau )=160$ kHz). (b) $\tau$-dependence of $V_X$ for the sensitivity estimate outlined in the main text (Eq. (5)). Black curve is a linear fit (see main text). Top axis shows predicted $\alpha$-dependence of $V_X$ ($\alpha =-\frac {1 - r_1r_2 + \delta }{L} + \frac {n}{c\tau }$). (c) Estimated absorption sensitivity for the cavity, including the cavity’s dynamical response to absorption changes (Appendix B.). Red curve is the raw sensitivity, and the orange curve shows the sensitivity after subtracting (in the time domain) the directly monitored laser noise. The raw measurement noise floor is $\sim 7\times 10^{-11}~\mathrm {cm}^{-1}/\sqrt {\mathrm {Hz}}$ at 100 kHz (a factor of 11 above the SNL), and $\sim 2\times 10^{-10}~\mathrm {cm}^{-1}/\sqrt {\mathrm {Hz}}$ at 1 MHz (a factor of 6 above the SNL). With subtraction, the noise floor decreases to a factor of 10 from the SNL at 100 kHz and a factor of 4 at 1 MHz.

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For the chosen components, the dominant noise is technical. As shown in Fig. 4(a), with the laser off, we can measure the noise from our readout electronics, but even with the addition of the theoretical shot noise (yellow curve) it lies well below the $V_X$ noise spectra observed when the laser is on and the cavity is locked or unlocked. When unlocked (teal curve), $V_X$ measures the electronic noise and the ($\sim$flat, classical) laser amplitude noise near the demodulation frequency $\Omega /(2\pi )\sim 1$ GHz away from the carrier. In contrast, when a sideband is locked on resonance (red curve), $V_X$ is sensitive to low-frequency amplitude noise from several sources. First, there is the laser’s low-frequency classical amplitude noise, combined with the VCO’s inherent amplitude noise. In addition, as the feedback loop adjusts the VCO frequency to keep the sideband on resonance, the (frequency-dependent) power of the VCO can change, introducing yet another source of noise. We suspect the latter contributes the majority of noise below our locking bandwidth ($\sim$2 MHz), where the feedback gain is still large enough to modulate the VCO setpoint. Nonetheless, as shown in Fig. 4(c), the sensitivity is $\sim 7.4\times 10^{-11}~\mathrm {cm}^{-1}/\sqrt {\mathrm {Hz}}$ at 100 kHz (a factor of 11 above the shot noise limit) for ${38\ \mathrm{\mu}}$W collected from the cavity; at 1 MHz, the measurement noise floor is a factor of 6 above the shot noise limit (SNL).

The above is the main result of this work, but we also report preliminary efforts to cancel known sources of classical noise. Appendix C details the noise sources we considered and the methods for canceling them. In most cases, cancellation is limited by a combination of technical noise in our auxiliary monitors and drift in the lock frequency. The most meaningful improvement comes from subtracting the laser’s classical amplitude noise. Specifically, the signal from the photodiode monitoring the input laser power is appropriately scaled, delayed, and subtracted from $V_X(t)$ in the time domain, resulting in a noise-reduced signal, the PSD of which is shown in Fig. 4(c) (orange). This subtraction is effective in eliminating the large peak in laser amplitude noise near 500 kHz, and results in a sensitivity of ${\sim 6.6\times 10^{-11}~\mathrm {cm}^{-1}/\sqrt {\mathrm {Hz}}}$ at 100 kHz, a factor of 10 above the shot noise limit.

5. Outlook

In summary, we demonstrated a simple, single-port, cavity-enhanced absorption readout with sensitivity ${\sim 10^{-10}\;\textrm{cm}^{-1}/\sqrt {\textrm {Hz}}}$ from 0.03 to 1 MHz, and a robustly locked, low-power probe.

The technique should readily adapt to smaller cavities or on-chip resonators; the only requirement is that the modulation frequency – which can be 10’s of GHz with standard components – is larger than the resonator linewidth. However, while shorter and lower-finesse cavities exhibit higher bandwidth, this comes at the expense of sensitivity (see Eq. (4)); this tradeoff fundamentally arises from the characteristic time required for light to be absorbed in a material.

One application of this technique is radiation dosimetry based on water radiolysis [16,17]: when pulses of ionizing radiation deposit energy in water, this produces solvated electrons that exhibit a broad optical absorption feature in the visible and near infrared [18,19]. Typically, this signal persists for microseconds, and can be observed by monitoring optical absorption over a meter-scale path [17]. Similar signals could be observed by folding the optical path within a cavity. Assuming the same circulating power and assuming a sensitivity 10 times the shot noise limit, such systems should be capable of resolving radiation pulses typically administered in clinical settings. With the option to measure in reflection, one could even envision constructing a small-footprint endoscopic probe with a fiber cavity [20] to measure local radiation dose at point-like locations. Achieving dosimetry with water, a nearly tissue-equivalent medium, is of interest for existing and future (especially FLASH [21]) radiation-based cancer treatments.

Other cavity absorption sensing techniques are optimized for different niches with wildly varied system parameters, so comparing sensitivities requires some caution. For example, NICE-OHMS achieved $10^{-14} \textrm { cm}^{-1}/\sqrt {\textrm {Hz}}$ with kHz cavity bandwidth and 300 W circulating in a 50-cm-long, finesse-100,000 cavity. As per Eq. (4), improved sensitivity is expected for longer, higher-finesse cavities and higher power. Additionally, while NICE-OHMS operates at the shot noise limit, the technique is actually less sensitive per circulating watt than an ideal transmission measurement. To get a sense of scale, if we performed our measurements with 300 W circulating in a 50-cm-long, finesse-100,000 cavity (realized, e.g., with a $|t_1|^{2}=60$ ppm input mirror and a few-ppm-loss back mirror, corresponding to 18 mW collected), the sensitivity would be $\sim 2\times 10^{-14} \textrm { cm}^{-1}/\sqrt {\textrm {Hz}}$, after assuming technical noise 10 times greater than shot noise. If we collected only 38 $\mathrm{\mu}$W as above (i.e., 630 mW circulating), the sensitivity would be $\sim 5\times 10^{-13} \textrm { cm}^{-1}/\sqrt {\textrm {Hz}}$. Ultimately, however, our equipment is optimized for detecting high-speed transients with low circulating power, and would likely not be appropriate for low-speed applications. Regardless, Appendix D provides a comparison with this and several relevant techniques.

While our technique is developed for measurements of broadband absorption transients, it could in principle be adapted to measure narrower absorption lineshapes. The sideband (probe) frequency can be rapidly adjusted by varying the lock point (red dot in Fig. 1) via an offset voltage applied to the feedback electronics; for larger frequency excursions, the cavity length could be varied while the sideband remains locked on resonance. During either sweep, $V_X$ would provide a record of the absorption lineshape. The achievable sweep speed would be limited by the least of the feedback bandwidth, the cavity bandwidth, and the dynamics of the absorber under study, potentially allowing one to exploit the excellent sensitivity observed in the 0.3-1 MHz range. Importantly, to perform such sweeps reliably would likely require additional system complexity such as laser stabilization.

Finally, the presented scheme may offer opportunities to further improve performance without requiring lower-noise components; for example, an approach in the vein of cavity ringdown or CEAMLAS could be realized by modulating the laser or sideband amplitude electronically, with the characteristic response time (phase and amplitude shift) of $V_X$ revealing the cavity lifetime.

Appendix A: extracting system parameters

We perform a set of measurements using the voltages of four photodetectors (the “pick-off” from the first beam splitter, cavity transmission, “diagnostic” (DC-coupled) slow reflection and fast (AC-coupled) reflection) to extract unknown system parameters. The input mirror’s power transmission ${e^{-\delta /2}t_1^{2} \approx t_1^{2}}$ (which gives $r_1=\sqrt {1-t_1^{2}}$), the cavity length $L$ and the transmission path gain factor $G_T = V_T/P_T$ (where $V_T$ is the transmission diode voltage reading and $P_T$ the power transmitted by the cavity) are directly measured. By rapidly sweeping the cavity length by means of a piezo-mounted back mirror, we observe ringdown measurements like the one in Fig. 3(a) to extract the cavity lifetime $\tau$. In addition, by sweeping the cavity length more slowly, we acquire a set of five measurements: cavity reflection $V_R$ and transmission $V_T$ without phase modulation, cavity reflection with phase modulation around the carrier $V_{R\textrm {c}}$ and the lower sideband resonance $V_{R\textrm {sb}}$, and the demodulated PDH amplitude quadrature $V_X$ around the lower sideband resonance. These five curves are then simultaneously fit to extract the remaining parameters (carrier and sideband relative power $\rho _{\mathrm {c}}$ and $\rho _{\mathrm {sb}}$, back mirror amplitude reflectivity $r_2$, power coupling parameter $\epsilon$ and the gain factors in the slow ($G_{\mathrm {slow}}$) and fast ($G$) reflection paths). To account for incoming power fluctuations, we normalize the voltage sweeps using the readings from the pick-off diode. The swept ringdown [15] measurements are fit with the function

(6)$$V_{\textrm{RD}} = a e^{{-}t/2\tau} \cos{((\omega + bt ) t + c )} + d + g t + h t^{2}$$

where $\tau$, $a$, $b$, $c$, $d$, $g$, and $h$ are fit parameters ($b$ accounts for acceleration due to mechanical noise, while $g$ and $h$ account for background drifts). The equations for the simultaneous fits are:

(7)$$\frac{ V_R }{P_\mathrm{in} } = G_{\mathrm{slow}} \left( (1-\epsilon)r_1^{2} + \epsilon R(\Delta L)\right)$$

(8)$$\frac{ V_T }{P_\mathrm{in} } = G_T \epsilon T(\Delta L)$$

(9)$$\frac{ V_{R\textrm{c}} }{P_\mathrm{in}} = G_{\mathrm{slow}} \left( (1-\epsilon)r_1^{2} + \epsilon ( \rho_{\mathrm{c}} R(\Delta L) + 2\rho_{\mathrm{sb}} r_1^{2}) \right)$$

(10)$$\frac{ V_{R\textrm{sb}}}{P_\mathrm{in}} = G_{\mathrm{slow}} \left( (1-\epsilon) r_1^{2} + \epsilon (\rho_{\mathrm{c}} r_1^{2}+\rho_{\mathrm{sb}} r_1^{2}+ \rho_{\mathrm{sb}} R(\Delta L)) \right)$$

(11)$$\frac{ V_X }{P_\mathrm{in} } = 2 G \epsilon \sqrt{\rho_{\mathrm{c}} \rho_{\mathrm{sb}}}\; r_1 \left( r_1+\Re{r(\Delta L)} \right)$$

with the cavity fractional amplitude reflection $r$, fractional power reflection $R$, and transmission $T$ given by the high finesse approximation expressions:

(12)$$r(\Delta L) \approx \frac{ t_1^{2} } { \frac{\pi}{\mathcal{F}} - 2ik_0\Delta Ln } - r_1$$

(13)$$R(\Delta L) \approx \frac{ t_1^{4} - 2\pi r_1 t_1^{2}/\mathcal{F} } { \frac{\pi^{2}}{\mathcal{F}^{2}} + (2 k_0 \Delta L n)^{2} } +r_1^{2}$$

(14)$$T(\Delta L) \approx \frac{ t_1^{2} t_2^{2} } { \frac{\pi^{2}}{\mathcal{F}^{2}} + (2 k_0 \Delta L n)^{2} },$$

where ${k_0 \approx \omega _l/c}$ is the resonant wavenumber in vacuum, the time-dependent displacement from resonance $\Delta L = v_a(v_b t^{2} + t - t_0)$, $v_j$ are fit parameters representing the sweep trajectory, $t_0$ is a fit time offset, and the empty cavity finesse $\mathcal {F}=\pi c \tau /(Ln)$. In these equations, the fit parameters are $\tau$, $\rho _{\mathrm {c}}$, $\rho _{\mathrm {sb}}$, $r_2$, $\epsilon$, $G$ and $G_{\mathrm {slow}}$.

Appendix B: cavity response to modulated absorption

Here we use a classical input-output formalism to derive the changes in $V_X$ (the collected and demodulated amplitude quadrature of the Pound-Drever-Hall reflection beat) due to time-varying intra-cavity absorption. We find that $V_X$ responds to modulated absorption as a low pass filter with cutoff frequency equal to the inverse of the cavity’s amplitude ringdown time $1/4\pi \tau$, a sensible result since $V_X$ itself is proportional to the electric field of the sideband.

We start by deriving an equation of motion for the right-moving intra-cavity field $E_{\mathrm {circ}}$ (defined just past the input mirror). At time $t$, this is equal to the incident field transmitted by the front mirror $t_1 E_{\mathrm {in}}(t)$ plus the field $E_{\mathrm {circ}}(t-\Delta t)$ from a roundtrip time ${\Delta t=2Ln/c}$ earlier ($L$ is the cavity length, $n$ the intra-cavity medium refractive index and $c$ the speed of light), after bouncing off both mirrors once and suffering amplitude losses $\delta +\alpha (t)L$:

(15)$$E_{\mathrm{circ}}( t ) \approx t_1 E_{\mathrm{in}}(t) + \left({-}1 +r_1 + r_2 - \delta - \alpha(t) L\right) E_{\mathrm{circ}} (t - \Delta t),$$

where we again assume lossless mirrors with amplitude reflectivity $r_i$ and transmissivity $t_i$, with $\delta$ capturing their internal losses, and $\alpha (t)$ representing the time-varying extrinsic absorption (power loss per unit length). We have also approximated $\ln (r_i) \approx -1 +r_i$ in the limit $r_i \to 1$. Assuming the roundtrip time is negligible compared to all timescales of interest (i.e., ${2Ln/c\ll \tau }$ and the absorption modulation frequency), in the limit $\Delta t \rightarrow 0$, Eq. (15) approximates the differential equation

(16)$$\frac{d E_{\mathrm{circ}}(t) } {d t} \approx{-} \frac{ \left( 2 - r_1 - r_2 + \delta + \alpha(t) L \right) c E_{\mathrm{circ}} (t ) } {2Ln} + \frac{ t_1 c E_{\mathrm{in}}(t) } {2Ln}$$

(17)$$={-} \frac{ \kappa_{\mathrm{in}} + \kappa_0 + \kappa_{\alpha} (t) } {2} E_{\mathrm{circ}}(t) + \sqrt{ \frac{ \kappa_{\mathrm{in}} c } {2Ln} } E_{\mathrm{in}}(t),$$

in a frame rotating at the cavity resonance frequency $\omega _c$, with total cavity power loss rate

(18)$$\kappa(t) = \frac{c t_1^{2} }{ 2 L n } + \frac{c \left(t_2^{2} + 2\delta\right) }{ 2 L n } + \frac{c \alpha(t) L }{L n}$$

(19)$$\equiv \kappa_{\mathrm{in}}+\kappa_0+\kappa_{\alpha}(t),$$

where we approximate ${r_i \approx 1-t_i^{2}/2}$ in the high-finesse limit ${t_i \to 0}$.

We now let the time-varying absorption be a sinusoidal function $\alpha (t) = \alpha _0 \cos (\omega _{\alpha } t)$ (for constants $\alpha _0$ and $\omega _\alpha$), which will in turn modulate the circulating and output fields. Assuming the fluctuations are small, the circulating field can be written as a mean value $\bar {E}_{\mathrm {circ}}$ at the laser frequency plus a small fluctuating term $\Delta E_{\mathrm {circ}}(t)$ due to the modulated absorption, yielding

(20)$$\frac{d \left( \bar{E}_{\mathrm{circ}}+\Delta E_{\mathrm{circ}}(t) \right) } {dt} ={-}\frac{\kappa_{\mathrm{in}}+\kappa_0+\kappa_{\alpha}(t)}{2} \left( \bar{E}_{\mathrm{circ}}+\Delta E_{\mathrm{circ}}(t) \right) +\sqrt{ \frac{\kappa_{\mathrm{in}}c}{2Ln} } \bar{E}_{\mathrm{in}}.$$

Keeping only time-varying terms, and neglecting quadratic terms in the fluctuation, we find:

(21)$$\frac{d \left(\Delta E_{\mathrm{circ}}(t) \right)} {dt} \approx{-} \frac{ \kappa_{\mathrm{in}} + \kappa_0 } {2} \Delta E_{\mathrm{circ}}(t) - \frac{ \kappa_{\alpha} (t) } {2} \bar{E}_{\mathrm{circ}}.$$

Substituting $\kappa _{\alpha }(t)=c \alpha _0 \cos (\omega _{\alpha } t)/n$ and working in Fourier space,

(22)$$\Delta E_{\mathrm{circ}}(\omega) ={-}\underbrace{\frac{1}{ -i\omega+ \frac{ \kappa_{\mathrm{in}} + \kappa_0 } {2} }}_{\chi(\omega)} \Big( \delta(\omega-\omega_{\alpha}) + \delta(\omega+\omega_{\alpha}) \Big) \frac{ c\alpha_0 }{4n} \bar{E}_{\mathrm{circ}}$$

where $\delta (\omega \pm \omega _{\alpha })$ is the Dirac delta function, the Fourier frequency $\omega$ is the detuning from resonance, and we defined the cavity susceptibility $\chi (\omega ) \equiv 1/\left ( -i \omega + \frac { \kappa _{\mathrm {in}} + \kappa _0 } {2} \right )$. We use $FT \{f(t)\} = f(\omega ) = \frac {1}{2\pi } \int _{-\infty }^{+\infty } f(t) e^{i \omega t} dt $ for the Fourier transform of a function $f(t)$.

Meanwhile, $\bar {E}_{\mathrm {circ}}$ can be found from the time-independent terms of Eq. (20):

(23)$$\bar{E}_{\mathrm{circ}} = \underbrace{\frac{2}{\kappa_{\mathrm{in}} + \kappa_0}}_{\chi(0)} \sqrt{ \frac{\kappa_{\mathrm{in}} c }{ 2 Ln } } \bar{E}_{\mathrm{in}},$$

which, upon substitution into Eq. (22), yields the modulation in the circulating power:

(24)$$\Delta E_{\mathrm{circ}}( \omega) ={-}\chi(\omega) \Big( \delta(\omega-\omega_{\alpha}) + \delta(\omega+\omega_{\alpha}) \Big) \frac{ c\alpha_0 }{4n} \chi(0) \sqrt{ \frac{\kappa_{\mathrm{in}} c }{ 2 Ln } } \bar{E}_{\mathrm{in}}.$$

The cavity reflected field, given by $E_{r}(t)= -r_1 \bar {E}_{\mathrm {in}} + t_1 \left ( \bar {E}_{\mathrm {circ}} +\Delta E_{\mathrm {circ}}(t)\right )$, becomes

(25)$$E_{r}(\omega) ={-}r_1 \bar{E}_{\mathrm{in}} + t_1 \left( \bar{E}_{\mathrm{circ}} +\Delta E_{\mathrm{circ}}(\omega)\right)$$

in the frequency domain. Substituting Eq. (23) and (24) and $t_1 = \sqrt {2 \kappa _{\mathrm {in}} Ln/c }$ (Eq. (19)) into Eq. (25), we find that there are only three frequencies with nonzero field, at DC and $\pm \omega _a$:

(26)$$E_{r}(0) =\left({-}r_1 + \kappa_{\mathrm{in}} \chi(0) \right) \bar{E}_{\mathrm{in}},$$

(27)$$E_r({\pm}\omega_\alpha) ={-} \kappa_{\mathrm{in}} \chi(0) \chi({\pm}\omega_{\alpha}) \frac{ c \alpha_0 }{ 4n } \bar{E}_{\mathrm{in}}.$$

Recalling that we are working in a frame rotating at $\omega _c$, the outgoing reflected field will have a carrier at the resonance frequency $\omega _c$ and two sidebands detuned by the absorber frequency at $\omega _c \pm \omega _{\alpha }$. In Fig. 5(a) and (b), we plot the magnitude and phase of ${E_{r}(\omega _{\alpha })}$ (Eq. (27)), and we observe that the cavity acts as a low-pass filter with cutoff frequency ${\omega _{\textrm {cutoff}}= (\kappa _{\mathrm {in}}+\kappa _0)/2 \equiv 1/2\tau }$ for intra-cavity modulated absorption, as would be expected from the cavity amplitude decay time.

 

Fig. 5. Modulated intra-cavity absorption. (a) Magnitude and (b) phase of the outgoing absorption modulation sideband $E_r(\omega _{\alpha })$ (Eq. (27)) as a function of the absorption modulation frequency $\omega _{\alpha }/2\pi$. We observe the cavity low-pass behaviour with cutoff frequency $1/4\pi \tau =160$ kHz (vertical dashed lines). Horizontal dashed lines mark the −3 dB point in magnitude and $3\pi /4$ phase-shift.

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To get an expression for the time-modulated $V_X(t)$, we have to account for the phase-modulation of the incoming laser beam, i.e., include the promptly reflected, off-resonant carrier and (upper) sideband, and lock the other (lower) sideband on resonance with the cavity ($\omega _c\to \omega _l-\Omega$, where $\omega _l$ is the carrier frequency and $\Omega$ is the modulation frequency). The incoming field can be written as

(28)$$E_{\mathrm{in,PDH}}(t)= E_0 e^{i\omega_l t}\sqrt{\rho_{\mathrm{c}}} + E_0 e^{i (\omega_l+\Omega)t}\sqrt{\rho_{\mathrm{sb}}} - E_0 e^{i (\omega_l-\Omega)t}\sqrt{\rho_{\mathrm{sb}}},$$

where now we let the incoming field be a fast-varying function of time with amplitude $E_0$. Upon reflection from the cavity, having the lower sideband locked on resonance and the intra-cavity field time-modulated by transient absorption, the outbound field becomes

(29)$$\begin{aligned} E_{r, \mathrm{PDH}}(t) =& - r_1 E_0\left( e^{i\omega_l t}\sqrt{\rho_{\mathrm{c}}} + e^{i (\omega_l+\Omega) t}\sqrt{\rho_{\mathrm{sb}}}\right)\\ &-\Big({-}r_1 E_0 + t_1 \left( \bar{E}_{\mathrm{circ}} +\Delta E_{\mathrm{circ}}(t) \right) \Big) e^{i(\omega_l-\Omega) t} \sqrt{\rho_{\mathrm{sb}}} . \end{aligned}$$

We can find $\Delta E_{\mathrm {circ}}(t)$ by taking the inverse Fourier transform of Eq. (24):

(30)$$\Delta E_{\mathrm{circ}}(t) ={-} \Big( e^{i\omega_{\alpha}t} \chi(-\omega_{\alpha}) + e^{{-}i\omega_{\alpha}t} \chi(\omega_{\alpha}) \Big) \frac{c\alpha_0}{4n}\chi(0) \sqrt{\frac{\kappa_{\mathrm{in}}c}{2Ln}} E_0.$$

Substituting Eq. (23) and (30) into Eq. (29), and noting that $-r_1+\kappa _{\mathrm {in}}\chi (0)=r_{\mathrm {res}}$,

(31)$$\begin{aligned}E_{r, \mathrm{PDH}}(t) = &- r_1 E_0 \left( e^{i\omega_l t}\sqrt{\rho_{\mathrm{c}}} + e^{i(\omega_l+\Omega)t}\sqrt{\rho_{\mathrm{sb}}} \right)- r_{\mathrm{res}}E_0e^{i(\omega_l-\Omega)t}\sqrt{\rho_{\mathrm{sb}}}\\ &+(r_1+r_{\mathrm{res}})E_0\sqrt{\rho_{\mathrm{sb}}} \frac{c \alpha_0}{4n} \left( e^{i(\omega_l-\Omega+\omega_{\alpha})t} \chi(-\omega_{\alpha}) + e^{i(\omega_l-\Omega-\omega_{\alpha})t} \chi(\omega_{\alpha}) \right). \end{aligned}$$

Finally, we find $V_X(t)$ by first calculating $P_{R}(t)=E_{r, \mathrm {PDH}}(t)^{*} E_{r, \mathrm {PDH}}(t)$, collecting it with a photodetector, shifting by $\pi /2$ and mixing with a reference signal at $\sin (\Omega t)$. Neglecting terms oscillating at $\geq \Omega$,

(32)$$\begin{aligned} V_X(t) = &- 2 G P_\mathrm{in} \sqrt{\rho_{\mathrm{c}}\rho_{\mathrm{sb}}} \; r_1 \left(r_1+r_{\mathrm{res}}\right) \left(1 - \Big( \chi(-\omega_{\alpha})+\chi(\omega_{\alpha}) \Big) \frac{ c\alpha_0 }{4n} \cos(\omega_{\alpha}t) \right.\\ &\left. - \Big( \chi(-\omega_{\alpha})-\chi(\omega_{\alpha}) \Big) \frac{ c\alpha_0 }{4n} i\sin(\omega_{\alpha}t) \right) \end{aligned}$$

where $G$ is the gain of the detector and mixer circuits and we define $P_\mathrm {in}\equiv E_0^{2}$. We conclude that intra-cavity time-modulated absorption adds a fluctuating term at the absorption modulation frequency $\omega _{\alpha }$ to $V_X$, which is filtered by the cavity behaving as a low-pass with cutoff frequency $1/4\pi \tau$, determined by the cavity susceptibility $\chi (\pm \omega _{\alpha })= 1/\left (\frac {1}{2\tau }\mp i\omega _{\alpha }\right )$.

Appendix C: subtraction of classical noise

To better understand and improve technical noise, we simultaneously monitor the error signal, VCO output power, and incident laser power, which provide some information about phase noise, changes in modulation frequency, and laser amplitude noise, respectively. If one of these quantities is strongly correlated with our excess noise, we can appropriately scale, delay, filter, and subtract these signals from $V_X$ in the time domain, thereby helping identify the most important noise sources and partially mitigating their impact. When possible, we determine the parameters for subtraction from our theoretical model of the system. The rest are the result of minimization of the average PSD after time-domain subtraction over a targeted frequency range. All subtraction parameters are estimated from a signal-free “calibration” data set, and fixed for the subtraction process performed on the “measurement” data set.

Fluctuations in the intensity of the incoming laser produce correlated noise in $V_X$; for small fluctuations, we expect a frequency-dependent, linear scaling $H(\omega )$ in the frequency domain, such that $P_X(\omega ) \propto H(\omega ) P_\mathrm {in}(\omega )$. We derived $H(\omega )$ in a manner similar to the derivation presented in Appendix B., considering sinusoidal modulation of the incoming beam’s intensity at frequency $\omega$ and calculating the appropriate quadrature of the PDH signal. The resulting transfer function for a cavity with a sideband on resonance is

(33)$$H(\omega) = \sqrt{\rho_{\mathrm{c}}\rho_{\mathrm{sb}}} r_1^{2} + \frac{1}{4}\sqrt{\rho_{\mathrm{c}}\rho_{\mathrm{sb}}}r_1\Big(r(-\omega)+r(0) + r^{*}(\omega)+r^{*}(0)\Big),$$

(34)$$\approx \frac{1}{2} \sqrt{\rho_{\mathrm{c}}\rho_{\mathrm{sb}}} t_1^{2} \Big( \frac{\mathcal{F}}{\pi}+ \frac{1} { \frac{\pi}{\mathcal{F}} + 2iLn\omega/c} \Big),$$

using the high finesse cavity fractional amplitude reflection $r(\omega ) \approx \frac { t_1^{2} } { \pi /\mathcal {F} - 2iLn\omega /c } - r_1$. This filter is applied to the pick-off photodiode voltage by multiplying in the Fourier domain then transforming back to the time domain (though this can also be done in real time), which produces a filtered intensity voltage $V_{FI}(t)$ that should be on equal footing with $V_X$.

In principle, fluctuations in the VCO output power and fluctuations in the error monitor (a proxy for cavity detuning) also have a frequency-dependent influence on $V_X$. But we found the results below to be insensitive to changes in the subtraction method, likely due to noise in the auxiliary measurements we wish to subtract.

Since the signals are measured with different devices having different gains and delays at different locations in the beam path, subtraction requires that we scale each voltage by a constant and find the appropriate delay. In the case of the filtered laser intensity and VCO amplitude, we can use the mean value of the signals to determine the scaling factor because

(35)$$V_X(t) = \bar{V}_X ( 1 + n_L(t))(1 + n_O(t))$$

(36)$$\approx \bar{V}_X ( 1 + n_L(t) + n_O(t))$$

is approximately proprotional to both, where $n_L(t)$ is the relative classical noise in the laser power and $n_O(t)$ is the relative fluctuation in VCO output amplitude. Specifically,

(37)$$n_L(t) = \frac{V_{FI}(t)}{\bar{V}_{FI}}-1$$

(38)$$n_O(t) \approx \frac{1}{2}\left(\frac{V_{OP}(t)}{\bar{V}_{OP}} -1\right),$$

where mean values are denoted with a bar and $V_{OP}(t)$ is a voltage proportional to the power of the VCO output (note that, for small fluctuations, the noise in the VCO amplitude is half the noise in its power). Ultimately we obtain a noise-reduced subtracted voltage $V_{\mathrm {sub}}$ via

(39)$$V_{\mathrm{sub}}(t) = V_X(t) - \bar{V}_X\left(\frac{V_{FI}(t - \Delta t_{FI})}{\bar{V}_{FI}}-1\right) - \frac{\bar{V}_X}{2}\left(\frac{V_{OP}(t - \Delta t_{OP})}{\bar{V}_{OP}} -1\right) - a_{E} V_{E}(t-\Delta t_{E}),$$

where we have now included measurements of the error monitor voltage $V_{E}$ with unknown scaling factor $a_{E}$, and added empirical time delays $\Delta t_{FI}$, $\Delta t_{OP},$ and $\Delta t_{E}$.

To determine the scaling factor for the error signal and the appropriate time delays for each noise source, we resort to empirical methods, minimizing the area under the PSD of $V_{\mathrm {sub}}$ over a different frequency range for each signal (chosen based on where noise is observed to be most correlated). Figure 6 shows the results of of subtracting first the input power, then the error monitor, then VCO noise, illustrating that (at least with our equipment) classical laser noise subtraction yields the greatest benefit for improving the sensitivity of our system. It is worth noting that this data set was taken with the cavity precisely tuned to sideband resonance; any static detuning (arising, for example, from drifts in the lock setpoint) makes the system highly sensitive to detuning noise, in which case we found that subtraction of the error monitor voltage largely cancels the additional noise.

 

Fig. 6. Noise reduction by time-domain subtraction. The red curve shows the PSD of locked $V_X$, while orange, blue, and purple show the signal PSD after successively subtracting the appropriately scaled and shifted input power monitor, error monitor, and VCO output. The input power is filtered with the appropriate transfer function, derived in the text, as well as delayed by $\Delta t_{FI}=50$ ns (likely not a real delay, and instead an empirical correction that partially compensates for differences in the equipment used to measure the transfer function). Time delays for the other subtractions were not found to improve the result and were left at zero. Horizontal bars show the bandwidth targeted by each subtraction.

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Appendix D: comparing the sensitivity of different techniques

Table 1 shows the reported sensitivity and system parameters (when provided) for a range of cavity-enhanced absorption sensing techniques. A broader review can be found in Refs. [3,6,7].

Table 1. Comparison of sensitivity and experimental parameters for different cavity-enhanced absorption sensing techniques. NICE-OHMS: noise-immune cavity-enhanced optical heterodyne molecular spectroscopy. CRDS: cavity ring-down spectroscopy. CEAMLAS: cavity enhanced amplitude modulated laser absorption spectroscopy. OF-CEAS: optical feedback cavity-enhanced absorption spectroscopy. CE-DCS: cavity-enhanced dual comb spectroscopy.

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Funding

Canada Foundation for Innovation (229003, 33488); Natural Sciences and Engineering Research Council of Canada (RGPIN 2018-05635, RGPIN 2020-04095, RGPIN 435554-13); Canada Research Chairs (231949, 235060); MUHC Foundation; Varian Medical Systems; TransMedTech; MEDTEQ+; Institut Transdisciplinaire d'Information Quantique (INTRIQ); McGill Centre for the Physics of Materials (CPM).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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