Continuous-wave cavity ringdown for high-sensitivity polarimetry and magnetometry measurements

Optical polarimetry is a fundamental technique for studying magnetometry and chirality and has its basis in the rotation of the polarization plane of linearly polarized light when passing through a magneto-optic or chiral medium, respectively. Magneto-optic rotational activity plays a key role in technologically important areas, such as the development of novel Faraday rotators, magnetometry, and controlling light polarization,^1–4 while chirality is of crucial importance across many areas of science, from fundamental physics and investigations of parity violation,^5,6 to pharmacology and biochemistry.⁷ 

Standard laboratory polarimeters have a typical detection precision no better than a few mdeg Hz^−1/2 that is limited by birefringence and are suitable only for liquid-phase observations. It is important to be able to make precise measurements of optical rotation for all phases of matter, and thus, there has been a continuous effort to develop novel, more sensitive, and precise polarimetric methods.^8–10 Particularly attractive in this regard is cavity-based polarimetry, which allows the interaction path length between the light and the magneto-optic/chiral medium to be increased manyfold, thus enhancing the detection sensitivity; for example, cavity ringdown techniques using a high-finesse linear cavity offer a large interaction path length between the light beam and sample of the order of a few kilometers, compared to the few tens of centimeters afforded by a single cell. However, linear cavities without intracavity optics cannot be employed for chirality observations because the optical rotation is dependent upon the propagation direction of the light beam and is, therefore, suppressed when the linearly polarized light passes back and forth between two cavity mirrors. Vaccaro et al. circumvented this problem by developing pulsed laser-based cavity ringdown polarimetry (CRDP) by inserting a pair of intracavity quarter-wave plates to amplify the chiroptical rotation by the number of cavity round trip passes—the stress-induced birefringence of the intracavity optics, however, is correspondingly increased.^11–13 This apparatus was successfully used in studying the gas-phase optical rotation of chiral molecules and utilized as a pulsed nanosecond laser source. More recently, Visschers et al. have demonstrated a continuous-wave (cw) variant of CRDP at 408 nm within a linear cavity for which the laser frequency is locked to a cavity resonance. Note, however, that this technique only applies for the measurement of the non-resonant Faraday effect of gases and magneto-optic crystals,¹⁴ where the Faraday rotation is independent of the light propagation direction.

Given the limitations associated with using linear cavity geometries, a variety of CRDP instruments based upon bow-tie cavities have recently been demonstrated using pulsed and cw lasers. The first pulsed laser CRDP methodology using a bow-tie cavity was demonstrated by Sofikitis and co-workers and employs two counter-propagating linearly (and orthogonally) polarized beams.^15–18 An intracavity magneto-optic crystal is inserted into one arm of the cavity to provide a large bias rotation angle—the intracavity anisotropies of opposite symmetry are suppressed by a signal reversing technique. The chiroptical rotation is then extracted from the ringdown decay signals of the two counter-propagating beams. These instruments have ringdown times, τ, between 0.5 and 1 μs and have been used for magnetometry measurements and chirality detection across a range of environments, including open air, pressure-controlled vapors, and the liquid phase, with precision $≥80$ μdeg per cavity pass. Recently, our group has demonstrated an implementation of bow-tie cavity-enhanced polarimetry with a cw laser source at 730 nm.^19,20 A key feature of the method is the controlled perturbation of the laser frequency, achieved by the addition of radio frequency noise to its injection current, which leads to significant improvements in detection sensitivity. In addition, the use of cavity mirrors with a reflectivity R ≥ 99.99%, and low-loss intracavity optical components, enables τ values of $∼9$ μs. The detection precision was $∼10$ μdeg per cavity round trip for gas-phase optical rotation measurements. A means of achieving μdeg-level detection of optical rotation on the second timescale has recently been demonstrated using a frequency metrology method;²¹ the chiroptical rotation is determined via the frequency splitting between the left- and right-circularly polarized cavity modes. Notably, this approach requires the laser frequency to be locked to the cavity resonance.

Most of the bow-tie cavity-based polarimeters mentioned above have employed intracavity Terbium Gallium Garnet (TGG) crystals to induce a large bias rotation or an “offset” angle.^15–20 However, TGG exhibits significant losses at wavelengths shorter than 600 nm, particularly in the spectral windows below 400 nm (due to the Tb³⁺⁵D₃ ← ⁷F₆, ⁵D₂ absorption bands) and around 490 nm (caused by the electronic transition ⁵D₄ ← ⁷F₆). These losses (significantly) limit the sensitivity of CRDP, and substitution for alternative magneto-optic materials is desirable. One such alternative, crystalline cerium fluoride (CeF₃), exhibits a lower absorption coefficient and a larger transparent spectral window (from 280 to 2500 nm) than TGG,^22,23 making it one of the best candidates for developing highly sensitive CRDP in the UV, visible, and near-infrared regions. Indeed, recent work by Xygkis et al.²⁴ has identified CeF₃ as a Faraday-rotating crystal of high “figure of merit” for application in cavity ringdown polarimetry (under the condition that the sample exhibits “ideal polishing”).

In this paper, we first present the experimental setup for our present implementation of CRDP using a cw laser operating at a fixed wavelength of 532 nm and show that the external frequency modulation of the laser output at the frequency splitting between the left- and right-circularly polarized cavity modes leads to an increased detection sensitivity. Then, we demonstrate the high-precision measurement of the Verdet constant of crystalline CeF₃ and then present the gas-phase optical rotation measurement of enantiomers of α-pinene and R-(+)-limonene. These highly precise measurements are compared with the state-of-the-art computations employing the TURBOMOLE quantum chemistry package, which elucidate the optical rotatory dispersion (ORD) curves exhibited by a relevant selection of individual limonene conformers. This paper concludes with two examples of liquid-phase CRDP. First, the kinetics of D-glucose mutarotation are quantified. Subsequently, the variation in the optical rotation of L-histidine with decreasing pH is presented. The latter data are interpreted with a simple thermodynamic model, which allows the optical rotation of the parent zwitterion, cation, and dication to be estimated; these observations showcase the particular utility of the apparatus for studies of aqueous solutions and the characterization of charge states that otherwise evade measurement.

Figure 1 shows the experimental setup for CRDP at 532 nm. The operating principles of the cavity-enhanced continuous-wave polarimeter are explained in detail in previous publications,^19,20 and here, we only highlight the necessary differences between the laser sources and the optical setups.

A bow-tie cavity is constructed with four high-reflectivity plano–concave mirrors (LAYERTEC, diameter of 12.7 mm). Each mirror has a reflectivity R ≥ 99.99% at 532 nm. The total cavity length is L = 557.8 ± 0.4 cm (width and length separations of $∼15$ and $∼139$ cm, respectively). For scanning the cavity length, one of the cavity mirrors is mounted on a piezoelectric actuator (PZT, Piezomechanik GmbH) driven by a K-cube high-voltage piezo-controller (Thorlabs, KPZ101). The polarimeter was addressed by a diode-pumped solid state (DPSS) laser at 532 nm (Verdi V5, Coherent) with a maximum power of 5 W. Typically, a laser power of $∼200$ mW was used. The laser beam is directed to an acousto-optic modulator (AOM, AA Opto-Electronic, MT80B30-A1.5-VIS) operated at a frequency of 80 MHz, which has a maximum fall time of $∼200$ ns. The AOM is driven by a radio frequency (rf) signal generated by a voltage-controlled oscillator (VCO, AA Opto-Electronic, DRFA10Y-B-0-50.110) with a tunable frequency range between 65 and 95 MHz and then power-boosted by an rf power amplifier (AA Opto-Electronic, AMPB-B-30-10.500) with a gain of $∼34$ dB. The AOM’s driving frequency can be adjusted by a voltage signal provided by a function generator (Keysight, 33509B). The VCO is also controlled by a home-made transistor–transistor logic (TTL) trigger box, which acts as the master trigger to generate cavity ringdown signals following intracavity power buildup. The resulting first-order diffraction beam from the AOM has a power of $∼10$ mW and is directed to the cavity. A half-waveplate (λ/2, B-Halle) and a polarizing beam-splitting cube (Thorlabs, PBS251) are used in tandem to separate the laser beam into two linearly polarized beams, p- and s-beams of clockwise (CW) and counterclockwise (CCW) propagations, respectively. These beams are then injected into the cavity where they continue to propagate in opposing directions. Two Glan–Taylor polarizers (Thorlabs, GT10-A, extinction ratio $≥105$⁠) are placed at the inputs of the cavity to filter and fix the polarization states of the incident beams. A set of lenses is used to match the Gaussian spatial profile of the CW and CCW beams with the TEM₀₀ mode of the cavity.

Due to the Faraday effect,²⁷ each cavity resonance is split into two circularly polarized modes, known as left- and right-resonant modes with a frequency separation of Δf = cϕ_F/πL (∼1 MHz for ϕ_F = 3.2°). This separation is comparable with the bandwidth of the laser, and, therefore, the ringdown signal may be dominated by only one of these resonances, leading to a reduction in the modulation depth, thus limiting the detection sensitivity. In our recent studies, we have perturbed the laser frequency by applying a Gaussian noise signal to the diode laser current, thus allowing the detection sensitivity to be improved by an order of magnitude.^19,20 For the current setup using a fixed-frequency DPSS laser, perturbation of the laser frequency can be achieved via the driving frequency of the AOM. However, the slow rise time of the AOM (∼200 ns) makes this method inefficient, and, therefore, we choose as an alternative, to modulate the laser frequency via the AOM at the frequency splitting between the left- and right-circularly polarized modes, Δf. This perturbs both modes equally, promoting equal, simultaneous mode excitation, and, therefore, increases the observed polarization-modulation depth, thus enhancing the CRDP detection sensitivity.

To study the effect of the frequency modulation of the AOM’s driving frequency on the polarization-modulation depth, we have fixed the B field applied to the CeF₃ crystal at $∼$0.15 T (corresponding to Δf ∼ 1 MHz) and recorded ringdowns over a range of modulation frequencies from 0.8 to 1.3 MHz. Figures 2(a) and 2(b) show the exemplar ringdown signals (dark points) from the CW and CCW beams, recorded at modulation frequencies f_mod = 1.2 and 1.03 MHz, respectively. The red and blue solid curves are the fits of the model given by Eq. (1) to the data. Figure 2(c) shows the variation in the polarization-modulation depth as a function of f_mod. C approaches zero when the modulating frequency approximately equals the cavity mode splitting, Δf, from 0.95 to 1.07 MHz, allowing the detection sensitivity to be improved by around 1 order of magnitude.²⁰ 

With a Verdet constant similar in magnitude to that for TGG, CeF₃ has been identified as a suitable Faraday rotator for the UV and visible regions.^22–24,26 In Fig. 3, we present measurements of the Faraday effect for crystalline CeF₃ at 532 nm. For these studies, only the CW beam is used, and its polarization direction is rotated by a Faraday rotation angle ϕ_F. The magnetic field is varied between 0.03 and 0.18 T by changing the distance between the two ring magnets and is precisely measured using a factory-calibrated Hall probe magnetometer (Hirst Magnetic Instruments, GM08) with a relative accuracy of $∼0.5%$⁠. For each B field, the modulation frequency f_mod has been chosen to be close to the cavity mode frequency splitting, allowing C to approach zero. This was achieved via a manual variation in f_mod until a maximum modulation depth had been attained. The Faraday rotation angle, ϕ_F = 2πLf/c, with c being the speed of light, is determined by fitting Eq. (1) to the data.

Figures 3(a)–3(c) show the sample ringdown signals (blue points) and the corresponding fits to the data (red solid curves) at fields of B = 0.033, 0.060, and 0.089 T. Figure 3(d) shows the Faraday rotation angle as the magnetic field ranges from 0.03 to 0.18 T. For each B field, six measurements were carried out and their mean values are shown in Fig. 3(e) (blue points), along with a linear fit to the data. The Verdet constant of CeF₃ is determined to be 188.4 ± 0.4 rad T⁻¹ m⁻¹ and is consistent with the value of $∼188$ rad T⁻¹ m⁻¹ derived from the Verdet constant dispersion model previously measured using a UV–IR spectrometer,²¹ although marginally larger than the value of 171 rad T⁻¹ m⁻¹ obtained via an alternative linear, two-mirror cavity ringdown setup.²⁴ 

Fused silica represents another dielectric medium capable of exhibiting Faraday rotation. The material’s broadband transparency and ubiquity in optical elements have motivated the characterization of its magnetically induced circular birefringence. In a manner analogous to the aforementioned study, a cavity ringdown magnetometry analysis was conducted on a 6.35 mm-thick fused silica window measuring 1 in. in diameter and possessing ion beam sputtered anti-reflection coatings of 0.035% maximum reflectivity (Edmund Optics). In this instance, an electromagnet comprising a U-core (LeyboldⓇ 562 11), a pair of coils (Leybold 562 13), and a brace of bored poles (Leybold 560 31) received current over a range of 1 A (R&SⓇNGL202), the resulting magnetic field being estimated using a transverse Hall probe (GM08 Gaussmeter, Hirst Magnetic Instruments Ltd.). The ringdown traces were recorded at intervals of 0.1 A—the corresponding Faraday rotation measurements are displayed in Fig. 3(f).

To measure the optical rotation of gas phase chiral molecules, a 1.1 m-long gas cell was inserted into one arm of the cavity. The cell was enclosed by two 1 in. diameter, 6.35 mm thick fused silica windows [Edmund Optics, ion beam sputtered antireflection (AR) coated with R < 0.035%]. Each window was placed in a home-made mount and rested on two O-rings to minimize forces acting on the window, thus reducing birefringence. The gas pressure in the cell was measured using a capacitance pressure gauge (Leybold, CTR100, pressure range of 100 Torr). The cell was evacuated below 10⁻⁶ mbar using a compact turbo-molecular pump station (Pfeiffer, HiCube 300). The chiral samples were contained in a reservoir, and their vapor was injected into the gas cell via a dosing valve. A B field of $∼0.15$ T was applied to the CeF₃ crystal, corresponding to a Faraday rotation angle of ϕ_F ∼ 3.2°. The empty cavity ringdown times for both beams are $∼4.7$ μs, corresponding to $∼255$ round trips and an effective path length through the gas cell of $∼280$ m.

The gas-phase chiroptical power of R-(+)-limonene (Sigma-Aldrich, ee = 99%) has also been quantified, and Fig. 5(a) shows its optical rotation per unit length as a function of pressure between 0 and 1.6 mbar. Due to a combination of optical absorption and scattering, τ is observed to decrease from 4.7 μs for zero sample pressure to $∼2.5$ μs at 1.6 mbar; the error bar for each data point correspondingly increases with pressure. The linear weighted fit to the data has gradient $dϕC/dp=0.6147±0.0024$ mdeg dm⁻¹ mbar⁻¹, from which the optical rotation is determined to be $110.49±0.47$ deg dm⁻¹ (g/ml)⁻¹ at 532 nm and 22 °C. The quoted uncertainty of the specific rotation includes a systematic uncertainty relating to the intracavity gas cell and cavity lengths. There is, to our knowledge, only one other gas-phase optical rotation measurement of R-(+)-limonene (assay $≥95%$⁠, unknown enantiomeric excess) at 532 nm¹⁶ with a reported specific rotation of $∼70$ deg dm⁻¹ (g/ml)⁻¹. The measurements of the optical rotation of R-(+)-limonene vapor have been performed at 355 and 633 nm at 25 °C,^11,12 at 421 nm at 21 °C,²¹ and 730 nm at 22 °C.²⁰ The findings of these studies are summarized in Table I and displayed in Fig. 5(b). Noting that there are also weak transitions at 174.5, 210, 218, and ∼400 nm in limonene,^31,32 if we assume that the optical rotation is dominated by the intense π → π* transition of the cyclohexane moiety at 186 nm, then the simple model given by Eq. (4) can be used to describe the ORD curve of R-(+)-limonene. The blue solid line in Fig. 5(b) is then the weighted fit to the data for which the electronic transition is fixed at λ₀ = 186 nm. Within this crude estimation, the resulting optical rotation amplitude is found to be $AT∼2.81±0.05×107$ deg dm⁻¹ (g/ml)⁻¹ nm². Once again, the gray area encompassing the blue line represents a 95% confidence level. Limonene is a cyclic monoterpene and, at room temperature, can exist in three distinct conformers (rotamers) defined by a relative moiety rotation about a single bond such that eclipsing interactions between the ring and isopropenyl moiety are minimized. The different conformers (denoted 1–3, with increasing enthalpy) are depicted as Newman projections in Fig. 6 (upper panes) and have relative energies 0, 0.44, and 1.91 kJ mol⁻¹, respectively. These energies were obtained from coupled-cluster single double triple [CCSD(T)] (F12*)/aug-cc-pVQZ single point energy calculations^33–36 at B3LYP/def2-TZVPP optimized geometries, accounting for vibrational zero-point energy at the B3LYP/def2-TZVPP level of theory.^37–39 Including a correction for the zero-point energy yields relative energies of 0, 0.89, and 2.28 kJ mol⁻¹ for the three conformers, which shifts the thermal average slightly, as shown in Fig. 6. The ORD values for each conformer were computed using a linear response⁴⁰ with the CAM-B3LYP functional⁴¹ in the length representation with the aug-cc-pVQZ basis set.³⁶ The calculations reveal that each of the conformers exhibits a different optical rotatory power, of which the interplay dictates the calculated (and experimentally observed) ORD curves shown in Fig. 6. Notably, two of the three conformers (1 and 3) display optical rotations that are smaller in magnitude and of the opposite sense to the other (2); at room temperature (295 K), the population-weighted rotation for conformer 2 dominates. The thermal average rotation is shown by the green unfilled triangles in Fig. 6 and is in reasonable agreement with experiment.

For measuring the chiroptical properties of aqueous samples, a home-made 5.3 mm-long flow cell was inserted into one arm of the cavity (see Fig. 1). The cell consists of two fused silica windows (LAYERTEC GmbH), which are AR-coated for the air/fused-silica interface (R < 0.1%) and non-coated for the water/fused-silica interface. These windows are separated using a perfluoroelastomer (FFKM) O-ring (Polymax, thickness of 5.30 ± 0.03 mm) and mounted on two kinematic cage-compatible mounts (Thorlabs, KC1/M). Chiral aqueous solutions were injected into the flow cell using a syringe pump (Harvard Apparatus, 11) with a flow rate of $∼0.2$ ml min⁻¹. Once again, a B field of $∼0.15$ T was applied to the CeF₃ crystal (ϕ_F ∼ 3.2°). Figure 7(a) shows the typical ringdown signals observed with the flow cell filled with water. Each signal is an average of 2000 events over 30 s. τ is $∼3$ μs and corresponds to $∼135$ round trips in the cavity and an effective path length through the flow cell of $∼0.72$ m.

We use this apparatus to probe the mutarotation of D-glucose via measurement of its specific rotation—where the dissolved chiral molecules eventually equilibrate between different anomers. Mutarotation manifests as a time-dependent change in the specific rotation (as the two anomers in this case have different specific rotations), and Fig. 7(b) shows the time evolution of the optical rotation angle for 3 × 10⁻³ g/ml D-glucose in water over a period of 250 min. For each time point, five measurements were carried out over a total observation duration of 75 s. Each data point and its error bar in Fig. 7(b) are given by the weighted mean and the standard error of the five measured rotation angles, respectively. The reaction kinetics are well described by a simple, first-order exponential model ϕ_C(t) = ϕ_C,t→∞ + Ae^−kt (dashed curve). Fitting this to the data reveals a rate constant for mutarotation of k = 1.298 ± 0.047 × 10⁻² min⁻¹ at 18 °C, in good agreement with recent broadband Mueller ellipsometry measurements conducted by Vala et al.⁴²  Figure 7(c) shows the specific rotation of D-glucose in water as the mutarotation approaches completion (⁠$∼5$ h observation period). Each data point and its error are the average and the standard error of ten data sets at a given concentration, respectively. The optical rotation of D-glucose is determined to be 62.96 ± 0.38 deg dm⁻¹ (g/ml)⁻¹ at 532 nm, where the quoted uncertainty of the specific rotation includes a systematic uncertainty relating to the intracavity gas cell and cavity lengths.

Under the experimental conditions used, it can be assumed that only the zwitterion, cation, and dication of L-histidine could be present in significant quantities in the solution. In the above equalities, the concentrations of these species are represented as $AA±$⁠, $AAH+$⁠, and $AAH22+$⁠, respectively. Of these, Eqs. (5) and (6) represent the acid dissociation equilibria of the cation and dication, respectively, where $pKa1$ (corresponding to the imidazole side chain of the histidine molecule) was taken to be 6.00, while $pKa2$ (representing the carboxylic acid moiety) was set equal to 1.82.⁴⁶ Equation (7) accounts for electroneutrality in the bulk solution, with Eq. (8) describing the system’s mass balance. The results of one such set of concentration profile evaluations are depicted in Fig. 8(b). With the concentration profiles in hand, a sequential least squares programming (SLSQP) method, fitting to the experimental findings, was employed to determine the optimized values for the individual optical rotations. This was conducted for all three environments simultaneously under the assumption that the optical rotation of an individual species does not change dramatically between each condition set. The results of these calculations are displayed in Table II. The total optical rotation is a concentration-weighted sum of the contributions from each species in the solution, and so the optimized individual rotations can be used to infer the predicted total observed rotation as a function of amino acid concentration at each acidity regime. This analysis was conducted, and the results are depicted in Fig. 8(a).

It is noteworthy that this simple, first-principles methodology shows a quite good agreement with both theoretical⁴⁷ and experimental⁴³ values, some of which are also displayed in Table II. For example, at 532 nm, we predict an 8.23 deg dm² mol⁻¹ rotation increase from zwitterion to dication, agreeing well with Greenstein’s experimental finding of 7.81 deg dm² mol⁻¹ at 589.3 nm. Notably, this analysis provides access to approximate optical rotations of intermediate charge states, in this case the monocation. These values are otherwise difficult to attain by experiment due to the individual species’ optical rotations being directly unobservable.

We have developed a new variant of continuous-wave CRD polarimetry using a fixed-wavelength laser source at 532 nm. The sensitivity of this apparatus is optimized via frequency modulation of the laser output to regularly and simultaneously excite both the non-degenerate left- and right-circularly polarized cavity modes. The CRDP technique has been demonstrated by evaluating the Verdet constants of CeF₃ and fused silica, as well as the gas-phase optical rotation of (+)- and (−)-α-pinene and (R)-(+)-limonene. The limit of precision for the gas-phase optical rotation measurements is $∼30$ μdeg per cavity pass, limited by the optical losses on the CeF₃ crystal, and the uncertainty on the specific optical rotation is better than 0.3 deg dm⁻¹ (g/ml)⁻¹. The measurements of limonene are interpreted via state-of-the-art computational evaluations of the optical rotations of the three thermally accessible conformers of this species in the gas phase. Their relative energies were determined to span a range of approximately 2 kJ mol⁻¹. In addition, the CRDP technique has been extended to measurements in the liquid phase with a limit of detection of $∼120$ μdeg per cavity pass, corresponding to a time-averaged sensitivity of $∼657$ μdeg Hz^−1/2. Specifically, the mutarotation of D-glucose has been tracked over 4 h, and a rate constant for mutarotation of k = 1.298 ± 0.047 × 10⁻² min⁻¹ at a temperature of 18 °C was determined. Finally, the optical rotation of L-histidine has been determined as a function of the total amino acid concentration and the acidity of the solution. These measurements allowed for the estimation of the optical rotatory powers of the aqueous L-histidine zwitterion, cation, and dication, thus presenting a promising means of evaluating the chiroptical properties of otherwise unobservable charge states in the solution. The results of these final investigations also provide accessible, empirical quantification of the time-tested Clough–Lutz–Jirgensons rule for amino acids.

This work was funded by the European Commission Horizon 2020, UltraChiral Project (Grant No. FETOPEN-737071).

The authors have no conflicts to disclose.

Dang-Bao-An Tran: Investigation (equal). Evan Edwards: Investigation (equal). David P. Tew: Investigation (equal). Robert Peverall: Investigation (equal). Grant A. D. Ritchie: Investigation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.