Magneto-optical contrast in liquid-state optically detected NMR spectroscopy

Abstract

We use optical Faraday rotation (OFR) to probe nuclear spins in real time at high-magnetic field in a range of diamagnetic sample fluids. Comparison of OFR-detected NMR spectra reveals a correlation between the relative signal amplitude and the fluid Verdet constant, which we interpret as a manifestation of the variable detuning between the probe beam and the sample optical transitions. The analysis of chemical-shift-resolved, optically detected spectra allows us to set constraints on the relative amplitudes of hyperfine coupling constants, both for protons at chemically distinct sites and other lower-gyromagnetic-ratio nuclei including carbon, fluorine, and phosphorous. By considering a model binary mixture we observe a complex dependence of the optical response on the relative concentration, suggesting that the present approach is sensitive to the solvent-solute dynamics in ways complementary to those known in inductive NMR. Extension of these experiments may find application in solvent suppression protocols, sensitivity-enhanced NMR of metalloproteins in solution, the investigation of solvent-solute interactions, or the characterization of molecular orbitals in diamagnetic systems.

Nuclear Magnetic Resonance is one of the leading analytical tools among material scientists, organic chemists, and structural biologists. One of its main advantages is the spectral dispersion resulting from system-specific chemical shifts, and from dipolar, quadrupolar, and J-couplings (1). In a typical inductively detected NMR spectrum the relative amplitude of a resonance is given by the fraction of nuclei associated with a particular functional group. The greater natural abundance and higher gyromagnetic ratio of protons make ¹H NMR the most sensitive, a feature that has led spectroscopists to develop various schemes of polarization or coherence transfer to more efficiently detect other less favorable nuclei. Conversely, selective deuteration and solvent suppression schemes are the traditional tools at hand to highlight weaker proton resonances from molecules in solution.

Unlike inductive detection, where nuclear spins interact directly with a pick-up circuit tuned to the Larmor frequency, optical schemes in general rely on electrons as intermediaries. The resulting signal amplitude depends on the hyperfine coupling between electrons and nuclear spins on the one hand, and on the interaction between electrons and optical photons on the other (the so-called oscillator strength of the system at a given illumination wavelength). Favorable conditions are typically met in select atomic vapors (2–4) and condensed matter systems (5, 6), which explains why optical schemes, normally designed to monitor the sample fluorescence or absorption, have not yet enjoyed a more widespread use. Recent experiments, however, indicate that optical Faraday rotation (OFR) can serve as a more general platform applicable, in principle, to all transparent condensed matter systems. Initial studies based on a continuous wave (cw) protocol demonstrated OFR-based detection of proton magnetic resonance in prepolarized water at approximately 5 G (7). Recent results at high magnetic field show that it is possible to optically probe nuclear spins in real time after resonant radio frequency (rf) excitation, to determine, after Fourier transform, the corresponding chemical-shift-resolved NMR spectrum (8). Further, ab initio calculations in small molecules predict that nuclear spin-induced Faraday rotation should convey distinct signals for nuclei in differing chemical environments (9).

Here we integrate OFR with pulsed, high-field NMR to monitor nuclear spins in a variety of diamagnetic transparent fluids, and show that OFR–NMR provides unique mechanisms of spectral contrast. We observe a correspondence between the fluid Verdet constant and the optical NMR signal in pure solvents, which we correlate with the detuning between the laser frequency and the system optical transitions. Crafted binary mixtures of variable composition yield component signal amplitudes that exhibit a markedly nonlinear dependence on its volume fraction. These amplitudes reflect solvent-induced changes in the solute molecular orbitals, implying that OFR-detected NMR could serve as a tool to probe intermolecular dynamics in multicomponent fluids. A set of sample fluids is used to illustrate the effect of hyperfine couplings on the relative amplitude of site-specific resonances. Comparison of spectra from different spin species shows that the optical response tends to grow with the atomic number of the species under study, which portends OFR–NMR to be a technique well-suited for the investigation of heavier, lower gyromagnetic ratio nuclei.

Results and Discussion

We start in Fig. 1 with a brief survey of our experimental setup: A linearly polarized laser beam (532 nm) propagates through an approximately 1.5-cm long sample fluid along the x axis, perpendicular to the direction of the static magnetic field (9.4 T, 400 MHz ¹H frequency). A solenoidal coil wound on the sample glass container and part of a homemade probe is used to generate an rf magnetic field collinear with the beam. We detect small changes in the laser polarization with the aid of a polarizing beam splitter and a broadband, balanced photodetector, whose rf output is fed into our NMR spectrometer for demodulation and processing. We operate our setup in two different modalities: The first class of experiments uses a 1-s long rf pulse near (but not necessarily coincident with) the NMR resonance frequency to determine the electronic Faraday rotation of the sample fluid. Ignoring for simplicity the presence of the static field (8), one expects a change in the beam polarization of order

[1]

where L is the sample length, μ₀ denotes the vacuum magnetic permeability, N is the system number density, c represents the speed of light, ω is the illumination frequency, and is the first-order dynamic polarizability. The angular rotation per unit magnetic field and unit sample length is known as the Verdet constant, more precisely, a complex function of the system molecular orbitals, and detuning Δω from an optical transition (see below). Fig. 1C shows the OFR signal, demodulated from 400 MHz, for a B₁ field of 50 μT; to rule out possible baseline (dc) drifts we use an optical chopper, which results in a square wave-like shape at the chopper frequency (∼115 Hz). The corresponding Fourier transform (Fig. 1D) shows the resulting symmetric pattern of central peaks and satellites whose amplitude we take as a measure of the electronic Faraday rotation in the sample fluid. Rather than duplicating information already known, each OFR signal so recorded allows us to empirically take into account the effect of our atypical experimental conditions (high-frequency B₁ collinear with the beam and strong, static B₀ fields) on the Verdet constant of the sample system (approximately a factor of 2–3 relative to accepted literature values at the present wavelength, see refs. 8 and 10 and refs. therein).

Fig. 1.

Schematics of the experimental setup. (A) Linearly polarized laser light at 532 nm is steered into the bore of a 9.4-T NMR magnet and through the sample along the x axis, perpendicular to the static field . Changes in the polarization of the outgoing beam are detected with a balanced photodetector, whose rf output is fed into our NMR spectrometer for demodulation and processing. (B) Electronic OFR detection schematics. Rotation of the beam polarization upon crossing the sample is induced by a magnetic field at 400 MHz. Homodyne detection via our NMR spectrometer down-converts the resulting rf signal at the B₁ frequency. A mechanical chopper synchronous with our detection protocol is used to modulate the beam amplitude and thus minimize errors from baseline drifts. (C) After demodulation the OFR signal has the form of a square wave at the chopper frequency. The result in the figure corresponds to a 1-s long B₁ field of 50 μT; the total number of repeats is 500. (D) Fourier transform of the signal in C.

Underpinning the OFR-detected NMR spectra we show next is a second class of experiments in which nuclear spins are resonantly excited and optical detection is conducted in the absence of applied rf. Formally, the effect of nuclear spins on the electronic cloud can be characterized via the susceptibility and the nuclear spin average polarization . When has a nonzero projection along the direction of beam propagation, one expects the nuclear field to generate a contribution to the electronic OFR signal (Eq. 1) given by

[2]

Thus, Larmor precession of nuclear spins on a plane that contains the laser beam must lead to a change in the laser polarization precisely at the nuclear precession frequency. Qualitatively, one can interpret the OFR–NMR signal as a manifestation of the nuclear field , where a_eff is a magnitude representative of the hyperfine coupling, g^∗ indicates the Landé factor, and μ_B is the Bohr magneton. In this approximation, one would expect the nuclear spin-induced rotation to be of order θ_N ∼ LVB_N, proportional to the fluid Verdet constant. Note that because high-field NMR detection typically probes a restricted bandwidth around the Larmor frequency, the observed polarization change is immune to static electronic OFR caused, for example, by residual components of the B₀ field along the beam propagation axis.

Most protonated solvents feature relatively weak hyperfine couplings and correspondingly exhibit a small nuclear spin Faraday rotation (for the conditions reported herein, θ_N typically amounts to a few nanoradians at 9.4 T, much smaller than θ_E, of order 1 μrad for an approximately 0.1-mT B₁ field). To attain sufficient detection sensitivity, we trail the excitation pulse with a train of inversion pulses separated by equal time intervals of free evolution [the so-called Carr–Purcell–Meiboom–Gill (CPMG) protocol (11), Fig. 2A]. Enabling the detector during these intervals only, we collect a set of nuclear spin Hahn-echoes, whose duration (approximately 8 ms) is determined by the B₀ field inhomogeneity over the sample volume. We then coadd these echoes to produce an average signal (8) whose amplitude will be a transverse coherence time-weighted measure of the system density; naturally, subsequent Fourier transform leads to the corresponding low-resolution (∼1 ppm) chemical shift-resolved spectrum. Although this strategy regains part of the sensitivity loss caused by field inhomogeneity and susceptibility broadening, we emphasize that the simpler scheme excitation-acquisition should suffice in a system designed to simultaneously suit the constraints of rf manipulation and optical sensing (8, 10).

Fig. 2.

(A) Detection of nuclear-spin-induced OFR–NMR. To improve sensitivity, we use a CPMG sequence which induces a train of nuclear spin echoes. The signal acquired during the interpulse intervals is added to produce a T₂-weighted average signal intensity. (B, Left to Right) Fourier transform of the average signal for water (H₂O), bromobenzene (C₆H₅Br), and bromonaphthalene (C₁₀H₇Br). Inductively and optically detected signals are shown in the upper and lower traces after 16 and 17 × 10³ repeats, respectively. The separation between inversion pulses in the CPMG train is 8 ms. (Upper Insets) Inductive spectra from a high-resolution 300 MHz, liquid-state NMR spectrometer after one-pulse excitation. (Lower Insets) Electronic OFR signal as determined using the protocol of Fig. 1. (C) Ratio between optically and inductively detected amplitudes as a function of the OFR signal for methanol (CH₃OH), ethanol (C₂H₅OH), water (H₂O), acetone [(CH₃)₂CO], toluene (C₆H₅CH₃), tri-ethyl-phosphite [P(OCH₂CH₃)₃], bromobenzene (C₆H₅Br), 2-methyl-benzothiazole (C₈H₇NS), and bromonaphthalene (C₁₀H₇Br). We define λ_eff ≡ 2πc/ω_eff. In cases where the spectrum exhibits more than one resonance, we consider the peak with the highest amplitude. The uncertainty of each data point (mainly determined by the noise level of the corresponding optically detected NMR spectrum) is of order 17%; the dashed green line is a guide to the eye.

Fig. 2B shows the Fourier transform of the OFR–NMR signals obtained using the above protocol for three different fluids. Maintaining the same acquisition scheme, the figure also displays the corresponding inductive signals (where the photoreceiver is disconnected from our spectrometer and detection during the CPMG intervals is conducted via the NMR probe). For completeness, the accompanying inserts display the corresponding one-pulse, high-resolution spectra obtained from an inductive-only, 300 MHz, liquid-state spectrometer we use herein as an independent, complementary reference. Sensitivity considerations not withstanding, we observe clear differences in the relative amplitudes of the resonance peaks obtained either inductively or optically. In going from water (Left), to bromobenzene (Middle), to bromonaphthalene (Right), coil-detected signals show a very noticeable decay, chiefly the result of the slower tumbling rate inherent to larger molecular size and its concomitant reduction of the nuclear spin coherence time T₂. Optical signals, on the other hand, feature comparable amplitudes in all three cases. Noting that both inductive and optical responses share identical dependence on the sample density and coherence time, we surmise that the optical encoding process itself is the source of the observed differences.

To formally bring into consideration some key parameters relevant in OFR–NMR, we start by considering the first-order dynamic susceptibility of a diamagnetic system (12)

[3]

where the sums extend over all excited molecular orbitals , and, for simplicity, we assume a nondegenerate ground state ; ω_eg denote the optical transition frequencies (greater than ω in all the systems studied herein) and , are functions containing products of the magnetic moment m_x and electric dipole moments μ_y, μ_z within the subspace of states . Although, in general, prediction of depends on a detailed knowledge of the compound molecular orbitals, i.e., of the functions and , we note that optical detuning typically plays a pivotal role in scaling the OFR response of most diamagnetic fluids (13). The latter is a direct consequence of Eq. 3 where, for far off-resonance illumination, the dynamic susceptibility takes the approximate form

[4]

where ω_eff is a frequency representative of the first intense dipole-allowed optical transition and we write in explicit form as a function of the difference between the molecular magnetic moments with . From 4, we conclude that molecular moieties with an absorption band near the illumination frequency are prone to exhibit stronger Faraday rotation.

When detecting nuclear spin resonances using OFR, a formula for the nuclear spin susceptibility can be obtained from Eq. 3 through the correspondence for each spin species I. Hence, for nuclear spins where the dispersion of hyperfine couplings at chemically distinct sites k is small, one expects optical detuning to play a major role in defining the overall amplitude of the OFR–NMR response. The results in Fig. 2C corroborate this idea: Here we plot the ratio between optical and inductive ¹H signals for several diamagnetic fluids as a function of the OFR amplitude as determined in our setup (Fig. 1). We find remarkable correlation with OFR–NMR signals growing as the electronic OFR response increases. Interestingly, however, we observe a seemingly nonlinear dependence, suggesting that the predicted proportionality between the nuclear Faraday rotation angle θ_N and the Verdet constant V may be inadequate to describe our data. Although much theoretical work remains to be done, we suspect that the sums in Eqs. 2 and 3 contain contributions from many (as opposed to few, select) excited states . In this limit, the characterization of nuclear spins in terms of an average field B_N is too crude and, consequently, the linear relation between θ_N and V breaks down.

The results presented above raise some interesting possibilities, particularly in the context of multicomponent mixtures where, for example, one could imagine using OFR–NMR to selectively reduce the solvent signal, thereby highlighting resonances from different molecular moieties in solution. As a first, exploratory step in this direction we conducted a set of observations in a binary mixture of methanol and bromobenzene. This system lends itself to a systematic characterization because the large chemical shift differences between the corresponding proton spin resonances allow us to unambiguously identify each component. Further, from Fig. 2C we conclude that the nuclear dynamic susceptibility of bromobenzene is substantially larger than that of methanol. Thus, in the approximation where either resonance peak changes linearly with the corresponding volume fraction, OFR–NMR should preferentially enhance the signal from aromatic protons over those of the methyl (CH₃) and hydroxyl (OH) groups in methanol.

Fig. 3A shows the results from a mixture where the fractional volume concentration of methanol (40%) was chosen so as to yield a CH₃ resonance peak slightly larger than that of aromatic protons (low-resolution inductive spectrum on the Upper Left). Surprisingly, we find that, rather than deemphasizing the CH₃ resonance, the OFR–NMR spectrum (Fig. 3A, Upper Right) further enhances the relative difference between the two peaks. Fig. 3B summarizes our observations in mixtures of various concentration exposing intriguing differences between inductive and optical signals: On the one hand, coil-detected resonances show a slightly nonlinear, convex dependence on volume fraction, very likely the result of slower molecular tumbling (i.e., shorter T₂) due to solvent-solute interactions. The optical response, on the other hand, does not exhibit a uniform trend: Although the aromatic signals virtually reproduce the inductive curve (except for an overall multiplicative constant), the OFR–NMR response of methyl protons exhibits a markedly nonlinear, concave dependence.

Fig. 3.

(A, Left to Right) Inductively and optically detected NMR spectra in a mixture of methanol and bromobenzene; the numbers of repeats are 16 and 17 × 10³, respectively; other conditions as in Fig. 2. In both cases, the volume concentrations x_CH3OH and x_C6H5Br are 0.4 and 0.6, respectively. The observed resonances correspond to aromatic (–CH), hydroxyl (–OH), and methyl (–CH3) protons. The upper right inset compares a fit to the observed optically detected spectrum (blue solid line) with the calculated spectrum (red solid line) assuming that the optical response of either pure compound scales linearly with its volume concentration. (B) Amplitude of the –CH and –CH3 resonances as a function of the volume concentrations in the inductively and optically detected spectra (Left and Right, respectively). Dashed and solid lines are guides to the eye. Faded lines in the right figure reproduce the corresponding curves in the left figure, properly rescaled to fit the optically detected amplitudes. (C) OFR signal of the binary mixture as a function of the volume concentration. (D) Chemical shift difference between the -CH₃ and –CH resonances in methanol/bromobenzene mixtures as determined from high-resolution, liquid-state NMR spectroscopy at 300 MHz. In B–D, bars around data points indicate the standard error deviation as determined from the measured resonance peak amplitudes.

Bromobenzene/methanol mixtures are one example within the broad class of solutions combining protic and hydrophobic groups, widespread in surfactants and proteins. A range of noncovalent, transient complexation processes take place in these systems, which, we think, underlie our observations. In particular, polar OH groups are known to form short-lived π-hydrogen bonds with the electronic cloud of aromatic rings, as determined in phenol/bromobenzene mixtures using ultrafast optical spectroscopy (14). Further, studies in benzene-(methanol)_m clusters indicate a gradual transition from a configuration where methanol molecules form chains π-H-bonded to the benzene molecule (m = 1–3), to other more complex, cyclic structures where the π-hydrogen bond plays a less relevant role (m = 4–6) (15). In this light, the results of Fig. 3 hint at the use of OFR–NMR as a tool to probe intermolecular dynamics in unique, complementary ways. It is worth noting that, when compared with either technique alone, the combined use of Faraday rotation and nuclear magnetic resonance appears to be particularly sensitive. For example, a monotonic, linear change of the electronic Faraday rotation signal is observed when gradually transitioning from bromobenzene to methanol (Fig. 3C). On the other hand, the CH-CH₃ chemical shift difference (as determined from the examination of reference spectra obtained in a separate, high-resolution NMR system) experiences a rather minor (∼100 ppb), linear frequency change over the whole concentration range (Fig. 3D).

Another facet of interest in OFR-detected NMR concerns the ability to make quantitative determinations of hyperfine couplings in diamagnetic fluids. Presently, most studies rely on EPR, and are therefore restricted to systems with unpaired electron spins. OFR-detected NMR could offer perhaps ancillary information, as hyperfine coupling constants (HFCCs) are central in determining the amplitude of the observed signal. Although an in-depth theoretical investigation escapes the scope of this work, our results can, nonetheless, expose some general trends. For example, the upper half of Fig. 4 shows the ¹H NMR spectra from three model compounds, methanol (CH₃OH), 2-methyl-benzothiazole (C₈H₇NS), and tri-ethyl-phosphite [P(OCH₂CH₃)₃], with resonances corresponding to hydroxyl (-OH), aromatic (-CH), and methylene (-CH₂) protons. All three compounds also contain at least one methyl (-CH₃) group, which can thus be used as a common reference.

Fig. 4.

(A) Comparison between ¹H (Upper traces) and ¹³C (Lower traces) NMR spectra of methanol (CH₃OH). The ¹³C spectra correspond to a 99% enriched sample. (B, Upper) Methyl (-CH₃) and aromatic (-CH) resonances in the ¹H inductively and optically detected spectra of 2-methyl-benzothiazole (C₈H₇NS). (Lower) The ¹⁹F NMR spectra of perfluorohexane in the vicinity of the -CF₃ resonance. (C, Upper) ¹H and ³¹P (Lower) NMR spectra of tri-ethyl-phosphite. In A–C, Top and Bottom traces in each half correspond to inductively and optically detected spectra, respectively. The detection protocol and CPMG conditions are those of Fig. 2. All inductive and optical traces are the result of 16 and 17 × 10³ scans, respectively. For presentation purposes, the optically detected ¹⁹F spectrum was scaled down by a factor of 0.5.

Consider first the NMR data of 2-methyl-benzothiazole, where, within experimental error, we find that the ratio between the optically detected aromatic and methyl resonance peaks coincides with that observed in the inductive spectrum . Because both optical and inductive signals depend identically on the concentration and transverse relaxation time of each nuclear spin species, this observation is only consistent with equal nuclear dynamic susceptibilities at each site, i.e., . Therefore, in the approximation of 4, where all optical transitions are described by a common effective frequency ω_eff, we conclude that the HFCC-dependent quantity

remains approximately unchanged at sites k = -CH, -CH₃ suggesting that hyperfine couplings are not substantially different. By the same token, the ¹H spectra in Fig. 4 C and A point to similar constraints at the methylene and hydroxyl sites [although the poorer signal-to-noise ratio (SNR) in the latter case makes the uncertainty considerably greater].

To extend this comparison to nuclei other than ¹H it is convenient to start by noting the dissimilar frequency response of either detection method. Both Faraday induction and nuclear polarization combine to produce a coil-detected signal S_ind dependent on the square of the static magnetic field B₀ and the third power of the nuclear gyromagnetic ratio γ. On the other hand, optical signals S_opt, directly proportional to , must grow linearly with the field and nuclear gyromagnetic ratio. Along with the corresponding proton data, the lower row in Fig. 4 A and C show the ³¹P and ¹³C NMR spectra from tri-ethyl-phosphite and a 99% ¹³C-enriched sample of methanol, respectively. For further comparison, Fig. 4B includes ¹⁹F NMR observations in perfluorohexane. Using the methyl proton signal as a reference and taking into account the relative nuclear density and transverse relaxation time of ³¹P spins in tri-ethyl-phosphite, we measure , a result that leads us to .

Our observations for ³¹P in tri-ethyl-phosphite are in stark contrast with those in methanol (Fig. 4A) where no optical ¹³C signal could be observed. We note that this negative result is neither a consequence of a lower ¹³C concentration [N^(13C)/N^(31P) ≅ 5] nor of a shorter relaxation time . Taking into consideration our present sensitivity and starting from the conditions , and we conclude that , which is likely to be indicative of hyperfine couplings substantially smaller than in phosphorous. Finally, a similar comparison between the ¹⁹F and ¹H NMR resonances corresponding to the -CF₃ and -CH₃ groups of perfluorohexane and methyl-benzothiazole, respectively, leads to (Fig. 4B). This same value serves as an approximation to the ratio Q^(19F)/Q^(1H) at these two sites, although we warn that differences in the optical transition frequencies and oscillator strengths of either molecular moiety add uncertainty compared to the ¹³C and ³¹P estimates.

The present results portend exploitation of the distinct optical response of a compound to enhance the information content of an NMR spectrum. Rather than using light to alter chemical shifts (16), an effect too small to be observed (17, 18), OFR–NMR relies on the detection process itself—governed by optical detuning, oscillator strengths, and hyperfine constants—to introduce the desired spectral contrast. In disentangling the range of mechanisms at play, the present results layout the ground for further experimental and theoretical work, including the modeling of the correspondence between OFR and NMR signals, a better understanding of the influence on the optical response of solvent-solute interactions, and the determination of hyperfine constants. From a methodological standpoint, one appealing future possibility is to use a second laser beam nearly resonant with an optical transition to drive molecules in the fluid away from the ground state. This cw pump-probe modality could provide yet another route to selectively alter the compound NMR signal depending on the oscillator strengths and hyperfine constants of the chosen excited state.

Extending these results to other nuclear spin species and solutes in low concentration rests on the ability to improve the SNR, presently below the limits of routine use. Unlike inductive detection, Faraday rotation grows with the sample length, not its volume, implying that, for ideal light transmission, an SNR comparable to that reported herein would be anticipated for samples confined to approximately 1-cm long, micrometer-wide channels. This trait makes OFR-detected NMR an approach well-suited to the emerging fields of microfluidics and chip-scale integrated optical sensing. In this context, it is worth noting that the OFR angle doubles if the beam, after crossing the medium once, is reflected back through it a second time, a property that points to optical cavities as a signal enhancement route (10, 19). Microfluidic distributed feedback grating (20) and liquid-liquid waveguide (21) architectures have already been used as the platform for lab-on-a-chip tunable fluidic lasers, and this approach could arguably be adapted to serve the purposes of OFR-detected NMR. On a final note, we mention that these same ideas could perhaps be extended to accommodate other geometries such as that of ring cavities, already exploited to conduct absorption measurements of nanoliter samples (22). Likewise, one could benefit from the high gain of whispering gallery mode resonators, shown to attain superb sensitivity in fluidic media (23).

Materials and Methods

In all experiments reported herein we use the linearly polarized beam from a diode-pumped laser operating at 532 nm (Coherent Compass 315). Samples are positioned at the sweet spot of a 9.4 T magnet (400 MHz ¹H frequency) stripped from its shim-stack to facilitate light alignment in and out of the 89-mm diameter bore. We rely on mirrors attached to each side of our homemade NMR probe-head to steer the beam through the sample container (a ∼ 1.5 - cm glass cylinder with optical windows on each end). Part of the probe tank circuit is an eight-turn, 1.5-cm long solenoid wound on the container surface, which we use alternatively to generate cw (Fig. 1) or pulsed rf fields (Figs. 2–4) over the sample volume.

Detection of Faraday rotation is carried out via a bridge configuration comprising a half-wave plate, a Glan-Laser polarizer, and a dual, 600-MHz bandwidth photoreceiver (New Focus 1607-AC), whose 50-Ohm matched rf output is fed directly into the preamp of our NMR spectrometer for final demodulation and processing. The half-wave plate is regularly adjusted to correct for any static changes in the electronic Faraday rotation signal introduced, for example, by the dielectric coatings of the many guiding mirrors, and the sample and its glass container (through residual components of the B₀ field along the beam propagation axis). We note that this practice is exclusively aimed at preserving the balance in our detection bridge (i.e., at maintaining detection sensitivity at its maximum) as OFR–NMR signals are insensitive to zero- and low-frequency changes in the beam polarization (see main narrative). When needed, we use a mechanical chopper to modulate the laser amplitude at a predetermined frequency. To allow for coherent averaging of successive scans in regular OFR experiments (Fig. 1), we trigger acquisition in our NMR spectrometer via the digital monitor signal from the chopper.

We accommodate most optical components in a vicinity of the magnet bore using an aluminum platform rigidly connected to the bottom plate of the magnet. However, to circumvent complications derived from the stray field (200–500 G), we keep the mechanical chopper and the laser far removed (∼10 m from the magnet site). Without sacrificing much light intensity or introducing noticeable divergence we guide the beam using a 20-m long fiber, optical couplers, and a collimator. Because the magnet sits on vibration-isolation legs, this configuration is adequate for the long measurement times required by the present experiments (typically lasting 3–4 d), simultaneously preventing any cross-talk between the optical and rf sources.

To further confirm the absence of unwanted rf pick-up, we contrasted our results against observations in the dark for almost every single data point reported herein. Such experiments keep all conditions unchanged except for the light beam, which is sometimes rerouted away from the sample (without preventing, however, illumination of the photosensor), and sometimes blocked (i.e., no light reaches the photodetector). For our present conditions, neither experiment yields an observable signal.

Slow, temperature-induced changes in the beam path tend to alter the coupling efficiency at the various fiber-optics ports, causing a drop in the OFR–NMR signal amplitude. To eliminate this problem we made frequent use of the monitor outputs in the photodetector (typically every 4 to 5 h, never exceeding 8 h) to compensate for small intensity fluctuations in each branch of the optical bridge. Throughout the time (∼3 mo) devoted to the collection of our full dataset, we also conducted periodic reproducibility checks via duplicate measurements of some of our samples as well as the repeated observation of the OFR–NMR signal from a standard (water). We attained a remarkable stability, with amplitude variations from one observation to the next falling well within the SNR level of a single measurement (10–20% maximum).

Not surprisingly, electronic OFR measurements over a broad frequency range (from 1 to 400 MHz) both inside the magnet bore and on an optical table in the absence of static magnetic field show that the optical signal grows linearly with B₁ amplitude. Verdet constants, however, are altered (to approximately a factor of 2–3) by the presence of B₀, a limitation that makes inadequate the direct comparison between the OFR signal amplitudes reported herein (Fig. 2C) and those available in the literature. We refer the reader to the supplementary material in ref. 8 for a fuller description of this effect.

As expected, OFR signals grow linearly with light intensity. To optimize sensitivity we invariably set the laser power on each branch of the photoreceiver slightly below its maximum, at 1.2 mW. In our CPMG experiments the typical duration of the π/2-pulses (π-pulses) is 20 μs (40 μs). Inversion pulses are separated by 8 ms, the time delay between successive scans ranges from 5 to 10 s (depending on the sample relaxation time), and the total number of repeats is of order 10⁴. All chemicals (99%, reagent grade, Sigma-Aldrich) including our ¹³C-enriched methanol sample (Cambridge Isotopes, 99% deuteration) are used without further purification.