High resolution spectral metrology leveraging topologically enhanced optical activity in fibers

Optical rotation, a form of optical activity, is a phenomenon employed in various metrological applications and industries including chemical, food, and pharmaceutical. In naturally-occurring, as well as structured media, the integrated effect is, however, typically small. Here, we demonstrate that, by exploiting the inherent and stable spin-orbit interaction of orbital angular momentum fiber modes, giant, scalable optical activity can be obtained, and that we can use this effect to realize a new type of wavemeter by exploiting its optical rotary dispersion. The device we construct provides for an instantaneous wavelength-measurement technique with high resolving power R = 3.4 × 106 (i.e., resolution < 0.3 pm at 1-μm wavelengths) and can also detect spectral bandwidths of known lineshapes with high sensitivity.


Supplementary Note 1: Vortex Fiber Index Profile
Cylindrically symmetric and isotropic optical fibers utilizing a ring-core design fall under a particular classification, known as vortex fibers. Specifically, these fibers are specially designed to mirror the field profile of an OAM beam, which has been found to maximize the effective index splitting (∆ eff ) between modes of a given |ℒ| but opposite circular polarizations, ̂± (i.e. SOa and SOaa) 1 . We illustrate an example of such a profile in Supplementary Figure 3, which has been measured using an interferometry-based fiber profiler (Interfiber Analysis IFA-100) at 633nm. The fibers used in this work possess a center air-filled hole at the inner boundary of the ring-core. This ∆ eff split is due to spinorbit coupling and the inhomogeneity of the ring-core, which, as predicted by perturbative theory, introduces polarization-dependent perturbations based on ̂± and is enhanced by OAM 2,3 . As such, a 4-dimensional Hilbert space of OAM / SAM states exist in vortex fiber [see Fig. 1(a)] comprised of degenerate pairs of SOa and SOaa modes, which have effective index differences of ∆ eff > 10 −4 (circular birefringence 4 ). A combination of this spin-orbit interaction ∆ eff splitting and conservation of angular momentum allows vortex fibers to avoid both non-degenerate and degenerate coupling effects 5,6 between all the modes in their Hilbert space, and have been well known to stably propagate OAM / SAM singlet states 7 , as described in Eq. 1.

Supplementary Figure 3 | Vortex fiber profile.
The vortex fiber (reproduced from supp. ref. [5], 2015, Optical Society of America) refractive index profile (red trace) is designed in the shape of an annular ring, mirroring the intensity profile of an OAM beam. Since the vortex fiber is cylindrically symmetric, the profile can be measured radially in 1-dimension. The ∆ refers to the profile's refractive index difference with respect to standard silica.

Supplementary Note 2: Wavelength to Polarization Mapping
The basis of the ORD Wavemeter (ORDW) is the ability to map changes in wavelength directly to a rotation in polarization. The basic optical activity (OA) relation, for a single wavelength, is described in Eq. 4, and explicitly reveals a wavelength-dependence. However, to track how our OA polarization state changes with wavelength from a tunable laser source, we must also account for that fact that our effective index splitting (∆ eff ) is wavelength dependent. As such, we take a simple first order Taylor expansion of ( ) around 1 : We see that from the first order Taylor expansion of our OA, Supplementary Equation 1, we obtain the linear ORD mapping between ∆ and ∆ from Eq. 5.

Supplementary Note 3: Modal Purity Analysis and Output Mode Conversion
The ORDW performance relies on our ability to propagate pure OAM states in vortex fiber 5 . To determine modal purity, the output OAM beam's intensity profile is imaged on a Thorlabs DCC1545M camera [see Supplementary Figure 1] and interferometric analysis is used to detect interference effects between vortex fiber modes 8 . One way to reduce modal cross talk is with precise input coupling of the beam to the vortex fiber. For this reason, a 6-axis stage (Thorlabs, MAX607L) and "walk-the-

Supplementary Note 4: Spectral Bandwidth Measurements and Simulations
The ORDW setup for measuring spectral bandwidth is the same as described in the Methods section, with a few alterations 10 . In order to control the linewidth of our laser, we add an electro-optic phase modulator (iXblue MPX-LN-10) controlled by an arbitrary waveform generator (AWG, Tektronix AWG7082C). Supplementary Figure 5 depicts a simplified setup, similar to Fig. 2(a), to illustrate where these additional components are inserted. The spectral bandwidth experiments also requires our PBS to be biased at an angle such that power is maximized ( max ) in one polarization bin, while minimized ( min ) in the other. In our case, we use an alternative method and left = 0°, aligned with (̂,̂) polarizations. Instead, we simply adjusted the operating wavelength until our power condition was satisfied. With either method, pre-processing of our setup to meet the visibility power condition is necessary, but could be accounted for during a calibration of the device. shows how for bandwidths > 0.7 GHz our visibility results converge to a Gaussian numeric model (with 13.7 dB noise), but < 0.7 GHz our visibility deviates and matches closer to a Lorentzian numeric model (with 17.5 dB noise). Note that the data presented in Supplementary Figure 6(b) is different than the data presented in Fig. 2(d). Noise is included to realistically model our photodetector measurements, as min ≈ 0 for a narrowband linewidth would be clearly unphysical. which results in, at most, a drop of ~15% visibility. Theoretically, the ORDW can measure spectral bandwidth up until ≈ 0, indicating there is a limit to the total spectral broadening it can detect.
However, we can overcome this limit and extend the total bandwidth detection. By shortening fiber length and/or using lower order modes the power distribution in Supplementary Equation 2 changes, encompassing a larger span of wavelengths. This would allow us to measure larger changes in for similar drops in visibility, hence extending the range over which we can broaden our spectrum.
Supplementary Figure 6 | Spectral bandwidth simulations. (a) Lorentzian (red) and Gaussian (yellow) fits applied to both an unbroadened (bandwidth < 0.7 GHz) and broadened (bandwidth > 0.7 GHz) spectrum, measured with a FP. In the unbroadened case, a Lorentzian functional form fits the experimental spectrum better. In the broadened case, a Gaussian functional form fits the experimental spectrum better, especially around the tails. (b) Visibility simulations using both Lorentzian and Gaussian numeric models, considering different noise parameters. When the spectrum bandwidth is < 0.7 GHz, the experimental visibility displays Lorentzian behavior, but as the spectrum is broadened, tends towards Gaussian behavior, albeit with a higher noise floor.

Supplementary Note 5: Mechanical and Thermal Perturbations
We determine stability of our ORD mapping by assuring that for a single wavelength ( 0 ) its corresponding polarization angle 0 does not drift over time. From Figure 3(a) we see that the ORDW is far more stable than the PM-fiber (which was spooled and not subjected to external perturbations) 11 .
We consider two cases, mechanical and thermal perturbations, where we used a layout identical to that presented in the Methods section, except that we laid out 40 m of vortex fiber on a stirring hotplate (Thermolyne Cimarec 2) and two Ge Thorlabs S122C photodetectors were used to sample power (hence ). For the mechanical perturbations, we laid out the fiber on a cardboard base, and used the maximum stirring function to vibrate the fiber. Over the course of 5 minutes we measured the OA angle [see Fig. 3 charge. Supplementary Figure 7 shows results for OAM modes ℒ = 10, 11, and 12, demonstrating excellent agreement of experiment with theory, as the derived ∆ g values from ORD calibration is clearly proportional to ℒ 2 .