Optical Feedback FM-to-AM Conversion with integrated Micro-Ring Resonator for Displacement Sensing Applications

In this study, we show the capability of integrated Micro-Ring Resonators (MRRs) to perform frequency-to-amplitude (FM-to-AM) conversion of optical feedback interferometry (OFI) signals with improved signal-to-noise ratio compared to conventional AM OFI signals. Further, contrary to traditional OFI FM-to-AM conversion techniques using gas cell-based edge filters and free-space or fiber Mach-Zehnder Interferometers (MZI), integrated photonic processing offers greater compactness and perturbation resilience, enhancing noise performance through improved temperature control and immunity to parasitic mechanical vibrations. The OFI FM-to-AM conversion was performed with a fabricated silicon nitride MRR of radius 120 μm and a quality factor of 130,000. The FM-to-AM conversion factor achieved was 0.61 GHz−1, with a noise equivalent displacement of only 4.9 nm for a 1 kHz bandwidth. This demonstration highlights the potential of integrated edge filters to replace traditional freespace and fiber architectures, more prone to environmental perturbations, in OFI signal processing for vibrometric applications.


I. INTRODUCTION
Optical feedback interferometry (or OFI), commonly known as the self-mixing (SM) effect in laser diodes (LDs), has garnered much attention in recent years as a cost-effective and self-aligned solution for various sensing applications, such as measuring displacement, velocity, and vibrations [1], [2].The quality of OFI-based displacement sensor measurements is typically dependent on the signal processing method [3]- [6].
OFI involves a portion of the laser beam being backscattered from a target and re-entering the active laser cavity, leading to modulation of the amplitude and frequency of the optical field within the cavity (as shown in Fig. 1 (a)) according to Lang and Kobayashi equations [7].The standard method of obtaining OFI information from the optical power or amplitude modulation (AM channel) is through monitoring the LD's emitted optical power using a photodiode.Nevertheless, a study in [8] found that the frequency modulated (FM) OFI signals exhibit a signal-to-noise ratio (SNR) that is significantly better than that of AM signals, generally with a two-order magnitude improvement.Subsequently, FM channel OFI demodulation could potentially be used to enhance sensitivity in applications that require low-noise for small displacement measurements, and in cases of non-cooperative, weakly back-scattering target surfaces.The use of edge optical filters for OFI FM-to-AM conversion has gained attention in recent years.In Fig. 1 (b), the conversion process is demonstrated.A study by Contreras et al. [9] utilized an acetylene gas cell as the converting filter and achieved a conversion factor of approximately 2.2 GHz −1 with a typical absorption line depth of -8 dB.
Further advances in OFI FM-to-AM conversion came with the introduction of free-space and fiberbased components such as Mach-Zehnder Interferometers (MZI) as a processing method.This approach, documented in [8], [10], offers increased sensitivity and flexibility over traditional methods, and allows for the easy tuning of the LD wavelength to one of the periodic signatures of the MZI.Furthermore, the sensitivity and response steepness can be regulated through adjustments to the optical path imbalance in the two arms of the MZI.To perform the OFI FM-to-AM conversion, we have previously implemented MZIs on silicon nitride (Si 3 N 4 ) process [11], as illustrated in Fig. 2. Si 3 N 4 is advantageous here as it has lower propagation loss and smaller thermo-optic coefficient than silicon photonics, with respective values of 2.45•10 −5 RIU/ • C and 1.8•10 −4 RIU/ • C at 1550 nm wavelength [12].With an MZI having a 2 cm imbalance in arm lengths, we previously obtained a filter with a conversion factor of 0.3 GHz −1 with 20 dB extinction ratio and a thermal sensitivity of -17 pm/K.In this work, we implement an integrated Micro-Ring Resonator (MRR) to be used as the edge filter, in order to drastically reduce footprint and increase the conversion factor.The paper is divided into two primary sections.The first, Section II, provides an overview of the FM OFI signal and the integrated OFI FM-to-AM conversion scheme proposed here.In Section III, the experimental setup and results are described, leading to a conclusion in the final section, Section IV.

II. INTEGRATED OFI FM-TO-AM DEMODULATION A. FM OFI overview
In the context of the AM channel OFI, fluctuations in the optical output power (OOP) of an LD can be induced by OFI, as defined by P (t) in equation ( 1) [1]: where P 0 is the emitted optical power when the laser is free-running, m the modulation index, and Φ F (t) the laser output phase in the presence of feedback.The laser output phase without feedback, Φ 0 (t), is related to Φ F (t) through equation ( 2) [1], [2]: with α the linewidth enhancement factor of LD and C the optical feedback factor.Based on the value of C, the laser can operate in three regimes: weak feedback (C < 1), moderate feedback (1 < C < 4.6), or strong feedback (C > 4.6).However, moderate feedback is generally preferred as it offers apparent simple saw-tooth shaped SM fringes [13] which provide directional indication of the target movement and simplified SM fringe detection processing [3].
The equation for the FM channel OFI in (2) can be written as: where ν 0 is the original LD frequency, ν F its frequency in the presence of feedback, and τ ext the external round trip time.The difference between the two frequencies, Δν = ν F − ν 0 , is limited to |Δν| ≤ C 2πτext , and the optical filter must take this maximum range into consideration to avoid folding.Fringe-locking to the half fringe, as described in [8], can broaden this range and make the filter even steeper.

B. The OFI FM-to-AM conversion
As mentioned earlier, the OFI FM-to-AM conversion is accomplished through the use of an MRR, which is depicted in Fig. 2. If the coupling is symmetrical and lossless, the output intensity transmission I out can be represented as [14]: , with φ = 2πLn ef f /λ (4) I in represents the optical spectrum of the light that is injected into the MRR, while λ indicates the wavelength in vacuum.The self-coupling coefficient of the resonator's coupler is denoted as τ , and the round trip field transmission with attenuation due to light scattering and absorption is expressed as α a .The single-pass phase shift of the ring-propagated wave is represented by φ, the circumference of the ring is denoted by L, and n ef f represents the effective refractive index of the ring's waveguide mode considered.When critical coupling is achieved [14], which happens when internal losses are equal to cross-coupling, complete destructive interference can be obtained, leading to maximum extinction ratios in the resonance spectrum.In this case, the Full-Width-Half-Maximum (FWHM) can be estimated as: where n g the group index is defined as n g (λ) = n ef f (λ) − λ ∂n(λ) ∂λ , and κ is the cross-coupling coefficient for which κ 2 + τ 2 = 1.Approximating the transmission spectrum as a Lorentzian function, the edge filter sensitivity can therefore be given as: around half-intensity transmission.

A. Experimental set-up
Figure 3 illustrates the experimental setup.The light source used in this experiment is a WSLD-1550-020m fiber pigtailed LD from Wavespectrum, which emits 8 mW of power at λ=1550 nm.The injected lightwave is then divided into two branches using a fiber coupler with a splitting ratio of 80/20, as shown in Fig. 2: (1) the measurement branch that connects LD to the target, and (2) the OFI FM-to-AM conversion photonic chip.A temperature controller (TEC) is used to set the PIC temperature with a resolution of 0.01 • .A PZT from Physics Instruments is employed as the target in this experiment.The output of the OFI FM-to-AM converted photonic chip is monitored by a Thorlabs PDA50B-EC photodetector.
The distance between LD and the target is approximately 2.5 m, which corresponds to a time delay of τ ext ≈ 24 ns.According to equation (6), in the moderate feedback regime, the maximum variation Δν that can be observed is around 30 MHz.The integrated photonic circuit is fabricated using a 300 nm Si 3 N 4 process with a refractive index of n Si 3 N 4 ≈ 1.7 (Fig. 2).An MRR with a circumference of 754 μm (see Fig. 3(c)) has been implemented.Prior to design and fabrication, propagation loss and waveguide coupling have been measured and analyzed for critical coupling to be obtained.The characterization and reporting of this MRR are performed using the setup illustrated in Fig. 2. The light from the fiber is injected into the integrated photonic circuit via integrated grating couplers using a fiber array (FA) with a 30 • polishing angle (see Fig. 3(b)).

B. MRR characterization
The optical transmission spectrum of the MRR, measured with an APEX206 optical spectrum analyzer, is shown in Fig. 4(a).The measured extinction ratio is ∼ 10 dB, and the FWHM is approximately 12 pm, corresponding to a sensitivity of ≈ 0.67 (GHz) −1 .The measured slope of the optical filter is approximately 0.61 GHz −1 .It should be noted that the MRR spectrum in Fig. 4(a) also accounts for the injection loss into the integrated photonic circuit via the grating couplers (≈ 6.5 dB/coupler) and the propagation loss (≈ 2 dB/cm) in the Si 3 N 4 waveguide.

C. OFI FM-to-AM noise performances
Figure4(b) presents a typical OFI signal obtained after FM-to-AM conversion using the integrated MRR.The photodiode gain was set at 20 dB.The NED can also be estimated as reported in [8]: where V pp denotes the peak-to-peak amplitude of the OFI signal and V RM S denotes the OFI RMS noise.The current NED achieved is approximately 4.9 nm for a 1 kHz bandwidth.
IV. CONCLUSION AND DISCUSSION Our experimental results demonstrate the successful integration of an OFI FM-to-AM conversion scheme for OFI applications with a sensitivity of 0.61 GHz −1 for a 754 μm long MRR.Although the achieved NED, of approximately 4.9 nm, for a 1 kHz bandwidth, is more than two orders of magnitude larger than in [8], there is potential for sensitivity enhancement through future designs featuring resonators with higher quality factors and shorter laser-target distances.Additionally, noise performance could also be improved by using edge couplers [15] instead of grating couplers and reducing waveguide propagation loss to lower dB/m values, as reported in [16].
With its compact size, electrical tunability, and ease of thermo-regulation, the integrated OFI FM-to-AM conversion technique provides a promising alternative to fiber-based approaches for OFI FM-to-AM conversion in sensing applications.Further improvement in sensitivity could also be achieved by exploring other types of filters based on integrated photonic filters.

Fig. 1 .
Fig.1.a) Typical fiber OFI sensing setup using an LD and a piezo-electric transducer (PZT) as a target, showing, in particular, the AM channel retrieved from the monitoring photodiode.b) OFI frequency modulation (FM) signal embedded in the laser frequency is converted into amplitude modulation (AM) by using the fringe edge of a Micro-Ring Resonator (MRR) serving as an optical filter.

Fig. 2 .
Fig. 2. Schematic diagram of one of the three OFI FM-to-AM OFI systems that includes one LD, two photodiodes (PD), one fiber array, an Integrated Photonic OFI FM-to-AM filter based on an MRR, and a loudspeaker as remote target.

Fig. 3 .
Fig. 3. a) Experimental setup incorporating fiber array and integrated photonic chip including MRRs and, b) zoomed image of FA-to-chip coupling and, c) microscope image of MRR and adjoint input/output grating couplers

Fig. 4 .
Fig. 4. a) Optical transmission spectrum of the MRR and, b) OFI output signal obtained after OFI FM-to-AM conversion using MRR for a target (PZT) vibrating at 50 Hz.