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Optical Harmonic Vernier Effect: A New Tool for High Performance Interferometric Fibre Sensors
André D. Gomes, Marta S. Ferreira, Jörg Bierlich, Jens Kobelke, Manfred Rothhardt, Hartmut Bartelt, Orlando Frazão
TL;DR
Optical Vernier sensing magnifies interferometric shifts, but conventional configurations use closely matched interferometer frequencies and can be difficult to reference. This paper introduces harmonic Vernier sensing in Fabry–Perot interferometers, with magnification increasing by harmonic order. The authors report enhanced magnification and resolution, alongside greater flexibility in sensor dimensioning and fabrication tolerances.
Problem
Optical fiber sensors need higher sensitivity and resolution, while conventional optical Vernier configurations rely on slightly shifted interferometer frequencies and can make it difficult to maintain a stable reference.
Source: §1 · p. 1Method
The authors introduce harmonic Vernier sensing in Fabry–Perot interferometers by setting one interferometer’s optical path length to a multiple of the other’s plus detuning.
Source: §1 · p. 2Results
The reported strain sensitivities were 27.6 ± 0.1 pm/με, 93.4 ± 0.6 pm/με, and 59.6 ± 0.1 pm/με for the first, second, and third harmonics, respectively.
Source: §3.2 · p. 10Takeaways & Limitations
The harmonic effect can increase magnification and resolution without compromising signal detection and monitoring, while allowing more flexible sensor dimensioning.
Source: §4 · p. 11Fabrication-related detuning error remains a limiting factor for the magnification factor, although its effect is smaller at higher harmonic orders.
Source: §3.1 · p. 8
Abstract
from arXiv · showhide
The optical Vernier effect magnifies the sensing capabilities of an interferometer, allowing for unprecedented sensitivities and resolutions to be achieved. Just like a caliper uses two different scales to achieve higher resolution measurements, the optical Vernier effect is based on the overlap in the responses of two interferometers with slightly detuned interference signals. Here, we present a novel approach in detail, which introduces optical harmonics to the Vernier effect through Fabry-Perot interferometers, where the two interferometers can have very different frequencies in the interferometric pattern. We demonstrate not only a considerable enhancement compared to current methods, but also better control of the sensitivity magnification factor, which scales up with the order of the harmonics, allowing us to surpass the limits of the conventional Vernier effect as used today. In addition, this novel concept opens also new ways of dimensioning the sensing structures, together with improved fabrication tolerances.
1. Introduction
Conventional optical Vernier sensing uses two interferometers with slightly shifted frequencies, but maintaining a stable reference can be difficult. The paper introduces harmonic Vernier sensing with Fabry–Perot interferometers, using different frequency relationships to increase magnification and offer more design flexibility.
1. Introduction
- The conventional effect overlaps two slightly shifted interferometer responses, but keeping one interferometer stable as a reference can be difficult when the devices are physically close.A parallel configuration using a 3dB fiber coupler had addressed this separation issue, but still relied on the standard Vernier effect.Source: §1 · pp. 1–2
- The harmonic approach sets one interferometer’s optical path length to a multiple of the other’s plus a detuning factor, allowing the interferometers to have very different frequencies.The authors introduce this as an extension of the optical harmonic Vernier effect for Fabry–Perot interferometers.Source: §1 · p. 2, Figure 1 · p. 2
- Magnification increases in proportion to harmonic order, while internal envelopes provide additional intersections for tracking spectral shifts.These internal envelopes also increase in number and free spectral range with harmonic order.Source: §1 · p. 2
- The authors present the approach as a way to tune sensing performance and adapt sensor dimensioning to an application.The introduction describes an experimental demonstration alongside the account of the effect’s properties.Source: §1 · p. 2
2. Theoretical Considerations
The theoretical treatment first describes the fundamental Vernier effect for two Fabry–Perot interferometers, then extends it to harmonic orders. Harmonics preserve the upper envelope while adding internal envelopes whose properties support increased magnification and spectral-shift tracking.
2.1. Fundamental Optical Vernier Effect
- A parallel 3dB-coupler configuration places one FPI in each arm, allowing the interferometers to remain independent and one to serve as a reference.The analysis assumes air cavities formed by silica tubes between single-mode fiber sections; the discussion says the theory can extend to other interferometer types.Source: §2.1 · p. 3
- The fundamental Vernier effect combines two FPI responses to produce a lower-frequency component determined by the difference between their optical path lengths.The analysis assumes coherent light and a two-wave approximation; it considers no propagation losses.Source: §2.1 · p. 3
- The fundamental magnification factor can be defined as the envelope FSR divided by the sensing FPI’s FSR, or as the envelope sensitivity divided by the sensing FPI’s sensitivity.Both definitions depend on the interferometer optical path lengths; the reference FPI can be adjusted to maximize enhancement.Source: §2.1 · p. 4
2.2. Optical Harmonic Vernier Effect
- Harmonics arise when one FPI’s optical path length is set to a multiple of the other’s plus detuning, avoiding the fundamental effect’s close-path-length requirement.The close optical path lengths can be challenging to fabricate in inline FPIs, while harmonic design expands sensor-dimensioning options.Source: §2.2 · p. 4
- The upper envelope regenerates with the same frequency and FSR across harmonic orders, and odd orders introduce a π-shift.Figure 2 compares the fundamental case with the first three harmonic orders and their FFTs.Source: §2.2 · pp. 4–5, Figure 2 · p. 5
- Although upper-envelope visibility decreases at higher orders, internal envelopes provide multiple intersections for tracking wavelength shifts and reduce errors from peak-intensity fluctuations.The internal-envelope FSR scales with harmonic order, and the number of internal envelopes is one greater than the order.Source: §2.2 · pp. 5–6
- For the same detuning, magnification scales linearly with harmonic order and is determined by the internal envelope rather than the upper envelope.A perfectly harmonic, zero-detuning configuration tends toward infinite magnification and an untrackable envelope, so detuning is required.Source: §2.2 · p. 6
- Even when the upper-envelope period exceeds the detector’s wavelength range, internal-envelope intersections can still track the wavelength shift.The reported 1 μm detuning-error comparison shows smaller magnification-factor variation at higher harmonic orders.Source: §2.2 · pp. 6–7
3. Results
Experiments with three reference FPIs demonstrate harmonic-order-dependent enhancement in strain sensing; compensated sensitivity rises with harmonic order, and higher orders tolerate greater fabrication error at the same M-factor.
3.1. Experimental Setup
- Strain was applied only to FPI1 in static measurements at room temperature, up to 600 με in 87.2 με steps.The sensing and reference interferometers were connected in parallel through a 50/50 fiber coupler.Source: §3.1 · p. 8, Figure 4 · p. 8
- Fabrication error was the limiting detuning source in this setup, and higher harmonic orders allowed larger fabrication tolerances at the same M-factor.The text identifies fabrication accuracy of 1 μm to a few micrometers; a 1 μm fixed detuning caused smaller M-factor variations at higher orders.Source: §3.1 · p. 8
3.2. Sensor Fabrication
- A 41 μm sensing cavity was paired with 72, 118, and 155 μm reference cavities to produce the first three harmonic orders.The reference lengths included detunings of −10, −5, and −9 μm, respectively.Source: §3.2 · p. 9
- The measured spectra show internal-envelope counts increasing linearly with harmonic order, while M-factors vary with reference-interferometer detuning.The first three M-factors derived from the spectra were 8.38, 28.42, and 18.33; the second harmonic had a larger upper envelope and M-factor than the less well-tuned third harmonic.Source: Figure 5 · p. 10, Figure 5a
- Compensated sensitivity increases linearly with harmonic order, and the M-factor improves by a factor of 1+i relative to the fundamental effect.Compensation permits comparison independent of reference-interferometer detuning.Source: §3.2 · pp. 10–11
- Strain sensitivities were 27.6 ± 0.1 pm/με, 93.4 ± 0.6 pm/με, and 59.6 ± 0.1 pm/με for the first, second, and third harmonics, respectively.The corresponding M-factors were 8.18, 27.7, and 17.7; sensitivity-based and FSR-based M-factors differed by at most 3.5%.Source: §3.2 · p. 10, Table 1 · p. 11
4. Discussion
Optical harmonics increase Vernier magnification and offer more flexible sensor dimensioning, while internal-envelope intersections provide sensing markers. Practical harmonic order is bounded by detection resolution, signal noise, and measurement demands; dynamic measurements also face wavelength-range and detector-speed constraints.
4. Discussion
- Harmonics can further increase sensitivity magnification without compromising signal detection, while internal-envelope intersections provide tracking points better suited to sensing.The authors also describe improved fabrication tolerances and new ways to dimension sensing structures.Source: §4 · p. 11
- Peak narrowing can put harmonic interference peaks below detector resolution, making them indistinguishable and preventing internal-envelope fitting.The maximum usable harmonic order depends on the application and detection system, and modulation visibility also deteriorates with signal noise.Source: §4 · pp. 11–12
- Multiple-parameter discrimination, such as strain and temperature, remains future work; the Fabry–Perot response combined with the envelope could support matrix-based discrimination or temperature compensation.Source: §4 · p. 12
- Dynamic measurements are constrained by the broad wavelength range needed to track the envelope, especially with a slow OSA.Suggested approaches include a fast optical interrogator with envelope-extraction software or a tunable laser that tracks the shift without measuring a full spectrum.Source: §4 · p. 12
- The authors present optical Vernier harmonics as a route to sensors for medicine, biology, and chemistry, and cite sub-nanostrain resolution with a simple silica-tube geometry as a possible future development.Source: §4 · p. 12
Appendix A. Derivation of the reflected light intensity for the fundamental optical Vernier effect
The appendices derive reflected-light intensity for coupled Fabry–Perot interferometers and develop the fundamental and harmonic-order Vernier envelope free spectral range and magnification relationships.
Appendix A. Derivation of the reflected light intensity for the fundamental optical Vernier effect
- The intensity derivation assumes physically separated FPIs in a parallel configuration with a 3dB coupler, equal input splitting, and a two-wave approximation for weak silica/air reflections.The silica/air Fresnel reflection is about 3.3% at 1550 nm, so only one reflection at each interface is considered.Source: Appendix A · p. 12
- The normalized reflected output intensity is derived by combining the fields from both interferometers, including their individual interference and mutual interference terms.The derivation uses the total output field and its complex conjugate to obtain the intensity expression.Source: Appendix A · p. 14
Appendix B. Derivation of the free spectral range (FSR) of the envelope and magnification factor (M-factor) for the fundamental Vernier effect
- The fundamental envelope FSR is derived from successive wavelengths where the two interferometers’ peaks return to phase alignment, using their respective FSRs.The interferometers’ peak wavelengths are written as offsets from a shared in-phase wavelength, and the envelope period follows from their FSR difference.Source: Appendix A · pp. 14–15
- The magnification factor is the envelope FSR divided by the sensing interferometer FSR, and can be expressed using the interferometers’ refractive indices and lengths.The appendix also gives the fundamental envelope FSR in terms of the two individual FSRs.Source: Appendix A · p. 15
Appendix C. Derivation of the free spectral range (FSR) of the envelope for the first optical Vernier effect harmonic and generalization for any harmonic order
- For the first harmonic, the second interferometer’s optical path length is increased by one first-interferometer optical path length, changing its FSR before the overlap analysis proceeds.The passage specifies OPL_2 = n_2L_2 + n_1L_1 for this setup.Source: Appendix A · p. 16
Appendix A. Derivation of the reflected light intensity for the fundamental optical Vernier effect
- The envelope FSR is the wavelength spacing between consecutive conditions in which both interferometers are in phase.The appendix derives this spacing from the two interferometers’ FSRs.Source: Appendix A · pp. 16–17
- For harmonic order i, interferometer 2’s optical path is increased by i times interferometer 1’s optical path, and its FSR is expressed in a general form.Source: Appendix A · p. 17
- The envelope FSR has a general expression as a function of harmonic order i and the interferometers’ FSRs.The appendix presents this as the result of its harmonic-order derivation.Source: Appendix A · p. 17
Appendix D. Example of the spectral shift
The example tracks an internal-envelope intersection in the second harmonic spectrum as strain changes. The intersection shifts toward longer wavelengths as applied strain increases.
Appendix D. Example of the spectral shift
- The tracked intersection around 1400 nm shifts toward longer wavelengths as applied strain increases.The example uses the second harmonic spectrum; Figure A3 shows spectra at 0, 348.8, and 610.5 με, with one of multiple internal-envelope intersections circled.Source: Appendix D · p. 17, Figure A3 · p. 17
- Figure A3 marks one of the multiple internal-envelope intersections with a red circle.Source: Figure A3 · p. 17
- The illustrated strain values are 0, 348.8, and 610.5 με.Source: Figure A3 · p. 17