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We make the case for minimizing cavity length of extrinsic Fabry-Pérot (FP) cavities for use in fiber-tip sensors. Doing so mitigates multiple challenges that arise from using multimode fibers: mode averaging, phase uncertainty, amplitude reduction, and spectral modal noise. We explore these effects in detail using modal simulations, and construct pressure sensors based on this principle. We discuss the multimodal effects that we observe in our fiber sensors, and use simple filtering of the spectral signal to more easily measure pressure sensitivity. The concept of short-cavity FP interferometry is important for ensuring high quality and performance of multimode fiber sensors.
© Optical Society of America
1. Introduction
Optical fiber sensors are known for their immunity to electromagnetic interference, operability in harsh environments, survivability in high temperatures, biocompatibility, compactness, and sensitivity. Multimode fiber (MMF) systems offer additional benefits of being less expensive compared to single-mode fiber (SMF) systems: the larger cores in MMFs permit easier coupling and a greater tolerance to misalignment, rendering expensive single-mode laser diodes unnecessary and relaxing the manufacturing constraints of fiber connectors.
Of fiber sensors, extrinsic Fabry-Pérot interferometers (EFPI) are among the simplest to build, and provide an inexpensive but reliable solution for probing displacement changes induced by measurands such as temperature or pressure. With proper calibration and assembly techniques, EFPIs can be used for absolute measurement as well. The MMF-EFPI combination thus enables a variety of sensors that are cost effective and easy to build.
Some reported MMF sensors use only an MMF segment in the sensor body while single-mode fiber is used elsewhere in the system [1,2], and therefore these sensors do not reap the benefits of an all-MMF system. Many sensors also require splicing and/or polishing during assembly [3,4], which are not easily scalable. In this paper, we present FP MMF pressure sensors fabricated using a CMOS-compatible process for scalability, and discuss how to mitigate multimode phenomena that adversely affect sensor operation: mode averaging [5], mode-power distribution (MPD) dependent phase [6], fringe amplitude reduction [7,8], and modal noise [5,9–11]. We investigate these detrimental multimode effects and demonstrate multimode pressure sensors designed with short cavities, which mitigate all of these effects.
2. Multimodal effects
2.1 Mode averaging induced sensitivity loss
One of the key issues in MMF-EFPIs operating in multimode is that of mode averaging, in which the sensitivity of the sensor is reduced because of the distribution of modal power. One can model the different mode groups traveling through a fiber as plane waves with different angles of propagation. These waves impinge on the Fabry-Pérot (FP) cavity at different angles, so the different modes do not all resonate at the same wavelength for a given cavity thickness. Consequently, the overall transmission peaks are widened and the FP fringes are reduced in amplitude (Fig. 1). For systems where the laser is operating at a fixed wavelength, a wider resonance means that there is less change in reflectance for a given change in cavity length. The longer the cavity is, the greater this loss of sensitivity, since the phase dispersion among the different incident angles is greater.
Fig. 1 The top plot shows mode averaging effects on the reflectance of an ideal Fabry-Pérot cavity for beams with different numerical apertures assuming equal distribution across angles up to the NA for cavity lengths of 0.5λ to 3.25λ. The bottom plot shows sensitivity defined as the slope of the reflectance. The NAs were chosen to represent common multimode fibers. The case where NA = 0 represents an ideal plane wave. FP mirror reflectances were assumed to be 0.8. The maximum sensitivity drops rapidly as cavity length increases.
For a low-finesse Fabry-Pérot cavity created by low-reflectance mirrors (such as formed by an air-glass interface), the sensitivity loss is less apparent (Fig. 2). However, since the resulting waves are sinusoid-like, one can think of them as having a “phase shift,” as discussed in the next section.
2.2 Phase dependence
The low-finesse Fabry-Pérot fringes shown in Fig. 2 illustrate another characteristic of the presence of multiple modes: the overall resonance (or fringe) shifts more from the ideal plane wave case as the cavity length increases. This effect becomes is more pronounced for higher NA fibers. In Fig. 1 and Fig. 2, we assumed uniform field contribution from all acceptance angles less than the NA. This is analogous to a uniform mode power distribution (MPD) in the wave model (ignoring degeneracy).
The effect of the MPD on the effective phase of the back-reflected light was studied for step-index multimode fibers and low-finesse FP cavities in [6]. The consequence of this MPD-dependent effective phase (the average of the accumulated phase of each mode group, or angle) is that the FP fringes in the transmittance or reflectance spectrum will shift depending on the power weighting of the modes in a multimode fiber. For example, a fiber with an NA of 0.39 can operate with just the fundamental mode excited, or it could guide a mix of modes, thus changing the shape (in the case of high-reflectance mirrors) or position (in the case of low-reflectance mirrors) of the FP resonances. Since MPD is not fully predictable in practical systems and changes according to varying environmental conditions, this effect introduces uncertainty in the interpretation of the FP spectrum.
Figure 3 shows the effect of different MPDs on the phase of the effective coupling coefficient of the reflected wave at λ = 1550 nm inside a 62.5 µm core parabolic graded-index fiber. The modes were approximated as Laguerre-Gaussian modes [12], with the radial and azimuthal indices calculated using the method in [13]. Waist size was computed according to Marcuse’s equation assuming a parabolic index profile [14].
Fig. 3 Left: Dependence of φeff of the reflected light from the distal mirror on the cavity length L for different MPDs inside graded-index multimode fiber with core diameter of 62.5 µm and NA of 0.275, for λ = 1550 nm. Right: Modal components of the different MPDs.
The phase shift φeff is related to the effective coupling coefficient ηeff and the individual mode coupling coefficients by
where k is the mode number, N is the total number of modes, pk is a complex coefficient of the kth mode in a given MPD, and ηk is the mode coupling coefficient of mode k coupling into itself after propagating 2L in free space [6]. In this model, cross coupling between modes is ignored under the assumption that all the modes have a random initial phase relationship and the number of modes is large [8]. However, the validity of this model is questionable for long cavities when the initial MPD contains few modes. We address modal cross coupling in more detail in Section 2.4 when discussing modal noise.From Fig. 3, we see that the range of the variation in φeff increases as the cavity length L increases. The absolute phase shift of a curve is less important than its deviation from the fundamental MMF mode, as our concern is with the spread of the φeff rather than its actual value. It is the variation in φeff that is responsible for shifting spectral peaks and valleys that correspond to a pressure reading, so uncertainty in φeff directly leads to uncertainty in the reading. It is therefore advantageous to use a short cavity that reduces the uncertainty caused by the MPD.
Gouy phase is not taken into account in Fig. 1 and Fig. 2, but is taken into account by the free space propagation modeling of the mode profiles used in Fig. 3, as evidenced by Fig. 4.
Fig. 4 Plot showing the phase accumulation of a plane wave with (dash-dot) and without (dash) Gouy phase, assuming a Gaussian beam waist calculated using Marcuse’s equation in [14] for our model 62.5-µm core graded index fiber used in simulations. The phase accumulation of the fundamental mode of a graded index fiber is also plotted (solid) to show that our freespace propagation simulation takes into account Gouy phase. The cavity plotted is about 12.5 wavelengths long, with λ = 1550 nm.
2.3 Fabry-Pérot fringe amplitude reduction
An effect closely related to the sensitivity loss induced by mode averaging is the reduction of fringe visibility for low-finesse MMF-EFPI, which is explored in detail in [7,8]. Using the ray-optics model in [7] and the wave-optics model in [8], we produce the plots shown in Fig. 5. We note that the effect is aggravated by an MMF with a larger NA. While having a larger NA is advantageous in coupling more light into the MMF, the tradeoff is that the sensor experiences greater reduction in the fringe amplitude. However, not only is sensitivity reduced in general, but at certain cavity lengths L, the fringe amplitude is at a local minimum. Minimizing the cavity length L of the EFPI preserves the advantage of high coupling as well as of mode averaging minimization.
Fig. 5 Fringe visibility as a function of cavity length for different NA. Mirror reflectances assumed to be 0.04. For the wave simulation, the fibers were assumed to be step index fibers of 50-µm diameter core with the following parameters: NA = 0.2, ncore = 1.454, nclad = 1.440; NA = 0.27, ncore = 1.465, nclad = 1.440; NA = 0.39, ncore = 1.492, nclad = 1.440.
2.4 Modal noise
Finally, a short cavity addresses what is known in the literature as modal noise [9–11], which is an effect of modal dispersion, interference, and mode dependent loss such as misalignment, partial light coupling, or partial light capture at the detector (in the case of an optical spectrum analyzer, the partial capture is caused by the presence of an aperture in the light path to the detector) [15]. The dispersion of modes with different propagation constants creates interference patterns (also known as speckle) at the output facet of a fiber segment that are dependent on length, temperature, laser wavelength, and stress. Changing interference patterns translate into fluctuations in power in both wavelength and time domains given spatially dependent loss. Commercial FP MMF sensors use white-light interferometry for absolute measurement, but the noise performance is not addressed in [16,17].
In the context of tracking peaks and valleys in the FP spectrum, simple filtering is sufficient to recover the Fabry-Perot. In our post-processing scheme, we took the discrete Fourier transform of the spectral data, computed with the Fast Fourier Transform (FFT). We then truncated the frequency-domain signal to a specified fraction of the original signal, and then computed its inverse FFT to get the “filtered” spectrum. When the free spectral range (FSR) of the FP oscillation is away from the frequency of the power fluctuations in the spectrum, we can filter the spectrum more aggressively while retaining the underlying FP oscillation (see the Results section for an experimental illustration). As a corollary, increasing the fiber length leading up to the FP cavity also makes it easier to filter out the modal noise (see the Results section), so the spectrum does not have to be filtered as aggressively, allowing for a larger portion of the FP fringes to be kept.
One significant source of modal noise (aside from imperfect fiber coupling) is the FP cavity itself. A sample FP spectrum is simulated in Fig. 6 for two different cavity lengths. By propagating the modes through a long length of fiber, the phase relations among the modes become random. Not only are the shifts in the FP fringes evident as a consequence of the effective phase described in Section 2.2, but the signal-to-noise ratio of the FP fringe is also significantly higher at lower L. This spectral modal noise is caused by intermodal coupling of the reflected wave from the distal mirror of the FP back into the fiber and is worse in the fibers we simulated when there are more modes in the system or when the cavity length increases. The diffraction of the reflected light causes the modes to spread out spatially, and so the overlap integral between reflected mode k and the fields of the other modes becomes non-zero. This effect is less prominent the shorter a cavity gets, as shown in Fig. 7.
Fig. 6 Spectrum of low-finesse EFPI with cavity lengths 10 µm (left) and 40 µm (right). This spectral modal noise is caused by intermodal coupling of the reflected wave from the distal mirror of the FP back into the fiber. The SMF fundamental mode waist size depends on wavelength and ranges from 4.1 µm to 4.3 µm in our simulation. We swept the wavelength in 0.5 nm increments, which limits the perceived frequency of the noise in our spectra.
Fig. 7 The magnitude of modal coupling from back-reflection at the distal mirror (sensor membrane) at different cavity lengths (columns). Five Laguerre-Gaussian modes are shown as examples of increased mode coupling at longer cavity lengths. The operating wavelength is 1550 nm.
While this modal cross coupling complicates the earlier-described phase effect in Section 2.2, the random phase relationship between each of the modes greatly reduces its overall phase-changing effect. The result of cross coupling manifests itself primarily in the spectral noise, which is filtered out from the experimental data. Nevertheless, the greater the cavity length, the higher this noise is, and the effective phase itself become more uncertain as filtering becomes less effective.
In the case of “all modes equally excited” in the example illustrated in Fig. 6, the signal to modal noise ratio for 10-µm cavity length is estimated to be 5.5, whereas that of the 40-µm cavity is about 1.1. The estimations are performed by dividing the RMS of the AC component of the fitted sine by the RMS of the noise, which is computed using the difference between the simulation result and the curve fit.
In a practical system, the MPD in a multimode fiber is more a function of the launching condition than of bends and imperfections [18,19]. Indeed, experimental evidence suggests that if a single-mode fiber laser source is coupled into a multimode fiber, the modes remain in the lower few modes as shown in Fig. 8. While other sources of modal noise can be minimized, the inherent mode coupling caused by the FP cavity itself becomes significant.
Fig. 8 (A) A 1521 nm laser beam is launched through a single mode fiber with an optical isolator into a multimode fiber. After propagating through the MM 1x2 50/50 coupler and through ~100 m of fiber, the mode profile remained in the lower few modes (B). Applying lateral pressure using the Newport FM-1 Mode Scrambler induced mode coupling. (B-E) Mode profiles of fiber under different mode power distributions. (B) Mode profile without applying lateral pressure. (C) Mode profile with the mode scrambler knob turned 6 minor tick marks from initial position. (D) Mode profile with the mode scrambler knob turned 7 minor tick marks from initial position. (E) Mode profile with the mode scrambler knob turned 8 minor tick marks from initial position; most of the lower order modes have been coupled into the higher order modes.
3. Sensor construction
3.1 Structure
Our pressure sensors consist of FP cavities at the tip of multimode fibers (Fig. 9). The silica-air interface at the fiber tip forms one mirror of the FP cavity. A microfabricated membrane (with its interfaces) is lumped together in the other mirror. We used two packaging methods as shown in Fig. 9. In package A, the fiber was inserted into the backside hole and affixed using epoxy. In package B, a ferrule was aligned to a membrane and bonded to the chip using epoxy before the fiber was adjusted and bonded. In both cases, the fiber-membrane distance was adjusted using a closed-loop DC stepper motor-driven translation stage (Physik Instrumente’s M-111). We tried to minimize the FP cavity length while still being able to discern a FP transmission dip on the optical spectrum analyzer (OSA). This mitigates the multitude of challenges introduced by multimode operation.
Fig. 9 Cross-sectional view of the two sensor package types. (A) Backside fiber insertion: 125 µm-diameter fiber is inserted into the backside of the chip and is affixed using epoxy. (B) Fiber insertion through ferrule: A ferrule was aligned to a membrane and bonded to the chip using epoxy before the fiber was adjusted and bonded.
In practice, we ended up with cavity lengths down to about 4 wavelengths. Our broadband source (AFC BBS1310) had a bandwidth of 1280 nm to 1420 nm, outside of which range interference effects could not be discerned.
3.2 Fabrication
To fabricate the membrane, a silicon-on-insulator (SOI) wafer with a device layer thickness of 2 ± 1 µm was thinned on the front-side using reactive ion etching (RIE) and released on the backside using Deep RIE (DRIE), producing a 135 µm-diameter membrane (Fig. 10). Finally, a 6:1 buffered oxide etch (BOE) was performed for about 10 min to thin the oxide to a nominal thickness of 1 µm. Due to the device layer thickness variation, the membrane consists either of pure silicon dioxide (where the device layer was fully removed by RIE) or a bilayer of Si on SiO2.
Fig. 10 Membrane fabrication processing steps. (A) Start with SOI wafer with device layer thickness of 2 ± 1 µm and buried oxide thickness of 2 µm. (B) Thin the sensor area using reactive ion etching (RIE) to a nominal thickness of 0.5 µm. (C) Perform deep RIE using the Bosch process; the buried oxide acts as the etch stop. (D) Buried oxide etching with 6:1 buffered oxide etchant (BOE) for 10 min to thin the oxide to about 1 µm.
4. Experimental results
4.1 Experiment
Five MMF and four SMF pressure sensors were tested in a pressure chamber (Fig. 11). In the experimental fiber set up, broadband light is coupled through a multimode (MM) fiber to a 1x2 MM combiner/splitter, where half the power is directed to the sensor arm. A fiber chuck holds the sensor head inside a pressure chamber made of ABS. Compressed air pressurizes the chamber according to the setting on the regulator, and the precision digital pressure meter monitors the chamber pressure. Light reflected from the sensor travels back up to the 2x1 MM splitter, and half of it is diverted to the optical spectrum analyzer (OSA), where the spectrum is recorded for different pressures.
The usable bandwidth of the broadband source was 140 nm, from 1280 nm to 1420 nm. The OSA resolution used for each measurement result is included with the spectra.
4.2 Results
The effect of modal interference is readily seen in the optical spectra of the MMF sensors (Figs. 12-14). To compensate, we post-processed the spectral data using an ideal low-pass filter (LPF) using the method described in Section 2.4. It is important for the FP fringes to have a longer oscillation period in the spectral plot than the modal interference fringes to facilitate filtering (Fig. 12). Using a long lead-in fiber is also beneficial, as described in Section 2.4 and shown in Fig. 13.
Fig. 12 Unfiltered and filtered spectra of two sensors with cavity lengths ~30 λ and ~6 λ. The OSA resolution used was 0.5 nm. For both spectra, the Fast Fourier Transform (FFT) was taken and truncated to what corresponds to a frequency of 0.03 nm–1, and then the inverse FFT was taken to get the filtered spectrum.
Fig. 13 Unfiltered and filtered spectra of a sensor using short and long lead-in fibers, where the round-trip fiber length was about 10 m and 150 m, respectively. The OSA resolutions used were 0.5 nm and 0.1 nm, respectively. Both spectra were low-pass filtered at 0.075 nm–1.
The first-order pressure sensitivity in nm shift per psi is calculated by finding the intersection between the set of filtered spectral curves and an appropriate horizontal line drawn near the middle of the sloped region (Fig. 14). Similarly, pressure sensitivity in percent reflectance per psi is calculated at an appropriate fixed wavelength (chosen differently for each sensor). The results are summarized in Table 1.
Fig. 14 (Top left) Unfiltered pressure spectra for sensor MMF-2 (OSA resolution of 0.2 nm). (Bottom left) Spectra filtered at 0.0625 nm–1. (Right) Shift of the filtered spectra vs. pressure at 30% reflectance point from 1345 nm to 1370 nm. For MMF-2, the sensitivity is about –1.5 nm/psi.
Table 1. Performance summary of five multimode fiber-tip pressure sensors (MMF 1 through 5) and four single mode fiber-tip pressure sensors (SMF 1 through 4). The spectral shift sensitivity and the mechanical sensitivity change depending on the low-pass filter (LPF) cut-off as well as the reflectance at which the spectral shift is measured. The reflectance sensitivity can also change depending on the LPF cut-off and the operating wavelength.
5. Discussion
The spectral modal noise observed in the OSA measurements is partly attributed to the interference pattern that is periodic as the wavelength is swept [11]. More specifically, the difference in propagation constants (Δβ) between two modes causes the accumulation of a phase difference Δϕ between the two modes. For a slight change in wavelength, the modes remain approximately the same, but Δϕ = lΔβ will change, where l is the length of the fiber. The interference pattern between two modes will repeat for every 2π increase in Δϕ, causing rise and fall in the light collected by the OSA (which is spatially filtered by an aperture). Over a range of wavelengths, one will therefore see oscillations in the spectrum. Because there may be many modes interfering at the same time, the modal noise can appear quite irregular. The underlying oscillation frequency components will be determined by the difference in propagation constants Δβ. Naturally, a longer fiber length l means that Δϕ changes more rapidly with λ. Δβ is generally not the same between adjacent mode groups. However, with parabolic graded-index fibers, β is evenly spaced between adjacent mode groups [13]. Therefore, once the oscillations caused by Δϕ becomes more rapid than that caused by the FP interference, they should be relatively easy to filter out.
The variation in the thickness of the SOI device layer ( ± 1 µm) in conjunction with additional fabrication variations explains the device-to-device variation in the mechanical sensitivity, because the spring constant of a membrane scales as the inverse cube of its thickness [20]. Furthermore, depending on whether the membrane consists of only SiO2 or a bilayer of Si on top of SiO2, the membrane stress would affect the sensitivity as well. In light of this, improvements in fabrication could be made by using more SOI wafers with tighter tolerances in their device layer thicknesses, or by using larger membranes such that the device layer variations have less impact on the mechanical behavior of the device. We can even obviate the use of SOIs by using a timed backside etch. In this way, we can avoid residual stress from the buried oxide layer.
In the course of our sensor assembly and experiments, we found that the sensors with ferrules (package B) were easier to assemble, had a higher yield, and were more structurally robust. Having shorter cavities showed marked improvement of the contrast ratio between bright and dark fringes in the Fabry-Pérot cavity as well as a better signal to noise ratio in the spectrum (Fig. 12), as expected from our modal simulations (Fig. 6). We also found that the multimode noise, though significant, can be filtered out. Ultimately, the required response time and accuracy of the application will determine the best post-processing scheme. In a practical system that uses a spectrometer, the resolution of the spectrometer may be the limiting factor on the precision of the system. In such a case, shortening the cavity allows better discernment of spectral shifts. A shift of ΔL in the cavity length will lead to a shift of ΔL/m in the resonant wavelength (which quantifies the spectral shift), where m is the number of wavelengths that can fit in the cavity. For a given membrane design, a shorter cavity will reduce the m, making the shift more discernable by the spectrometer. For a practical system that operates at a single wavelength to measure the change in the reflected power, small shifts in wavelength would lead to larger variations in output signal power in longer cavities, so shorter cavities mitigate the effects of wavelength stability.
In a practical system, the environment may also undergo significant changes in temperature. The focus of this paper is on the optical benefits of short cavities, so the temperature sensitivity of our devices has not been studied in detail. Insofar as the variation in the measurement environment may change the FP fringe location, we have kept measurement times long enough for the temperature in the chamber to reach room temperature as we slowly increased the pressure. The room temperature of the lab is fairly stable as well. The major sources of temperature dependence of the device are the type of epoxy we used to seal the sensor as well as the air sealed within our cavities. In a production system, the sensors would be assembled in a vacuum environment using thermally stable adhesive. Additional temperature compensation can be provided during data processing using information from temperature sensors in the system.
6. Conclusion
Multimode fiber sensors are advantageous over their single mode brethren because they have larger cores, and therefore are more tolerant to misalignment, making connectors cheaper. Because they support more modes, it is easier to couple light into these fibers, so cheaper light sources such as LEDs may be used instead of high-quality lasers.
With these advantages come some challenges for FP fiber-tip sensors: loss of sensitivity due to mode averaging, MPD-dependence of the FP reflectance spectra, and modal noise. Short FP cavities help to overcome all of these: in addition to reducing diffraction and spectral averaging over incident angles, a short cavity reduces smearing of resonance peaks for high-finesse cavities, increases FP fringe visibility for low-finesse cavities, reduces the effect of the mode power distribution on the peak or dip position of the Fabry-Pérot fringes in the spectral domain, and facilitates the filtering process to better deal with modal noise.
The conclusion is that multimode fiber-based sensors can be practical in spite of mode averaging and modal interference by going to very small cavity lengths (on the order of a wavelength) and simple signal processing, which can be performed in real-time in production systems. The simplicity of the system enables the inexpensive mass production of sensors that integrate easily with existing multimode fiber infrastructure. Furthermore, using the short-cavity design principle, sensor designers and manufacturers can greatly improve their sensors in terms of sensitivity, noise performance, and reproducibility.
Acknowledgments
This work was sponsored by the Defense Advanced Research Projects Agency (DARPA) N/MEMS S&T Fundamentals program under grant no. N66001-10-1-4004 issued by the Space and Naval Warfare Systems Center Pacific (SPAWAR) and the Boeing Company under contract #33130. All cleanroom work was performed at Stanford Nanofabrication Facility.
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