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Abstract
Speckle patterns produced by the interference between multiple optical fiber modes carry detailed information regarding the waveguide characteristics. However, reflection-type setups implemented with fiber coupler and mirror generate oscillating specklegrams that cycle consistently around well-defined states. To study such a peculiar phenomenon, an analytical model derived from a multimode fiber Fabry-Pérot interferometer investigates the contributions of the cavity length, refractive index, and laser wavelength, revealing a sinusoidal modulation of the correlation coefficient. Experiments confirm the effect of the fiber-mirror distance on the specklegram changes: computing the correlation and tracing the spatiotemporal evolution of a light granule through active contour models yielded a phase sensitivity of 15.71 rad/µm. Furthermore, the presented system works as a precise displacement sensor enhanced by the extended correlation algorithm, achieving 3 nm resolution with linear response over a 70 µm range.
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1. Introduction
Multimode optical fibers (MMF) excited with coherent light generate a typical speckle-like pattern (specklegram) due to the interference between several guided modes. Variations in mode coupling characteristics and phase deviations—induced by disturbances in the optical setup—produce light intensity fluctuations that compromise optical communication systems [1]. Despite this undesirable modal noise, specklegrams carry dense, precise, and repeatable information regarding the fiber state [2]. Such remarkable feature motivates recent applications in imaging [3], signal multiplexing [4], spectral analysis [5], and physical/chemical sensing.
Fiber specklegram sensors (FSS) quantify a measured variable from the relative changes observed in the output speckle field. FSS present high sensitivity and intrinsic quasi-distributed sensing capability [6,7]. Besides, compared with the widespread intensity and spectral systems, specklegram sensors are implementable with visible lasers and ordinary cameras, avoiding expensive optical interrogators and spectrometers. Successful examples of FSS encompass the assessment of strain [8], temperature [9], magnetic field [10], and refractive index [11].
Most FSS operate in transmission mode, wherein the source and detector locate in the opposite extremities of a guiding/sensing fiber. Despite being straightforward, applications like medical probes and process monitoring require remote access to the measurement environment by isolating the interrogation hardware from the probe. Reflection-type sensors accomplish such tasks using a fiber coupler or beamsplitter to investigate the reflected light. For example, one may implement a speckle-based reflectometer by immersing one termination of a multimode fiber coupler inside the analyzed sample to investigate its chemical concentration or temperature [12,13]. Another approach consists of a contactless distance sensor that detects the light reflected at a polished surface, enabling displacement, vibration, and surface roughness measurements [14]. Ultimately, probe shape estimators retrieve the magnitude and direction of bending events along the fiber probe by addressing each fiber state to a speckle pattern image [15,16].
One may project the incident specklegram over a polished surface or coat the fiber tip with selected materials to enhance the reflected signal. However, a parallel mirror creates a peculiar speckle pattern that cycles consistently around well-defined states [17]. These oscillations also persist if the waveguide experiences bending or thermal disturbances, offsetting the specklegram while preserving its periodic motion. Previous work reported this effect in refractive index measurements [13], suggesting an AC component in the correlation signal superposing the intensity drop caused by the Fresnel reflection. Later, the same effect manifested in a displacement sensor [14], wherein the authors vaguely attributed such cyclic behavior to phase drift. Apparently, the gap between the fiber tip and mirror forms an optical cavity noticeable in extrinsic Fabry-Pérot interferometers (FPI) [18,19]. Indeed, a detailed analytical and experimental study is necessary to understand this phenomenon and develop strategies for controlling or suppressing the sinusoidal response. Nevertheless, to the best of our knowledge, no systematic approaches or application proposals are available regarding such specklegram oscillations.
Therefore, this paper investigates oscillatory speckle patterns in reflection-type FSS. Firstly, an analytical model derived from an extrinsic MMF FPI explains the occurrence of waveguide- and cavity-induced specklegram changes. Secondly, experiments confirm the numerical analyses by evaluating the speckle field deviations versus the cavity length according to the correlation function and tracing with active contour models. Lastly, data processing techniques available for FSS enhance the proposed setup by unwrapping the sinusoidal behavior, creating a displacement sensor provided with an unambiguous, monotonic response.
2. Fundamentals
2.1 Fiber specklegram sensors
Speckle fields originate from the interference between multiple guided modes in an MMF. The quantity of light granules approximates the number of core modes, which depends on the fiber geometry, numerical aperture, and laser wavelength [1].
The intensity $I(x,y)$ of a specklegram projected over the $xy$-plane is
where $M$ is the number of core modes, $a_m$ and $a_n$ are the amplitude distributions of the $m$th and $n$th modes, respectively, $\phi _m$ and $\phi _n$ are the phase distributions of the $m$th and $n$th modes, respectively [7]. Disturbing the fiber with a physical or chemical stimulus modulates $I(x,y)$ by mode coupling and phase deviation effects [20]. Since these spatiotemporal changes are reversible under controlled conditions, one may compare the specklegram images acquired for different fiber states to quantify the magnitude of a measurand and its distribution along the waveguide.A straightforward metric to compute speckle field variations is the zero-mean normalized cross-correlation coefficient (ZNCC). Given $I(k)$ the intensity distribution for the $k$th acquisition frame and $I_0=I(k_0)$ the specklegram for a neutral state. External stimuli modulate the spatiotemporal distribution of fiber modes and gradually deviate the current speckle pattern image $I(k)$ from the reference $I_0$. The ZNCC quantifies the subtle differences by evaluating the correlation between $I(k)$ and $I_0$, the ZNCC is
2.2 Multimode fiber Fabry-Pérot interferometer
In the reflection-type FSS setup depicted in Fig. 1(a), a three-port multimode fiber coupler delivers the optical signal from the laser source to the probe tip. Part of the light reflected at the mirror surface couples back to the fiber; after traveling to the detection port, it produces an output specklegram $I$. Indeed, Fig. 1(b) is similar to an extrinsic MMF Fabry-Pérot interferometer supporting multiple reflections inside its cavity. Assuming parallel interfaces of similar reflectivity with a perpendicularly incident light, the total reflected intensity $I_R$ of the FPI approaches
where $M$ is the number of modes, $a_m$ and $\phi _m$ are the amplitude and phase distributions of the $m$th mode, $r$ is the reflection coefficient of the interfaces, $b_m$ and $c_m$ are coupling factors, and $\Delta \phi _0$ is the phase shift of the FPI, denoted by wherein $n_0$ and $d$ are the cavity refractive index and length, respectively, and $\lambda$ is the laser wavelength [19].
Fig. 1. Reflection-type fiber specklegram sensor: (a) measurement setup and (b) detail of the fiber tip-mirror interface forming a cavity of resonance.
Supposing $I = I_R$ by disregarding the coupling losses, one may conclude that the output specklegram carries the contributions of the waveguide and the cavity characteristics. Whilst the former depends on individual features of each guided mode, the latter shifts the speckle pattern ensemble following the temporal evolution of $\Delta \phi _0$. As to ZNCC, mechanical disturbances over the fiber length produce a DC offset typical of specklegram decorrelation events, whereas $\Delta \phi _0$ imposes an AC behavior stimulated by the cavity parameters.
2.3 Numerical analyses
For the sake of simplicity, numerical analyses reduce the specklegram image to a 1D intensity distribution $I(x)$ with $M = 1$, $a_m = 1$, $r = 1$, $b_m = c_m =1$, and $0 \leq \phi _m(x) <2\pi$. To investigate the effect of cavity parameters, the phase shift term $\Delta \phi _0$ of $I(x)$ varies as a function of the refractive index $\Delta n_0$, gap length $\Delta d$, and wavelength $\Delta \lambda$ deviations, yielding
Figure 2(a,b) presents the ZNCC and phase shift as a function of $\Delta d$ with $1 \leq (d+\Delta {}d) \leq 2$ µm. Changes in $\Delta \phi _0$ modulate the correlation coefficient as expected, creating a sinusoidal pattern. The phase sensitivity is $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}d = 2\pi /0.316 = 19.88$ rad/µm, i.e., subtle deviations in the fiber tip-mirror distance introduced during displacement or vibration measurements affect the output specklegram significantly, corroborating the previous works [14]. According to Fig. 2(c,d), the refractive index produces a similar effect for $1.0 \leq (n_0+\Delta {}n_0) \leq 1.5$ RIU (refractive index units), yielding $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}n_0 = 2\pi /0.32 = 19.64$ rad/RIU. Therefore, variations in the concentration and temperature of the cavity medium modulate the speckle pattern in a periodical fashion [13]. Conversely, the ZNCC fluctuations are moderate concerning the laser wavelength, resulting in a phase sensitivity of $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta \lambda = 2\pi /194 = 0.03$ rad/nm for $533 \leq (\lambda +\Delta {}\lambda ) \leq 733$ nm, as shown in Fig. 2(e,f). Albeit drastic $\Delta \lambda$ changes are unexpected in practical setups, these results illustrate how the output specklegram responds to wavelength shifts [22].
Fig. 2. Specklegram changes in response to the FPI cavity parameters. ZNCC $Z$ and phase shift $\Delta \phi _0$ as a function of (a,b) the gap length $\Delta {}d$, (c,d) refractive index $\Delta {}n_0$, and (e,f) laser wavelength $\Delta \lambda$ deviations. Initial conditions adopt $d = 1$ µm, $n_0 = 1$, and $\lambda = 633$ nm.
It is worth noticing that the simulations ignore optical losses introduced by the cavity and waveguide. In practice, the average speckle intensity may decrease with the gap length, refractive index, and attenuation of the cavity medium [12,14]. Such systematic specklegram deviations would result in a decaying envelope modulating the ZNCC curve. Recalling Fig. 1, for an incident Gaussian beam, part of the reflected light may couple the cladding region instead of the fiber core, imposing an optical loss that depends on the numerical aperture of the MMF and the gap length [23]. Furthermore, the simultaneous variation of cavity parameters may affect the phase sensitivity and the period of the ZNCC signal. Supposing displacement measurements, shorter $\lambda$ reduces the period of $Z$ and increases $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}d$, so the experimentalist can optimize the dynamic range by deciding the laser wavelength or excite the system with multiple sources in a multiplexed approach for improving the statistics of the measurements. A similar effect occurs for $Z$ and $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}d$ by increasing $n_0$. Deviations in the refractive index of the cavity manifest due to temperature changes and affect the period of the ZNCC waveform, which demands compensation of thermal drifts for practical measurements.
3. Materials and methods
3.1 Measurement setup
Figure 3 portrays the experimental setup. A continuous-wave, randomly polarized, and self-contained HeNe laser source (633 nm wavelength and 1 mW power) illuminates the launching silica MMF (step-index, 62.5/125 µm core/cladding diameters, $\sim$2 m length) through a 20$\times$ objective. After passing a mode scrambler (MS) to balance the power distribution, the coherent light excites a $1\times 2$ (10:90) fiber coupler comprising three arms of $\sim$0.2 m. The probe (straight tip connector with plane-polished zirconia ferrule) faces a plane silicon mirror mounted in a mechanical stage. Light strikes the mirror perpendicularly through the air cavity ($n_0 = 1$) and recouples to the fiber probe. The coupler delivers the optical signal to the detector arm, projecting an output specklegram $I(x,y)$. Subsequently, a CCD camera ($1024\times 768$ pixels resolution, frame rate of 15 fps, equipped with a 1.4/8 mm focal distance lens) acquires the subjective, near-field speckle pattern images for further processing in a computer.
Fig. 3. Experimental setup. The light reflected at the mirror couples back to the MMF and produces the output specklegram. A goniometer controls the cavity length $d$.
The moving stage consists of a goniometer (10 µrad resolution) supporting the mirror through the sample holder (Fig. 3). By neglecting the subtle misalignment between the fiber end-face and the polished surface, one may approximate the effective gap deviation as $\Delta {}d = L \sin {(\Delta \theta )}$, where $L \approx 5$ mm is a lateral offset between the probe tip and the rotation axis, $\Delta \theta$ is the controlled angular displacement, yielding a $\Delta {}d$ resolution of $\sim$0.05 µm.
The experiments proceed with the optical setup assembled on a vibration-free table at room temperature and in the absence of environmental illumination. Moreover, the optical fiber cables remain stationary during the tests to avoid curvature-induced specklegram fluctuations. Preliminary tests with the undisturbed system indicated no severe random noise introduced by thermal or mechanical effects, i.e., the assessed speckle pattern changes are due to the cavity displacement.
3.2 Data processing
A program developed in MATLAB captures the speckle field images, selects a square $201\times 201$ region-of-interest (ROI), and performs an RGB to grayscale conversion. Complimentary preprocessing steps may include intensity normalization and filtering. Firstly, the algorithm obtains the specklegram for the reference state $I_0$ and evaluates its average value. Then, for each acquisition frame $k$, the routine gets the current speckle image $I(k)$, calculates the mean value, and computes the ZNCC between $I$ and $I_0$ using Eq. (2). Lastly, the program stores the correlation coefficient $Z(k)$ and proceeds to the subsequent frame $k+1$.
Despite robust to intensity fluctuations, the original ZNCC algorithm exhibits an intrinsic limitation imposed by the saturation of the correlation function [21]. To overcome this constraint, one may compute the extended version of the ZNCC score (namely, EZNCC) denoted by
where $EZ_0$ is a cumulative offset, $Z$ is the ZNCC value, $\tau _{-} = 0.7$ and $\tau _{+} = 0.9$ are thresholds adjusted empirically. Let a decreasing ZNCC with $I$ departing from $I_0$. Starting with $EZ_0 = 1$, the routine computes $Z$ for the current frame and evaluates $EZ$ using Eq. (6). If $Z$ drops below $\tau _{-}$, the algorithm makes $EZ_0 = EZ$ and renews the reference specklegram to $I_0 = I$, resetting the ZNCC and ensuring a continuous, non-saturating response. Now suppose an increasing ZNCC with $I$ diverging from $I_0$ in the opposite direction. When $Z$ exceeds $\tau _{+}$ the minus signal before the brackets in Eq. (6) inverts to preserve the continuity of the $EZ$ curve for reverse cavity displacements. Therefore the EZNCC extends the sensor response for positive and negative gap length changes [24].3.3 Speckle tracing
Albeit the correlation metrics evaluate the overall speckle field changes, these approaches cannot track the motion of individual light granules. Therefore, an active contour model (snakes) algorithm performs on selected speckle entities to describe their cyclic behavior in response to $\Delta \phi _0$.
Snakes are parametric curves that adapt to the contour of an image according to internal and external energy constraints [25]. Assuming the snake $\mathbf {x}(s) = [x(s),y(s)]$ a parametric function of $0 \leq s \leq 1$, the total energy of $\mathbf {x}(s)$ is
$E_\mathrm {int}$ is the internal energy given bySolving the Euler equations provides the snake model $\mathbf {x}(s)$ that minimizes $E$,
Figure 4 exemplifies the data processing pipeline for a speckle entity. The MATLAB routine generates sub-windows enclosing a particular granule to investigate its dynamic behavior. A binary mask segregates the speckle from the background. Then, the program computes the gradient $E_\mathrm {ext}$ ($\sigma = 5$) and the edge map $\mathbf {f}$ (using the Laplacian of Gaussian method). The snake begins as a circular balloon assuming $\alpha = 1\times 10^{-4}$, $\beta = 1\times 10^{-14}$, and a parametric step of $\Delta {}s = 2\times 10^{-3}$. The balloon inflates (its radius increases) until it reaches the gradient field [27]. Afterward, the routine solves Eq. (9) by considering $\mathbf {x}(s)$ as a function of time to employ an iterative method (step size of $\Delta {}t = 1\times 10^{-3}$) [25], yielding the preliminary snake model. The succeeding step improves $\mathbf {x}(s)$ with the GVF method, choosing $\mathbf {v}(x,y)$ as a time-varying function and evaluating Eq. (10) ($\mu = 0.1$, $\Delta {}t = 1\times 10^{-3}$) [26]. Lastly, the snake computation resumes with $E_{\mathrm {ext}} = \mathbf {v}(x,y)$ to generate the optimized version of $\mathbf {x}(s)$.
Fig. 4. Specklegram processing: (a) original near-field speckle field image; (b) ROI enclosing a single speckle; (c) gradient field (blue arrows) and initial snake model (red line); (d) GVF (blue arrows) and enhanced snake model (red line).
From the image analysis perspective, speckles may eventually appear, vanish, or transform. The algorithm processes consecutive images and obtains the snakes $\mathbf {x}(s,k)$ regarded to the $k$th video frame to evaluate the spatiotemporal evolution of light granules. Subsequently, the routine retrieves the centroid and area $A(k)$ enclosed by the parametric curves to quantify the relative translation $\Delta {}x(k)$ and shape changes, respectively.
4. Results and discussion
The experiments initiate with the fiber tip aligned with the mirror surface, ensuring a minimum gap to avoid direct contact and provide a detectable reflection signal. The static characterization proceeds with controlled displacement steps of 0.1 µm by maintaining the $\Delta {}d$ condition during the acquisition of specklegram images. Figure 5(a) shows the ZNCC ($I_0$ referenced to $\Delta {}d = 0$) as a function of the relative cavity length. Each data point contains an average of ten repetitions regarding 100 sequential acquisitions, summarizing 1000 values, wherein the error bars indicate the measurement uncertainties with a 95% probability level.
Fig. 5. (a) Variation of the ZNCC as a function of the cavity displacement. The solid line is a cosine function. (b) Near-field specklegrams for selected displacements (indicated above the images). Colors depict the normalized intensity levels.
The sensor response corroborates the predicted sinusoidal behavior and exposes the speckle field modulation by the FPI cavity parameters. Indeed, the speckle pattern images displayed in Fig. 5(b) restore their original characteristics after completing an oscillation period, i.e., the speckle image for $\Delta {}d = 0.0$ µm is similar to $\Delta {}d = 0.4$ µm, while $\Delta {}d = 0.1$ µm matches $\Delta {}d = 0.1+0.4 = 0.5$ µm and $\Delta {}d = 0.3$ µm agrees with $\Delta {}d = 0.7$ µm. The phase sensitivity for Fig. 5(a) is $\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}d = 15.71$ rad/µm, diverging from the simulation results ($\mathrm {d}\Delta \phi _0/\mathrm {d}\Delta {}d = 19.88$ rad/µm). The differences probably account for experimental errors like the misalignment between interfaces, or due to simplifications in the analytical model (coupling efficiency, optical losses in the cavity, and number of excited modes) [18,19,23]. Concerning the ZNCC, the sensitivity within the linear range ($0.05 \leq \Delta {}d \leq 0.15$ µm) is $\mathrm {d}Z/\mathrm {d}\Delta {}d = 3.77$ µm$^{-1}$. Assuming a conservative, detectable ZNCC variation of $\Delta {}Z = 0.01$ gives a maximum displacement resolution of 3 nm. This value exceeds the goniometer resolution ($\sim$50 nm) because it depends on the specifications of the interrogation system rather than the applied input displacements. However, the systematic uncertainties introduced during the data acquisition and reduction procedures may degenerate the effective resolution compared to the theoretical prediction.
Figure 6(a) presents the variation of ZNCC when the mirror holder advances continuously at a 0.05 µm/s rate regarding a 2 µm displacement range. The specklegram cycles in periods of 0.4 µm, as expected, and remains correlated despite the gap length increase, suggesting that the cavity losses are still negligible for this distance. However, the sinusoidal behavior makes the sensor response ambiguous once a unique $Z$ value addresses different $\Delta {}d$ conditions. Fortunately, the EZNCC circumvents this limitation by unwrapping the decorrelation curve into a monotonic decrease as shown in Fig. 6(b), providing a sensitivity of $\textrm {d}EZ/\textrm {d}\Delta {}d = 4.28$ µm$^{-1}$ ($\sim$2 nm resolution) over the 2 µm range. Moreover, Fig. 6(c) exploits the sensor response to an augmenting displacement of 2 µm followed by a reverse movement, emphasizing the ability of EZNCC to track both increasing and decreasing cavity length changes. Nevertheless, one may amend the residual oscillations in the EZNCC curve by tuning up the threshold values to anticipate the reference resetting events, forcing a linear behavior [28].
Fig. 6. (a) ZNCC and (b) ENZCC as a function of the cavity displacement within the 2 µm range. The red line in (b) is a linear curve fitting. Inset: EZNCC response for the 0.4 µm range. (c) Variation of ZNCC and (d) ENZCC for positive and negative displacements of 2 µm.
As the specklegram spatiotemporal evolution, Fig. 7(a) displays a ROI comprising a pair of speckles enclosed by snakes $\textbf {x}_1$ and $\textbf {x}_2$ for $\Delta {}d = 0.1$ µm. Figure 7(b) traces the frame-by-frame transformation of $\textbf {x}_1$ as a function of the cavity displacement, indicating that the light granule appears for $\Delta {}d = 0$, develops its maximum size with $\Delta {}d = 0.25$ µm, then vanishes after achieving $\Delta {}d = 0.70$ µm. The contour model exhibits an overall translation of $\Delta {}x = 50$ pixels with a maximum area of $A = 1045$ pixels, as shown in Fig. 7(c, d). Besides, another speckle (snake $\textbf {x}_2$) emerges at $\Delta {}d \approx 0.4$ µm and overlaps the trajectory of $\textbf {x}_1$ with a temporal shift, confirming the periodical behavior disclosed by the correlation function.
Fig. 7. Light speckle tracing: (a) snakes $\textbf {x}_1$ and $\textbf {x}_2$ over a specklegram image for $\Delta {}d = 0.1$ µm (see Visualization 1); (b) spatial evolution of $\textbf {x}_1$ for a cavity displacement of 0 to 0.7 µm; (c) variation of relative translation $\Delta {}x$ and (d) area $A$ as a function of $\Delta {}d$.
Parametric contour models are convenient for characterizing continuous speckle field changes and predicting their behavior to a certain extent. Though, it is worth noticing that speckles are not wandering entities but represent the result of constructive and destructive interference between fiber modes. Notwithstanding the selected case of Fig. 7, some granules are harder to track once they merge into other speckles rather than waning. Such an aspect may require refinements to the binary mask and gradient field parameters. Furthermore, the presented technique depends on user-assisted screening to select areas-of-interest and exhibits limited convergence regarding concave shapes—the latter demands alternative snake energy models [26].
Regarding the input image, this experiment uses saturated near-field speckles to feature their edges from the background. Conversely, unsaturated entities characterized by smooth intensity profiles produce weaker external energy terms that cannot trap the snake functions adequately [26]. To avoid this problem, one may choose $E_{\mathrm {ext}}$ models based on alternative image features to enhance faint speckles [29] or process the acquired specklegrams through threshold and texture analyses with spatial segmentation [30].
Lastly, Fig. 8(a) shows the correlation coefficient for large displacements ($\Delta {}d = 70$ µm). Even though the ZNCC still sustains a sinusoidal waveform, the reflected intensity reduction caused by the gap length prevails over the correlation signal. The attenuation modulates the oscillatory pattern with an exponential decay typical of specklegrams decorrelation. Based on the upper-envelope, the ZNCC exhibits a high sensitivity range for $0 \leq \Delta {}d \leq 20$ µm, yielding $\mathrm {d}Z/\mathrm {d}\Delta {}d = 0.02$ µm$^{-1}$ ($\sim$0.5 µm resolution), but the saturation impairs the sensor reliability for $\Delta {}d > 40$ µm. Nevertheless, the EZNCC algorithm ensures a linear response within the full-scale range, achieving $\mathrm {d}EZ/\mathrm {d}\Delta {}d = 3.29$ µm$^{-1}$ with $\sim$3 nm resolution, as portrayed in Fig. 8(b). Compared to Fig. 6(b), the sensitivity degenerated probably due to the light attenuation and cavity losses, making the details in speckle field images less discernible even for diverging fiber states.
Fig. 8. (a) ZNCC as a function of the cavity displacement. The red line is an exponential curve fitting. Inset: ZNCC for $0 \leq \Delta {}d \leq 10$ µm. (b) EZNCC as a function of $\Delta {}d$. The red line is a linear curve fitting.
Table 1 summarizes the performance of selected displacement sensors based on the extrinsic FPI. Despite its elementary interrogation setup, the proposed specklegram system exhibits a resolution comparable to single-mode fiber (SMF) interferometers realized through sophisticated phase demodulation techniques [32,33]. Moreover, the ENZCC algorithm improves the sensor response over the available speckle-based MMF FPI [14], providing a noticeable dynamic range extension and preserving the measurement resolution. For instance, the EZNCC requires fewer calibration images than other methods applicable to FSS, such as morphological analysis [34] and deep-learning processing [9]. It is worth noticing that the maximum displacement detectable by the FSS depends on the misalignment between fiber tip and mirror, as well as on the optical losses due to the cavity medium [18,19]. In practice, the measurement range can exceed 70 µm by optimizing the laser power and the numerical aperture of the MMF to circumvent the coupling losses and ensure a discernible output speckle pattern [35]. Concerning the resolution, one may use sensitive metrics like phase-only correlation and mutual information to quantify minimal changes in the speckle field images and improve the detection limit by sacrificing the dynamic range [21].
Table 1. Comparison of optical fiber extrinsic Fabry-Pérot interferometers. NA: not available.
The maximum random uncertainty obtained in Fig. 5 is 1.69% (ZNCC units), indicating acceptable repeatability under controlled experimental conditions. Disregarding its merits, the FSS is vulnerable to mechanical and thermal effects. Mode coupling and phase modulation may shift the speckle patterns from their reference condition and frustrate the initial calibration. In this context, shorter launching fiber lengths are preferable to improve the visibility of light granules and make the system less vulnerable to the modal noise induced by extraneous events [36,37]. Protecting the optical fibers inside semi-rigid cables is effective for amending bending and vibration [38], while temperature drifts demand recalibration through reference probes [39]. Software-based approaches by deep-learning are also available to correct specklegram changes produced by temperature [40] and fiber bending [41]. Besides, laser power fluctuations and misalignments in the launching setup also affect the modal distribution and may result in unpredictable decorrelation [1,35]. Thus, a stable light source is essential to guarantee repeatable measurements. Nevertheless, the ZNCC and EZNCC algorithms require a single image (reference state) for calibrating the system instead of a comprehensive dataset covering multiple fiber conditions, making this task less laborious than deep-learning approaches.
Concerning the assessment of refractive index, one may expect cycling specklegrams in response to variations in the cavity medium. Albeit previous works confirmed such a phenomenon [13], the intensity modulation due to the Fresnel reflection at the fiber-sample interface predominates over the periodic behavior introduced by the cavity of resonance. Thus, the system works as an ordinary reflectometer with average intensity readout rather than a FSS [12,13]. Alternatively, the FPI setup may be suitable for detecting the temperature according to the slight refractive index changes.
Ultimately, the period of the ZNCC function varies with the laser wavelength, as stated by the analytical model. A possible application comprises a specklegram-based spectrometer for evaluating subtle $\Delta \lambda$ deviations by scanning the cavity length with controlled steps. Furthermore, one may excite the fiber interferometer with different visible wavelengths and perform complementary displacement and refractive index measurements by investigating the correlations in the color space [6].
5. Conclusion
This paper evaluated the oscillating speckle patterns observed in reflection-type FSS. The analytical model of a MMF extrinsic FPI exposed the differences in modal phase deviations induced by the waveguide and cavity parameters. Experimental results confirmed the numerical predictions regarding the gap length changes, while active contour models evidenced the periodical behavior of individual light granules. Lastly, the setup operated as a micrometric displacement sensor supported by an enhanced correlation algorithm, yielding a 3 nm resolution over the 7 µm range.
It is worth noticing that the cyclic speckle field behavior introduces AC components in reflection-type FSS regardless of their applications. For example, supposing a static mirror coupled to the fiber probe tip, vibration and thermal expansion effects are sufficient to disturb the gap between the fiber and the reflecting surface, changing the output specklegram and sweeping the correlation signal. Therefore, understanding and controlling such oscillations is fundamental for implementing robust FSS and enhancing their measurement capabilities.
Funding
Fundação de Amparo à Pesquisa do Estado de São Paulo (2017/25666-2); Conselho Nacional de Desenvolvimento Científico e Tecnológico (403418/2021-6); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (001).
Acknowledgments
Portions of this work were presented at the JSAP-Optica-SPP Joint Symposia in 2022, paper 21a-C304-7.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicity available at this time but may be obtained from the authors upon reasonable request.
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