1. Introduction

Coherent optical frequency domain reflectometry (COFDR) was originally proposed as an alternative to optical time domain reflectometry (OTDR) [1–3]. Since it offers a spatial resolution several orders of magnitude better than OTDR, it also allows the analysis of small-scale components and integrated optical waveguides [4–7]. The high resolution and sensitivity is useful in many different sensing applications, for example, distributed temperature or strain sensing based on Rayleigh scattering fingerprints [8–10].

In many sensing situations there is also a great need for dynamic measurements. Although COFDR systems for detecting vibrations or acoustic signals have been reported [11,12], they have not yet been extended to dynamic measurements of moving targets. Here, we report for the first time a Doppler COFDR system in which the instantaneous position and speed of moving objects are measured with µm-scale spatial resolution and speed accuracy in the µm/s-range. Using a custom-built dynamic OFDR system, the position and speed of optically levitated flying particles in a hollow-core photonic crystal fiber (HC-PCF) were continuously monitored, permitting realization of a versatile distributed temperature sensor with high spatial resolution.

2. Principle of Doppler OFDR

Figure 1 shows a sketch of the Doppler COFDR system, which is based on a Mach-Zehnder topology. Analogously with radio-frequency-modulated radar systems, the laser frequency was linearly swept so as to remove the distance ambiguity inherent to conventional interferometers. The laser output was split up into a reference and a measurement arm incorporating the reflecting target. First, assuming a steady target with reflectivity R, the time varying signal (after filtering out the DC and sum-frequency components) collected by a square-law detector can be written as:

 

Fig. 1. COFDR setup in Mach-Zehnder topology.

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Frequency analysis of Eq. (1) yields a beat frequency:

For static objects, assuming negligible dispersion over the sweep bandwidth, the distance can be readily calculated from the beat frequency:

The two-point spatial resolution of COFDR is determined by the target main lobe width of the Fourier transform [4]:

Note that the exact target position can be determined much more precisely by approximating the peak with a second-order polynomial and calculating its maximum position. Using a conventional COFDR system, we have previously demonstrated localization of optically trapped particles with a precision of ∼2 µm, corresponding to ∼1/15 of the two-point resolution [13].

In order to analyze the effects of (uniformly) moving targets, τ in Eq. (1) is replaced by:

A phase-gradient autofocus algorithm, initially proposed for synthetic aperture radar images [14], was used to refocus the target peak. First, coarse peak detection and windowing of the reflection profile were applied around the peak of interest. The window width was set wide enough to cover the full peak energy and yet small enough to exclude noise. Although this algorithm can be applied iteratively, a fixed window size and a single iteration were sufficient for our application. The windowed section of the reflection profile containing the peak of interest was shifted to the origin and the phase error derivative was determined using the relation given in [14,15]. The phase errors were then determined by numerical integration and any linear slope was compensated for, since this would cause a target position error. Eventually, the blur was mitigated by multiplying the windowed peak spectrum with the conjugate phase term and shifting back the peak to its original position in the reflection profile. Naturally, this correction not only compensates the chirp term (V) but also removes the phase distortions remaining after imperfect sweep nonlinearity correction (see Appendix A.1). Figure 2 depicts the reflection profiles (up and down sweep) of a moving target (2-µm-diameter silica particle trapped and propelled inside HC-PCF with a moving speed of ∼30.4 mm/s) before and after applying the autofocus algorithm. It can be seen that peak broadening due to the phase chirp effect can be effectively suppressed by re-focusing.

 

Fig. 2. Calculated reflection profile for the up (red) and down (blue) ramp for a propelled 2-µm-diameter silica particle through the HC-PCF with (saturated colors) and without (faint colors) refocusing.

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Once the chirp term is compensated, two consecutive independent measurements are combined to determine the position and speed of the target. Figure 3 depicts the impact of the Doppler frequency shift and the time delay related to the target distance on the measured beat frequency.

 

Fig. 3. Influence of Doppler frequency shift on the measured beat frequency caused by a moving target.

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Assuming ideal up and down sweeps with the same absolute sweep rate |k| (but opposite signs) for both measurements, the target position and speed can be calculated from the beat frequencies |f_b,up| and |f_b,down|:

Note that in contrast to conventional Doppler velocimetry based on fixed-wavelength laser [16], the system can measure the instantaneous absolute position of the moving objects and is not subject to integration error over time.

3. Tracking of optically trapped and propelled particles in HC-PCF

The Doppler COFDR system was used to track optically trapped particles in HC-PCF. Flying particle sensors in HC-PCF have inherent advantages such as galvanic isolation and coverage of long distances, while additionally offering reconfigurability, improved spatial resolution and multi-parameter sensing [17]. The experimental setup is shown in Fig. 4. The 1030 nm trapping laser emitted ∼200 fs pulses at a repetition rate of 70 MHz. The average total trapping power was ∼2 W. The use of a pulsed laser is necessary to suppress inter-modal beating between fundamental and higher order core modes [18]. The measured modal walk-off distance is ∼20 cm (see Appendix A.3). The high repetition frequency of the pulse train ensures that gravity-related particle displacement between pulses is negligible. The COFDR probe beam at 1550 nm wavelength was coupled into the beam path using a dichroic mirror. The power ratio of the trapping beams was set using a motorized half-wave plate to adjust the polarization state of the trapping laser in front of a polarizing beam-splitter. We used a commercially available swept-wavelength laser source (Luna Phoenix 1400 PM) with a maximum tuning range of 50 nm around a central wavelength of 1540 nm at a maximum tuning rate of 2000 nm/s (i.e. a sweep rate of approx. 2.5×10¹⁴ Hz s⁻¹). The inset in Fig. 4 shows a scanning electron micrograph of the single-ring HC-PCF. It has a core diameter of ∼30 µm and can guide both the trapping beam and the COFDR probe beam with low loss. The microparticles in a suspension were sprayed into a trapping chamber in front of the end face of the HC-PCF, where they were trapped by two counter-propagating laser beams and then propelled into the hollow core by adjusting the power ratio of the trapping beams. The trapping of single (as opposed to multiple) particles in the hollow core was verified by monitoring the scattered light from the side of the fiber using an infrared viewer.

 

Fig. 4. Experimental setup for tracking optically trapped microparticles in a HC-PCF using Doppler COFDR and the configuration for remote temperature sensing.

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We first tested the achievable precision of position measurement for silica microparticles with different sizes at the maximum sweep rate. In each configuration, a series of 100 repeat measurements was conducted for which the measured standard deviation is plotted in Fig. 5(a) as a function of sweep bandwidth. It can be seen that the measurement data fits excellently with the inverse proportionality suggested in Eq. (4). Since the sweep bandwidth is directly linked to the measurement duration, it is possible to balance position precision against measurement rate (see upper axis). Furthermore, the achievable precision is related to the signal-to-noise ratio (SNR) of the target reflection peak, which is higher for larger particles and greater sweep bandwidths. In practice, the maximum measurement rate will be reached once the SNR of the target peak is too low to be detected reliably. For highly reflective targets, it is limited by the minimum sweep duration of the tunable laser source. With the laser used, measurement rates up to approximately 250 Hz could be achieved. At the cost of reduced precision, measurement rates in the kHz range are feasible using commercially available sources.

 

Fig. 5. (a) Measured positional precision versus sweep bandwidth for different particle sizes. The solid lines are fits using inverse proportionality. (b) Instantaneous position (blue) and speed (red) of a tracked 2-µm-diameter silica particle in HC-PCF. The inset is a zoom into the first 5 s, during which the particle accelerates.

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Figure 5(b) shows the tracking of a moving silica particle with 2 µm diameter that is propelled along the fiber for around 38 s (see Table 1 for detailed COFDR parameters). We tentatively attribute the small-scale speed variations to fluctuations in trapping laser power (see Appendix A.4) that could be reduced using active power stabilization. The speed fluctuations may be also partly caused by fiber non-uniformities or static charges on the core wall, while the overall exponential speed reduction is due to fiber attenuation. The accuracy of the speed measurement (a few tens of µm/s) was assessed using a mirror on a linear stage moving at a known speed (see Appendix A.5). Note that both positive (forward) and negative (backward) particle speeds can be measured by the system. The maximum target speed that can be measured is limited by the Nyquist sampling theorem for highly reflective targets and by the SNR for weakly reflecting targets.

Table 1. COFDR measurement parameters

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4. Remote temperature sensing

Next, we used flying particles in HC-PCF in combination with Doppler COFDR to build a distributed temperature sensor. In the experiment, silica particles of diameter 2 µm were once again propelled by adjusting the power ratio of the counter-propagating trapping beams. The terminal velocity of the particle is determined by the balance between the scattering force F_opt and the Stokes drag force F_drag [17]:

A 2-meter-long single-ring HC-PCF was used in the measurement. As shown in Fig. 4, a section of 2.5 cm was fed through an aluminum block and heated to temperatures up to 200 °C while the rest of the fiber was kept at room temperature. The temperature of the metal block was measured using an embedded temperature sensor with an accuracy of 2.5 K (IEC 584, type K, class 2). The particle was propelled through a ∼50 cm long fiber section containing the heater while its absolute position and speed were measured. The COFDR parameters are summarized in Table 1.

So as to mitigate the effects of any overall speed drift with time, all the experimental measurements of particle speed were normalized to the average speed in a reference fiber section held at room temperature (see Fig. 6(a)). The estimated spatial resolution for the COFDR configuration was 120 µm, corresponding to the ultimate theoretical spatial resolution, while the mean particle speed in the reference section was measured to be 29.1 mm/s, yielding position and speed data at intervals of 0.47 mm on average. Since this corresponds to the spatial sampling distance, it sets the upper bound for the spatial resolution of the temperature sensor. By further adjusting the particle speed it is possible to balance the spatial resolution and the time required to propel the particle along the fiber length. Note that the steep temperature gradients at the edges of the heater cause thermal creep flow of air within the hollow core (related to Knudsen pumps [22]), leading to visible undershoots and fluctuations in speed in the vicinity of these points. Based on Eq. (13) and (14), a speed reduction of ∼20% for a temperature increase of 100 K starting from room temperature is expected. This in good agreement with the experimental results. The observed decrease in speed over distance within the heated section we attribute tentatively to the thermal creep effect, pending a full investigation.

 

Fig. 6. (a) Measured particle speed versus location for different heater temperatures.

(b) Measured average particle speed in the heater section (squares) and curve fit using Eq. (15) (red line).

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To calibrate the sensor, the average particle speed in the heater section was first measured at seven different metal block temperatures (Fig. 6). The error bars in Fig. 6(b) are given by the standard deviation of the speed fluctuations over distance within the heater section (see Fig. 6(a)). Combining Eq. (13) and (14), the temperature dependence of particle speed can be written as:

 

Fig. 7. (a) Calculated and set temperatures for measurements during heating and cooling of the heater. (b) Deviation between calculated temperature and set temperature.

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5. Conclusion

These proof-of-principle experiments demonstrate that a Doppler OFDR system can track the instantaneous absolute position and speed of moving objects with high accuracy. Such a system can be used to probe the motion dynamics of macroscopic objects with micrometer spatial resolution. It can also be used to probe the viscoelasticity of the gases or liquids in the hollow-core of a HC-PCF. The integration of Doppler OFDR with particle guidance in HC-PCF opens up novel possibilities for flying particle sensors capable of measuring multiple physical quantities with high-spatial resolution. Since the COFDR system can also track the positions of multiple particles [13], it may be exploited to investigate the dynamics of levitated particles or bound particle arrays in HC-PCF [18].

Appendix

A.1. Sweep linearization of COFDR

Even for highly optimized sweeping lasers, there always remains a residual deviation from a perfect frequency ramp, resulting from environmental influences or the internal sweep feedback mechanisms. It is intuitively clear that any deviation from an ideal sweep will lead to a time-varying beat signal and therefore distort the reflection profile. As a result, the localization precision will be reduced or, in the case of oscillating deviations, side-lobes will occur and corrupt the reflection profile.

To analyze this effect, the ideal laser sweep signal is extended with a phase error function ϕ_e(t):

(16)$$s(t) = \textrm{exp} ({j2\pi ({f_0}t + {{k{t^2}} / 2}) + j2\pi {\phi_\textrm{e}}(t)} ),$$

where ϕ_e(t) consists of a deterministic (but unknown) error term ϕ_nl(t) caused by the sweep nonlinearity and a stochastic process ϕ_pn(t) used to model the laser phase noise:

(17)$${\phi _\textrm{e}}(t) = {\phi _{\textrm{nl}}}(t) + {\phi _{\textrm{pn}}}(t).$$

This yields the beat signal in complex notation after down-mixing, low-pass filtering and scaling (assuming an interferometer delay τ and neglecting any losses):

(18)$${b_\textrm{e}}(t,\tau ) = \sqrt R \;\textrm{exp} ({j2\pi (kt\tau + {f_0}\tau - {{k{t^2}} / 2})} )\cdot \textrm{exp} ({j2\pi \Delta {\phi_\textrm{e}}(t,\tau )} ),$$

where Δϕ_e(t,τ) = Δϕ_nl(t,τ) + Δϕ_pn(t,τ) with Δϕ_x(t,τ) = ϕ_x(t) − ϕ_x(t−τ). For a semiconductor laser, ϕ_pn(t) is normally modelled as a Wiener process, meaning that Δϕ_pn is a stationary Gaussian random process and therefore only depends on τ. For delays much smaller than the laser coherence time τ_c, Δϕ_pn ∼ 0 holds and the impact of the laser phase noise can be neglected.

To correct the effect of a deterministic phase error term ϕ_nl(t), an auxiliary interferometer was used to interpolate the actual laser frequency for each sample. An absolute wavelength reference (an absorption cell) is available from trigger signals provided by the laser source at predefined wavelengths. The measurement interferometer signal was then resampled at equidistant source frequencies.

Figure 8(a) plots a section of the auxiliary interferometer signal (after subtracting a slowly varying DC component). First, the exact zero crossings (red markers) were determined by linear interpolation between data-points on either side of the sign changes. In a second step, the mean distance between zero crossings with rising and falling slopes was calculated (blue markers) and used for interpolation. Figure 8(b) shows auxiliary signal period length (resembling equidistant frequency intervals) determined from the markers for a laser sweep from 1535 nm to 1545 nm in ∼5 ms sampled at 20.8 MS/s. The reference delay τ_ref was chosen small enough to allow direct monitoring of the laser frequency. Even though a longer delay leads to a smaller free spectral range in the auxiliary interferometer, and thus a finer “sampling” of the laser frequency, the limit for τ_ref is the maximum sampling frequency since oversampling is required to precisely determine the positions of the zero crossings. Table 2 lists the parameters of the auxiliary interferometer in the COFDR system. The delay fiber in the auxiliary interferometer was placed in a styrodur box so as to shield it from environmental influences. Furthermore, polarization maintaining fibers were used in the auxiliary interferometer to avoid polarization-related drift.

 

Fig. 8. Working principle of the auxiliary-based resampling technique. (a) Section of the auxiliary interferometer signal. (b) Period length of the beat signal during the sweep.

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Table 2. Auxiliary interferometer interpolation

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The correction term extracted from the auxiliary interferometer with delay τ_ref is given by:

(19)$${c_\textrm{a}}(t) = \textrm{exp} ({ - j2\pi \Delta {\phi_{\textrm{nl}}}(t,{\tau_{\textrm{ref}}})} )= \textrm{exp} ({ - j2\pi ({{\phi_{\textrm{nl}}}(t) - {\phi_{\textrm{nl}}}(t - {\tau_{\textrm{ref}}})} )} ),$$

where we assume τ_ref, τ_meas ≪ τ_c so that the influence of laser phase noise is neglected. Using this term to correct the measurement signal b_e(t, τ_meas) we obtain:

(20)$${b_{\textrm{corr}}}(t,{\tau _{\textrm{meas}}}) = {c_\textrm{a}}(t) \cdot {b_\textrm{e}}(t,{\tau _{\textrm{meas}}}) = b(t,{\tau _{\textrm{meas}}})\textrm{exp} ({j2\pi [{{\phi_{\textrm{nl}}}(t - {\tau_{\textrm{ref}}}) - {\phi_{\textrm{nl}}}(t - {\tau_{\textrm{meas}}})} ]} ).$$

Obviously, a perfect correction is only achieved when τ_ref = τ_meas. In most cases, however, the error can still be significantly reduced for differing delays Δτ = τ_meas – τ_ref. Assuming a harmonic phase error function ϕ_nl(t) = m sin(2πf_nlt) with frequency f_nl and modulation index m yields a remaining phase error of the form:

(21)$$\begin{array}{l} {\phi _{\textrm{nl}}}(t - {\tau _{\textrm{ref}}}) - {\phi _{\textrm{nl}}}(t - {\tau _{\textrm{meas}}}) = 2m\sin ({\pi {f_{\textrm{nl}}}({\tau_{\textrm{meas}}} - {\tau_{\textrm{ref}}})} )\cdot \cos ({\pi {f_{\textrm{nl}}}(2t - {\tau_{\textrm{ref}}} - {\tau_{\textrm{meas}}})} )\\ = {A_\textrm{e}}(\Delta \tau ) \cdot m\cos ({\pi {f_{\textrm{nl}}}(2t - {\tau_{\textrm{ref}}} - {\tau_{\textrm{meas}}})} ), \end{array}$$

which resembles a phase modulation at frequency f_nl with a phase shift that depends on Δτ. More importantly, the amplitude is scaled by the factor A_e(Δτ) = 2 sin(πf_nlΔτ), which is smaller than unity when |f_nlΔτ| < 1/6. The parameters chosen for the setup (Table 2) yield a beat frequency of ∼1 MHz, allowing aliasing-free sampling of sweep-linearity deviation frequency components up to 1 MHz. Suppression of those components would be effective up to a target distance of 25 m in air. Figure 9 compares the measured reflectivity of the anti-reflection coated FC/PC fiber connector at a relative distance of ∼2.8 m with and without auxiliary resampling. It can be seen that resampling reduces the peak width by a factor of ∼450 while increasing the signal-to-noise ratio by more than 20 dB.

 

Fig. 9. Comparison of reflection from an anti-reflection-coated FC/PC fiber end when auxiliary resampling is applied (blue) and absent (red).

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A.2. Derivation of Eq. (10) and (11)

Starting from the approximation in Eq. (7), the beat frequencies resulting from an up and down frequency ramp are:

(22)$${f_{\textrm{b,up}}}(t )= \frac{{2{f_{01}}{v_0}}}{c} + \frac{{2{k_1}{z_0}}}{c} + \frac{{2{k_1}{v_0}{T_1}}}{c},$$

(23)$${f_{\textrm{b,down}}}(t )= \frac{{2{f_{02}}{v_0}}}{c} + \frac{{2{k_2}({{z_0} + {v_0}\Delta t} )}}{c} + \frac{{2{k_2}{v_0}{T_2}}}{c},$$

where f₀₁ and f₀₂ are the start frequencies, k₁ and k₂ are the sweep rates, and T₁ and T₂ are the durations of the up and down frequency ramps, respectively. Without loss of generality, the down ramp is assumed to start at a time delay Δt after the up ramp. The last term in Eq. (22) and (23) is a frequency offset of 4k₁v₀T₁/2/c and 4k₂v₀T₂/2/c, respectively, which remains after refocusing the peak to the center of the “blur”. Since the sign of the beat frequency cannot be determined, the sum and difference of the detected peak frequencies are:

(24)$${\Delta _ + }{f_\textrm{b}} = |{{f_{\textrm{b,up}}}} |+ |{{f_{\textrm{b,down}}}} |= \frac{{2{f_{01}}{v_0} + 2{k_1}{z_0} + 2{k_1}{v_0}{T_1} - 2{f_{02}}{v_0} - 2{k_2}({{v_0}\Delta t + {z_0}} )- 2{k_2}{v_0}{T_2}}}{c},$$

(25)$${\Delta _ - }{f_\textrm{b}} = |{{f_{\textrm{b,up}}}} |- |{{f_{\textrm{b,down}}}} |= \frac{{({({2{T_2} + 2\Delta t} ){k_2} + 2{T_1}{k_1} + 2{f_{01}} + 2{f_{02}}} ){v_0} + 2{z_0}({{k_1} + {k_2}} )}}{c}.$$

Equation (24) and (25) can be combined to determine the position and speed of the target yielding Eq. (10) and (11).

A.3. Effect of intermodal beating on particle speed

Figure 10 plots the measured particle speed close to the input face of the HC-PCF (at 3.568 m). The periodic fluctuations in speed due to intermodal beating are clearly visible. The period of ∼2.3 mm matches the beat period between the LP₀₁- and LP₀₂-like core modes. The beat pattern decays after ∼20 cm propagation due to walk-off between the different modal pulses.

 

Fig. 10. Periodic oscillation in the instantaneous particle speed observed close to the fiber end face, caused by intermodal beating effect. The inset shows a zoomed-in section where the beat period can be determined.

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A.4. Measured particle speed and trapping power fluctuation

Figure 11 depicts the normalized particle speed deviation and the trapping power fluctuation for a ∼10 s measurement interval. The trapping power propagating towards the PCF was probed with a photodiode placed after a beam splitter while the particle was propelled in the backward direction with an average speed of 32.7 mm/s. It can be seen that the speed variation mostly follows the trapping power fluctuation.

 

Fig. 11. Measured normalized particle speed and trapping power variation over time.

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A.5. Characterization of the speed measurement accuracy

The accuracy of the position and speed measurements was estimated using a gold mirror mounted on a linear stage (Zaber X-LDQ0600C). The stage was set to a speed of +/–10 mm/s. The linear stage outputs a trigger signal every 500 µm as it travels. This was used as a reference.

 

Fig. 12. (a) Moving speed of stage measured by the COFDR system (red) and calculated from the stage trigger signal (blue). (b) Deviation between measured speed and reference (excluding outliers during acceleration/deceleration).

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Aligning the COFDR speed measurements (sweep rate: 2000 nm/s, sweep range: 1518 nm to 1562 nm) with the speed calculated from the trigger signal (see Fig. 12(a)) allowed the deviations to be calculated by linearly interpolating the reference speed at the times when the COFDR measurements were made (see Fig. 12(b)). The standard deviation (excluding outliers during acceleration and deceleration) was estimated as 49.5 µm/s. Note that due to limited internal sampling frequency (10 kHz) of the stage, the trigger signal uncertainty is 0.1 ms, causing a speed uncertainty of ∼20 µm/s in the reference trace.

Funding

Deutsche Forschungsgemeinschaft (418737652); Max-Planck-Gesellschaft; Erlangen Graduate School of Advanced Optical Technologies.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented here may be obtained from the authors upon reasonable request.

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