Introduction. When the polarization state of light traverses a path on the Poincaré sphere, it acquires a geometric phase in direct proportion to half the solid angle subtended at the center of the sphere by the path traversed [1]. This may happen in the spatial domain when light itself traverses a circular path, for example, when light travels through a helically wound optical fiber [2] or through a quarter- or half-wave plate [3]. The geometric phase and its manipulation may be used in opto-electronics setups as a design parameter [3]. Especially relevant to our work is the use of the geometric phase to design optical fiber distributed sensors [4–6]. The geometric phase may also exist in the time domain, i.e., without rotating the path of light, as long as its state-of-polarization (SOP) undergoes rotation; as was recently theorized and demonstrated in a double pinhole experiment [7], in two-beam interference [8], as well as in interference between two frequency-offset beams [9]. We focus on the latter case where the phase of the beat signal of two frequency-offset beams acquires a periodically varying geometric component, as long as the two beams have different SOPs [9]. When this condition is true, the SOP of the resultant beat signal traverses a complete circle on the Poincaré sphere in each beat period, thus giving rise to a geometric phase in the time domain. This phase is a function of the relative intensity and SOP, respectively, of the two interfering beams. Moreover, the total phase of light with angular velocity, $\omega$, in each beat period, $T$, remains constant, equal to $\omega T \pm \pi$, divided into the geometric and the remaining dynamic components [9].

The coupling between the geometric and the dynamic phases motivates one to measure the geometric phase in a distributed fiber sensing setup, where the dynamic phase is used as a measurand [10], and evaluate its utility as an alternative to or to augment the information obtained from the dynamic phase. This is in turn motivated by the fact that the condition for the existence of the geometric phase is complimentary to that of the dynamic phase. Dynamic phase measurement using coherent heterodyne detection assumes that the state-of-polarization of the two beams is identical [11,12], whereas for the existence of the geometric phase, they must not be so [9]. In fact, the condition of identical SOPs of the two beams becomes a challenge for measuring the dynamic phase, given that the polarization of light in an optical fiber does not necessarily remain constant beyond a few meters [11–13]. Given these considerations, it is anticipated that a sensing mechanism based on the geometric phase will be free from polarization-mismatch fading. Phase-sensitive optical time-domain reflectometry, $\phi$-OTDR, is one of the current state-of-the-art technologies for distributed fiber sensing [14]. It is currently being investigated for better performance in terms of higher dynamic range, spatial resolution, and other respects, for applications in seismic sensing, structural health, and infrastructure monitoring, among others [10,14]. Coherent heterodyne detection [11] is one of the methods used in $\phi$-OTDR for demodulation of the dynamic phase [12]. In fact, $\phi$-OTDR based on coherent heterodyne detection [12] provides a context in which the geometric phase may exist, i.e., coherent interference of light beams with a frequency offset, subject to the condition that the SOPs of the interfering beams are not exactly the same [9]. We note that the original work on the measurement of the geometric phase in the beating of light waves is done using free-space optics with beams of constant intensity [9].

In this work, we measure the geometric phase in the optical fiber medium to use it in distributed fiber optic sensing [10], such that one of the interfering beams has changing intensity (i.e., backscattered signal) while the other is the constant-intensity local oscillator (LO) signal [10]. In this Letter, we first analyze the expression [9] for calculating the geometric phase, followed by a description of our experimental setup designed for its measurement. Two cases of stimulation of a fiber-under-test by an acoustic signal, applied to a polarization scrambler and a piezoelectric transducer, are presented and discussed.

Theory and experimental setup. The expression for the geometric phase, $\phi _g$, is given as [9]

Equation (1) shows that to calculate $\phi _g$, we require individual beam intensities, $S_0^\prime$, $S_0^{''}$, and the normalized beat signal, $\gamma _0$[9]. The envelope signal, $|\gamma _0|$, is extracted by taking the Hilbert transform of the beat signal, $S_0$, after normalizing it as per Eq. (2). Our proposed setup to calculate the geometric phase is based on standard coherent heterodyne $\phi$-OTDR, which also relies on $S_0$ to extract the dynamic phase [12]. The difference from the standard setup is the measurement of individual beam intensities; half of the backscatter signal is directly detected using a single photodetector and only half is sent for interference with the LO, whose intensity is measured with a power meter.

Our experimental setup is depicted in Fig. 1. Light from a narrow linewidth laser at 1550 nm is split using a 90:10 splitter into a probe and LO branch, respectively. The probe branch is modulated into 100-ns duration pulses at 20-kHz repetition rate by an acousto-optic modulator (AOM) that also provides a 110-MHz frequency upshift to the beam. This probe signal is then amplified using a pulsed EDFA and sent to a 4.4-km fiber-under-test (FUT) using the transmission port of a three-port circulator. The circulator collects the backscattered signal (BSS) via its reflected port and sends it to an EDFA preamplifier. In a standard $\phi$-OTDR setup based on coherent heterodyne detection [12], at this point, the BSS is interfered in a $2 \times 2$ 50:50 coupler with the LO. However, in our approach, we first split the BSS using a $1 \times 2$ 50:50 coupler; half of this light is sent to a $2 \times 2$ 50:50 coupler, as in the standard case, while the other half is sent for direct detection by a single photodetector, thus giving us $S_0^\prime$. As for the intensity of the LO, $S_0^{''}$, it remains constant and can be measured one time using a power meter. The interference of LO with $S_0^\prime$ takes place in the $2 \times 2$ 50:50 coupler. Its output is fed to a balanced photodetector that downshifts the frequency to 110 MHz, thereby providing the beat signal, $S_0$. Outputs of both photodetectors are sampled at 250 MS/s each via a high-speed data acquisition system. The $N$ in Eq. (1) translates to the number of samples in each beat period. Here, $N=2$ because the 110-MHz beat signal, sampled at 250 MS/s, gives 2.27 samples for each period.

Results and discussion. The geometric phase changes with a change in either the SOP or the intensity of one of the interfering beams [9]. The setup shown in Fig. 1 is used to see the effect of varying both the SOP and the intensity of the probe signal by using a polarization scrambler (PSM) and a piezoelectric transducer (PZT) in the FUT, respectively. The distance–time plots of the geometric phase, its power spectral density (PSD), and behavior of $|\gamma _0|$ for both cases are discussed below.

 

Fig. 1. Experimental setup of the proposed $\phi$-OTDR used to measure the geometric phase, $\phi _g$. PZT, piezoelectric transducer; P-EDFA, pulsed erbium-doped fiber amplifier; EDFA, erbium-doped fiber amplifier.

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For the first test scenario, a 160-m-long patch cable coming from the transmission port of the three-port circulator, connects an inline polarization scrambler to the 4.4-km fiber-under-test (FUT). A PSM modulates the SOP of the probe signal in the FUT in response to a 100-Hz sinosoidal signal provided by a signal generator for 0.5 s. Distance–time contour plots of the geometric phase, $\phi _g$, and the corresponding backscatter intensity, $S_0^\prime$, are plotted in Figs. 2(a) and 2(b), respectively. Figure 2(a) shows that $\phi _g$ indeed corresponds to the test signal in frequency and amplitude, implying that SOP modulation caused by the PSM is reflected in $\phi _g$. It is observed in Fig. 2(b) that the SOP modulation of the probe signal also affects the intensity of backscatter, $S_0^\prime$. Thus, we are not able to see the impact of only the SOP changing without a corresponding change in the intensity of light. Nonetheless, $\phi _g$ is not a direct function of $S_0^\prime$ as seen clearly at the end of the fiber at approximately 4.5 km, shown in Fig. 3 where $\phi _g$ and $S_0^\prime$ are plotted on the left and right axis, respectively. Thus, the modulation of $\phi _g$ carried forward along the length of the fiber with higher fidelity to the test signal, as compared to $S_0^\prime$, can be attributed to the change in SOP, independent of the change in intensity, of the backscatter signal. The above mentioned trend of $\phi _g$ can be observed more clearly in its time series plot given in Fig. 2(c) on the left axis at the location of the PSM at 0.16 km [marked by a horizontal yellow line in Figs. 2(a) and 2(b)]. On the right axis, $S_0$ is plotted overlapping with $\phi _g$ to show that the frequency of $S_0$ along the time axis corresponds to the high-frequency components in $\phi _g$. Now, we look at the plot of $|\gamma _0|$ in Fig. 4, which is an important variable in Eq. (1), reflective of the beat signal, $S_0$, obtained from coherent heterodyning of $S_0^\prime$ with the LO, overlapped with $S_0$ itself along the distance axis. Here, $|\gamma _0|$ varies with the beat signal amplitude but does not closely follow it especially around the location of the PSM at 164 m. In the standard $\phi$-OTDR based on coherent heterodyne [12], the envelope of $S_0$ effectively reconstructs the backscatter intensity. Thus, the deviation of $|\gamma _0|$ from $|S_0|$ may signal the effect of the change in SOP of the probe signal, independent of its intensity. Figure 5 shows the PSD of $\phi _g$, $S_0$, and $S_0^\prime$ at 0.16 km. The frequency component of the applied acoustic signal at 100 Hz as well as its higher-order harmonics are seen in PSDs of both $S_0^\prime$ and $\phi _g$. The aliasing in the PSD of $\phi _g$ here is attributable to the small $N$ and may be improved by having more samples in each beat period. This is possible by increasing the sampling rate or frequency-offset of the AOM. In addition, $S_0$ shows modulation along the time axis at approximately 500 Hz – an alias of the pulse repetition frequency of 20 kHz which is also the sampling frequency along the time axis – discussed before in the time domain in Fig. 2(c). In fact, the PSD of $\phi _g$ in Fig. 5 also contains the second harmonic at approximately 1 kHz which is not seen in the PSD of $S_0$ itself. Thus, the experiment with PSM shows that the geometric phase is indeed modulated by the SOP of the probe signal, independent of a corresponding change in $S_0^\prime$.

 

Fig. 2. Inline polarization scrambler (PSM) located at 0.16 km of the fiber-under-test, marked by the horizontal yellow line in the distance–time contour plot of: (a) geometric phase $\phi _g$; (b) backscatter intensity, $S_0^\prime$; (c) geometric phase $\phi _g$ (blue, left) and backscatter intensity $S_0^\prime$ (cyan, right) versus time at 0.16 km.

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Fig. 3. Geometric phase $\phi _g$ (blue, left) and backscatter intensity $S_0^\prime$ (cyan, right) near the end of the fiber-under-test at 4.5 km.

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Fig. 4. Beat signal $S_0$ in mW (green) and normalized envelope signal $|\gamma _0|$ (black) versus distance for the case of an inline polarization scrambler (PSM) at a fiber distance of 0.16 km.

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Fig. 5. Power spectral density (PSD) of the geometric phase, $\phi _g$, beat signal, $S_0$, and backscatter intensity, $S_0^\prime$, for an inline polarization scrambler (PSM) at 0.16 km of the fiber-under-test.

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In the second test scenario, approximately 375 m into the 4.4-km FUT, 15 m of fiber is wound around a cylindrical PZT. The PZT diameter expands in proportion to the applied voltage causing the fiber section wound around it to experience strain. The strain affects the intensity and phase of the backscatter signal corresponding to the test signal applied to the PZT by the signal generator at 100 Hz for 0.5 seconds. Distance–time contour plots of the geometric phase, $\phi _g$, and the corresponding backscatter intensity, $S_0^\prime$, are plotted in Figs. 6(a) and 6(b), respectively. It is observed in Fig. 6(a) that $\phi _g$ indeed behaves correspondingly to a test signal, implying that the phase and intensity modulation caused by the PZT is reflected by $\phi _g$. Figure 6(b) shows that $S_0^\prime$ is also modulated by the strain caused by the PZT to the probe branch. Time series of $\phi _g$ around the location of the PZT at 0.373 km, marked by a horizontal yellow line in Figs. 6(a) and 6(b), is plotted in Fig. 6(c). In this case, the relative SOP of the two interfering beams, $S_0^\prime$ and $S_0^{''}$, is relatively constant (this being the underlying assumption/condition for coherent heterodyne detection), so we can observe the effect of a change in intensity only on $\phi _g$. In Fig. 7, $|\gamma _0|$, which is the normalized amplitude of $S_0$ as per Eq. (2), is plotted against $S_0$ itself. In contrast to Fig. 4, the envelope closely follows $S_0$. This behavior is similar to the standard coherent heterodyne [12], as mentioned already in the discussion on Fig. 4. This further strengthens the assumption that given the relatively stable SOP of the probe signal with respect to LO, $\phi _g$ is affected by $S_0^\prime$ only. Figure 8 shows the power spectral density of $\phi _g$, $S_0$, and $S_0^\prime$ at 0.373 km. As in the case with the PSM, the test signal frequency of 100 Hz can be seen in both $\phi _g$ and $S_0^\prime$ along with higher order harmonics. In this case also, the PSD of $S_0$ shows frequency modulation at approximately 500 Hz along the time axis, an alias of the 20-kHz sampling frequency along the time axis. However, in this case $S_0$ is itself modulated by $S_0^\prime$, as shown by peaks at 200 Hz and the following harmonics (the test signal frequency of 100 Hz is not seen, possible due to relatively small sampling rate of 250 MS/s of the 110-MHz beat signal). This is in contrast to the first test scenario with PSM, where $S_0$ has a single-frequency component at approximately 500 Hz. Thus, with a PZT in the FUT, we have demonstrated that the geometric phase is indeed influenced by a change in intensity of $S_0{'}$ when the SOP of the interfering beams is relatively constant.

 

Fig. 6. Inline piezoelectric transducer (PZT) located at 0.373 km of the fiber-under-test, marked by a horizontal yellow line in the distance–time contour plot of: (a) geometric phase $\phi _g$; (b) backscattered intensity $S_0^\prime$; (c) geometric phase $\phi _g$ (left) and backscattered intensity $S_0^\prime$ (right) versus time at 0.373 km.

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Fig. 7. Beat signal, $S_0$ (mW), and normalized envelope signal, $|\gamma _0|$, versus distance for inline piezoelectric transducer (PZT) at 0.373 km of the fiber-under-test.

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Fig. 8. Power spectral density (PSD) of the geometric phase, $\phi _g$, beat signal, $S_0$, and backscattered intensity, $S_0^\prime$, for inline piezoelectric transducer (PZT) at 0.373 km of the fiber-under-test.

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In conclusion, we have measured the geometric phase using a novel phase-sensitive optical time-domain reflectometer setup based on simultaneous direct detection and coherent heterodyne detection. By testing the setup with a polarization scrambler and a piezoelectric transducer in the fiber-under-test, we have shown that the geometric phase is affected by both a change in polarization state and in intensity of the backscatter signal with respect to the constant local oscillator, as anticipated [9]. To our knowledge, this is the first measurement of the geometric phase in the time domain in an optical fiber medium as well as the first with varying beam intensities. Thus, the results are important for characterizing the behavior of the geometric phase as applied to distributed fiber sensing. Further research will be directed toward developing sensing technology based on these results and its evaluation in comparison with the current state-of-the-art. The results should also be applicable to optical communication.