## Abstract

In Chirped Pulse *φ*-OTDR systems used for sensing temperature or strain along an optical fiber, the largest disturbance between two single-shot measurements that can reliably be detected depends on the range of frequencies swept by the chirped pulse. If electrical modulation is used to generate the laser frequency sweep, the achievable sweeping range is limited by the electrical components, leading to a narrow measurement range for static measurements. In this work, we demonstrate the extension of the frequency range of a chirped laser pulse by all-optical means using evenly spaced frequency sidebands generated via the Kerr effect to improve the Chirped Pulse *φ*-OTDR measurement range. We report chirp extensions by factors up to 13 and apply the effect to achieve a sixfold increase in the measurement range of a Chirped Pulse *φ*-OTDR system measuring the temperature of a random fiber grating array. The method described in this paper can be applied to other optical systems utilizing chirped laser pulses and allow for variable extension of their chirping range.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Compared to devices based on electrical signals, environmental sensors utilizing optical fibers have a number of advantages. Apart from durability and immunity to electromagnetic interference, optical fibers allow for distributed sensing, where information is recorded along their entire length. To measure a quantity of interest, such as local temperature, strain or vibration, a technique referred to as phase sensitive Optical Time Domain Reflectometry ($\varphi$-OTDR) is often employed. By launching a monochromatic pulse of light into a fiber and measuring the Rayleigh back-scattered intensity as a function of time, one can determine the state of the fiber along its length in a continuous manner [1]. However, since the relationship between the magnitude of an applied disturbance and the change in the intensity of the back-scattered Rayleigh traces is not linear, conventional amplitude demodulation cannot be used in $\varphi$-OTDR to measure the magnitude of the disturbance [2] [3] [4]. Measuring the magnitude of disturbances requires the use of various phase demodulation techniques. These involve comparing the relative phase change of two different segments of fiber [5], utilizing 90$^{\circ }$ optical hybrid based homodyne detection [6], dual-pulse phase modulation [7], or the phase-generated carrier demodulation algorithm [8]. One drawback of phase demodulation techniques is their reliance on optical interferometers, which are limited by phase recovery range and fringe contrast.

To obtain accurate, single-shot measurements of the magnitudes of applied disturbances based on back-scattered Rayleigh traces, one can instead utilize chirped pulses, where the instantaneous frequency of the pulse is linearly increased or decreased within pulse duration. This technique called Chirped Pulse $\varphi$-OTDR (CP $\varphi$-OTDR) relies on the interference of a range of frequencies back-scattered by different parts of the fiber due to the Rayleigh effect. By calculating the cross correlation between subsequent Rayleigh traces, such sensing systems have achieved resolutions of 1 mK for temperature and 4 $n\varepsilon$ for strain [9]. When using Rayleigh traces from chirped pulses scattered by a random fiber grating array (RFGA), lower pulse powers are required and a spatial resolution of 1.2 m can be achieved [10]. Since CP $\varphi$-OTDR with an RFGA is a single-shot system which does not require heterodyne detection, one section of the sensor without disturbance could be utilized for compensating time delay errors induced by laser frequency drifts. This technique enables the detection of ultra-low frequency vibrations [11]. While CP $\varphi$-OTDR is highly sensitive because the Rayleigh traces undergo large changes from small applied disturbances, the static measurement range is limited because subsequent traces become decorrelated if the external perturbation is too large. In practice, two Rayleigh traces recorded before and after a disturbance is applied will be approximately identical (with high correlation coefficient) apart from a time delay, $\Delta t$, if this delay is smaller than 10$\%$ of the pulse duration. To conduct dynamic measurements, large perturbations could be detected by oversampling using a high repetition rate for the pulses. However, in distributed sensing systems, there is a trade off between pulse repetition rate and sensor length. To extend the limited measurement range, one method based on multi-frequency database demodulation has been implemented [12]. By establishing a database consisting of Rayleigh traces collected from chirped pulses with different initial optical frequencies, the time delays $\Delta t$ due to large strains can be partially compensated by the initial frequency difference of the chirped pulses, thereby extending the static measurement range by a factor of 3. Additional extension of the measurement range using this method requires building a database for a larger number of frequencies, which is time consuming. Another direct way to extend the measurement range is to increase the range of frequencies swept by each chirped pulse.

When two frequencies of light are launched into a fiber with a large nonlinear parameter, $\gamma$, the Kerr effect causes frequency sidebands to be generated. Since the output power of these sidebands depends on the input power, the Kerr effect has been used to enhance the extinction ratio of optical pulses for OTDR by all-optical means [13], and to enhance the resolution of fiber optical sensors [14]. Furthermore, changing the frequency spacing between the input lasers causes the central frequencies of the higher order sidebands to shift by integer multiples of this change. When used in Optical Frequency Domain Reflectometry (OFDR), this method can extend the range of a laser frequency sweep taking place on the scale of seconds, thereby improving the resolution of an OFDR measurement [15]. This frequency chirp magnification (FCM) induced by the Kerr effect has also been used for magnifying the wavelength drift of an FBG sensor subject to strain and temperature changes, thereby enhancing its sensitivity [16]. Because the Kerr effect is fundamentally caused by the reaction of bound electrons in the glass fiber to variations in the power of an external optical field, the smallest duration of a sweep extendable using the Kerr effect is determined by the electron response time, which is on the scale of femtoseconds. Therefore, it should be possible to extend frequency sweeps, which are much shorter in duration than ones typically used in OFDR.

In this paper, we apply the method of extending frequency sweeps using the Kerr effect to broaden the chirp range of a nanosecond optical pulse by factors up to $13$ through all-optical signal processing. First, the measurement principle behind CP $\varphi$-OTDR, its advantages and limitations are discussed. A theoretical model describing sideband generation using the Kerr effect is then presented, showing how the frequency shift of a sideband depends on its order. An experimental setup used for extending the chirping range of a pulse is described, and it is shown how the static temperature measurement range of a system based on an RFGA can be increased by factors up to $6$. Finally, the limitations of our model and experiment are discussed along with potential application to other optical techniques relying on chirped pulses.

## 2. Theory

#### 2.1 Chirped Pulse $\varphi$-OTDR measurement principle

In traditional $\varphi$-OTDR, a monochromatic pulse is employed as the interrogation signal to investigate the status of the fiber. The lack of a linear relationship between the applied external disturbance and the intensity of Rayleigh traces means that the magnitude of the disturbance cannot be determined directly. In order to create a linear relationship between applied disturbances and the intensity profile of the Rayleigh traces, a chirped pulse is utilized instead of a monochromatic pulse. The instantaneous optical frequency profile of the chirped pulse is expressed as [12]

where $v_{0}$ is the initial laser frequency, $\Delta v$ is chirping range of the chirped pulse, $W$ is the pulse duration, and $\operatorname {Rect}$ is the rectangular function with unit height. When the chirped pulse is sent into the fiber under test, the interference of back-scattered electrical fields occurring in the fiber section corresponding to the half pulse width will form the time domain Rayleigh traces. At the time $t$, when the interference signal arrives at the detection end, the optical power can be expressed as:#### 2.2 Sideband generation using the Kerr effect

The following description of the generation of sidebands in a Kerr medium is based on the approach used in [17]. Sending laser light with two different frequencies, $\omega _p<\omega _c$ into a Kerr medium consisting of an optical fiber of length, $L$, with a high nonlinear parameter, $\gamma$, the normalized electric field amplitude is initially given by $A_{in}=\left [\sqrt {P_p}\exp (-i0.5\omega _d t)+\sqrt {P_c}\exp (i0.5\omega _d t) \right ]$. Here, $P_{p/c}$ referes to the power of the pump field/chirped pulses, $\omega _d=\omega _c-\omega _p$, and the power of the field as a function of time is $P_{in} = \left [P_p + P_c + 2\sqrt {P_p P_c}\cos {\omega _d t} \right ]$. Assuming that dispersion, loss, and polarization effects are negligible, the input field evolves according to the Nonlinear Schrödinger Equation, $dA/dz = i\gamma |A|^2A$ [18]. Solving this differential equation for the field at the output of the Kerr medium yields $A_{out}=A_{in} \exp \left [i\gamma L(P_p + P_c) \right ]\exp \left [ i\gamma L 2\sqrt {P_p P_c} \cos {\omega _d t} \right ]$. In the following, the exponential factor, $\exp \left [i\gamma L(P_p + P_c) \right ]$, is ignored, as it does not affect the output frequency if the input powers remain constant. Chirps induced by self-phase modulation (SPM) are only present near the pulse edges constituting less than 5$\%$ of pulse duration and do not contribute significantly to the scattered signal. The Jacobi-Anger expansion [19], $\exp \left [iM\cos {B t}\right ]=\sum _{n=-\infty }^{\infty } i^n J_n(M)\exp (inB t)$, where $J_n$ indicates the $n^{th}$ order Bessel function of the first kind is then applied to the output field, thereby expressing it as an infinite sum of frequency sidebands spaced $\omega _d$ apart:

## 3. Experimental setup and results

The experimental setup for temperature measurement with chirped pulses is shown in Fig. 1. A distributed feedback laser (DFB) (CQF938/500, JDS Uniphase) with a linewidth of 1MHz emitting at 193506.25 GHz is periodically modulated by a saw-tooth pattern with a rising edge of 20 ns from a signal generator (8130A, Hewlett Packard). The so-called "bias-T structure" converts the applied electrical modulation to a change in the drive current supplied to the DFB. When the saw-tooth signal periodically increases(decreases) the supplied current, the instantaneous frequency emitted by the DFB is increased(decreased). Figure 1(a) depicts the laser signal at the output of the DFB. In the time domain, the laser signal is a continuous wave (CW) and its power is approximately constant because the change in the drive current is small compared to the total current supplied. The spectrum shown in Fig. 2(b) is measured with an optical spectrum analyzer (OSA) (AP201x, APEX) and has a 5dB width of 746 MHz, indicating that the changing drive current causes the central frequency to vary in this range. The signal generator creating the saw-tooth pattern also generates rectangular signals at a rate of 1 MHz activating a custom solid state optical amplifier (SOA) driver circuit board. When activated by a pulse, this circuit board switches on the SOA (SOA-S-OEC-1550, CIP) allowing it to serve as a gate for pulsing the signal from the DFBs with a duration of 6ns. As illustrated in Fig. 1(b), once the modulation of the DFB and the trigger for the SOA are synchronized, a linearly chirped optical pulse is produced.

Once generated, the chirped optical pulses are split into two different branches of the setup by a 50:50 coupler. The branch, which sends pulses through a delay line and an erbium doped fiber amplifier (EDFA) (AEDFA-PA-25-B-FA, Amonics) without applying nonlinear chirp extension is marked with a red fiber in Fig. 1. We refer to pulses from this branch as "pure pulses". The delay line ensures that the pure pulses and the pulses generated by the other branch arrive at the RFGA with a 40 ns time delay, allowing for an accurate comparison of their response to changes in temperature.

The pulses in the other branch are first amplified by an EDFA (APEDFA-C-10-B-FA, Amonics) and then sent through an isolator into one input port of a 50:50 coupler. A CW light wave from a narrow linewidth laser (NLL) (PS-NLL-1550.12, Teraxion) with a linewidth of 5 kHz emitting at 193480.00 GHz is amplified by an EDFA (AEDFA-33-B-FA, Amonics), passed through an isolator and sent into the other input port of the 50:50 coupler. The chirped pulses and the light from the NLL used as a pump are combined and launched into a highly nonlinear fiber with a non-uniform core referred to as a "Kerr medium" (418SG 04611A, Draka Comteq). The non-uniform core continuously changes the Brillouin frequency of the fiber, which prevents the CW light of the narrow linewidth laser from losing power to Stimulated Brillouin Scattering. The combined spectrum of the chirped pulses and the narrow linewidth laser signal before entering the Kerr medium is visualized in Fig. 1(c). At this stage, the power of the signal in the time domain consists of a constant offset due to CW light from the NLL and a pulse with a sinusoidally varying optical power due to interference. Because of the nonlinear phase modulation caused by the interference of the NLL and the chirped pulses described in Subsection 2.2, frequency sidebands spaced 26.25 GHz apart are generated. Figure 1(d) illustrates three higher order sidebands generated by this process. Figure 2(a) shows a recorded optical spectrum after the Kerr medium with sidebands ranging from -4 to 13. Figures 2(b)-2(d) shows that the chirp range for the $n^{th}$ order sideband is $n$ times the chirp range of the input pulse as predicted in Subsection 2.2. Individual sidebands are extracted using a tunable filter (XTM-50, EXFO). This produces a chirped pulse, referred to as a "Kerr pulse", with a wider chirping range than the pure pulses, but with the same shape and duration.

The extracted Kerr pulses for the $n^{th}$ order sideband are amplified by an EDFA (AEDFA-PA-25-B-FA, Amonics) and recombined with the pure pulses in a 50:50 coupler. They are then launched through a circulator to an RFGA placed inside an insulating, temperature controlled styrofoam box. The RFGA has a reflection coefficient of -30 dB, and a length of 0.8m, which is used to provide a locally enhanced Rayleigh scattering signal. It consists of 8 alternating sub-gratings of lengths 10mm and 5mm respectively. The periods of the sub-gratings are randombly distributed between 0.5180µm and 0.5464µm. The reflected Rayleigh patterns are passed back through the circulator, through a tunable filter (Lambda Commander OSP 9100, Newport) and finally to a photodiode (PDB435C, Thorlabs) connected to an oscilloscope (infiniium DSO81204B, Agilent) sampling at 40 GSa/s.

To determine the temperature static measurement range for different Kerr pulses, the heating element inside the styrofoam box is activated, raising the temperature of the RFGA to 40°C. The intensities of the two Rayleigh traces from the pure and Kerr pulses are equalized by adjusting the amplifiers. The heater is then switched off, allowing the RFGA to gradually cool down, causing the Rayleigh traces to shift accordingly. For every 0.1°C step down to 34°C, a Rayleigh trace is recorded. The duration of Rayleigh traces is about 20ns, and the total number of samples for each traces is 800. The oscilloscope is set to average 16 traces to reduce the impact of intensity fluctuation and electrical noise. A custom python script is used to determine the magnitude of the time delays between the traces recorded at different temperatures by numerically calculating the correlation between them. First, pulses for sideband orders from -2 to 6 were used to measure temperature changes in steps of 0.1 °C, demonstrating the feasibility of using higher order sideband pulses. Figure 3 shows the measured time delay as a function of the temperature change for different Kerr pulses in steps of 0.1°C. The slope corresponding to pulses from the $n^{th}$ order sideband is $n$ times smaller than the slope for the pure pulses as predicted in Subsection 2.2. The sign of the slope is negative for sidebands where $n<0$, corresponding to pulses for which the direction of the chirp is reversed.

Next, the static measurement range of pulses from each sideband order are investigated. Figure 4 depicts the reflected traces from the RFGA before and after the application of a temperature change. For pure pulses with a short chirping range, a temperature change of 0.2°C causes the traces from before and after the change to lose correlation. This is in accordance with Subsection 2.1 where it was predicted, that a short chirping range would imply high sensitivity and low static measurement range. For the 2$^{nd}$ order Kerr pulses, the same temperature change of 0.2°C introduces a smaller time delay due to the extended frequency chirping range. As shown in Fig. 4(f) in which the 6$^{th}$ order Kerr pulses are used, two traces collected before and after a temperature change of 0.6°C remain highly correlated. Recovering the correct time delay between two subsequent traces is not possible for temperature changes that cause the cross correlation coefficient to drop below 0.8. For example, when 1$^{st}$ order chirped pulse is used to measure temperature variations of 0.2°C as shown in Fig. 4(b), the coefficient decreases to 0.505. In this case, the demodulation cannot recover the correct time delay, making temperature measurements impossible. Therefore, the largest change in temperature that can be measured between two single shots can be found by determining which temperature shift is required for the traces to decorrelate. The relationship between the cross-correlation coefficient and the temperature variation range for different order Kerr pulses is shown in Fig. 5. Choosing a correlation coefficient of 0.8 as a threshold, it is seen that the static temperature measurement range for the 6$^{th}$ order sideband is improved by a factor 6 compared to the pure pulses.

To investigate how the resolution of the temperature measurements changes when different Kerr pulses are used, 500 Rayleigh traces generated by averaging 16 raw traces are collected at constant room temperature over 0.8s for different orders. Fluctuations in the time delays between consecutive pulses are calculated and the equivalent temperature variations are determined using the slopes in Fig. 5. The raw fluctuation data for pure pulses and 6$^{th}$ order Kerr pulses are shown in Fig. 6(a). A histogram of this data is shown in Fig. 6(b). The calculated standard deviation is 2.82 mK for the pure pulses and 25.1 mK for pulses from the 6$^{th}$ order sideband. The temperature measurement precision for different sideband orders is shown in Fig. 6(c). The dashed lines in Fig. 6(c) indicate how the standard deviation of the arrival time and the smallest discernable temperature change should behave in an ideal case. We attribute the discrepancy between the measured resolution and the ideal one to the decreasing Signal to Noise Ratio (SNR) for higher values of $n$. As mentioned in Subsection 2.2, the power profile of the $n^{th}$ order Kerr pulse will be the power profile of the pure pulse raised to the $n^{th}$ exponent. This potentiation magnifies small power fluctuations and decreases the SNR [21]. Furthermore, the Kerr pulses are affected by power fluctuations for both the input pulses and the amplified NLL used as a pump. The magnified intensity noise causes subsequent traces for higher order sidebands to vary more in shape leading to the larger measured variation in arrival time shown in Fig. 6(c).

## 4. Discussion

Several simplifying assumptions are used in modelling the extension of the chirping range. The pure pulses launched into the Kerr medium are assumed to be quasi-CW, which is a valid assumption if the duration of the pulses is much longer than the period of the beat with the CW light from the NLL ($\omega _d\gg 2\pi /W$). In this experiment, the frequency difference of 26.25 GHz implies a beat period of 38 ps, which is much shorter than the pulse duration of 6ns, meaning that the quasi-CW assumption is acceptable. Loss was ignored, and dispersion in the Kerr medium was assumed to be zero. The latter assumption is valid for operation close to the zero-dispersion wavelength, and one can ensure that the sidebands experience similar dispersion by making the frequency difference between the input pulses and the CW light from the NLL small. Effects related to the relative polarization of the input pulses and CW light were also ignored. A comprehensive model accounting for all these effects is being researched, but is beyond the scope of this paper.

By calculating the power of each of the sidebands in Eq. (6), and expressing the Bessel functions in terms of their Taylor expansions, it can be shown that their power decreases as $1/(n!)^2$. Thus, utilization of sidebands beyond $n=6$ is limited by their rapidly decreasing power. Furthermore, because the Kerr effect magnifies power fluctuations, accurately calculating the correlation between traces for sidebands higher than $n=6$ proved unfeasible. This limitation due to the lasers and amplifiers can be addressed by using more stable devices. To minimize the effect of power fluctuations, one could use a single, highly stable laser to generate both the chirped pulses and the pump. Another way to reduce the impact of intensity fluctuations is to utilize pulse powers that cause the argument of the $n^{th}$ order Bessel function in Eq.(6) to equal its first maximum [22]. In this regime, power fluctuations are attenuated by the Kerr effect. These possibilities will be investigated in future work.

The method described in this paper implies a trade-off between sensitivity and static measurement range for higher order Kerr pulses. However, as the experimental approach demonstrates, it is possible to mitigate this trade-off by using both highly sensitive pure pulses and Kerr pulses with a large static measurement range simultaneously. The extension of chirped pulses described in this work was only applied to a single RFGA, but could be applied to distributed measurements using CP $\varphi$-OTDR. More generally, it can be applied to extend the chirping range of pulses in a wide variety of optical systems. For example, the ability to broaden or reverse chirped pulses by all-optical means without altering the pulse duration could be useful in systems relying on Chirped Pulse Amplification [23] or Dispersive Fourier Transforms [24].

## 5. Conclusion

All-optical chirp extension based on the Kerr effect is demonstrated for sidebands ranging from $n=-3$ to $n=6$. When applied to a CP $\varphi$-OTDR system, the effect allows the measurement range to be scaled up by integer multiples of the sideband number as predicted by theoretical calculations. The chirp extension effect can be applied to existing system relying on chirped pulses to extend their chirping range.

## Funding

China Scholarship Council (201808330421); Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-06302, STPGP 506628-17); Canada Research Chairs (950231352).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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