## Abstract

We demonstrate for the first time the use of digital range-gating in OFDR to allow for orders of magnitude reduction in the required sampling rates. This allows for sensing over long lengths of fiber with fast sweeps of the optical source frequency, without requiring impractical sampling rates. The range-gating is achieved using digitally enhanced interferometry (DI), which isolates individual sections of OFDR signal bandwidth. The reductions in sampling rates permitted by the bandwidth-division are demonstrated both numerically and experimentally.

©2013 Optical Society of America

## 1. Introduction

Coherent optical frequency domain reflectometry (OFDR), otherwise known as frequency modulated continuous wave reflectometry (FMCW), allows for high resolution sensing over short to intermediate lengths of fiber. The technique's high spatial resolution makes it ideal for optical fiber network maintenance and diagnostics [1,2]. However, a major limiting factor on the spatial resolution of OFDR is the presence of *environmental perturbations*. Fan *et al.*, have shown that an effective approach for reducing the effect of environmental perturbations is to *sweep the lightwave frequency at higher rates* because the ratio of the OFDR beat frequencies to the acoustic noise band (which is of the order of kHz) increases [3,4]. The larger beat frequencies resulting from the faster sweep however necessitates greater sampling frequencies in accordance with Nyqusit sampling theory. The sampling frequencies then required for sensing over long lengths of fiber may make real-time signal processing impractical.

To reduce the sampling rate required in OFDR, Fan *et al.* have proposed a *bandwidth-division* technique [3]. This involves dividing the total signal bandwidth into *N* subsections using band-pass filters in hardware. The individual isolated bandwidth sections are then down-converted by mixing down to baseband [3]. This therefore means that the required sampling rate can be reduced by a factor equal to the bandwidth-division. Simultaneous measurement of *N* bandwidth sections would however require *N*-1 band-pass filters and mixers, and *N* low pass filters and ADCs. Such an approach would be cumbersome and impractical for large levels of bandwidth-division.

In this paper an alternative technique based on digitally enhanced interferometry (DI) [5, 6], is demonstrated. This is the first time DI has been used in conjunction with OFDR. The technique avoids additional hardware requirements, allowing for real-time *digital bandwidth-division*. The technique de-spreads and isolates individual bandwidth sections digitally and therefore only a single ADC is required, whilst there is no need for band-pass filters in hardware and all bandwidth sections can be measured simultaneously. Digitally enhanced interferometry is essentially a spread spectrum technique that isolates interferometric signals in the time domain. This is accomplished by attaching a time-stamp onto the lightwave using high frequency phase modulation [5]. Since the beat frequencies of OFDR reflections are proportional to their time delays allows DI to isolate signals in the frequency domain. The application of DI to OFDR can thus be thought of as having the effect of a discretely variable digital band-pass filter. Since individual sections of OFDR signal bandwidth can be isolated, the OFDR signal can in effect be undersampled by a factor equal to the level of bandwidth-division *N*. This reduction in the sampling rate will assist in lessening the practical and economic constraints imposed on the performance and range of OFDR.

## 2. Principles of digital bandwidth-division

The maximum beat frequency in OFDR is given by *f _{bmax}* =

*γτ*, where

_{max}*γ*(Hz/s) is the slope ofthe frequency sweep and

*τ*(s) is the maximum fiber delay. Bandwidth-division involves dividing this signal bandwidth up into

_{max}*N*sections each of bandwidth

*f*/

_{bmax}*N*. Therefore undersampling the OFDR signal by an equivalent factor

*N*, results in the isolated bandwidth sections aliasing down to within 0 ≤

*f*≤

*f*/

_{bmax}*N*bandwidth with no beat note ambiguity [3]. In other words for a bandwidth-division factor

*N*, the required OFDR sampling rate can be reduced from

*f*= 2

_{s}*f*to

_{bmax}*f*= 2

_{s}*f*/

_{bmax}*N*. This principle of bandwidth-division is shown in Fig. 1 , with

*N*discrete inline fiber reflections modeled. There is however no fundamental limit on the number of reflections permissible within each bandwidth section. The bandwidth-division is achieved by using digitally enhanced homodyne interferometry, as is outlined in the following sections.

#### 2.1 Digitally enhanced homodyne OFDR (DI-OFDR)

To understand how bandwidth-division is achieved using DI, consider the basic digitally enhanced homodyne OFDR configuration of Fig. 2
. Here a coupler splits the laser output into a *local oscillator* (LO) and a *probe beam*. An electro-optic modulator (EOM) is then used to attach the time-stamp (i.e. 0 and π phase shifts according to the pseudorandom noise (PRN) code, *c*(*t-τ _{0}*)) onto the probe beam before being coupled into the test fiber [5].

*τ*represents the time at which PRN phase modulation commences. The model presented here assumes only three discrete inline fiber reflectors for simplicity. The reflected light from the three reflectors (

_{0}*R*,

_{1}*R*and

_{2}*R*) is then combined with the local oscillator using a third coupler and the interferometric signal is measured using the photodetector. This optical setup differs from conventional OFDR in that the probe beam is time-stamped using the PRN phase modulation.

_{3}The electric field of the reflected probe beam at the photodetector is given by,

*E*is the reflected electric field amplitude (at the photodetector) from the

_{Rj}*j*

^{th}reflector. The reflection has a delay and phase of

*η*and

_{j}*ϕ*, respectively, at the photodetector. The phase depends on the optical path length associated with the reflector. In addition,

_{Rj}*ω*(

*t + η*) represents the source angular frequency at the photodetector (with delay

_{j}*η*) of the

_{j}*j*

^{th}reflector. In this paper triangular frequency modulation of the lightwave is assumed. The beat note associated with triangular frequency modulation is represented in Fig. 3 . Note that when acquiring data over more than one modulation period the beat notes consist of envelopes of harmonics. Now, the PRN phase shifts (0 or π) attached to the probe signal are represented by multiplication with the code

*c*(

*t-τ*) which has individual chip values of ± 1 and a modulation frequency of

_{0}*f*. The discrete delays of the individual PRN chips at the photodetector are given by

_{PRN}*τ*.

_{j}The time interval of a PRN chip is *ΔT = 1/f _{PRN}*, which corresponds to a fiber section of length

*ΔL = v*, where

_{g}ΔT/2*v*is the carrier group velocity. This means that any reflections within

_{g}*ΔL*lengths of fiber (i.e. delay intervals of

*ΔT*) can be isolated or recovered when decoding with the appropriately-delayed code sequence,

*c*(

*t-τ*). This is because when using maximal length [7] based PRN codes

_{j}*c*(

*t*-

*τ*)

_{j}*c*(

*t*-

*τ*) = 1 and

_{j}*c*(

*t*-

*τ*)

_{j}*c*(

*t*-

*τ*) remains a random code. The delay of each reflection at the photodetector

_{k≠j}*η*, is assumed here to lie within a unique PRN chip time interval (i.e. each having a different PRN code sequence) as shown in Fig. 2. The specific delay of a reflection within this time interval is however unimportant. In addition, multiple reflections within a specific PRN chip can be distinguished in the frequency domain since the OFDR beat frequency of a given signal with the local oscillator, is proportional to the delay.

_{j}The electric field of the local oscillator is given by,

Here*E*represents the electric field amplitude at the photodetector,

_{L}*ϕ*is the phase of the signal and

_{LO}*η*is the delay. The photodetector output voltage

_{LO}*V(t)*is proportional to the optical power, which is given by,

*not amplified*by the local oscillator, the signal to first-order approximation becomes,

*c*(

*t*-

*τ*), and since

_{1}*c*(

*t*-

*τ*)

_{j}*c*(

*t*-

*τ*) = 1, whilst

_{j}*c*(

*t*-

*τ*)

_{j}*c*(

*t*-

*τ*) remains a random code,

_{k≠j}The phase modulation of reflection 1 is therefore inverted. However the signals from reflectors 2 and 3 (i.e. *ε _{R2}* and

*ε*) are randomly inverted and their spectra remain spread, contributing small levels of residual

_{R3}*broadband*background noise. Furthermore, the principles presented here also apply in the case of multiple reflections occurring within a given bandwidth section. In this case all the reflections of that particular bandwidth section are decoded when the signal is multiplied by the PRN code sequence of appropriate delay.

#### 2.2 Residual broadband noise suppression

This section outlines the residual broadband noise suppression achievable for a given data acquisition time, PRN frequency and PRN code length. The residual broadband noise terms *ε _{Rj}*, that contribute to the noise floor of the selected reflection are given by:

*f*by increasing the acquisition time

_{PRN},*t*(i.e. increasing the number of PRN chip periods over which data is acquired). This is however only true for acquisition times less than or equal to the time interval of the entire PRN code sequence. Therefore the noise floor can be reduced indefinitely, in principle, by increasing both the acquisition time and the PRN code length. Note that since the PRN code length ultimately dictates the total span possible in DI-OFDR, it may be desirable to have a long PRN code sequence even when sampling for much shorter periods. In the frequency domain the residual noise terms are broadband, which means that they are spread across the signal bandwidth. The Fourier Transform of the residual noise terms is given by,

_{s}*−1, where*

^{n}*n*is the number of linear feedback shift register stages used for generating the maximal length sequence [7]. Since the convolution of a delta function with white noise in Eq. (7) is white noise, it can be shown that the relationship between the noise floor level and the acquisition time is characterized by,

*f*is equal to the sampling frequency

_{PRN}*f*, because it is assumed that one datum is taken per PRN chip. This result also assumes that

_{s}*t*>> 1. Moreover, when sampling over an entire PRN code sequence time interval it can be shown that,so that increasing the PRN code length reduces the noise floor. Note that Eq. (8) and (9) are not exclusive to maximal length sequences and also apply to other pseudorandom codes. This theory assumes a perfect broadband spread of the reflections outside the selected PRN chip. As discussed in Section 3C, the performance is however somewhat limited in practice by constraints such as the bandwidth of the analog electronics used for amplifying the PRN signal. This may result in imperfect signal isolation implying that reflections from neighboring PRN chips/bandwidth sections may also be partially de-spread, contributing noise which could lie above the ideal noise floor.

_{s}f_{PRN}#### 2.3 Level of digital bandwidth-division

The maximum bandwidth-division/undersampling achievable in DI-OFDR is determined in this section. Assuming that high residual noise suppression can be achieved and taking the Fourier Transform of the decoded signal of Eq. (5) then yields the usual beat note between the selected reflector (within the isolated/de-spread PRN chip) and the local oscillator.

*ϕ*−

_{Rj}*ϕ*is constant, although in practice this value will vary due to phase noise and small perturbations in the reflector positions, resulting in the broadening of the beat notes [4].

_{LO}Since the beat notes within individual PRN chips can be isolated, the original OFDR signal can be undersampled. The undersampling factor is given by,

*M*=

*N*=

*2Lf*when

_{PRN}/v_{g}*f*=

_{s}*γ/f*Therefore the optimal bandwidth-division factor is given by,

_{PRN}.## 3. Demonstration of principle

#### 3.1 Numerical simulations

This section demonstrates the full potential of digital bandwidth-division in DI-OFDR, using numerical simulations that are based on a previously reported high-performance OFDR system [3]. The parameters used are given in Table 1
. Since the bandwidth-division here is *N* = 670, the required sampling frequency of the OFDR is reduced by an equivalent factor. So instead of requiring a sampling frequency of 2.32 GHz to achieve a spatial resolution of approximately 8 mm over 40 km with the moderately high, low-noise sweep rate of *γ* = 3 THz/s, the sampling rate can be reduced to just 1.73 MHz. The numerical simulations verifying the bandwidth-division are shown in Fig. 4
. Figure 4(a) shows the original OFDRbeat spectrum with four reflections along the length of the 40 km fiber. Here, reflections 2 and 3 occur within the same PRN chip. Figures 4(b-d) demonstrate the bandwidth-division, with the noise floor for each isolated bandwidth section being ~0.5% of the total power in the remaining PRN chips/bandwidth sections. This means that the noise floor is considerably worse when suppressing intense reflections from other bandwidth sections. So therefore problems may arise if very dissimilar intensity reflections are to be isolated. This is clearly seen in Fig. 4(b-d) where the noise floor of the first isolated reflection is lower than that of the remaining weaker reflections. As outlined in Section 2B, the residual PRN noise floor can however be decreased almost indefinitely by increasing both the PRN code length and acquisition time.

In these simulations propagating losses from the reflections are modeled, as is fiber attenuation (attenuation coefficient of α = 4.6 × 10^{−5} m^{−1}), Gaussian-distributed fluctuations in the reflector positions as well as 1/f^{2} phase noise (red noise) due to the finite linewidth of the laser. Note that an equal power split between the local oscillator and probe is assumed, whilst Rayleigh backscatter (RBS) is not simulated. In the absence of strong discrete reflections RBS would however be measurable in DI-OFDR since the residual PRN noise floor is proportional to the signal power.

In this simulation the number of PRN chips along the length of fiber far exceeds the number of reflections modeled. If all sections of the fiber are to be analyzed *N* = 670 *FFT* computations are necessary, although these can all be computed in parallel and from the same data. When analyzing only select bandwidth sections the data required decreases by a factor *N*. So these simulations demonstrate the potential of DI-OFDR for sensing discrete reflections over long lengths of fiber with significant reductions in the required sampling rate. The reduction in the required sampling rate also accommodates for higher tuning rates γ, which reduce the influence of acoustic noise, and furthermore lowers the minimum acquisition time required to achieve the theoretical spatial resolution (i.e. half the period of the triangular frequency modulation, *F _{s}*/

*γ*). Note however that for DI-OFDR the acquisition time will be greater than that of standard OFDR if very high levels of residual noise suppression are required.

#### 3.2 Experimental proof of concept

This section provides an experimental proof of concept for bandwidth-division in DI-OFDR. The setup used was nearly identical to Fig. 2. However, in order to allow for a relatively large linear sweep of the lightwave frequency, whilst still allowing for a highly-coherent source, double-sideband suppressed carrier (DSB-SC) *external modulation* was implemented. Note that fluctuations in the beat signal intensities were observed as a function of distance along the length of the fiber as predicted in Ref [10]. This could be mitigated by instead using single-sideband modulation as in Ref [10]. The spectra obtained were averaged (by a factor of 250), to average out the intensity variations caused by micron-scale fluctuations in the reflector positions. The frequency was swept over *F _{s}* = 0.5 GHz using a voltage controlled oscillator and an arbitrary waveform function generator. The laser used was a 400 Hz linewidth fiber laser (

*Orbits Lightwave Ethernal*). Moreover, PRN encoding/decoding and direct memory access (DMA) transfers were implemented on a field-programmable gate array (FPGA), and FC/PC connector and flat fiber end face reflections were used for the reflectors.

^{TM}Without the PRN phase modulation, the optical setup is identical to conventional homodyne OFDR. Figure 5(a)
shows the standard beat spectrum of two reflections with offsets ~40 m and ~110 m relative to the path length of the local oscillator. The beat notes here are actually the envelopes of harmonics because the data acquisition time far exceeds the modulation period (see Fig. 3(a)). The beat notes occur at approximately 0.55 MHz and 1.6 MHz, respectively. However since the sampling frequency is only *f _{s}* =

*f*= 2.5 MHz, the second beat note aliases down to 0.85 MHz and may then result in beat note ambiguity.

_{PRN}However with the application of digitally enhanced interferometry the beat of each reflector can be isolated (Fig. 5(b), 5(c)) which means that there is no impending beat note ambiguity despite the undersampling. Using DI, the first and second reflections (Fig. 5(b), 5(c)) are isolated by 18 dB and 15 dB, respectively. This is despite the use of low bandwidth analog electronics as discussed further in Section 3C. The parameters used for both the PRN phase modulation and the triangular frequency modulation are given in Table 2 . The level of the PRN noise floor for an acquisition time of 38/3 ms is ~1% of the suppressed signal power.

The results were found to agree with numerical simulations, and therefore confirm bandwidth-division in DI-OFDR. This is despite the use of a very simple OFDR setup (i.e. without linearization of the frequency sweep, polarization diversity schemes, etc.). The potential for significant reductions in the required sampling rate of higher performance OFDR systems is however evident in section 3A. Moreover, in the absence of strong discrete reflections DI-OFDR could also be used to measure weak Rayleigh backscatter since the residual PRN noise floor is proportional to the return signal power.

#### 3.3 Practical considerations

The performance of DI-OFDR can be corrupted, for example, by errors in the PRN phase modulation arising from the finite bandwidth of the analog electronics required to amplify the PRN signal. This can occasionally result in a single reflection being partially decoded in neighboring chips [6], as seen for example in Fig. 5(c). This minor effect could however be lessened in DI-OFDR by reducing the level of bandwidth-division below the theoretical, *N*.

Furthermore, the time-offset of the PRN code sequence must be adjusted/calibrated such that each integer *ΔL* segment along the test fiber coincides with a unique PRN chip. In other words, the PRN time delay at the photodetector, for the start of the test fiber, should be an integer PRN chip period. This is necessary in order to avoid any signal overlap issues when undersampling. Note however that an appropriate post measurement demodulation of the signal could be used instead of actually calibrating the time-offset of the PRN code sequence. The aim of the demodulation would then be to achieve bandwidth-division resembling that of Fig. 2 where the beat frequency of the start of the test fiber corresponds to an integer multiple of *f _{bmax}*/

*N*.

## 5. Summary

In this paper the application of digitally enhanced interferometry to OFDR has been shown for the first time, with the purpose here being digital bandwidth-division. The application of this range-gating technique can allow for orders of magnitude reduction in the required sampling rate of OFDR. The sampling rate in traditional OFDR is proportional to both the length of the fiber and the frequency sweep rate. So therefore, digitally enhanced or time-resolved OFDR permits the use of very low sampling rate sampling cards, without reducing the total measurable span when using low-noise high frequency sweep rates.

## Acknowledgments

N. Riesen is the recipient of Research Scholarships from both The Australian National University, Canberra, Australia and the Commonwealth Scientific and Industrial Research Organisation, Lindfield, Australia. The authors also thank John W. Arkwright, Andrew Sutton and Daniel A. Shaddock for useful discussions and acknowledge the Australian Research Council for research support.

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