Compensation of laser frequency tuning nonlinearity of a long range OFDR using deskew filter

Zhenyang Ding; X. Steve Yao; Tiegen Liu; Yang Du; Kun Liu; Junfeng Jiang; Zhuo Meng; Hongxin Chen

doi:10.1364/OE.21.003826

1. Introduction

Optical frequency-domain reflectometry (OFDR) [1] has been attracting considerable attention for a number of applications, including optical communication networks monitoring [2], and distributed optical fiber sensing [3,4]. In an OFDR system, the interference signals are collected as a function of optical frequency of a tunable laser source (TLS). A Fast Fourier transform (FFT) is then used to convert this frequency domain information to a desired spatial information, where it requires that the interference signals are sampled at an equal interval of the optical frequency. However, any frequency tuning nonlinearity of a TLS gives a rise of non-uniform sampling interval of the optical frequency when the signal is sampled by an equal time interval which, in turn, results in spreading of the reflection peak energy, deteriorating the spatial resolution and reducing the peak’s amplitude [5,6].

Two class methods are generally adopted to solve the problem of the TLS frequency tuning nonlinearity. The first class method is a frequency-sampling method, which samples the interference signal at the equidistant instantaneous optical frequency points. It is realized by applying an auxiliary interferometer output as a clock signal to trigger data acquisition [5,6]. However, a drawback of this method is that the maximum measurement range of an OFDR is limited by a path difference between two arms of the auxiliary interferometer in order to satisfy the Nyquist Law [7]. By using this method [2], an OFDR can achieve a high spatial resolution up to be micrometers, but a measurement range is limited to a few tens of meters. Although a frequency multiplication method was proposed to increase a measurement range without increasing the path difference of an auxiliary interferometer [7], an increased measurable range is still limited by the frequency multiplication hardware and phase noise (jitter) of a clock signal. The second class method is to use algorithms to suppress a TLS nonlinearity after the data acquisition. In this class method, several types of algorithms are proposed to compensate the laser frequency tuning nonlinearity effect by using obtained optical frequency or phase information of a TLS from an auxiliary interferometer. One of those algorithms is the re-sampling technique that re-samples these main beating interference signals with an accurate equidistant optical frequency grid based on the optical frequency information of the TLS by the interpolation algorithms (such as linear and cubic spline interpolations) [6,8,9] and non-uniform FFT [9,10]. A major advantage of those re-sampling algorithms is that an OFDR’s maximum measurement range is independent on auxiliary interferometer’s path length difference, but the method cannot be very effective to compensate TLS’s phase noise or nonlinear phase completely and also a measurement range is short, for example, about a few tens of meters [6–10]. The other algorithms of the phase noise compensation are also reported [11,12]. M. Froggatt et al. presented a method that uses main interference signals to mix with a predicted phase of the TLS to achieve an OFDR with a measurement range of 2 km and a spatial resolution of 1 mm [11,13]. It's worth mentioning that F. Ito et al. introduced a novel OFDR system with the measurement range of 40 km and the spatial resolution of 5 cm [12]. Such a high resolution that was realized over tens of kilometers is unique among existing reflectometry techniques. The key technologies of this novel OFDR are high linear optical frequency tuning based on an external single sideband modulator [14] and the phase noise or nonlinear phase compensating algorithm based on different concatenation-generated phases [12]. In this algorithm [12], the spatial resolution is improved a factor of 20 times than that of without phase noise compensation. Although those phase noise compensation algorithms [11,12] greatly improved both the measurement range and spatial resolution, they are very complex and require long signal processing time, for example, a signal process time of the algorithm in [12] is about 10 minutes, which cannot satisfy a requirement for the real-time measurement and monitoring for the field application that generally requires a short measurement time, e.g. a few seconds or less. In addition, a measurement range still requires to be further extended in order to test the long optical commutation fiber links and to be applied for the long range sensing applications.

In this paper, we present a simple but effective technique to reduce a degradation of the optical frequency tuning nonlinearity of a TLS in a long range OFDR by using the deskew filter. The deskew filter (also called Residual Video Phase filter) was originally applied to compensate a frequency tuning or frequency chirp nonlinearity in a Frequency Modulated Continuous Wave Synthetic Aperture Radar (FMCW SAR) [15–17]. For our proposed technique an auxiliary interferometer is implemented to obtain the required nonlinear phase of a TLS. The nonlinear laser frequency or phase noises of the beating signals that were acquired from the main OFDR interferometer can be compensated by the deskew filter for the entire spatial domain beating signals by one time calculation so as to have a high computational efficiency. In theory the proposed method could eliminate any laser frequency tuning nonlinearity completely provided that the nonlinear phase of a TLS can be estimated accurately. Using the proposed method, we demonstrated that the reflection peaks that were undetectable due to the frequency tuning nonlinearity of the TLS can be measured with a significantly improved spatial resolution. In particular, in our OFDR system we observed a spatial resolution of 20 cm and 1.6 m at distances of 10 km and 80 km, respectively, that is about 93 times enhancement when compared with that of the same OFDR system without nonlinearity compensation. If the TLS has a better tuning linearity and wider tuning frequency range, a spatial resolution of our OFDR could be further improved. Moreover this method doesn’t require numerous measurements for the data averaging and thereby a total signal processing time is short, e.g. less than 1 s for data size of 5 × 10⁶. The reported approach in this paper can result a significantly improvement for an OFDR instrument system that could be developed as a powerful tool for a real-time fiber optical network testing and optical fiber sensing which has both the high spatial resolution and long measurement range.

2. Algorithm principle

2.1 Basic theory of the OFDR

An OFDR interferometer provides beating signals that are produced by the optical interference between two light signals originating from the same linearly chirped high coherent light source. One signal $E_{s} (t)$ is reflection or backscattering light from an optical path of a fiber under test (FUT) while another $E_{r} (t)$ follows a reference path of the interferometer. For a TLS having a linear optical frequency tuning speed $γ$ , an optical field $E_{r} (t)$ can be written as

E_{r} (t) = E_{0} \exp {j [2 π f_{0} t + π γ t^{2} + 2 π e (t)]},

where

f_{0}

is the optical frequency,

e (t)

is the nonlinear phase or phase noise of the TLS.

By assuming that a reflection reflectivity is $r (τ)$ at a delay time $τ$ and $α$ is the fiber attenuation coefficient, a reflectivity with the fiber attenuation can be written as $R (τ) = r (τ) \exp (- α τ c / n)$ , where $c$ is the light speed in vacuum and n is the refractive index of fiber. Then $E_{s} (t)$ of reflection with the reflectivity can be expressed as

E_{s} (t) = \sqrt{R (τ)} E_{0} \exp {j [2 π f_{0} (t - τ) + π γ {(t - τ)}^{2} + 2 π e (t - τ)} .

The $E_{r} (t)$ and $E_{s} (t)$ can also be considered as the local oscillator (LO) signals and received signals from the FUT, respectively. Thus, an AC coupled beat signal $I (t)$ generated by the interferences of the LO light $E_{r} (t)$ and received signals $E_{s} (t)$ can be written as

I (t) = 2 \sqrt{R (τ)} E_{0}^{2} \cos {2 π [f_{0} τ + f_{b} t + \frac{1}{2} γ τ^{2} + e (t) - e (t - τ)]},

where the beat frequency

f_{b} = γ τ

. Because different beat frequencies

f_{b}

correspond to the different locations (i.e. fiber distances) in a spatial domain, the last term

e (t) - e (t - τ)

is a source of the laser frequency tuning nonlinearity effect, i.e. a nonlinear phase or phase noise of the TLS. This could also be understood as a beating between the LO light nonlinear phase

e (t)

and received signal’s nonlinear phase

e (t - τ)

.

A complex exponential expression is transformed from Eq. (3) by Hilbert transform that can be expressed as

I (t) = 2 \sqrt{R (τ)} E_{0}^{2} \exp [j 2 π (f_{0} τ + f_{b} t + \frac{1}{2} γ τ^{2})] S_{e} (t) S_{e}^{*} (t - τ),

where

S_{e} (t) = \exp [j 2 π e (t)]

and

S_{e} (t - τ) = \exp [j 2 π e (t - τ)]

. The symbol * represents the complex conjugate.

2.2 Principle and signal processing of the deskew filter algorithm

The nonlinear phase term $e (t) - e (t - τ)$ results in a spreading of the reflection peak energy that deteriorates the spatial resolution and reduces the peak’s amplitude. In addition, the nonlinear phase’s influence is greater for a longer distance as shown in Fig. 1(a) . It is difficult to remove the nonlinear phase directly because it is dependent on the distance of the FUT. Fortunately this problem can be solved by processing the LO light’s nonlinear phase $e (t)$ and received test signal’s nonlinear phase $e (t - τ)$ separately by using a deskew filter. The similar method was reported to solve a frequency chirp nonlinearity in the FMCW SAR [16].

Fig. 1 (a) The beat signals with the tuning nonlinearity effects. The upper figure shows the instantaneous optical frequency of the LO signal (blue line) and two received test signals (red lines). The lower figure depicts those corresponded two beat signals. The beat frequencies are not constant and their shapes vary with the distance of the reflections. The spreading of the beat signal in a frequency domain is greater for the reflections at a larger distance than that of at the shorter distance. (b) Diagram blocks of the nonlinearity compensation algorithm. The diagrams on the right represent the behavior of the instantaneous beat signals at the different steps for the compensation algorithm.

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The linear beating signals can be recovered via a three-step procedure as shown in Fig. 1(b).

Step one: The beat signals with laser frequency tuning nonlinearities are acquired as described in Eq. (4). Assuming a nonlinear function S_e(t) is known (its detailed estimation will be discussed in below next section), a distance independent LO light nonlinear phase in the beat frequency signal can be eliminated by following multiplication

I_{1} (t) = I (t) S_{e}^{*} (t) = 2 \sqrt{R (τ)} E_{0}^{2} \exp {j 2 π [f_{0} τ + f_{b} t + \frac{1}{2} γ τ^{2} - e (t - τ)]} .

In Eq. (5) the received signal’s nonlinearity

e (t - τ)

is dependent on distance. Due to many distributed reflections, e.g. Rayleigh backscattering and multiple Fresnel reflections, occur in the spatial domain, the received signal’s non-linearity term at the different distances cannot be removed by a single reference function. To solve this problem, a distance-dependent time shift (i.e. a round-trip time delay τ) must be implemented to the beat frequency signals and this can be achieved by using the deskew filter that can be expressed as

\exp (j π f^{2} / γ)

[17]. After the signal passes though the deskew filter, the received signal’s nonlinearity can be transformed to be distance independent .

Step two: The deskew filter is applied to $I_{1} (t)$ in a frequency domain

I_{2} (t) = F^{- 1} {F {I_{1} (t)} \exp (j π f^{2} / γ)},

where

F

and

F^{- 1}

denotes the Fourier transform and inverse Fourier transform, respectively. Inserting Eq. (5) into Eq. (6),

I_{2} (t)

can be expressed as

I_{2} (t) = 2 \sqrt{R (τ)} E_{0}^{2} \exp {j 2 π [f_{0} τ + f_{b} t]} F^{- 1} {S_{e}^{*} (- f) \exp (j π f^{2} / γ)} (t),

where

F^{- 1} {} (t)

denotes an inverse Fourier transform with time variable 𝑡 and

S_{e}^{*} (- f)

is Fourier transform of

S_{e}^{*} (t)

. Assuming that

S (t) = F^{- 1} {\exp (j π f^{2} / γ) S_{e}^{*} (- f)} (t)

,

S (t)

is distance-independent. From Eq. (7), the linear beat signals in any arbitrary delay

τ

can be recovered after removing

S (t)

.

Step three: The linear beat signals $I_{3} (t)$ can be obtained by removing $S (t)$ , i.e. multiplying with $S^{*} (t)$ . $S (t)$ can also be obtained by $S_{e}^{*} (t)$ passing though the deskew filter.

S (t) = F^{- 1} {F {S_{e}^{*} (t)} \exp (j π f^{2} / γ)}= F^{- 1} {S_{e}^{*} (- f) \exp (j π f^{2} / γ)},

and then

I_{3} (t) = I_{2} (t) \cdot S_{}^{*} (t) = 2 \sqrt{R (τ)} E_{0}^{2} \exp {j 2 π [f_{0} τ + f_{b} t]} .

From the above signal processing algorithm, one may find that theoretically any nonlinearity effect could be eliminated completely provided that the LO nonlinear phase $e (t)$ of the TLS could be accurately estimated.

2.3 Nonlinear phase estimation

In above analysis, it is assumed that the LO nonlinear phase $e (t)$ is known. In this section we will remove this assumption and describe how to estimate $e (t)$ from the acquired data of an auxiliary unbalanced Michelson interferometer with a constant reference time delay. The similar estimation method is also described for FMCW SAR in [16]. The normalized beat signal $I_{r e f} (t)$ corresponding to a reference time delay $τ_{r e f}$ is

I_{r e f} (t) = \cos {2 π [f_{0} τ_{r e f} + γ τ_{r e f} t + \frac{1}{2} γ τ_{r e f}^{2} + e (t) - e (t - τ_{r e f})]} .

Using the Hilbert transform with simple signal processing algorithms, the nonlinear term $e (t) - e (t - τ_{r e f})$ can be obtained.

For a small $τ_{r e f}$ , we can approximate $e (t)$ for the Eq. (10) by using Taylor series as

e (t) - e (t - τ_{r e f}) \approx e (t)' τ_{r e f},

where

e (t)'

denotes a derivative of

e (t)

with respect to time t. Thus an estimation of

\tilde{e} (t)

can be obtained as

\tilde{e} (t) = \int_{}^{t} \frac{e (μ) - e (μ - τ_{r e f})}{τ_{r e f}} d μ .

3. Experimental results and discussion

3.1 Setup

The experimental setup of our OFDR is shown as in Fig. 2 . The main measurement interferometer is a modified fiber-based Mach-Zehnder interferometer. The TLS has a linewidth of ~1 kHz at a center wavelength 1550 nm, an optical power about 10 mw, a laser frequency tuning speed of 5 GHz/s and a tuning range of 1 GHz. A polarization diversity detection is employed to eliminate any polarization sensitivity in our main measurement interferometer. A FUT is 80 km standard single mode fiber. The sampling rate of data acquisition card (DAQ) is 25 MS/s. An auxiliary Michelson interferometer is used to obtain the nonlinear phase of the TLS, where two Faraday rotating mirrors (FRMs) are implemented to reduce any polarization fading. A length of the reference delay fiber in the auxiliary interferometer is 10 km.

Fig. 2 Configuration of the OFDR system. The main interferometer is a modified fiber-based Mach-Zehnder interferometer. The auxiliary interferometer is an unbalanced Michelson interferometer with the 10km reference delay fiber. C1, C2, C3and C4 are 2 × 2 couplers, where C1 is a 1:99 coupler and C2, C3 and C4 are 50:50 couplers. TLS is tunable laser source. FRMs are Faraday rotating mirrors, PC is a polarization controller, PD is a photo-detector, PBS is a polarization beam splitter and DAQ is a data acquisition card.

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3.2 Nonlinear phase estimation

The LO nonlinear phase e (t)is estimated by using a method as mentioned above from measured beat signal of an auxiliary interferometer with 50 m and 10 km reference delay fiber as shown in Figs. 3(a) and 3(b). The estimated phases using 10 km reference delay fiber are shown in Fig. 3(e).

Fig. 3 (a) and (b) are measured signals from the auxiliary interferometer in the time domain with 50 m and 10 km reference delay fiber, where only a part of the entire signals are shown. (c) and (d) are local zoom-ins for figures (a) and (b). The signals from 10 km reference delay fiber interferometer is more smooth than that of from 50 m delay fiber interferometer. (e) is the estimated results of the local oscillator lights' nonlinear phase using 10 km reference delay fiber.
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An estimation of the LO nonlinear phase $e (t)$ determines the results of nonlinearity compensation. In theory our nonlinearity compensation algorithm could completely eliminate any nonlinearity from a TLS, provided that the nonlinearity can be estimated accurately. If $e (t)$ cannot be estimated accurately, the beat signals of the main interferometer may still contain some residual nonlinear laser phase that could deteriorate the spatial resolution and reduce the peak amplitude of the OFDR signals.

There are two conflicting requirements for the nonlinear phase estimation. On one hand, we need to choose a small reference time delay $τ_{r e f}$ so that it could yield a good estimation of the derivative $e (t)'$ . On the other hand, if the $τ_{r e f}$ is too small a differential phase $e (t) - e (t - τ_{r e f})$ is also too small to be easily swamped by those stochastic noises, as shown in Fig. 3(c). For example, when a length of the reference delay fiber $L_{r e f}$ is selected to be short, e.g. of 50 m, the $e (t) - e (t - τ_{r e f})$ is too small to be detected due to a low signal to noise ratio (SNR) that would induce measurement errors of $e (t)$ . In contrast, if $L_{r e f}$ is selected to be very long, we will lost these high frequency information of $e (t)$ [18] that would affect the nonlinearity compensation at a far distance. Comparing to Fig. 3(c), the curve shown in Fig. 3(d) does not have much glitch (noise) and also these small and fast changes of $e (t)$ (high frequency information) can be filtered out. Overall we have chosen an optimized value of $L_{r e f}$ to achieve a more accurate estimation of $e (t)$ . The more accurate the estimation of $e (t)$ is, the better spatial resolution and the higher the peak amplitude are.

By performing multiple experimental tests with different lengths of $L_{r e f}$ , e.g. 50 m, 200 m, 2 km, 6 km, 8 km,10 km, 12 km, and 16 km, we found that when $L_{r e f}$ is selected as 10 km, the result for the nonlinearity compensation is the best. The detailed experimental results will be discussed in the next section. Although the estimation error of $e (t)$ can be reduced by choosing an optimized $L_{r e f}$ , a method for obtaining a more accurate nonlinear phase estimation independent of the length $L_{r e f}$ is attractive in the future investigations.

3.3 Tuning nonlinearity compensation of main interferometer

In order to demonstrate an OFDR performance with our nonlinearity compensation algorithm, we applied our algorithm to process the acquired data from our OFDR which was measuring a 80 km single mode fiber. Three APC connections and one open APC connector are connected together in the FUT as shown in Fig. 2. Our experimental measurement results are shown in Fig. 4 . As shown in Fig. 4, a spatial resolution of the Fresnel reflection can be greatly enhanced after the nonlinearity compensation. Figure 4(a) shows that those reflections of the APC connections could not be detected due to a nonlinearity of the TLS frequency tuning . However, in contrast, those APC connection reflections can clearly be identified after the laser nonlinear frequency tuning compensation as displayed in Fig. 4(b). Comparing the reflection of a far-end open APC connector of the FUT as shown in Figs. 4(c) and 4(f), a spatial resolution by our method can increase about 93 times than that of without using any nonlinearity compensation. Our OFDR system’s theoretical spatial resolution $Δ z$ is about 10 cm based on a relationship $Δ z = c / 2 n Δ F$ [19], where $c$ is the light speed in vacuum, n is the refractive index of fiber and $Δ F$ is a frequency tuning range of TLS ( $Δ F$ = 1 GHz). A $Δ z$ for a reflection at 10 km is about 20 cm, which is a factor of 2 higher a theoretical value of 10 cm. The $Δ z$ sat 40 km and 80 km are 50 cm and 1.6 m, respectively, as shown in Figs. 4(d)–4(f). It should be noted that a spatial resolution of the Fresnel reflection is worse at a longer distance than that of at a shorter distance after the nonlinearity phase compensation. The reason for this degradation is that the measured beat signals from a main measurement interferometer still contain some residual “nonlinear laser phase” after the nonlinearity compensation due to the loss of high frequency information in the estimation of $e (t)$ . In order to further improve our OFDR performance, it is important to develop a more accurate estimation method of the nonlinear phase.

Fig. 4 Measured Rayleigh backscattering and Fresnel reflections for a FUT length of 80-km with APC connections and open APC connector without (a) and with (b) the nonlinearity compensations. Four Fresnel reflections are at locations of 10 km, 30 km, 40 km and 80 km. The Fresnel reflections from APC connections and connector without any nonlinearity compensation cannot be detected. After a nonlinearity compensation, a spatial resolution of these reflections are significantly improved, e.g. by more than 93 times for the far-end Fresnel reflection caused by open APC connector at 80 km as shown in (c) and (f). Their spatial resolutions are 20 cm at 10 km, 50 cm at 40 km and 1.6 m at 80 km, respectively. (c)-(f)'s y axis are transformed a logarithm to a normalized linear coordinate.
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Measured OFDR curves shown in Fig. 4 also included Rayleigh speckle noise, residual phase noise, and so on [20] that could be reduced by using a moving median filter in the signal processing. Our proposed method has a very high computational efficiency and also doesn’t require numerous measurements for the data averaging, for example, it takes less than 1 second to complete a total data processing for 5 × 10⁶ data points in a personal computer with Intel Core i7 2600 CPU and a 8 GB cache memory. This method indicates a powerful tool for a real-time optical fiber network monitoring and optical fiber sensing applications with a high spatial resolution and a long measurable range.

4. Conclusion

A simple and effective method to compensate a laser frequency tuning nonlinearity from a TLS in a long range OFDR system is presented and discussed. This method can be operated directly for the beating signals generated from a main OFDR measurement interferometer by using detected nonlinear phase information from an auxiliary interferometer and a deskew filter is used to compensate the laser frequency tuning nonlinearity effect for the entire spatial domain signals by one computation procedure with high efficiency. By using our proposed method we demonstrated a 93 times improvement for a spatial resolution and also achieved a measurable range of 80 km for our OFDR system. Spatial resolutions of 20 cm at 10 km and 1.6 m at 80 km were observed, respectively, in our laser frequency tuning nonlinearity compensated OFDR. It is also possible to further improve a spatial resolution by applying a wider frequency tuning range and a more accurate nonlinear phase estimation. A signal processing time is less than 1 s for a computed data size 5 × 10⁶. The reported technique allows an OFDR as a powerful tool for a real-time long haul optical fiber network monitoring and a long range optical fiber sensing applications that has both the high spatial resolution and long measurable range and short acquisition time. This method may also be effective for a nonlinearity compensation of a swept source optical coherence tomography (SS-OCT) and FMCW laser radar, etc.

Acknowledgments

This work is supported by National Basic Research Program of China (973 Program, grant 2010CB327806), National Natural Science Foundation of China under Grant No. 11004150&No. 61108070, Tianjin Science and Technology Support Plan Program Funding under Grant No. 11ZCKFGX01900, China Postdoctoral Science Foundation under Grant No. 201003298, International Science & Technology Cooperation Program of China under Grants No. 2009DFB10080&No. 2010DFB13180.

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Compensation of laser frequency tuning nonlinearity of a long range OFDR using deskew filter

Abstract

1. Introduction

2. Algorithm principle

2.1 Basic theory of the OFDR

2.2 Principle and signal processing of the deskew filter algorithm

2.3 Nonlinear phase estimation

3. Experimental results and discussion

3.1 Setup

3.2 Nonlinear phase estimation

3.3 Tuning nonlinearity compensation of main interferometer

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (12)

Optics Express