Abstract

The spatial resolution of OFDR is normally degraded by the laser phase noise, deviations from linear frequency scan and acoustic noise in the fibers. A method for mitigating these degradation mechanisms, without using an auxiliary interferometer, via inline auxiliary points, is presented and demonstrated experimentally. Auxiliary points are points that are a priori known to have (spatial) impulse reflectivities. Their responses are used for compensating the phase deviations that degrade the response of points that are further away from the source.

© 2012 OSA

1. Introduction

Optical reflectometry techniques are widely used and are attracting a lot of attention in recent years. Some of their main applications are: characterization and monitoring of optical communication networks, fiber sensor arrays and distributed optical fiber sensors. Optical time domain reflectometry (OTDR) is one of the most well-known techniques in this family. It is very useful for long range measurements but it suffers from an inherent tradeoff between spatial resolution and dynamic range (sensitivity) which results in a typical spatial resolution of more than one meter [1, 2]. In contrast, coherent optical frequency domain reflectometry (C-OFDR) techniques offer excellent spatial resolution along with high sensitivity, enabling detection of closely separated reflection sites and Rayleigh backscattering in the fiber under test [36]. However, the requirement for high phase correlation of these techniques limits the measurement range to the coherence length of the light source. As the linewidth of the light source is narrower, the coherence length is greater and the phase noise of the source is reduced, resulting in a high signal to noise ratio (SNR) [7]. Recently, a technique has been proposed and developed to overcome the C-OFDR measurement range limitation, i.e., measuring beyond the source coherence length, by compensating for its phase noise. This technique is called phase-noise-compensated optical frequency domain reflectometry (PNC-OFDR). It employs an auxiliary interferometer, whose length,L, is comparable with the coherence length of the source. The output of the auxiliary interferometer is sampled and used according to the concatenative reference method (CRM) to generate reference sequences which emulate the use of multiple auxiliary interferometers with lengths that are multiples of L [8, 9]. PNC-OFDR achieves near perfect compensation of the source phase noise at discrete locations along the measured fiber and partial compensation between these locations. In addition, any other deviation from the desired linear scan of the instantaneous frequency of the source, which is not related to phase noise, is also compensated for by this method. With the source induced phase deviations compensated for, the main remaining mechanism for degradation in spatial resolution is the effect of acoustic phase noise in the fiber under test [10].

In this paper we propose a novel approach for correcting source-related phase deviations as well as acoustic phase noise via inline auxiliary points. The inline auxiliary points are points which are known in advance to possess spatial impulse reflectivities. Such points can be formed by reflection from fiber ends, connectors or any other feature in the fiber with longitudinal dimensions shorter than the spatial resolution of the measurement. To demonstrate the method we implement it in a fiber with multiple point reflectors (10 short and weak fiber Bragg gratings of nominally the same center frequency). It is shown how the response of one point reflector can be used to compensate for the acoustic noise which degrades the responses of the following reflectors. Moreover, it is shown how a reflection from a connector at the fiber input can be used instead of an auxiliary interferometer for compensating for the source phase deviations. After compensation using these auxiliary points, the tested system nearly achieves the theoretical resolution limit, as dictated by the frequency scan range, without using an off-line auxiliary interferometer.

2. Theory

Consider a fiber-optic system as described in Fig. 1 . Light from a laser source is launched into a measurement arm and the back-reflected beam is mixed with a reference and detected by a coherent I/Q receiver. In OFDR the frequency of the laser is varied at a nominally constant rate, γ, over a frequency range Δf. The laser output field can be expressed as:

E(t)=E0exp{j[ω0t+πγt2+θ(t)]}
where ω0 is the nominal radial frequency at t=0 and θ(t) accounts for phase noise and any other deviation from the linear frequency scan. The measurement fiber can be considered as a distributed reflector with z-dependent reflectivity r(z). The output of the coherent receiver, V(t)I(t)+jQ(t), can be expressed as:
V(t)=a0Lr(z')exp{j[2πγ2z'vt+φ(t,z')]}dz'
where, a is a constant describing the responsivity and gain of the receiver and φ(t,z) denotes the phase of the light reflected from position z due to phase noise, non-linear frequency scan and acoustic noise in the fiber, referenced to the phase in the reference arm. An approximated expression for r(z) can be obtained by taking the Fourier transform of V(t)/a and substituting f=2zγ/v where v is the group velocity of light in the fiber:
r˜(z)=1a0TV(t')exp[j(2πγ2zvt')]dt'=0Lr(z')g˜(z,z')dz'.
where
g˜(z,z')=0Texp{j[2πγ2(z'z)vt'+φ(t',z')]}dt'.
is the response of the measurement system for a spatial impulse at z'. It describes the degradation in spatial resolution due to the phase noise, deviation from linear frequency scan and acoustic noise. It can be easily shown that for the ideal case where φ(t,z)=0 Eq. (3) reduces to r˜(z)=r(z). As will be described below it is possible to extract from the data an approximation of φ(t,z) using discrete reflection points in the fiber that we term auxiliary points. In this case a better estimation of r(z) can be obtained by calculating:
r˜˜(z)=1a0TV(t')exp{j[2πγ2zvt'+φ(t',z)]}dt'=0Lr(z')g˜˜(z,z')dz'.
where
g˜˜(z,z')=0Texp{j[2πγ2(z'z)vt'+φ(t',z')φ(t',z)]}dt'.
Note that g˜˜(z,z') attains a maximum value at z=z' for all φ(t,z) where g˜(z,z') does not. The significant improvement in the spatial impulse response, which result from calculating r˜˜(z) rather thanr˜(z), is demonstrated experimentally in sec. 4.

 

Fig. 1 The experimental setup.

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2.1 Compensating for source phase deviations

The use of auxiliary points for compensating the phase-related degradation mechanisms can be implemented in stages. For example, in the first stage the source phase deviations are compensated and then the acoustic phase noise is mitigated. In fact, as we demonstrate by experiment, in some situations it is necessary to compensate first for the source phase deviations for resolving the auxiliary point of the acoustic noise compensation stage. To implement compensation of the source related degradation mechanisms it is needed to obtain θ(t) or approximated version of it. To that end an auxiliary reflection point can be intentionally formed at the input of the measurement arm. The auxiliary reflection point is positioned at sufficient distance from the input of the measurement arm so its response, V˜(t), can be spectrally filtered from the total response of the measured fiber, V(t). Denoting by zaux the position of the auxiliary point, its response can be expressed as:

V˜(t)=ar(zaux)exp{j[2πγ2zauxvt+θ(t)θ(t2zauxv)]}.
where it was assumed that the acoustic noise which affect the reflection from this point can be minimized and was hence neglected. The harmonic signal V˜(t) is centered at frequency faux=2γzaux/v and is phase-modulated by the signal θ(t)θ(t2zaux/v). In order to extract θ(t)θ(t2zaux/v) from the raw data, V(t) (see Eq. (2)) is Fourier transformed and the broadened peak which corresponds to V˜(t) is identified. Next, a band-pass filter centered at faux is used to isolate the response of the auxiliary point and inverse Fourier transform yields V˜(t). Then θ(t)θ(t2zaux/v) is readily obtained from calculating the phase of V˜(t)exp(j2πfauxt). As τaux2zaux/v is chosen to be smaller than the coherence time of the laser, τC, and smaller than the characteristic variation time of the deviations from linear sweep, θ(t)θ(t2zaux/v) can be used to estimate θ˙(t)dθ(t)/dt and θ(t) as follows:
θ˙(t)θ(t)θ(tτaux)τaux
θ(t)=0tθ˙(t)dt0tθ(t)θ(tτaux)τauxdt
Once the approximated version of θ(t) is found it is possible to estimate θ(t)θ(t2z/v) for every z and to use it for calculating r˜˜(z) according to Eq. (5). This constitutes a high resolution approximation of the fiber reflectivity.

2.2 Compensating for acoustic phase noise

As lightwave propagates in the fiber its phase is affected by acoustic noise. The phase modulations that affect two beams which were reflected from two neighboring points are highly correlated. The compensation of acoustic phase noise using an auxiliary reflection point is based on the observation that the acoustic phase noise term, at a sufficiently close point, following the auxiliary point can be expressed as:

ψ(t,zaux+Δz)ψ(t2Δz/v,zaux)+Δψ(t,zaux,zaux+Δz).
where ψ(t,z) is the acoustic phase noise which corresponds to reflection from position z and Δψ(t,z1,z2) denotes the part of the acoustic phase noise, of light reflected from z2, which is imparted on it in the fiber segment between z1 and z2. It is assumed that the acoustic noise correlation time is much longer than 2Δz/v. Thus, given an auxiliary reflection point which is sufficiently far from other strong reflections its response can be spectrally filtered and ψ(t,zaux) can be obtained from phase demodulation. Once ψ(t,zaux) is known, its properly delayed versions can be used in Eq. (10) for z>zaux, to compensate for the acoustic phase noise which was accumulated up to the auxiliary reflection point.

3. Experiment

An experimental OFDR system with an array of discrete reflectors, as depicted in Fig. 1, was constructed. A tunable laser source (orbits lightwave) with center wavelength of 1550 nm was swept in optical frequency over a frequency range of Δf0.5GHz. The light emitted from the laser was fed to a 50/50 PM splitter. One output of the splitter (the reference arm) was connected to the Local Oscillator (LO) port of a dual-polarization 90° optical-hybrid (Kylia). The second output of the splitter was connected to the measurement arm using an optical circulator. The output of the circulator (port 3) was fed into the Signal port of the dual-polarization 90° optical-hybrid. The optical hybrid's outputs were detected in pairs by four balanced photoreceivers as described in Fig. 1. The photoreceivers yielded electric signals proportional to the four quadrature components of the signal field (two for each polarization). These were sampled and stored by a data acquisition (DAQ) system. Port 2 of the circulator was used to connect the measurement arm. The measurement arm comprised an array of 10 FBGs at equal center wavelengths and 10 m spacing. It was terminated with an open to air APC connector. The total length of the FBG fiber (including one long patch-cord) was ~850 m. The first auxiliary reflection point is at the fiber adaptor between the FC-APC connector to the FC-PC connector. The choice of the position of the first auxiliary point was made according to two considerations. First, to use the approximation in Eq. (8) it is required that τaux will be smaller than the characteristic variation time of the deviations from linear sweep. Second, it is desired to keep the auxiliary beat frequency sufficiently far from the elevated response at zero beat frequency (a result mainly of non-ideal balanced detection) and from the nearest reflector. This is done to ensure that the broadened spectral response of the auxiliary reflector is sufficiently isolated from other responses. The required excess fiber length preceding and following the auxiliary reflector can change if the source induced resolution degradation is changed. In case of overlap between the spectral response of the auxiliary point and other near spectral responses, the compensation of the non-linear frequency scan is compromised. In this work the first auxiliary point was positioned 30 m from the point of zero differential delay. The reflected light from the FBG fiber was directed by the circulator to the signal port of the optical hybrid. The acquired data was then processed in order to compensate for the source phase deviations initially, and then for the acoustic phase noise as well.

4. Results

Application of Fourier transform (Eq. (3)) and averaging |r˜(z)|2 over 70 independent measurements yielded the graph shown in Fig. 2 . The spatial resolution is severely degraded mainly due to source related phase deviations.

 

Fig. 2 Reflectivity without compensation.

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A (digital) low pass filter with a bandwidth of 9.82 KHz was used to filter out the response of the first auxiliary point. After phase demodulation and numerical integration according to Eq. (9) the estimated source phase deviation, θ(t), was found. Next, θ(t) was used for calculating θ(t)θ(t2z/v) as a function of z and for finding r˜˜(z) according to Eq. (5). The improved reflectivity graph is shown in Fig. 3 . The Bragg reflectors are clearly resolved as well as the reflection from the angled connector at the fiber end.

 

Fig. 3 Reflectivity after source phase deviation compensation (blue) and before (grey).

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While using the first auxiliary point managed to compensate well for the source phase deviations, the responses of the Bragg reflectors were also broadened by the presence of acoustic phase noise in the fiber. To mitigate that another auxiliary point was used for extracting the acoustic phase that was accumulated between the first auxiliary point and the Bragg reflectors. Since the first reflector was found to be relatively weak the second auxiliary point was chosen to be the second Bragg reflector. Again, a (digital) band-pass filter with a bandwidth of 2 KHz was used for extracting the response of the auxiliary point and its phase, ψ(t,zaux), was found by demodulation. With ψ(t,zaux) known, the integral in Eq. (5) was calculated once again but this time V(t) was replaced with the inverse Fourier transform of the one-time compensated reflectivity, r˜˜(z), and ψ(t,zaux) substituted for φ(t,z). The twice compensated reflectivity is plotted in Figs. 4 and 5 . All reflection peaks following the auxiliary point exhibit additional narrowing which is due to the compensation of a significant part of the acoustic phase noise that affected them. Also plotted in Fig. 5 is a theoretical reflection peak from a spatial impulse at the same position as the last FBG. The theoretical response is broadened due to numerical reasons as its frequency is not an integer multiple of the fundamental frequency step, df=1/T (T being the time duration of the response).

 

Fig. 4 The reflectivity of the Bragg reflectors after two compensations.

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Fig. 5 The reflectivity of a Bragg reflector after source phase deviation compensation (black), two compensations (blue) and the theoretical limit (red).

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5. Discussion

In reflectometry of optical networks the theoretical response of many of the features is a spatial impulse. Moreover, in many situations there is preliminary knowledge about the expected positions of such reflectors. In these cases it is possible to take advantage of the expected responses and use it for improving the spatial resolution and quality of other positions in the network. The spatial resolution of OFDR is limited by the frequency scan range Δf according to: Δzv/(2Δf) [9]. In our experimental setup the optical source and its frequency scan module was limited to a frequency scan range of Δf0.5GHz which determined a moderate spatial resolution of Δz200mm. It is expected, however, that the proposed method can maintain similar compensation abilities when implemented with optical sources with wider frequency scan ranges. In this case it will be possible to increase Δf by increasing the scan time or the scan rate, γ. Both actions do not have a fundamental effect on the applicability of the proposed method. An increase in the scan time may require additional memory for storing the added sample points while an increase in γ may require increasing the sampling rate. With these modifications the proposed compensation method should exhibit similar performance independent of the value of Δf. Following the compensation of the source phase deviations, the Full Width Half Maximum (FWHM) widths of all the reflection peaks (Figs. 3-5) were comparable with the z sampling step which was 193.5 mm. The acoustic noise was manifested mainly as elevated response values in the regions outside the −3 dB range. The application of acoustic phase noise compensation reduced these elevated responses and produced significantly narrower peaks. The twice compensated peaks were closer to the theoretical limit as can be seen in Fig. 5. In this example the compensated peak exhibited similar width as the theoretical limit down to −22 dB from the peak maximum. The deviation from the theoretical reflectivity is attributed to acoustical noise in the section between the second auxiliary point and the last FBG and to numerical errors inherent to the compensation method.

6. Conclusions

A method for mitigating resolution degradation mechanisms in OFDR was introduced and demonstrated experimentally. The method is based on in-line auxiliary points with pre-known reflectivities and does not require a separate auxiliary interferometer. By filtering out the responses of these auxiliary points it is possible to estimate the undesired phase terms which result from mechanisms such as the non-linear scan of the laser, the phase noise of the laser and acoustic phase noise in the fibers. Once these phase terms are extracted they are used for obtaining a compensated response with enhanced resolution. In addition to resolution enhancement in static OFDR measurements, the proposed method can be used for extracting the acoustical signals in dynamic OFDR acoustic sensors.

References and links

1. M. K. Barnoski and S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt. 15(9), 2112–2115 (1976). [CrossRef]   [PubMed]  

2. M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7(8), 1217–1224 (1989). [CrossRef]  

3. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981). [CrossRef]  

4. H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol. 7(1), 3–10 (1989). [CrossRef]  

5. B. J. Soller, D. K. Gifford, M. S. Wolfe, and M. E. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express 13(2), 666–674 (2005). [CrossRef]   [PubMed]  

6. G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett. 32(10), 926–927 (1996). [CrossRef]  

7. S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol. 11(10), 1694–1700 (1993). [CrossRef]  

8. X. Fan, Y. Koshikiya, and F. Ito, “Phase-noise-compensated optical frequency domain reflectometry with measurement range beyond laser coherence length realized using concatenative reference method,” Opt. Lett. 32(22), 3227–3229 (2007). [CrossRef]   [PubMed]  

9. F. Ito, X. Fan, and Y. Koshikiya, “Long-range coherent OFDR with light source phase noise compensation,” J. Lightwave Technol. 30(8), 1015–1024 (2012). [CrossRef]  

10. Y. Koshikiya, X. Fan, and F. Ito, “Influence of acoustic perturbation of fibers in phase-noise compensated optical frequency domain reflectometry,” J. Lightwave Technol. 28(22), 3323–3328 (2010).

References

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  1. M. K. Barnoski and S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt.15(9), 2112–2115 (1976).
    [CrossRef] [PubMed]
  2. M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol.7(8), 1217–1224 (1989).
    [CrossRef]
  3. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett.39(9), 693–695 (1981).
    [CrossRef]
  4. H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol.7(1), 3–10 (1989).
    [CrossRef]
  5. B. J. Soller, D. K. Gifford, M. S. Wolfe, and M. E. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express13(2), 666–674 (2005).
    [CrossRef] [PubMed]
  6. G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
    [CrossRef]
  7. S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol.11(10), 1694–1700 (1993).
    [CrossRef]
  8. X. Fan, Y. Koshikiya, and F. Ito, “Phase-noise-compensated optical frequency domain reflectometry with measurement range beyond laser coherence length realized using concatenative reference method,” Opt. Lett.32(22), 3227–3229 (2007).
    [CrossRef] [PubMed]
  9. F. Ito, X. Fan, and Y. Koshikiya, “Long-range coherent OFDR with light source phase noise compensation,” J. Lightwave Technol.30(8), 1015–1024 (2012).
    [CrossRef]
  10. Y. Koshikiya, X. Fan, and F. Ito, “Influence of acoustic perturbation of fibers in phase-noise compensated optical frequency domain reflectometry,” J. Lightwave Technol.28(22), 3323–3328 (2010).

2012 (1)

2010 (1)

2007 (1)

2005 (1)

1996 (1)

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

1993 (1)

S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol.11(10), 1694–1700 (1993).
[CrossRef]

1989 (2)

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol.7(8), 1217–1224 (1989).
[CrossRef]

H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol.7(1), 3–10 (1989).
[CrossRef]

1981 (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett.39(9), 693–695 (1981).
[CrossRef]

1976 (1)

Barfuss, H.

H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol.7(1), 3–10 (1989).
[CrossRef]

Barnoski, M. K.

Brinkmeyer, E.

H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol.7(1), 3–10 (1989).
[CrossRef]

Eickhoff, W.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett.39(9), 693–695 (1981).
[CrossRef]

Fan, X.

Froggatt, M. E.

Gifford, D. K.

Gisin, N.

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

Horiguchi, T.

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol.7(8), 1217–1224 (1989).
[CrossRef]

Ito, F.

Jensen, S. M.

Koshikiya, Y.

Mussi, G.

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

Passy, R.

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

Soller, B. J.

Sorin, W. V.

S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol.11(10), 1694–1700 (1993).
[CrossRef]

Tateda, M.

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol.7(8), 1217–1224 (1989).
[CrossRef]

Ulrich, R.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett.39(9), 693–695 (1981).
[CrossRef]

Venkatesh, S.

S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol.11(10), 1694–1700 (1993).
[CrossRef]

von derWeid, J. P.

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

Wolfe, M. S.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett.39(9), 693–695 (1981).
[CrossRef]

Electron. Lett. (1)

G. Mussi, N. Gisin, R. Passy, and J. P. von derWeid, “−152.5 dB sensitivity high dynamic-range optical frequency-domain reflectometry,” Electron. Lett.32(10), 926–927 (1996).
[CrossRef]

J. Lightwave Technol. (5)

S. Venkatesh and W. V. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol.11(10), 1694–1700 (1993).
[CrossRef]

F. Ito, X. Fan, and Y. Koshikiya, “Long-range coherent OFDR with light source phase noise compensation,” J. Lightwave Technol.30(8), 1015–1024 (2012).
[CrossRef]

Y. Koshikiya, X. Fan, and F. Ito, “Influence of acoustic perturbation of fibers in phase-noise compensated optical frequency domain reflectometry,” J. Lightwave Technol.28(22), 3323–3328 (2010).

H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optic systems,” J. Lightwave Technol.7(1), 3–10 (1989).
[CrossRef]

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol.7(8), 1217–1224 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1

The experimental setup.

Fig. 2
Fig. 2

Reflectivity without compensation.

Fig. 3
Fig. 3

Reflectivity after source phase deviation compensation (blue) and before (grey).

Fig. 4
Fig. 4

The reflectivity of the Bragg reflectors after two compensations.

Fig. 5
Fig. 5

The reflectivity of a Bragg reflector after source phase deviation compensation (black), two compensations (blue) and the theoretical limit (red).

Equations (10)

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E( t )= E 0 exp{ j[ ω 0 t+πγ t 2 +θ( t ) ] }
V( t )=a 0 L r( z' )exp{ j[ 2πγ 2z' v t+φ( t,z' ) ] } dz'
r ˜ ( z )= 1 a 0 T V( t' )exp[ j( 2πγ 2z v t' ) ]dt' = 0 L r( z' ) g ˜ ( z,z' ) dz'.
g ˜ ( z,z' )= 0 T exp{ j[ 2πγ 2( z'z ) v t'+φ( t',z' ) ] }dt' .
r ˜ ˜ ( z )= 1 a 0 T V( t' )exp{ j[ 2πγ 2z v t'+φ( t',z ) ] }dt' = 0 L r( z' ) g ˜ ˜ ( z,z' ) dz'.
g ˜ ˜ ( z,z' )= 0 T exp{ j[ 2πγ 2( z'z ) v t'+φ( t',z' )φ( t',z ) ] }dt' .
V ˜ ( t )=ar( z aux )exp{ j[ 2πγ 2 z aux v t+θ( t )θ( t 2 z aux v ) ] }.
θ ˙ ( t ) θ( t )θ( t τ aux ) τ aux
θ( t )= 0 t θ ˙ ( t )d t 0 t θ( t )θ( t τ aux ) τ aux d t
ψ( t, z aux +Δz )ψ( t 2Δz /v , z aux )+Δψ( t, z aux , z aux +Δz ).

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