^{A phase-sensitive optical time-domain refle}ctometer with dual-pulse diverse frequency probe signal.

A E Alekseev^{1, 2}, V S Vdovenko², B G Gorshkov³, V T Potapov¹ and D E Simikin²

¹ Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Vvedensky Sq. 1, Fryazino, Moscow region, Russia

²Petrofibre Ltd., Klinskiy proezd, 7, Novomoskovsk, Tula region, Russia

³Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str., 38, Moscow, Russia

E-mail: aleksey.e.alekseev@gmail.com

Abstract

In the present letter we propose a novel approach for realization of a phase sensitive optical time-domain reflectometer (OTDR) which is capable of precise reconstruction of a phase signal which impacts the arbitrary point of a fiber-optic line. The method implies using a dual-pulse probe signal with diverse carrier optical frequency within each half of the double pulse. The quasi-periodic intensity pattern which emerges as a result of double frequency backscattered signal interference contains the information of the external action over the fiber. The phase signal is extracted with an aid of I/Q quadrature demodulation scheme, realized on the receiving side of the OTDR. The feasibility and limitations of the proposed scheme are theoretically proved and experimentally demonstrated.

Keywords: optical time-domain reflectometer, fiber optic sensor, power spectral density, I/Q demodulation.

1. Introduction

Currently there has been proposed several realizations of a phase-sensitive OTDR, which enable reconstruction of the phase signal which impacts the optical fiber. One of these methods implies using a six-port optical hybrid that is placed in the receiving part of the OTDR in conjunction with an unbalanced Mach–Zehnder or Michelson interferometer [1-3]. The hybrid is conventionally represented by a symmetric 3 x 3 coupler, which shifts the phases of interfering signals relative to each other and enables to perform a phase diversity technique for external signal reconstruction. The limitations of this scheme are (related with the)\(coming from the) necessity of using of three independent photodetectors and consequently three independent detection schemes which have to be fully synchronized. The unbalanced interferometer in the receiving part of OTDR is sensitive to environmental perturbations and requires serious thermally and vibration isolation, moreover, to avoid polarization fading of detected signal the additional means for polarization matching have to be taken.

In the previous paper we proposed another realization of OTDR with the possibility of external impact signal reconstruction; the scheme implied using double pulse probe signal which had different phase shifts for the first and the second pulses of the pair [4]. The required phase difference between the pulses within the probe pulse pair was reached via a phase modulator after the double pulse source, the detected backscattered signals at the output of the OTDR consequently became phase shifted relative to each other, which enabled to perform the phase diversity technique for external impact signal reconstruction. In comparison with previously mentioned one, the advantage of this scheme is absence of applying of optical hybrid, unbalanced interferometer and three independent photodetectors in the receiving end, which significantly simplifies the scheme and makes it more sustainable to environmental fluctuations. The scheme limitations include requirement to use three independent pulse pairs with different relative phase shifts for correct registration of the single external impact signal time sample. Consequently the maximum detectable frequency or effective external signal bandwidth, capable of registration, is three times less than in hybrid schemes, described above.

Another approach to realization of a phase-sensitive OTDR, first mentioned in [5], is applying a double pulse probe signal, with the carrier frequencies of pulses, constituting the pair, shifted relative to each other over a certain value, unlike the previous scheme [4] where only the phases of these pulses are shifted. This solution with diverse frequency dual-pulse probe signal enables to reduce the number of independent pulse pairs required for correct registration of the single time sample of an external impact to one, thus three times enlarging the maximum frequency or effective external signal bandwidth, capable of registration, comparing with phase shift method [4]. As a result, this scheme is free from drawbacks of both the scheme with optical hybrid and the scheme with phase manipulation and looks promising for realization of high performance phase-sensitive OTDR.

Despite the method with two carrier frequencies was firstly mentioned in [5], it was not elaborated and somehow demonstrated experimentally. This article (paper), apparently, is the first detailed theoretical and experimental description of the phase-OTDR with dual-pulse diverse frequency probe signal.

2. Средняя пространственная частотная характеристика рефлектограммы с одинарным зондирующим импульсом The Average spatial backscattered intensity spectrum of the otdr with single-pulse probe signal

The operation of proposed phase-sensitive OTDR is based on the usage of double-pulse probe signal with different carrier frequencies of the first and the second parts of the pair. The carrier frequency difference must be aligned with the spatial spectrum of OTDR signal in a certain way. Thus firstly consider the spatial spectral characteristics of backscattered intensity OTDR traces, formed at the output, when a single probe pulse of a certain form, duration and fixed carrier frequency is used. In other words, define the spatial harmonics, constituting backscattered OTDR spatial traces. Then we will define the spectral characteristics of OTDR traces obtained when utilizing a double pulse with different carrier frequencies.

Extended optical fiber can be modeled as a set of scattering centers, randomly distributed along the length of the fiber. Consider an optical fiber of a length with embedded random scattering centers axial distribution , this distribution can be characterized by the set of complex amplitude backscattering coefficients , where is complex amplitude backscattering coefficient of the scattering center with longitudinal coordinate , note that this distribution can change to another statistically equivalent distribution under environmental influence. In the introduced model of scattering medium we consider as zero-mean circular complex Gaussian random variable (CCG) [6-10], which means that real and the imaginary parts of :, are both have Gaussian ensemble distributions with equal variances. The phase of such random variable obeys uniform statistics on the interval . Physically these assumptions represent the backscattering process as multiplication of complex amplitude of the source field to random values, having Gaussian distributions of a real and imaginary parts with zero-mean. Let us also consider complex amplitude backscattering coefficients as statistically uncorrelated [8, 9]. Mathematically the lack of correlation between complex backscattering coefficients with coordinates and and equality of real and imaginary parts variances can be expressed in two expressions:

11\* MERGEFORMAT (),

22\* MERGEFORMAT (),

where - denotes the average over the ensemble of independent distributions of scattering centers (or over the ensemble of independent scattering mediums ) i.e. ensemble average, , - are axial coordinates of scattering centers, - is the variance of real or imaginary parts of backscattering coefficients , - is the delta function.

Let us assume, for simplicity, that the source of optical field is fully monochromatic and that the state of optical field polarization is preserved during field propagation along the fiber as well as in the act of scattering. The analytic representation of source field can thus be written as:

33\* MERGEFORMAT (),

where - is the source field intensity, - is the angular frequency of the source field radiation. The complex amplitude of the field incident on the scattering center located at the distance can be written as:

44\* MERGEFORMAT (),

where - is the intensity attenuation coefficient, - is field propagation constant, - is field group velocity. The complex amplitude of backscattered field in the point of scattering is given by [8]:

55\* MERGEFORMAT ().

Let us consider that optical pulse is characterized by amplitude function of time or equivalent spatial coordinate with full width at half maximum (FWHM) equal to :, in the particular case this could be a rectangular pulse:

66\* MERGEFORMAT (),

or a Gaussian pulse

77\* MERGEFORMAT ().

The complex amplitude of the field backscattered by the region of the optical fiber, occupied by the optical pulse at the distance from the beginning can be expressed us:

88\* MERGEFORMAT (),

Thus, the complex amplitude of the backscattered field at the output of the OTDR with the amplitude probe pulse form and FWHM for every fixed realization of scattering centers distributionis represented by the convolution of the complex amplitude function of the optical field , backscattered by the centers with longitudinal coordinates and the function, which defines the shape of the probe pulse. In other words the random function of spatial coordinateis filtered by the linear filter with impulse response function. When filtering on spatial coordinate, the spatial spectrum of the signalis changed. Every new realization of the scattering centers distribution in the fiber leads to appearance of a new realization of the spatial intensity distribution and consequently a new spatial spectrum of OTDR trace. For practical purposes spectral characteristics over the ensemble of random OTDR intensity traces attract major interest.

To determine the average spatial spectral characteristics of the OTDR traces ensemble, consider the complex Gaussian random process with the sample function represented by single random distribution of complex scattering coefficients along the fiber. Each new distribution of scattering centers leads to a new sample function of considered random process . The autocorrelation function (ACF) of the regarded process has the form 2.

Multiplication of the random Gaussian process with the sample functionon the deterministic function 4 leads to another Gaussian random process with the sample function 5.

The power spectral density (PSD):of the linearly filtered random process with the sample function 8 is determined according to [10]:

99\* MERGEFORMAT (),

where denotes the Fourier transform of the impulse response i.e. the transfer function, is the PSD of the random process .

To calculate the PSD of the random processconsider its ACF, neglecting the intensity attenuation we obtain:

1010\* MERGEFORMAT ().

From 10 it is seen that the ACF depends only on the coordinate difference, thus the random process is wide-sense stationary and Wiener–Khinchin theorem can be applied [10], according to which the Fourier transform of the ACF 10 results in the PSD: of the random process :

1111\* MERGEFORMAT (),

The PSD of the process is then has the form:

1212\* MERGEFORMAT (),

in case of the rectangular probe pulse shape the PSD is:

1313\* MERGEFORMAT (),

in case of the Gaussian probe pulse shape the PSD is:

1414\* MERGEFORMAT ().

Found expressions for the PSD 12, 13 and 14 shows what spatial harmonics on average over the ensemble constitute the function which represents the spatial distribution of complex backscattered fields amplitudes over the length of the fiber in the assumption of very long fiber so the spectral leakage caused by finite fiber length can be ignored.

Let us now define the average spectral characteristic of the OTDR intensity traces, formed when the probe pulse with the time shape is used. The intensity of the field 8, backscattered from the fiber region with spatial coordinate is as follows:

1515\* MERGEFORMAT (),

- denotes the time average, in 15 we used high coherence assumption for the source field and so the intensity does not fluctuate in time and time averaging can be omitted. Random spatial function can also be regarded as the sample function of the random process, originated from the sample function, which changes when a new distribution of scattering centers occurs. To calculate the PSD of this random process define its ACF:

1616\* MERGEFORMAT ().

Note that linearly filtered process with the sample function is also Gaussian random process [10] and in compliance with 1 and 2 it is circular as well. Let us then use the Gaussian moment theorem [11] for the complex scattering coefficients , which claims:

1717\* MERGEFORMAT (),

in case of four complex Gaussian values it takes the form:

1818\* MERGEFORMAT ().

The fourth moment 16 can then be written in the form:

1919\* MERGEFORMAT (),

where the square brackets contain the ACF of the linearly filtered random process , which depends only on the coordinate difference , thus the random process with the ACF 19 is also wide-sense stationary and Wiener–Khinchin theorem can be applied again. The PSD of the random process with the sample function will be as follows:

2020\* MERGEFORMAT ().

Found expression shows what spatial harmonics on average over the ensemble constitute the spatial distribution of backscattered intensity over the length of the fiber in the assumption of very long fiber to neglect the spectral leakage effect. In other words 20 shows the average spatial power spectrum of the OTDR intensity trace.

In the case of the rectangular probe pulse, the convolution in 20 could, for instance, be calculated with using of convolution theorem by the following steps: the Fourier transform, multiplication of the results and inverse Fourier transform:

2121\* MERGEFORMAT (),

Taking 13 into account for the desired PSD we obtain:

2222\* MERGEFORMAT ().

To estimate the FWHM of this average spatial power spectrum, one have to solve the transcendental equation, the final result is, as it follows:

2323\* MERGEFORMAT (),

where is the FWHM of the average spatial power spectrum of the OTDR intensity trace. Note that the OTDR trace could be interpreted either as intensity-to-distance dependence or intensity-to-time dependence, the spectral width in time domain is obtained after multiplication of 23 by the group velocity of light: 2E8 m/s, as a result we have:

2424\* MERGEFORMAT (),

is the duration of the rectangular probe pulse amplitude characteristic . Thus the rectangular probe pulse with amplitude characteristic duration equal to 100 ns produces the OTDR trace with the FWHM of the average power spectrum equal to 12 MHz. Note that for the rectangular probe pulse the duration of amplitude and intensity characteristic coincide, the spectral width of the initial intensity probe pulse power spectrum is equal to:

2525\* MERGEFORMAT (),

comparing 24 and 25 one can see that FWHM of the average OTDR trace power spectrum exceeds the width of the power spectrumof the initial rectangular probe pulse as much as 1.36 times, the average power spectrum of the OTDR intensity trace broadens.

In the case of the Gaussian probe pulse less tedious calculations lead to the following PSD:

2626\* MERGEFORMAT (),

the FWHM of this average spatial power spectrum can be calculated as:

2727\* MERGEFORMAT (),

the FWHM in time domain is obtained in the same way, multiplying 27 by :

2828\* MERGEFORMAT (),

is the duration of the Gaussian probe pulse amplitude characteristic . Thus the Gaussian probe pulse with amplitude characteristic duration equal to 100 ns produces the OTDR trace with the FWHM of average power spectrum equal to 8.8 MHz. For the Gaussian probe pulse the duration of amplitude exceeds the duration of intensity characteristic as much as times, the spectral width of the initial intensity probe pulse power spectrum is equal to:

2929\* MERGEFORMAT (),

comparing 28 and 29 one can see that FWHM of average OTDR trace power spectrum now equals to the width of power spectrumof the initial Gaussian probe pulse, the average power spectrum of the OTDR intensity trace preserves it width. In intermediate cases, when the probe pulse has, for instance, Super-Gaussian shape the FWHM of average power spectrum of the OTDR intensity trace will be within the limits defined by 28 and 24.

3. The OTDR intensity trace for dual-pulse diverse frequency probe signal

Suppose, the dual-pulse probe signal with different carrier frequencies of its first and the second parts: and is launched into the OTDR fiber line, the corresponding field propagation constants are and , figure 1. Denote the first and the second parts of the probe pulse by the letters and. We assume the time duration of pulses in the double pulse and the time interval between them equal to , that corresponds to the spatial duration equal to . Consider an arbitrary time instance when the pulse pair has moved to position from the beginning of the fiber line. The electric fields, backscattered by the first and the second parts of the probe pulse, with the assumption of monochromaticity and polarization preservation of the pulse fields can be written in the form:

3030\* MERGEFORMAT (),

3131\* MERGEFORMAT (),

where and are defined by 8. Note that the optical fields backscattered by the second part of the pulse pair and the first part will have equal spatial positions near the point with coordinate, the first part of the probe pulse should then pass the distance forward while the backscattered fields of the second part of the probe pulse should pass the same distance in the reversed direction. The corresponding electric backscattered fields near the point of their spatial superposition are, as it follows:

3232\* MERGEFORMAT (),

3333\* MERGEFORMAT (),

where the complex amplitudes of the backscattered fields and are equivalently denoted by and , where, ,and , are the random amplitudes and the phases of the backscattered fields, corresponding to the parts an of the dual-pulse. When two spatially superimposed parts of backscattered pulse reach the beginning of the fiber line, they both pass double distance , however this additional phase could be measured only with the accuracy up to and therefore does not change the uniform statistics of the fields random phase distributionsand on the interval . For this reason this additive phase could be omitted.

The sum intensity of backscattered fields at the beginning of the OTDR fiber line is found to be:

3434\* MERGEFORMAT ().

Thus at the output of the OTDR the quasi-harmonic time-dependent bandpass signal is detected with the carrier frequency equal to , the power spectrum of this signal is broadened due to the random changes of the amplitudes and the phases of the backscattered fields:, ,and along the fiber. The cosine in 34 along with the regular time-dependent phase component:contains the random component:which depends on the certain realization of the fiber scattering centers distribution . The expression 34 shows that OTDR trace, which emerges when a dual-pulse with different carrier frequencies is used as a probe signal, has quasi-harmonic structure with a central frequency equal to:. Since the time dependence of the backscattered intensity 34 is interpreted by the OTDR as the distance dependence, the change of variable could be made: , that leads to the following:

3535\* MERGEFORMAT (),

where there is only time-dependence on the spatial coordinate along the fiber exists. Figure 2 shows the experimental OTDR traces, obtained with using dual-pulse probe signal, where each part of the double pulse has the Gaussian shape with the time duration equal to100 ns and the time interval between these parts also equal to100 ns, with equal carrier frequencies of the two pulse parts figure. 2a) and with different carrier frequencies of the two pulse parts, where 100 MHz figure 2 b). As one can see, frequency spacing of the dual-pulse parts leads to the oscillations of the OTDR intensity trace with a characteristic frequency, determined by this carrier frequency spacing . At the same time the power spectrum of the resulting OTDR trace broadens due to random amplitude and phase modulation of the carrier , which is also seen on figure 2 b).