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Polarization maintaining optical fiber multi-intruder sensor
Optics & Laser Technology 44 (2012) 2026–2031
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Polarization maintaining optical fiber multi-intruder sensor A.R. Bahrampour a, M. Bathaee a,n, S. Tofighi a, A. Bahrampour b, F. Farman a, M. Vali a a b
Physics Department of Sharif University of Technology, Azadi street, Tehran, Iran Dipartemento di scienze Fisiche, Universita diNapoli Federico II, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 February 2012 Received in revised form 21 March 2012 Accepted 22 March 2012 Available online 11 April 2012
In this paper, an optical fiber multi-intruder sensor based on polarization maintaining optical fiber (PMF), without any interferometric fiber loop, is introduced. To map the local coordinates of intruders on the beating spectrum of the output modes, radiation from a ramp frequency modulated laser is injected at the input of PMF optical fiber sensor. It is shown that the local coordinates and some characteristics of intruders can be obtained by the measurement of the frequencies and amplitudes of the output mode beating spectrum. Generally the number of beating frequencies is more than the number of intruders. Among the beating frequencies, a group with maximum signal to noise ratio is chosen. The short Fourier denoising method is employed to increase the sensor resolution. Because the output signal is the superposition of finite numbers of discrete frequencies this method is a powerful tool for denoising even for negative signal to noise ratio. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Intruder detector Fiber sensor Short Fourier transform denoising
1. Introduction The distributed optical fiber intruder sensors that are used to sense and locate the intruders have been widely studied by many researchers in the last three decades [1–8]. Most familiar intruder detectors are based on the optical time domain reflectometer (OTDR) [9], the Brillouin optical time domain reflectometer (BOTDR) [10] and phase sensitive OTDR (F-OTDR) [11]. Complicated optical fiber intruder detectors are based on the optical fiber interferometery. To detect the intruder crossing point often a combination of Sagnac Interferometer (SI) and other interferometers such as Sagnac–Mach–Zehnder [12], Sagnac–Sagnac [13] and Sagnac–Michelson [14] are employed. In all these compound interferometers a frequency modulation technique is applied to a polarization maintaining fiber Sagnac loop. The beating frequency of the two forward coupled beams and corresponding amplitude determine the location and amplitude of an applied stress respectively [15,16]. In this paper a simple multi-intruder detector on the basis of PMF optical fiber without any interferometric loop is presented. Radiation from a ramp frequency modulated laser is used as the input signal to map the intrusion positions to the output beating frequencies. At the intruder crossing point, the stress causes 11 the mode coupling between the HE11 x and the HEy modes of the PMF. The beating frequencies are functions of the intrusion positions and their amplitudes are functions of stress and size of the intruders. The resolution of the intruder positions are limited by the transmitter, receiver and PMF noises. The residual stress along the buried
n
Corresponding author. E-mail address: [email protected] (M. Bathaee).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.03.032
fiber causes a random mode coupling and consequently an additional noise is appeared at the output. The noise corresponding to the residual stress is saved and eliminated from the output signal, while such an operation for other noises is impossible. To improve the resolution of the intruder sensor the short Fourier denoising method is employed. The remaining part of this paper is organized as follows. In Section 2, theoretical method for finding intruder positions and some characteristics of them by analyzing the output mode beating spectrum is occurred. To check the validity of theoretical method, a numerical simulation for a specific situation of three intruders is presented in Section 3. Finally the paper is enclosed with some conclusion in Section 4.
2. Theoretical model As shown in Fig. 1(a) radiation from a ramp frequency modulated x-polarization laser is injected to the polarization maintaining fiber as the intruder fiber sensor. Variation of the laser frequency is shown in Fig. 1(b) and it is assumed that the amplitude of laser is maintained constant. The x and y axes are chosen along the principle axes of the PMF. In the presence of an intruder at the cross point of the optical fiber sensor and the intruder, due to the elasto-optic effect a part of Ex -polarization mode couples to the Ey-polarization in the remaining part of PMF (Z1 ¼ L Z0). L is the total length of PMF and Z0 is the location of the intruder cross point. It is assumed that the input laser frequency versus time is T-periodic and is linear on the time interval [0 and T].
oðtÞ ¼ o0 þ o1 t
ð1Þ
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Fig. 1. (a) Schematic of a birefringent fiber intruder detection system. FM is the frequency modulator, LD is a 10 mw laser diod, x-po is a x-polarizer, PMF is a 30 km birefringent optical fiber, FPBS is a fiber polarization beam splitter, FPR is a fiber polarization rotator, OFC is an optical fiber coupler, APD is an avalanche photo-diode detector, ALPF is an active electronic low pass filter, A/D is an analog to digital convertor, Com. a computer system for signal processing and denoising. (b) Ramp input to the FM system, the center frequency o0 ¼1.2 1015 Hz and the slope of the ramp o1 ¼ 2p 1011 Hz/s.
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Fig. 2. (a) Intruder system with two intrusions and (b) graph of traveling times of system indicated in (a).
where o0 and o1 are the initial laser frequency and ramp slope respectively. The birefringent property of the PMF causes a delay time t between the x and y polarization modes on the length Z1 of the fiber sensor. The time delay t1 produces a constant frequency 11 between the x (HE11 x ) and the y (HEy ) modes at the receiving end of the PMF:
Do ¼ 2o1 Z 1 ð1=vgx 1=vgy Þ
ð2Þ
where vgm(m ¼x, y) is the group velocity of the m mode and Do is 11 the frequency differences of the HE11 x and the HEy modes at the end point of the PMF. At the receiving end of the PMF, the HE11 x and the HE11 y are separated by a polarization beam splitter (PBS). The y-polarization is p/2 rotated by a polarization rotator. Both beams are directed to the avalanche photo-diode (APD). The output beating frequency is the difference between the frequencies of HE11 and HE11 modes. The amplitude of the output is x y related to the strength of the intruder. Due to the motion of the intruder a vibrational wave in the acoustic range is produced on the fiber sensor which has important information about the intruder. In this paper the effect of seismic waves is neglected. The effects of acoustic and seismic waves on the output of intruder sensor are under investigation by our group and results will appear in near future. By increasing the numbers of intruders, the numbers of coupling 11 points also increase The HE11 x and HEy modes are coupled to each other at each intruder cross points. Because it is assumed that the input light is x-polarized at the first intruder cross point the energy is 11 only coupled from the HE11 x to the HEy modes. Fig. 2(b) shows the graph of possible paths for the coupling of HE11 and HE11 modes to each other in the presence of two x y intruders. The coupling points on the graph are called nodes. The line connecting two nearest nodes is a branch. The x and y branches are corresponding to the Ex and Ey polarization. A path is a set of branches directly connecting the first node to one of the output nodes. Corresponding to each pair of paths there is a time delay and hence a beating frequency. By employing the graph shown in Fig. 2(b) it is easy to count the numbers of independent output beating frequencies (M): M ¼ 1=2ð3N 1Þ
ð3Þ
Fig. 3. General traveling time graph of an intruder system with N-intrusions.
where N is the number of intruders. The proof of Eq. (3) is presented in Appendix A. The beating frequencies are related to the delayed times (Dom ¼2o1tm). The delayed times are given in the following: X X tm ¼ ð1=vgx Z i 1=vgy Z j Þ ð4Þ i
j
where Zi is the distance of the ith intruder from the nearest left neighbor intruder (Fig. 3). For each m, Zi and Zj are unrepeatable and randomly selected. In (3) the repeated tm is eliminated and only one of them is saved. Obviously corresponding to each symmetries, the numbers of repeated tm increase and hence the numbers of beating frequencies decrease. However as shown in (3) for N 41, the numbers of beating frequencies are greater than the numbers of unknown Zi(i¼1,y, N). To obtain the positions of intruders it is enough to choose only N beating frequencies with higher amplitude among all M beating frequencies. Since the coupling coefficients ki(i¼1,y, N) are less than unity (9ki9o o1), the larger amplitudes correspond to the paths with less alternating branches between HE11 and HE11 modes. As an example x y consider a sensor with two intruders as shown in Fig. 2(a). The corresponding graph is presented in Fig. 2(b). The traveling time to the ith node is denoted by Ti(i¼1, 2, 3, 4): T 1 ¼ ðZ 0 þZ 1 þ Z 2 Þ=vgx
ð5Þ
T 2 ¼ ðZ 0 þZ 1 Þ=vgx þ Z 2 =vgy
ð6Þ
T 3 ¼ ðZ 0 þZ 2 Þ=vgx þ Z 1 =vgy
ð7Þ
T 4 ¼ Z 0 =vgx þðZ 1 þZ 2 Þ=vgy
ð8Þ
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As the total length of PMF is known, the independent unknown lengths are only Z1 and Z2 while we have four beating frequencies. To maximize the signal to noise ratio it is necessary to eliminate the end nodes with minimum strength. The strength at the end qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 nodes 1, 2, 3 and 4 are proportional to ð19k1 9 Þ, ð19k2 9 Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð19k1 9 Þ9k2 9,9k1 99k2 9 and ð19k2 9 Þ9k1 9 respectively. As mentioned above (9ki9o o1, i¼1, 2) hence node 3 has minimum strength even both intruders have the same coupling coefficients and is eliminated in our calculations. The beating frequencies corresponding to suitable branches are as follows:
o12 ¼ 2o1 Z 2 ð1=vgx 1=vgy Þ
ð9Þ
o24 ¼ 2o1 Z 1 ð1=vgx 1=vgy Þ
ð10Þ
o14 ¼ 2o1 ðZ 1 þZ 2 Þð1=vgx 1=vgy Þ
ð11Þ
Eqs. (9) and (10) are enough to obtain the unknowns Z1 and Z2. The APD output is amplified by a wideband audio frequency amplifier and the Fourier analysis of output signal can be carried out by an audio frequency spectrum analyzer. Also the Fourier transform can be obtained by the FFT algorithm. The first two frequencies with highest amplitude can be selected to obtain the intruder positions. The resolution of the intruder sensor is determined by the beating frequency bandwidths which depend on the laser bandwidth, laser noise and frequency modulation nonlinear effect. The time delays are continuous functions of the intruder positions, hence an intruder of finite width (d) produces a bandwidth do o1 ðnx ny Þd=c around the corresponding beating frequency. The temperature effects are neglected in this work. By increasing o1, both resolution and intruder size magnification (the ratio of signal band width to the intruder size) are increased. The Nth first high amplitude beating frequencies can also be employed to obtain the strength of intruders. Sakai’s analysis [17,18] is employed to obtain the coupling 11 coefficients from the HE11 x to the HEy modes kx and vice versa ky. To find these coefficients, two equal and opposite external forces F that compress a fiber at an angle y with respect to the x (or y)axis over the region with length (d) are assumed. When an x-polarized light with amplitude E0 passes through a perturbed region of the fiber, the output polarization components are given as follows: ExðyÞ ¼ E0 ½cosðdbd=2Þi cosðaÞsinðdbd=2exp½iðN xx þ Nyy Þd=2
ð12Þ
EyðxÞ ¼ E0 ½isinðaÞsinðdbd=2exp½iðNxx þN yy Þd=2
ð13Þ
By employing Eqs. (12) and (13) and assuming the coupling coefficient kx(y) is obtained:
kxðyÞ ¼ EyðxÞ =ExðyÞ ¼ isinðaÞsinðdbd=2Þ=cosðdbd=2ÞicosðaÞsinðdbd=2Þ ð18Þ For small external forces ðF o o p=4Bd=cÞ and at an angle ðy ¼ p=4Þ the coupling coefficient kx(y) is as follows:
kxðyÞ ¼ ð4iFC=pBdÞsinðpBd=lÞexpðipBd=lÞ:
db ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNxx N yy Þ2 þð2N xy Þ2
tan a ¼ 2N xy =ðN xx Nyy Þ
ð14Þ ð15Þ
The Nnm(m ¼x, y; n ¼x, y) are components of the coupling matrix (N) [19] and are related to the elasto optic constant C and the fiber modal birefringence factor B by the following relations: Nxx N yy ¼ 2p=l½BCð4F=pdÞcosð2yÞ
ð16Þ
Nxy ¼ Nyx ¼ ð4FC=ldÞsinð2yÞ:
ð17Þ
ð19Þ
As it is expected the absolute value of kx(y) is less than one (9kx(y)9o o1) and linearly increases by increasing the strength of the external forces. The coupling coefficients are determined from the magnitude of highest amplitudes of the beating frequencies. In general case with N intruders, each end node i of the graph presented in Fig. 3 is related to a traveling time Ti, a phase ji ¼ o1 T 2i o0 T i and a strength amplitude Ai. The delay time Ti is X X Z m þ 1=vgy Zn T i ¼ 1=vgx ð20Þ where the first and second summations are over the x and y branches of a path from the first node to the desired end node respectively. The strength of each end node is defined as the product of coupling coefficients of each branch in the path between the input and the end nodes: Y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 1jkm j2 Ai ¼ kn expðji Þ: ð21Þ m
Here the first product is over the branches that polarization does not change relative to its previous neighbor and the second product is over the other branches between the input and the desired end node. Corresponding to each pair of end nodes there is a beating frequency at the output detector. Amplitude of each beating frequency is proportional to the product of the strength of the corresponding end nodes. The beating frequency between the ith and jth end nodes is denoted by oij and is given by
oij ¼ 2o1 ðT i T j Þ:
ð22Þ
The amplitude of degenerate frequencies by consideration of the phase difference jij ¼ ji jj between two components is the phasor addition of corresponding amplitudes. It is easy to show that the beating frequencies Oj,k ¼ okð2Nj þ kÞ ðk ¼ 1,. . .,2Nj Þ are independent of the index k and are proportional to the intruder distance Zj (j¼ 1, y, N). The complex amplitude of the beating frequency Oj,k is denoted by Ajk ¼ ðAk Ak þ 2Nj expðijk,k þ 2Nj ÞÞ. The total complex amplitude of the beating frequency proportional to the distance Zj (Oj) is denoted by X X Cj ¼ Ajk þ A2N1 þ k,2N1 þ 2Nj þ k ð23Þ k
where db is the propagation constant difference and a are defined by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 19k1 9 1,
k
The Oj(j ¼2, y, N) does not have the maximum beating amplitude. The maximum amplitudes corresponding to the beating frequency between the first and 2j (j ¼1, y, N) nodes are named the first order modes. The amplitude of the jth first order mode is proportional to the coupling coefficient kN j þ 1. The modes with amplitude proportional to the product of mcoupling coefficients are called the mth order modes. Obviously the amplitude of mth order modes is less than the amplitude of m 1th modes. So we can distinguish these beating frequencies from others. In this calculation (1 9k92) is approximated by one. The beating frequency between the first node and the 2jth node is
o1,2j ¼ 2o1 ð1=vgy 1=vgx Þ
N X
Zm :
ð24Þ
m ¼ Nj þ 1
By considering Eq. (24), it is clear that o12 o o14 o o18 o o o12N . So after finding all highest amplitudes of beating frequencies, by sorting corresponding frequencies intruder positions are determined uniquely. By measuring the amplitudes
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of the first Nth high amplitude beating frequencies the corresponding coupling coefficients up to proportionality constant can be obtained. By introducing a standard intruder as a reference the proportionality constant is also determined. The detailed calculation is appeared in Section 3.
3. Simulation results As shown in Fig. 1(a) a 30 km Panda (PMF) fiber with birefrengent parameter B ¼4.4 10 4 and elasto-optic coefficient C ¼3.3 10 5 is employed as an intruder sensor. A 10 mW frequency modulation semiconductor laser at the wavelength l0 ¼1.55 mm and o1 ¼2p 1011 Hz/s is used as the sensor driver. The ramp period must be greater or equal to the time that light needs to pass through the fiber length. In this simulation we choose the ramp slope equal to 2p 1011 so that the beating frequencies are in the audio frequency range. It is assumed that three intruder with sizes (d1 ¼0.1 m, d2 ¼0.1 m and d3 ¼0.1 m) and forces (F1 ¼0.08 N, F2 ¼0.07 N and F3 ¼0.06 N) are presented at the positions (Z0 ¼10 km, Z1 ¼5 km, Z2 ¼8 km and Z3 ¼7 km) on the intruder sensor. From Eqs. (14 to 19) the coupling coefficients corresponding to the intruders are obtained (9k19¼0.0716, 9k29 ¼0.0627, 9k39 ¼0.0537). The field intensities of HE11 x ðEx Þ and HE11 y ðEy Þ at the output of the sensor are as follows: Ex ¼ E0 fexp½iðo0 ðtT 1 Þ þ o1 ðtT 1 Þ2 Þ þ k2 k3 exp½iðo0 ðtT 3 Þ þ o1 ðtT 3 Þ2 þ k1 k2 exp½iðo0 ðtT 5 Þ þ o1 ðtT 5 Þ2 þ k1 k3 exp½iðo0 ðtT 7 Þ þ o1 ðtT 7 Þ2 þ c:c:g
ð25Þ
Ey ¼ E0 fk3 exp½iðo0 ðtT 2 Þ þ o1 ðtT 2 Þ2 Þ þ k2 exp½iðo0 ðtT 4 Þ þ o1 ðtT 4 Þ2 þ k1 k2 k3 exp½iðo0 ðtT 6 Þ þ o1 ðtT 6 Þ2 þ k1 exp½iðo0 ðtT 8 Þ þ o1 ðtT 8 Þ2 þ c:c:g
ð26Þ
In the presence of isotropic optical fiber loss (a a0), it is easy to show that in (25) and (26), E0 is replaced by E0 expðaLÞ where L is the total fiber length. The panda optical fibers have isotropic loss while birefringent photonic crystal or quasiphotonic crystal fibers can have anisotropic losses, which are out of the scope of this work. By polarization beam splitter and the polarization rotator, Ey is also oriented in the x-direction and the field intensities are added algebraically. The light intensity is measured by the APD detector and filtered by an active electronic low pass filter. To consider the effect of noise on the output of the system, the Gaussian distribution white noise (GDWN) in the audio frequency range with several noise powers is added to the output signal. The noisy signal with signal to noise ratio (SNR) of 15 dB is depicted in Fig. 4. The absolute values of the Fourier transform of the outputs are shown in Fig. 5(a e). In this simulation we consider qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 19ki 9 1ði ¼ 1, 2, 3Þ. As illustrated by Fig. 5(a e), the three beating frequencies with highest amplitudes are detectable even in the presence of the noise with negative SNR. It is expected thirteen peaks can be found in the output spectrum but the computational and experimental noises fade some frequencies with small value amplitude. The bandwidth of each beating frequency is due to the laser, fiber, detector, electronic and computational noises. Effects of the intruder size, the finite ramp width and the acoustic wave generated by the moving intruder under investigation by our group and are neglected in this calculation. According to (24) the intruder positions can be
Fig. 4. Solid curve represents the noiseless signal at the output of intruder detector for three intrusions at (Z1 ¼ 5 km, Z2 ¼8 km, and Z3 ¼ 7 km) positions. The coefficients are assumed (9k19¼ 0.0716, 9k29¼ 0.0627, 9k39¼ 0.0537). The dashed curve represents the noisy signal with SNR¼ 15 dB.
determined from these three frequencies. Thus from beating frequencies (o12 ¼1867, o14 ¼ 4000, o18 ¼5333, and do E 710 Hz) the desired results are obtained (Z1 ¼5 km, Z2 ¼8 km, Z3 ¼7 km, and dZ E 710 m), which are in good agreement with our simulation data. The accuracy of beating frequency measurement (do) is proportional to the inverse of the system characteristic time, which is the minimum of ramp period and intruder transmission time on the fiber sensor. The above results are based on the 100 ms system characteristic time. As a result of Eq. (2) the error of distance measurement (dZ) is proportional to the beating frequency error and inversely proportional to o1 and birefringence fiber parameter. o1 and fiber birefringence parameter are limited by bandwidth of laser gain medium and fiber structure respectively. The resolution of the multi intruder sensor is defined by the minimum detectable distance between two adjacent intruders, which is two times of the system accuracy plus maximum size of the intruders. If we assume that maximum size of intruders is of the order of several meters, the resolution of the sensor is of the order of several 10 m ( E50 m) (Fig. 6). To obtain coupling coefficients we propose three different methods: 1. Stabilizing the input laser power and considering the fiber loss in calculations. 2. Introducing a standard object as a reference intruder. 3. Using the second order amplitudes signals.
In the first method the input power is known and the loss of the fiber can also be measured with an OTDR system. So the ratios of three highest amplitudes to the output power in the absence of intruders are exactly equal to the coupling coefficients. In the second method the object with known coupling coefficient is inserted on the specific position of the fiber line as a standard reference intruder. Our simulation is concentrated on this method. In this case the three highest amplitudes are proportional to the coupling coefficients. From the position of the reference intruder, the corresponding beating frequency can be recognized. Since the ratios of highest amplitudes to the reference amplitude are proportional to the ratios of unknown to the reference coupling coefficient, the coupling coefficients of other intruders are determined. In Fig. 5(b) the highest frequency is related to o18, the medium is related to o14 and the smallest is related to o12. Three highest amplitudes and related beating frequencies are A12 ¼0.0538007 7 10 6, A14 ¼0.0614071 10 5, A18 ¼0.0697071 10 5 and o12 ¼2p 1867 Hz, o14 ¼2p 4000 Hz and o18 ¼ 2p 5333 Hz respectively. By calculation the ratio of the amplitudes to the first beating frequency amplitude, the relative coupling coefficients are
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Fig. 5. Spectrum of the output signal: (a) noiseless signal spectrum, (b) SNR ¼ 15 dB, (c) SNR¼ 10 dB, (d) SNR ¼0 dB, and (e) SNR¼ 10 dB.
Fig. 6. Spectrum of denoised intruder output signal for (a) noiseless signal, (b) SNR¼ 15 dB, (c) SNR¼ 10 dB, (d) SNR ¼0 dB, and (e) SNR¼ 10 dB.
obtained: ðA14 =A18 ¼ 9k2 9=9k1 9 ¼ 0:8830 7 0:0051Þ, ðA12 =A18 ¼ 9k3 9=9k1 9 ¼ 0:7735 70:0033Þ: The variance variety of relative coupling coefficient versus SNR is depicted in Fig. 7. By multiplying relative coupling coefficients to reference intruder coupling coefficient k1, the other coupling results are k2 ¼0.0633072 10 5 and k3 ¼ 0.0554071 10 5. These data are achieved after running our program for hundred times. In a real intruder sensor this repetition corresponds to the many time measurements. For this purpose several frequency ramps should propagate in the fiber sensor during time that an intruder is crossing over the intruder sensor. Since the transmission time is limited by the intruder velocity, the number of measurements can be controlled by the ramp period. Decreasing of the ramp period causes to increase the number of possible measurements and so the SNR enlarges. On the other hand due to increasing of the ramp period, the beating frequency measurement accuracy increases, hence there is an optimum value for the ramp period, which depends on the intruder velocity and size. In the third method we do not have any reference and input amplitude is unknown but by use of other beating frequencies,
Fig. 7. Variance of the coupling coefficients of unknown intruders (k2 and k3) relative to the reference coupling coefficient (k1) versus the signal to noise ratio (SNR). The calculation values corresponding to the k2 and k3 are shown by m and K respectively.
the coupling coefficients can be determined. As mentioned in the previous section, amplitude of the first order modes is proportional to one of the coupling coefficient kj, while amplitude of the
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second order modes is proportional to the product of two coupling coefficients. In our example with three intruders the second order modes are o13 ¼ o24, o15 ¼ o48 and o17 ¼ o28. These frequencies are more visible in denoised signal spectrum as shown in Fig. 6(b–e) which is denoised signal of Fig. 5. Since Z1, Z2 and Z3 have been acquired by o12, o14 and o18 in the previous step, the frequencies o13, o15 and o17 are distinguishable from each other. Amplitude of the beating frequency o13 is proportional to the product of coupling coefficients k2 and k3. A1,3 p9k3 k2 99expðij13 þ ijÞ þ expðij24 ijÞ9
ð27Þ
where j is the phase of k3. Depending on the intruder characteristics, F and d are correlated. As the amplitude and phase of coupling coefficients are functions of F and d by solving Eq. (28) A1,3 =A1,4 ¼ 9k3 99expðij13 þ ijÞ þ expðij24 ijÞ9
4. Conclusion On the basis of mode coupling on the polarization maintaining optical fiber in the presence of intruder, an intruder sensor is proposed. The position and intruder characteristics are mapped into the frequency and amplitude of output beating frequencies respectively. The number of beating frequencies is more than the number of intruders. The highest amplitude of beating frequencies to achieve maximum signal to noise ratio are obtained. The transceiver and fiber noises cause to reduce the sensor resolution. It is shown that by the short Fourier denoising method we can improve the sensor resolution. Moreover the acoustic waves generated by moving intruder have much information about the intruder characteristics so they can improve our analysis.
Acknowledgment Authors thank N. Bathaee for her good discussion in the denoising subject.
Appendix A In this paper we illustrated that the beating frequencies are proportional to 1=vgx Si Zi 1=vgy Sj Z j where, i, j CL¼{1, y, N}, i\j ¼ f and i, ja f. It is clear that oij and oij are the same beating frequencies. Therefore, it is enough to count every configuration of the following equation: i ¼ 1,:::,N
ðA:2Þ
By use of Newtonian expansion M can be evaluated as follows: M ¼ 1=2ð3N 1Þ
ðA:3Þ
References
ð28Þ
the intruder characteristics are obtained. The coupling coefficients depend on the intruder size and weight simultaneously. This effect makes it difficult to specify the type of intruder from the measuring data. However moving intruder produces an acoustic vibrational wave on the PMF sensor which has much information about the intruder characteristics. The effects of intruder acoustic waves on the output of the intruder sensor are under investigation by our group and results will appear in near future.
Z 1 7Z 2 7 7Z i
The result is N N N N M¼ þ2 þ22 þ þ 2N1 1 2 3 N N N N þ2 1 ¼ 1=2 20 þ þ 2N 0 N 1
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ðA:1Þ
[1] Cogdell GB. Fiber optic sensors for intruder detection. In: Proceedings of the IEEE conference security technolology; 1988. p. 19–23. [2] Papagiannkis T, Phong WA, Woodrooffe JHF, Bergen AT, Haas RCG. Accuracy of weigh-in-motion scales and piezoelectric cables. Paper 880449, presented at the 68th annual meeting of the transportation research board. January 1989. [3] Park J, Lee W, Taylor HF. A fiber optic intrusion sensor with the configuration of an optical time domain reflectometer using coherent interference of Rayleigh back scattering SPIE 355S.0277-786X; 1998. [4] Park Jaehee. Reflection-type optical-fiber intrusion sensor based on speckle detection. Journal of the Korean Physical Society 2007;50(2):529–31. [5] Juarez JC, Maier EW, Kyoo Nam Choi, Taylor HF. Distributed fiber-optic intrusion sensor system. Journal of Lightwave Technology 2005;23(6): 2081–7. [6] Juarez Juan C, Taylor HF. Field test a distributed fiber-optic intrusion sensor system for long perimeters. Applied Optics 2007;46(11):1968–71. [7] Fang N, Wang L, Jia D, Shan C, Huang Z. Walking intrusion signal recognition method for fiber fence system. SPIE 7634.76341C 2009. [8] Huijuan Wu, Yunjiang Rao, Cheng Tang, Yu Wu, Yuan Gong. A novel FbGbased security fence enabling to detect extremely weak intrusion signals from nonequivalent sensor nodes. Sensors and Actuators A 2011;167(2): 548–55. [9] Surhone LM, Timpledon Miriam T, Marseken Susan F. Optical Time_Domain Reflectometer, (VDM Verlag Dr. Mueller AG & Co. Kg; 2010). [10] Kurashima T, Tateda M, Horiguchi T, Koyamada Y. Performance improvement of a combined OTDR for distributed strain and loss measurement by randomizing the reference light polarization state. IEEE Photonics Technology Letters 1997;9(3):360–2. [11] Choi KN, Taylor HF. Spectrally stable Er-Fiber laser for application in phasesensitive optical time-domain reflectrometry. IEEE Photonics Technology Letters 2003;15(3):1041–135. [12] Chtcherbakov Anatoli A, Swart Pieter L, Spammer Stephanus J, Lacquet Beatrys M. A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing. In: Proceedings of the SPIE, vol. 3489;1998. p. 60–4. [13] Fang XJ. A variable-loop sagnac interferometer for distributed impact sensing. Journal of Lightwave Technology 1996;14:2250–4. [14] Spammer SJ, Swart PL, Chtcherbokov AA. Merged Sagnac–Michelson interferometer for distributed disturbace detection. Journal of Lightwave Technology 1997;15:972–6. [15] Campbell M, Zheng G, Wallace PA, Holmes-Smith AS. A distributed frequency modulation continuous wave fiber stress sensor based on a Birefringent Sagnac Ring configuration. Optical Review 1997;4(1A):114–6. [16] Campbell M, Zheng G, Wallace PA, Holmes-Smith AS. Distributed stress sensor based on a birefringent fiber Sagnac ring. Proceedings of SPIE 1996;2838:138–42. [17] Sakai J, Kimura T. Birefringence and polarization characteristics of singlemode optical fiber under elastic deformations. IEEE Journal of Quantum Electronics 1981;QE-17:1041. [18] Sakai J, Kimura T. Polarization behaviour in multiply perterbed single-mode fibers. IEEE Journal of Quantum Electronics 1982;QE-18:59. [19] Tsubokawa M, Higashi T, Negishi Y. Mode couplings due to external forces distributed along a polarization-maintaining fiber: an evaluation. Applied Optics 1988;27(1).
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Polarization maintaining optical fiber multi-intruder sensor A.R. Bahrampour a, M. Bathaee a,n, S. Tofighi a, A. Bahrampour b, F. Farman a, M. Vali a a b
Physics Department of Sharif University of Technology, Azadi street, Tehran, Iran Dipartemento di scienze Fisiche, Universita diNapoli Federico II, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 February 2012 Received in revised form 21 March 2012 Accepted 22 March 2012 Available online 11 April 2012
In this paper, an optical fiber multi-intruder sensor based on polarization maintaining optical fiber (PMF), without any interferometric fiber loop, is introduced. To map the local coordinates of intruders on the beating spectrum of the output modes, radiation from a ramp frequency modulated laser is injected at the input of PMF optical fiber sensor. It is shown that the local coordinates and some characteristics of intruders can be obtained by the measurement of the frequencies and amplitudes of the output mode beating spectrum. Generally the number of beating frequencies is more than the number of intruders. Among the beating frequencies, a group with maximum signal to noise ratio is chosen. The short Fourier denoising method is employed to increase the sensor resolution. Because the output signal is the superposition of finite numbers of discrete frequencies this method is a powerful tool for denoising even for negative signal to noise ratio. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Intruder detector Fiber sensor Short Fourier transform denoising
1. Introduction The distributed optical fiber intruder sensors that are used to sense and locate the intruders have been widely studied by many researchers in the last three decades [1–8]. Most familiar intruder detectors are based on the optical time domain reflectometer (OTDR) [9], the Brillouin optical time domain reflectometer (BOTDR) [10] and phase sensitive OTDR (F-OTDR) [11]. Complicated optical fiber intruder detectors are based on the optical fiber interferometery. To detect the intruder crossing point often a combination of Sagnac Interferometer (SI) and other interferometers such as Sagnac–Mach–Zehnder [12], Sagnac–Sagnac [13] and Sagnac–Michelson [14] are employed. In all these compound interferometers a frequency modulation technique is applied to a polarization maintaining fiber Sagnac loop. The beating frequency of the two forward coupled beams and corresponding amplitude determine the location and amplitude of an applied stress respectively [15,16]. In this paper a simple multi-intruder detector on the basis of PMF optical fiber without any interferometric loop is presented. Radiation from a ramp frequency modulated laser is used as the input signal to map the intrusion positions to the output beating frequencies. At the intruder crossing point, the stress causes 11 the mode coupling between the HE11 x and the HEy modes of the PMF. The beating frequencies are functions of the intrusion positions and their amplitudes are functions of stress and size of the intruders. The resolution of the intruder positions are limited by the transmitter, receiver and PMF noises. The residual stress along the buried
n
Corresponding author. E-mail address: [email protected] (M. Bathaee).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.03.032
fiber causes a random mode coupling and consequently an additional noise is appeared at the output. The noise corresponding to the residual stress is saved and eliminated from the output signal, while such an operation for other noises is impossible. To improve the resolution of the intruder sensor the short Fourier denoising method is employed. The remaining part of this paper is organized as follows. In Section 2, theoretical method for finding intruder positions and some characteristics of them by analyzing the output mode beating spectrum is occurred. To check the validity of theoretical method, a numerical simulation for a specific situation of three intruders is presented in Section 3. Finally the paper is enclosed with some conclusion in Section 4.
2. Theoretical model As shown in Fig. 1(a) radiation from a ramp frequency modulated x-polarization laser is injected to the polarization maintaining fiber as the intruder fiber sensor. Variation of the laser frequency is shown in Fig. 1(b) and it is assumed that the amplitude of laser is maintained constant. The x and y axes are chosen along the principle axes of the PMF. In the presence of an intruder at the cross point of the optical fiber sensor and the intruder, due to the elasto-optic effect a part of Ex -polarization mode couples to the Ey-polarization in the remaining part of PMF (Z1 ¼ L Z0). L is the total length of PMF and Z0 is the location of the intruder cross point. It is assumed that the input laser frequency versus time is T-periodic and is linear on the time interval [0 and T].
oðtÞ ¼ o0 þ o1 t
ð1Þ
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Fig. 1. (a) Schematic of a birefringent fiber intruder detection system. FM is the frequency modulator, LD is a 10 mw laser diod, x-po is a x-polarizer, PMF is a 30 km birefringent optical fiber, FPBS is a fiber polarization beam splitter, FPR is a fiber polarization rotator, OFC is an optical fiber coupler, APD is an avalanche photo-diode detector, ALPF is an active electronic low pass filter, A/D is an analog to digital convertor, Com. a computer system for signal processing and denoising. (b) Ramp input to the FM system, the center frequency o0 ¼1.2 1015 Hz and the slope of the ramp o1 ¼ 2p 1011 Hz/s.
2027
Fig. 2. (a) Intruder system with two intrusions and (b) graph of traveling times of system indicated in (a).
where o0 and o1 are the initial laser frequency and ramp slope respectively. The birefringent property of the PMF causes a delay time t between the x and y polarization modes on the length Z1 of the fiber sensor. The time delay t1 produces a constant frequency 11 between the x (HE11 x ) and the y (HEy ) modes at the receiving end of the PMF:
Do ¼ 2o1 Z 1 ð1=vgx 1=vgy Þ
ð2Þ
where vgm(m ¼x, y) is the group velocity of the m mode and Do is 11 the frequency differences of the HE11 x and the HEy modes at the end point of the PMF. At the receiving end of the PMF, the HE11 x and the HE11 y are separated by a polarization beam splitter (PBS). The y-polarization is p/2 rotated by a polarization rotator. Both beams are directed to the avalanche photo-diode (APD). The output beating frequency is the difference between the frequencies of HE11 and HE11 modes. The amplitude of the output is x y related to the strength of the intruder. Due to the motion of the intruder a vibrational wave in the acoustic range is produced on the fiber sensor which has important information about the intruder. In this paper the effect of seismic waves is neglected. The effects of acoustic and seismic waves on the output of intruder sensor are under investigation by our group and results will appear in near future. By increasing the numbers of intruders, the numbers of coupling 11 points also increase The HE11 x and HEy modes are coupled to each other at each intruder cross points. Because it is assumed that the input light is x-polarized at the first intruder cross point the energy is 11 only coupled from the HE11 x to the HEy modes. Fig. 2(b) shows the graph of possible paths for the coupling of HE11 and HE11 modes to each other in the presence of two x y intruders. The coupling points on the graph are called nodes. The line connecting two nearest nodes is a branch. The x and y branches are corresponding to the Ex and Ey polarization. A path is a set of branches directly connecting the first node to one of the output nodes. Corresponding to each pair of paths there is a time delay and hence a beating frequency. By employing the graph shown in Fig. 2(b) it is easy to count the numbers of independent output beating frequencies (M): M ¼ 1=2ð3N 1Þ
ð3Þ
Fig. 3. General traveling time graph of an intruder system with N-intrusions.
where N is the number of intruders. The proof of Eq. (3) is presented in Appendix A. The beating frequencies are related to the delayed times (Dom ¼2o1tm). The delayed times are given in the following: X X tm ¼ ð1=vgx Z i 1=vgy Z j Þ ð4Þ i
j
where Zi is the distance of the ith intruder from the nearest left neighbor intruder (Fig. 3). For each m, Zi and Zj are unrepeatable and randomly selected. In (3) the repeated tm is eliminated and only one of them is saved. Obviously corresponding to each symmetries, the numbers of repeated tm increase and hence the numbers of beating frequencies decrease. However as shown in (3) for N 41, the numbers of beating frequencies are greater than the numbers of unknown Zi(i¼1,y, N). To obtain the positions of intruders it is enough to choose only N beating frequencies with higher amplitude among all M beating frequencies. Since the coupling coefficients ki(i¼1,y, N) are less than unity (9ki9o o1), the larger amplitudes correspond to the paths with less alternating branches between HE11 and HE11 modes. As an example x y consider a sensor with two intruders as shown in Fig. 2(a). The corresponding graph is presented in Fig. 2(b). The traveling time to the ith node is denoted by Ti(i¼1, 2, 3, 4): T 1 ¼ ðZ 0 þZ 1 þ Z 2 Þ=vgx
ð5Þ
T 2 ¼ ðZ 0 þZ 1 Þ=vgx þ Z 2 =vgy
ð6Þ
T 3 ¼ ðZ 0 þZ 2 Þ=vgx þ Z 1 =vgy
ð7Þ
T 4 ¼ Z 0 =vgx þðZ 1 þZ 2 Þ=vgy
ð8Þ
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As the total length of PMF is known, the independent unknown lengths are only Z1 and Z2 while we have four beating frequencies. To maximize the signal to noise ratio it is necessary to eliminate the end nodes with minimum strength. The strength at the end qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 nodes 1, 2, 3 and 4 are proportional to ð19k1 9 Þ, ð19k2 9 Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð19k1 9 Þ9k2 9,9k1 99k2 9 and ð19k2 9 Þ9k1 9 respectively. As mentioned above (9ki9o o1, i¼1, 2) hence node 3 has minimum strength even both intruders have the same coupling coefficients and is eliminated in our calculations. The beating frequencies corresponding to suitable branches are as follows:
o12 ¼ 2o1 Z 2 ð1=vgx 1=vgy Þ
ð9Þ
o24 ¼ 2o1 Z 1 ð1=vgx 1=vgy Þ
ð10Þ
o14 ¼ 2o1 ðZ 1 þZ 2 Þð1=vgx 1=vgy Þ
ð11Þ
Eqs. (9) and (10) are enough to obtain the unknowns Z1 and Z2. The APD output is amplified by a wideband audio frequency amplifier and the Fourier analysis of output signal can be carried out by an audio frequency spectrum analyzer. Also the Fourier transform can be obtained by the FFT algorithm. The first two frequencies with highest amplitude can be selected to obtain the intruder positions. The resolution of the intruder sensor is determined by the beating frequency bandwidths which depend on the laser bandwidth, laser noise and frequency modulation nonlinear effect. The time delays are continuous functions of the intruder positions, hence an intruder of finite width (d) produces a bandwidth do o1 ðnx ny Þd=c around the corresponding beating frequency. The temperature effects are neglected in this work. By increasing o1, both resolution and intruder size magnification (the ratio of signal band width to the intruder size) are increased. The Nth first high amplitude beating frequencies can also be employed to obtain the strength of intruders. Sakai’s analysis [17,18] is employed to obtain the coupling 11 coefficients from the HE11 x to the HEy modes kx and vice versa ky. To find these coefficients, two equal and opposite external forces F that compress a fiber at an angle y with respect to the x (or y)axis over the region with length (d) are assumed. When an x-polarized light with amplitude E0 passes through a perturbed region of the fiber, the output polarization components are given as follows: ExðyÞ ¼ E0 ½cosðdbd=2Þi cosðaÞsinðdbd=2exp½iðN xx þ Nyy Þd=2
ð12Þ
EyðxÞ ¼ E0 ½isinðaÞsinðdbd=2exp½iðNxx þN yy Þd=2
ð13Þ
By employing Eqs. (12) and (13) and assuming the coupling coefficient kx(y) is obtained:
kxðyÞ ¼ EyðxÞ =ExðyÞ ¼ isinðaÞsinðdbd=2Þ=cosðdbd=2ÞicosðaÞsinðdbd=2Þ ð18Þ For small external forces ðF o o p=4Bd=cÞ and at an angle ðy ¼ p=4Þ the coupling coefficient kx(y) is as follows:
kxðyÞ ¼ ð4iFC=pBdÞsinðpBd=lÞexpðipBd=lÞ:
db ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNxx N yy Þ2 þð2N xy Þ2
tan a ¼ 2N xy =ðN xx Nyy Þ
ð14Þ ð15Þ
The Nnm(m ¼x, y; n ¼x, y) are components of the coupling matrix (N) [19] and are related to the elasto optic constant C and the fiber modal birefringence factor B by the following relations: Nxx N yy ¼ 2p=l½BCð4F=pdÞcosð2yÞ
ð16Þ
Nxy ¼ Nyx ¼ ð4FC=ldÞsinð2yÞ:
ð17Þ
ð19Þ
As it is expected the absolute value of kx(y) is less than one (9kx(y)9o o1) and linearly increases by increasing the strength of the external forces. The coupling coefficients are determined from the magnitude of highest amplitudes of the beating frequencies. In general case with N intruders, each end node i of the graph presented in Fig. 3 is related to a traveling time Ti, a phase ji ¼ o1 T 2i o0 T i and a strength amplitude Ai. The delay time Ti is X X Z m þ 1=vgy Zn T i ¼ 1=vgx ð20Þ where the first and second summations are over the x and y branches of a path from the first node to the desired end node respectively. The strength of each end node is defined as the product of coupling coefficients of each branch in the path between the input and the end nodes: Y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 1jkm j2 Ai ¼ kn expðji Þ: ð21Þ m
Here the first product is over the branches that polarization does not change relative to its previous neighbor and the second product is over the other branches between the input and the desired end node. Corresponding to each pair of end nodes there is a beating frequency at the output detector. Amplitude of each beating frequency is proportional to the product of the strength of the corresponding end nodes. The beating frequency between the ith and jth end nodes is denoted by oij and is given by
oij ¼ 2o1 ðT i T j Þ:
ð22Þ
The amplitude of degenerate frequencies by consideration of the phase difference jij ¼ ji jj between two components is the phasor addition of corresponding amplitudes. It is easy to show that the beating frequencies Oj,k ¼ okð2Nj þ kÞ ðk ¼ 1,. . .,2Nj Þ are independent of the index k and are proportional to the intruder distance Zj (j¼ 1, y, N). The complex amplitude of the beating frequency Oj,k is denoted by Ajk ¼ ðAk Ak þ 2Nj expðijk,k þ 2Nj ÞÞ. The total complex amplitude of the beating frequency proportional to the distance Zj (Oj) is denoted by X X Cj ¼ Ajk þ A2N1 þ k,2N1 þ 2Nj þ k ð23Þ k
where db is the propagation constant difference and a are defined by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 19k1 9 1,
k
The Oj(j ¼2, y, N) does not have the maximum beating amplitude. The maximum amplitudes corresponding to the beating frequency between the first and 2j (j ¼1, y, N) nodes are named the first order modes. The amplitude of the jth first order mode is proportional to the coupling coefficient kN j þ 1. The modes with amplitude proportional to the product of mcoupling coefficients are called the mth order modes. Obviously the amplitude of mth order modes is less than the amplitude of m 1th modes. So we can distinguish these beating frequencies from others. In this calculation (1 9k92) is approximated by one. The beating frequency between the first node and the 2jth node is
o1,2j ¼ 2o1 ð1=vgy 1=vgx Þ
N X
Zm :
ð24Þ
m ¼ Nj þ 1
By considering Eq. (24), it is clear that o12 o o14 o o18 o o o12N . So after finding all highest amplitudes of beating frequencies, by sorting corresponding frequencies intruder positions are determined uniquely. By measuring the amplitudes
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of the first Nth high amplitude beating frequencies the corresponding coupling coefficients up to proportionality constant can be obtained. By introducing a standard intruder as a reference the proportionality constant is also determined. The detailed calculation is appeared in Section 3.
3. Simulation results As shown in Fig. 1(a) a 30 km Panda (PMF) fiber with birefrengent parameter B ¼4.4 10 4 and elasto-optic coefficient C ¼3.3 10 5 is employed as an intruder sensor. A 10 mW frequency modulation semiconductor laser at the wavelength l0 ¼1.55 mm and o1 ¼2p 1011 Hz/s is used as the sensor driver. The ramp period must be greater or equal to the time that light needs to pass through the fiber length. In this simulation we choose the ramp slope equal to 2p 1011 so that the beating frequencies are in the audio frequency range. It is assumed that three intruder with sizes (d1 ¼0.1 m, d2 ¼0.1 m and d3 ¼0.1 m) and forces (F1 ¼0.08 N, F2 ¼0.07 N and F3 ¼0.06 N) are presented at the positions (Z0 ¼10 km, Z1 ¼5 km, Z2 ¼8 km and Z3 ¼7 km) on the intruder sensor. From Eqs. (14 to 19) the coupling coefficients corresponding to the intruders are obtained (9k19¼0.0716, 9k29 ¼0.0627, 9k39 ¼0.0537). The field intensities of HE11 x ðEx Þ and HE11 y ðEy Þ at the output of the sensor are as follows: Ex ¼ E0 fexp½iðo0 ðtT 1 Þ þ o1 ðtT 1 Þ2 Þ þ k2 k3 exp½iðo0 ðtT 3 Þ þ o1 ðtT 3 Þ2 þ k1 k2 exp½iðo0 ðtT 5 Þ þ o1 ðtT 5 Þ2 þ k1 k3 exp½iðo0 ðtT 7 Þ þ o1 ðtT 7 Þ2 þ c:c:g
ð25Þ
Ey ¼ E0 fk3 exp½iðo0 ðtT 2 Þ þ o1 ðtT 2 Þ2 Þ þ k2 exp½iðo0 ðtT 4 Þ þ o1 ðtT 4 Þ2 þ k1 k2 k3 exp½iðo0 ðtT 6 Þ þ o1 ðtT 6 Þ2 þ k1 exp½iðo0 ðtT 8 Þ þ o1 ðtT 8 Þ2 þ c:c:g
ð26Þ
In the presence of isotropic optical fiber loss (a a0), it is easy to show that in (25) and (26), E0 is replaced by E0 expðaLÞ where L is the total fiber length. The panda optical fibers have isotropic loss while birefringent photonic crystal or quasiphotonic crystal fibers can have anisotropic losses, which are out of the scope of this work. By polarization beam splitter and the polarization rotator, Ey is also oriented in the x-direction and the field intensities are added algebraically. The light intensity is measured by the APD detector and filtered by an active electronic low pass filter. To consider the effect of noise on the output of the system, the Gaussian distribution white noise (GDWN) in the audio frequency range with several noise powers is added to the output signal. The noisy signal with signal to noise ratio (SNR) of 15 dB is depicted in Fig. 4. The absolute values of the Fourier transform of the outputs are shown in Fig. 5(a e). In this simulation we consider qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 19ki 9 1ði ¼ 1, 2, 3Þ. As illustrated by Fig. 5(a e), the three beating frequencies with highest amplitudes are detectable even in the presence of the noise with negative SNR. It is expected thirteen peaks can be found in the output spectrum but the computational and experimental noises fade some frequencies with small value amplitude. The bandwidth of each beating frequency is due to the laser, fiber, detector, electronic and computational noises. Effects of the intruder size, the finite ramp width and the acoustic wave generated by the moving intruder under investigation by our group and are neglected in this calculation. According to (24) the intruder positions can be
Fig. 4. Solid curve represents the noiseless signal at the output of intruder detector for three intrusions at (Z1 ¼ 5 km, Z2 ¼8 km, and Z3 ¼ 7 km) positions. The coefficients are assumed (9k19¼ 0.0716, 9k29¼ 0.0627, 9k39¼ 0.0537). The dashed curve represents the noisy signal with SNR¼ 15 dB.
determined from these three frequencies. Thus from beating frequencies (o12 ¼1867, o14 ¼ 4000, o18 ¼5333, and do E 710 Hz) the desired results are obtained (Z1 ¼5 km, Z2 ¼8 km, Z3 ¼7 km, and dZ E 710 m), which are in good agreement with our simulation data. The accuracy of beating frequency measurement (do) is proportional to the inverse of the system characteristic time, which is the minimum of ramp period and intruder transmission time on the fiber sensor. The above results are based on the 100 ms system characteristic time. As a result of Eq. (2) the error of distance measurement (dZ) is proportional to the beating frequency error and inversely proportional to o1 and birefringence fiber parameter. o1 and fiber birefringence parameter are limited by bandwidth of laser gain medium and fiber structure respectively. The resolution of the multi intruder sensor is defined by the minimum detectable distance between two adjacent intruders, which is two times of the system accuracy plus maximum size of the intruders. If we assume that maximum size of intruders is of the order of several meters, the resolution of the sensor is of the order of several 10 m ( E50 m) (Fig. 6). To obtain coupling coefficients we propose three different methods: 1. Stabilizing the input laser power and considering the fiber loss in calculations. 2. Introducing a standard object as a reference intruder. 3. Using the second order amplitudes signals.
In the first method the input power is known and the loss of the fiber can also be measured with an OTDR system. So the ratios of three highest amplitudes to the output power in the absence of intruders are exactly equal to the coupling coefficients. In the second method the object with known coupling coefficient is inserted on the specific position of the fiber line as a standard reference intruder. Our simulation is concentrated on this method. In this case the three highest amplitudes are proportional to the coupling coefficients. From the position of the reference intruder, the corresponding beating frequency can be recognized. Since the ratios of highest amplitudes to the reference amplitude are proportional to the ratios of unknown to the reference coupling coefficient, the coupling coefficients of other intruders are determined. In Fig. 5(b) the highest frequency is related to o18, the medium is related to o14 and the smallest is related to o12. Three highest amplitudes and related beating frequencies are A12 ¼0.0538007 7 10 6, A14 ¼0.0614071 10 5, A18 ¼0.0697071 10 5 and o12 ¼2p 1867 Hz, o14 ¼2p 4000 Hz and o18 ¼ 2p 5333 Hz respectively. By calculation the ratio of the amplitudes to the first beating frequency amplitude, the relative coupling coefficients are
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Fig. 5. Spectrum of the output signal: (a) noiseless signal spectrum, (b) SNR ¼ 15 dB, (c) SNR¼ 10 dB, (d) SNR ¼0 dB, and (e) SNR¼ 10 dB.
Fig. 6. Spectrum of denoised intruder output signal for (a) noiseless signal, (b) SNR¼ 15 dB, (c) SNR¼ 10 dB, (d) SNR ¼0 dB, and (e) SNR¼ 10 dB.
obtained: ðA14 =A18 ¼ 9k2 9=9k1 9 ¼ 0:8830 7 0:0051Þ, ðA12 =A18 ¼ 9k3 9=9k1 9 ¼ 0:7735 70:0033Þ: The variance variety of relative coupling coefficient versus SNR is depicted in Fig. 7. By multiplying relative coupling coefficients to reference intruder coupling coefficient k1, the other coupling results are k2 ¼0.0633072 10 5 and k3 ¼ 0.0554071 10 5. These data are achieved after running our program for hundred times. In a real intruder sensor this repetition corresponds to the many time measurements. For this purpose several frequency ramps should propagate in the fiber sensor during time that an intruder is crossing over the intruder sensor. Since the transmission time is limited by the intruder velocity, the number of measurements can be controlled by the ramp period. Decreasing of the ramp period causes to increase the number of possible measurements and so the SNR enlarges. On the other hand due to increasing of the ramp period, the beating frequency measurement accuracy increases, hence there is an optimum value for the ramp period, which depends on the intruder velocity and size. In the third method we do not have any reference and input amplitude is unknown but by use of other beating frequencies,
Fig. 7. Variance of the coupling coefficients of unknown intruders (k2 and k3) relative to the reference coupling coefficient (k1) versus the signal to noise ratio (SNR). The calculation values corresponding to the k2 and k3 are shown by m and K respectively.
the coupling coefficients can be determined. As mentioned in the previous section, amplitude of the first order modes is proportional to one of the coupling coefficient kj, while amplitude of the
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second order modes is proportional to the product of two coupling coefficients. In our example with three intruders the second order modes are o13 ¼ o24, o15 ¼ o48 and o17 ¼ o28. These frequencies are more visible in denoised signal spectrum as shown in Fig. 6(b–e) which is denoised signal of Fig. 5. Since Z1, Z2 and Z3 have been acquired by o12, o14 and o18 in the previous step, the frequencies o13, o15 and o17 are distinguishable from each other. Amplitude of the beating frequency o13 is proportional to the product of coupling coefficients k2 and k3. A1,3 p9k3 k2 99expðij13 þ ijÞ þ expðij24 ijÞ9
ð27Þ
where j is the phase of k3. Depending on the intruder characteristics, F and d are correlated. As the amplitude and phase of coupling coefficients are functions of F and d by solving Eq. (28) A1,3 =A1,4 ¼ 9k3 99expðij13 þ ijÞ þ expðij24 ijÞ9
4. Conclusion On the basis of mode coupling on the polarization maintaining optical fiber in the presence of intruder, an intruder sensor is proposed. The position and intruder characteristics are mapped into the frequency and amplitude of output beating frequencies respectively. The number of beating frequencies is more than the number of intruders. The highest amplitude of beating frequencies to achieve maximum signal to noise ratio are obtained. The transceiver and fiber noises cause to reduce the sensor resolution. It is shown that by the short Fourier denoising method we can improve the sensor resolution. Moreover the acoustic waves generated by moving intruder have much information about the intruder characteristics so they can improve our analysis.
Acknowledgment Authors thank N. Bathaee for her good discussion in the denoising subject.
Appendix A In this paper we illustrated that the beating frequencies are proportional to 1=vgx Si Zi 1=vgy Sj Z j where, i, j CL¼{1, y, N}, i\j ¼ f and i, ja f. It is clear that oij and oij are the same beating frequencies. Therefore, it is enough to count every configuration of the following equation: i ¼ 1,:::,N
ðA:2Þ
By use of Newtonian expansion M can be evaluated as follows: M ¼ 1=2ð3N 1Þ
ðA:3Þ
References
ð28Þ
the intruder characteristics are obtained. The coupling coefficients depend on the intruder size and weight simultaneously. This effect makes it difficult to specify the type of intruder from the measuring data. However moving intruder produces an acoustic vibrational wave on the PMF sensor which has much information about the intruder characteristics. The effects of intruder acoustic waves on the output of the intruder sensor are under investigation by our group and results will appear in near future.
Z 1 7Z 2 7 7Z i
The result is N N N N M¼ þ2 þ22 þ þ 2N1 1 2 3 N N N N þ2 1 ¼ 1=2 20 þ þ 2N 0 N 1
2031
ðA:1Þ
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