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Single-shot distributed Brillouin optical time domain analyzer

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Abstract

We demonstrate a novel single-shot distributed Brillouin optical time domain analyzer (SS-BOTDA). In our method, dual-polarization probe with orthogonal frequency-division multiplexing (OFDM) modulation is used to acquire the distributed Brillouin gain spectra, and coherent detection is used to enhance the signal-to-noise ratio (SNR) drastically. Distributed temperature sensing is demonstrated over a 1.08 km standard single-mode fiber (SSMF) with 20.48 m spatial resolution and 0.59 °C temperature accuracy. Neither frequency scanning, nor polarization scrambling, nor averaging is required in our scheme. All the data are obtained through only one-shot measurement, indicating that the sensing speed is only limited by the length of fiber.

© 2017 Optical Society of America

1. Introduction

As an important approach of distributed fiber sensing, Brillouin optical time domain analyzer (BOTDA) has achieved remarkable progress during the past decades due to its capability of monitoring the strain or temperature along the fiber, with high sensing accuracy and long measurable range [1–3]. In the conventional BOTDA configuration, a pulse pump and a continuous wave (CW) probe counter-propagate in the same fiber. If the frequency difference between the pump and the probe is near the Brillouin frequency shift (BFS), the probe will be amplified or attenuated via stimulated Brillouin scattering (SBS) process. By scanning the frequency difference between pump and probe, the distributed Brillouin spectrum (BGS) can be obtained for estimating the BFS, which has a linear relationship with the change of strain or temperature.

Conventional BOTDA requires taking a large number of measurements for distributed fiber sensing each time, including scrambling the pump polarization, averaging the received traces and scanning the probe frequency. Therefore the sensing process is usually time-consuming, ranging from tens of seconds to tens of minutes. Furthermore, it assumes that the fiber status (strain and temperature) shall remain unchanged during these measurements; otherwise, the distributed Brillouin spectra may not be correctly reconstructed. Thus it is applicable for monitoring some relatively slow changes along the fiber. In the past few years, distributed dynamic sensing (DDS), which records fast strain/temperature variation along the fiber, has become a hot topic driven by the growing demand in gas and oil industries, geophysical science and structural health monitoring [4]. Therefore it is desirable for BOTDA technique with boosted sensing speed. To reduce the measuring time, several methods have been proposed, including the slope-assisted method [5, 6], the fast frequency-swept method [7], the radio-frequency (RF) phase demodulation method [8] and the digital optical frequency comb method [9]. All of these methods have shown some encouraging improvements and applications. Nevertheless, either frequency scanning, or polarization scrambling, or averaging is still required, which limits the ultimate measuring speed.

Recently, we proposed a novel single-shot technique to measure the distributed Brillouin stimulated spectrum [10]. Here the word ‘single-shot’ means that all the information of the fiber is gathered by sending only one pump pulse and receiving the corresponding probe light. In this paper, we expand that work to develop a single-shot BOTDA (SS-BOTDA) system and systematically evaluate its performance. In our scheme, double sideband probe is modulated with orthogonal frequency-division multiplexing (OFDM) signal, recording both Brillouin gain and Brillouin loss spectra simultaneously. The polarizations of upper and lower sideband are orthogonal to each other, which is used to eliminate the polarization effect. Polarization diversity coherent receiver is adopted to detect the complex field of probe light, as well as to drastically enhance the signal-to-noise ratio (SNR) and sensitivity. Distributed temperature sensing experiment is then conducted over a 1.08 km standard single-mode fiber (SSMF) with 20.48 m spatial resolution and 0.59 °C temperature accuracy. There is no polarization scrambling, averaging or frequency scanning in our scheme, indicating that the ultimate sensing speed is only limited by the fiber length, which can be a promising approach in distributed dynamic sensing.

2. Principle of SS-BOTDA

The principle of SS-BOTDA is shown in Fig. 1. Instead of the CW probe used in conventional BOTDA, we use the orthogonal sideband probe with orthogonal frequency division multiplexing (OFDM) modulation in SS-BOTDA scheme. A baseband OFDM symbol s(t) can be expressed as [11]:

s(t)=i=+k=1Nckisk(tiTs)
sk(t)=(t)ej2πvkt
(t)={1,(0<t<Ts)0,(t0,t>Ts)
where cki is the ith complex information at the kth subcarrier, sk is the waveform for the kth subcarrier and vk is the frequency of the kth subcarrier [12]. N is the number of subcarriers, Π(t) is the pulse shaping function and Ts is the symbol period. If an identical complex sequence c = [c1, c2, …, cN ] is repeatedly sent as the information symbol, then in time domain the baseband signal becomes:
s(t)=k=1Nckej2πvkt

 figure: Fig. 1

Fig. 1 Principle of single-shot BOTDA. (a) Subcarriers of OFDM signal in frequency domain. (b) Dual-polarized double-sideband OFDM probe. LSB: lower sideband, USB: upper sideband. (c) Pump and probe interaction and Brillouin spectrum extraction. FFT: fast Fourier transform, DSP: digital signal processing. (d) The Brillouin gain spectrum (BGS) and the Brillouin loss spectrum (BLS) after the stimulated Brillouin scattering process.

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The frequency of subcarriers can be expressed as a vector v = [v1, v2, …, vN ], as shown in Fig. 1(a). After modulating on an optical carrier with amplitude A0 and frequency v0, the time-domain electric field Et (t) of transmitted optical OFDM is written as:

Et(t)=A0k=1Nckej2π(vk+v0)t
Taking the Fourier transform of the above equation, we get:
E^t(v)=A0k=1Nckδ[2π(vvkv0)]
where δ(·) is the Dirac delta function.

In order to cancel the polarization fading, the optical OFDM signal Et is then double sideband (DSB) modulated by a RF sine wave with frequency vRF in an intensity modulator working at its null point. Then the output electric field turns to:

E^DSB=a0A0k=1Nckδ[2π(vvkv0+vRF)]E^s++a0A0k=1Nckδ[2π(vvkv0vRF)]E^s
where a0 is a factor which contains the amplitude scaling and the common phase change. The polarizations of the upper sideband (USB) and lower sideband (LSB) are then rotated to be orthogonal to each other. Finally the orthogonal sideband probe can be represented as a Jones vector Es = Es+ex + Esey, where ex and ey are the basis vectors for x-polarization and y-polarization, respectively, as depicted in Fig. 1(b).

As shown in Fig. 1(c), the probe and pump counter-propagate along the fiber. When they meet each other at the position z, stimulated Brillouin scattering (SBS) process happens. If the RF frequency vRF is near the BFS vB, the lower sideband will be amplified while the upper sideband will be attenuated, which is shown in Fig. 1(d) and can be expressed as:

Er++h+(t,z)Es+
Er=h(t,z)Es
where ‘⊗’ is the convolution operator h+(t, z) and h(t, z) are the impulse responses of the Brillouin gain and loss at position z, respectively. In frequency domain Eq. (8) and Eq. (9) become:
E^r+=H+(v,z)E^s+
E^r=H(v,z)E^s
where H+(v, z) and H(v, z) are the complex Brillouin gain and Brillouin loss spectra at position z, respectively. Assuming that the pump width is longer than the lifetime of acoustic phonon, then these spectra can be given by [13]:
H±(v,z)=exp[±η±g0ΔvBΔvB+2j(vvp±vB(z))]
where g0 is the local Brillouin gain, ΔvB is the Brillouin linewidth, vp is the frequency of pump light. η+ and η are the real mixing efficiency factors of the LSB probe and the USB probe. According to [14], the mixing efficiency factors are written as:
η±=12(1+s1ps1s±+s2ps2s±s3ps3s±)
where sp = [s1p, s2p, s3p], ss+ = [s1s+, s2s+, s3s+], ss = [s1s, s2s, s3s] are the normalized Stokes vectors of the pump, the LSB probe and the USB probe, respectively. Since the LSB probe and the USB probe are orthogonal in polarization, i.e. ss+ + ss = 0, one has η+ + η = 1.

The probe light is then received by a polarization diversity coherent receiver, which mixes probe with a local oscillator (LO) light ELO = E0 exp(j2πvLOt) in a 90 degree optical hybrid. The received complex signals are written as:

R+=12γE+ELO*
R=12γEELO*
where γ is the detector responsivity. If LO, probe carrier and pump has the same frequency, i.e. (vLO = v0 = vp), the received frequency will be down-converted. Then the received time-domain signals R+ and R are divided into many segments, each of which has the identical length to OFDM symbol period Ts. The subcarrier amplitude of each segment can be obtained through Fourier transform, as shown in Fig. 1(d). Here we assign the subcarrier frequencies of LSB and USB as vectors v+ = vvRF and v = v + vRF, respectively. Since the transmitted symbol c is known, we can easily derive BGS and Brillouin loss spectrum (BLS), which can be described by the logarithmic gain vector Γ+(z) = Γ+(v+, z) and loss vector Γ(z) = Γ (v, z) where:
Γ±(v,z)=±2η±g0ΔvB2ΔvB2+4(v±vB(z))2

The next step is to combine BGS and BLS and eliminate the polarization effect. Here we flip the vector Γ(z) as Γ¯(z) and define ΔΓ(z)=Γ(z)Γ¯(z). If the subcarrier frequency of the baseband OFDM satisfies vk = −vN+1−k, k = 1, 2, …, N, then ΔΓ(z) = G(v, z), where G(v, z) is the Lorentzian shape Brillouin gain profile that:

G(v,z)=2g0ΔvB2ΔvB2+4(vvRF+vB(z))2
According to the Eq. (17) the gain profile is independent of polarization, indicating that the polarization fading has been eliminated. After calculating the Brillouin spectrum vector ΔΓ of the entire segments, we can combine them as the distributed Brillouin spectrogram. Therefore the BFS vB can be estimated by curve fitting with the vector v and ΔΓ.

It is worth noticing that the polarization-diversity coherent detection in our scheme can not only acquire the complete electric field of the probe light, but can increase the SNR and sensitivity as well. Compared with the direct detection in conventional BOTDA which can only detect the probe intensity, the LO in coherent detection can amplify the probe by a factor of |ELO|/|Es |, which can significantly enhance the SNR and make it easier to detect the weak probe light. Thus the desired SNR can be achieved by coherent detection, instead of the averaging process used in conventional BOTDA.

3. Experimental setup

The experimental setup is depicted in Fig. 2. An external cavity laser (ECL) with 1542.72 nm center wavelength and 100 kHz linewidth is split into three branches as the probe, the pump, and the local oscillator by two 3dB beam splitters (BS). The probe part is first modulated by an OFDM signal in an optical in-phase quadrature (IQ) modulator driven by an arbitrary waveform generator (AWG) operating at 10 GSa/s. The OFDM frame is initially designed in frequency domain with N = 256 subcarriers and 2048 total sample points, corresponding to a symbol period of 204.8 ns and a spatial resolution of 20.48 m. In order to reduce the peak-to-average power ratio (PAPR), the subcarriers are mapped by a Zadoff-Chu sequence c given by:

ck=exp(jπk2/N),k=1,2,,N

 figure: Fig. 2

Fig. 2 Experimental setup of single-shot BOTDA. ECL: external cavity laser, BS: beam splitter, AWG: arbitrary waveform generator, MSS: microwave synthesized sweeper, EOM: electro-optic modulator, EDFA: Erbium-doped fiber amplifier, DWDM: dense wavelength division multiplexer, PC: polarization controller, PBS: polarization beam splitter, BPD: balanced photo-detector, DSO: digital storage oscilloscope.

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Due to the large symbol period, intra-symbol phase noise between LO and probe may occur. Thus we add a pilot tone at the right side of c to track the phase as depicted in Fig. 3(a), and to compensate the phase noise in the following section. Then the frequency domain signal is transformed to time domain OFDM frame by a 2048 points inverse fast Fourier transform (Inverse FFT). The real and imaginary parts of an OFDM frame in time domain are shown in Fig. 3(b). According to the PAPR definition that:

PAPR=max[|s(t)|2]/E[|s(t)|2]
, the PAPR of our designed OFDM signal is only 3.04 dB, which is almost identical to the PAPR of an ideal sinusoidal waveform (3dB), indicating that our OFDM signal power is nearly uniformly distributed in both time domain and frequency domain.

 figure: Fig. 3

Fig. 3 (a) Generation of the complex baseband OFDM signal. (b) The real part (I) and imaginary part (Q) of one generated OFDM frame in time domain. (c) The electric spectrum of the baseband OFDM probe. (d) The optical spectrum of the double-sideband OFDM probe with orthogonal sideband polarizations.

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Then the OFDM frame is repeatedly modulated on the probe as the OFDM signal Et (t). The electrical spectrum of the OFDM signal is depicted in Fig. 3(c). The 256 subcarriers occupy a bandwidth from −625 MHz to +625MHz with 4.88 MHz frequency spacing and almost identical amplitude. The small fluctuation of the flat top in Fig. 3(c) is due to the electric distortion of cables and digital-to-analog converters (DACs). Then a vRF = 11 GHz RF sine wave from the microwave synthesized sweeper (MSS) modulates the OFDM baseband probe in an electro-optic modulator (EOM). The upper sideband and lower sideband are then separated by a 50 GHz dense wavelength division multiplexer (DWDM) with a sharp slope (30 dB over 10 GHz) shown in the inset of Fig. 2. The polarizations of USB and LSB are then adjusted to be orthogonal via two polarization controllers (PCs) and are combined in a polarization beam splitter (PBS). Fig. 3(d) depicts the optical spectrum of DSB probe with orthogonal polarization sidebands. The USB and LSB are assigned as x-polarization and y-polarization, respectively. The dual-polarized DSB probe is sent into the fiber through an optical isolator. The total transmitted probe power is −16 dBm.

The pump pulse is generated in another EOM driven by a pulse signal with 150 ns pulse width and 50 kHz repetition rate. Here 150 ns is chosen as the pump pulse width since it provides sufficient pump power. Other pulse widths are also available except those beyond the spatial resolution. Then the pump pulses are amplified by an EDFA to the average power of 8 dBm and sent into the fiber under test (FUT) via an optical circulator. The FUT is composed of a 983.14 m SSMF spool and a 98.85 m SSMF segment, which the first 20 m is heated by a water bath. The total length is around 1.08 km and the round-trip time is about 10.8 μs. FC/APC connectors are used to reduce the Fresnel reflection.

After the SBS process, the probe signal is mixed with a 6 dBm CW LO in a polarization diversity coherent detector, which consists of a 90-degree dual-polarization optical hybrid (Kylia) and four balanced photo-detectors (BPDs). The outputs of four BPDs are acquired by a digital storage oscilloscope (DSO) operating at 50 GSa/s. The received data are then treated in the following signal processing part.

4. Data processing

The procedure of data processing is illustrated in Fig. 4(a). The received outputs from four detectors are Ix, Qx, Iy, and Qy. After the skew alignment and power normalization, they form the complex received signals as Rx = Ix + jQx and Ry = Iy + jQy, which correspond to the x- and y-polarization, respectively. After the timing synchronization, frequency offset compensation and polarization rotation compensation using the training frames shown in Fig. 4(b), a bandpass filter is utilized to filter out the pilot tone and estimate its relative phase drift to the LO, as shown in Fig. 4(c). Since the pilot tone and the OFDM signal are generated simultaneously and propagate along the same fiber, their phases are inherently locked. Therefore the phase drift of OFDM signal can be compensated by multiplying the conjugate of the pilot phase and relative frequency offset. The time-domain data are then down-sampled by 5 and divided into segments, each of which has 2048 sample points, corresponding to 20.48 m fiber length. Fast Fourier transform (FFT) is conducted on each segment to get the spectra of both polarizations. The first several segments which are not affected by SBS process are used to estimate the frequency-domain channel distortion, as shown in Fig. 4(d). The channel distortion of other segments in SBS region can be compensated by inversing this distortion shape.

 figure: Fig. 4

Fig. 4 (a) Procedure of data processing. (b) Received time-domain signal. (c) The phase drift between the probe and LO in the SBS region. (d) Channel distortion of the OFDM subcarriers.

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Fig. 5(a) depicts the logarithmic subcarrier amplitudes of both x- and y-polarizations of one OFDM frame, corresponding to the BGS vector Γ+(z) and BLS vector Γ (z), respectively. We can find that after the previous signal processing steps, the channel distortion is almost removed. Some subcarrier amplitudes of y-polarization are amplified while some subcarrier amplitudes of x-polarization are attenuated. Then we flip the BLS vector Γ (z) and subtract it from the BGS vector Γ+(z), resulting in a logarithmic gain profile vector ΔΓ, as shown in Fig. 5(b). The inset figure illustrates the data points from the 98th to 118th subcarriers with a good Lorentzian shape.

 figure: Fig. 5

Fig. 5 (a) Logarithmic gain of BGS vector Γ+(z) and BLS vector Γ(z) of a OFDM frame in the SBS region. (b) Logarithmic gain profile of ΔΓ(z). Inset is the data points which used for curve fitting and BFS estimation.

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We then calculate the BLS and BGS vectors for each segment and draw them as the reconstructed Brillouin spectrogram for x-polarization, y-polarization and the combined dual-polarization in Fig. 6(a) (b) and (c), respectively. Due to the polarization effect, there are drastic amplitude fluctuations in Fig. 6(a) and (b). However, the Brillouin peaks in Fig. 6(c) are quite stable, meaning that the polarization fading has been successfully mitigated. We also notice that there is a small fluctuation of Brillouin peaks in the combined spectrogram due to measurement noise. To prove that this undulation is insignificant, we choose the maximal peak position, the minimal peak position and the maximal BFS shift position marked as A, B and C in Fig. 6(c). The original data points and the Lorentzian fitting curves of these three positions are shown in Fig. 6(d). According to the method described in [15], the SNR of position A is 31.41 dB and the SNR of position B is 29.26 dB. The bandwidths of Brillouin gain spectrum in Fig. 6(d) are 32.6 MHz (A), 35.1 MHz (B) and 24.2 MHz (C) respectively according to the fitting results. It can be found that all the data points for A, B and C are adequate for the curve fitting, meaning that all the data in the reconstructed spectrogram are sufficient for BFS identifying

 figure: Fig. 6

Fig. 6 Reconstructed Brillouin spectrogram for (a) x-polarization (b) y-polarization and (c) combined dual-polarization. (d). Measured data and Lorentzian fitting curves of markers A, B and C in (c).

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5. Results and discussion

In order to test our SS-BOTDA performance, distributed temperature sensing experiment is carried out by increasing the temperature of water bath from room temperature to over 75 °C and taking several single-shot BOTDA measurements. Then we treat the recorded data by the signal processing method described in section 4 and compute the distributed BFS. Fig. 7(a) shows the calculated BFS for the 1.08 km FUT. The slight variation around the far end of 983 m fiber spool is because of the strain change from manual spooling [16]. The inset in Fig. 7(a) depicts the variation of BFS in the hotspot. With temperature goes higher, the BFS shifts upwards. We then record the BFS of the hotspot position and evaluate its relationship with the temperature via linear fitting, as shown in Fig. 7(b). The temperature coefficient for the FUT is 1.03 MHz/°C. The coefficient of determination R2 is 99.96%, indicating that good linear relationship is achieved for temperature sensing.

 figure: Fig. 7

Fig. 7 (a) Estimated BFS along the fiber with different water bath temperature. The inset figure shows the increment of BFS of the heated fiber segment. (b) BFS of the hotspot as a function of temperature. Blue line is the linear curve fitting.

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To validate that our SS-BOTDA is replicable and reliable, the single-shot measurement is consecutively performed 25 times when the temperature of hotspot is set to 65 °C. Fig. 8(a) shows the BFS of each measurement. The blue line denotes the mean BFS values and the dots are the BFS data for each segment. We can find that the BFS data points are well superposed, indicating that the results are almost the same. Notice that in our approach the temperature resolution or accuracy is not directly related to the frequency spacing of the OFDM signal (4.88 MHz), since the temperature is determined by the BFS which is estimated by the curve fitting with all the OFDM subcarrier amplitudes. To demarcate the accuracy of temperature sensing, we calculate the BFS deviation to the mean value of each segment. According to the histogram in Fig. 8(b), the probability density of BFS deviation approximates a Gaussian distribution. We then perform Gaussian fitting N(μ,σ2) on the measured BFS and obtain the results as μ = 0 MHz and σ = 0.57 MHz. Since the temperature coefficient is 1.03 MHz/°C, the accuracy of temperature sensing is only 0.59 °C, indicating that our measurement result is accurate and reliable.

 figure: Fig. 8

Fig. 8 (a) BFS of 25 times of measurements along the fiber when the water bath is set to 65°C. Blue line is the mean value for each segment. Dots are BFS data points. Red lines are the error bars. (b) Normalized probability density distribution of BFS deviation. Blue line denotes the Gaussian curve fitting of the probability density function (PDF).

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In our configuration, the pump pulse repetition rate is set to 50 kHz. However, since all the data are acquired with one-time measurement, the ultimate sensing speed is only limited by the fiber round-trip time. The time consumption of the digital signal processing in our scheme might be larger than that in the conventional BOTDA due of the coherent complex signal processing and additional FFT steps. Since the real-time OFDM transmission containing these DSP steps has already been demonstrated [17], the DSP time is not a critical restriction of our scheme. Moreover, since the spectrum of OFDM probe can cover several GHz, the sensing speed can be further increased by sequentially launching the pump pulses with different frequencies, as used in [16]. Therefore it is possible to detect some fast-change variations, such as the vibrations or the sound waves. From this point of view, the proposed SS-BOTDA shows the potential dynamic applications in the structural health monitoring such as bridges, tunnels and dams, as well as the disaster remote alerts such as earthquake, landslides and tsunami. Furthermore, since the subcarriers within the Brillouin bandwidth are amplified or attenuated simultaneously, the pump depletion effect in conventional BOTDA, which has a hollow of the Brillouin spectrum, can also be avoided.

The spatial resolution in our experiment is 20.48 m. Different from conventional BOTDA, the spatial resolution of SS-BOTDA is determined by the subcarrier frequency. For example, in our experiment the frequency spacing Δ f is 4.88 MHz (2048 points over 10 GHz), therefore the spatial resolution is vg/(2Δ f) = 20.48 m, where vg is the group velocity in SSMF. Higher resolution can be achieved by increasing the Δ f. However, since the Brillouin linewidth ΔvB is fixed, a larger frequency spacing means less data points in the stimulated Brillouin profile, which may increase the measurement error during the curve fitting process [18]. This problem can be solved by increasing the Brillouin linewidth, which can be realized by adding proper modulation on the pump [19], and will be presented in our future work.

6. Conclusion

In conclusion, we have demonstrated a novel single-shot BOTDA method, which can monitor the distributed fiber status with only one-time measurement. Instead of the CW probe used in classic BTODA, we adopt an orthogonal-sideband OFDM probe to acquire stimulated Brillouin spectrum and to diminish the polarization fading. By carefully designing the OFDM symbol, we achieve a low PAPR signal which uniformly distributes the power in both time domain and frequency domain. Polarization-diversity coherent detection is used to collect the full field information and to enhance the signal-to-noise ratio. We then conduct the distributed temperature sensing experiment in a 1.08 km standard single-mode fiber with a spatial resolution of 20.48 m. Sensing results show that the distributed Brillouin frequency shift can be successfully located by one-time measurement, with a temperature coefficient of 1.03 MHz/°C and 0.59 °C accuracy. It is worth stressing that our SS-BOTDA scheme needs no frequency scanning, or polarization scrambling, or averaging. The sensing speed is only limited by the fiber length and can be further improved, indicating that our approach can significantly boost the sensing speed.

Funding

Australian Research Council (ARC) Discovery Project (DP150104815); National Key Scientific Instrument and Equipment Development Project of China (2013YQ040815); National Natural Science Foundation of China (NSFC) (61575052, 61308004).

Acknowledgments

The authors would like to acknowledge the support from Australian Government Research Training Program (RTP) Scholarship. Portions of this work were presented at the conference IEEE Sensors 2016 [20]

References and links

1. A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016 [CrossRef]  ).

2. X. Bao and L. Chen, “Recent Progress in Brillouin Scattering Based Fiber Sensors,” Sensors (Basels) 11(12), 4152–4187 (2011 [CrossRef]  ).

3. L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010 [CrossRef]  ).

4. A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016 [CrossRef]   [PubMed]  ).

5. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011 [CrossRef]   [PubMed]  ).

6. D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017 [CrossRef]  ).

7. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012 [CrossRef]   [PubMed]  ).

8. J. Urricelqui, M. Sagues, and A. Loayssa, “BOTDA measurements tolerant to non-local effects by using a phase-modulated probe wave and RF demodulation,” Opt. Express 21(14), 17186–17194 (2013 [CrossRef]   [PubMed]  ).

9. C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015 [CrossRef]   [PubMed]  ).

10. J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016 [CrossRef]  ), AT5C.3.

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12. Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85 [CrossRef]  .

13. J. Urricelqui, F. López-Fernandino, M. Sagues, and A. Loayssa, “Polarization Diversity Scheme for BOTDA Sensors Based on a Double Orthogonal Pump Interaction,” J. Lightwave Technol. 33(12), 2633–2638 (2015 [CrossRef]  ).

14. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994 [CrossRef]  ).

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16. A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011 [CrossRef]  ).

17. N. Kaneda, T. Pfau, H. Zhang, J. Lee, Y.-K. Chen, C. J. Youn, Y. H. Kwon, E. S. Num, and S. Chandrasekhar, “Field Demonstration of 100-Gb/s Real-Time Coherent Optical OFDM Detection,” J. Lightwave Technol. 33(7) 1365–1372 (2015 [CrossRef]  ).

18. A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016 [CrossRef]   [PubMed]  ).

19. M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14(4), 1395–1400 (2006 [CrossRef]  ).

20. J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

References

  • View by:

  1. A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
    [Crossref]
  2. X. Bao and L. Chen, “Recent Progress in Brillouin Scattering Based Fiber Sensors,” Sensors (Basels) 11(12), 4152–4187 (2011).
    [Crossref]
  3. L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
    [Crossref]
  4. A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016).
    [Crossref] [PubMed]
  5. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
    [Crossref] [PubMed]
  6. D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
    [Crossref]
  7. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
    [Crossref] [PubMed]
  8. J. Urricelqui, M. Sagues, and A. Loayssa, “BOTDA measurements tolerant to non-local effects by using a phase-modulated probe wave and RF demodulation,” Opt. Express 21(14), 17186–17194 (2013).
    [Crossref] [PubMed]
  9. C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
    [Crossref] [PubMed]
  10. J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), AT5C.3.
    [Crossref]
  11. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
    [Crossref] [PubMed]
  12. Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85.
    [Crossref]
  13. J. Urricelqui, F. López-Fernandino, M. Sagues, and A. Loayssa, “Polarization Diversity Scheme for BOTDA Sensors Based on a Double Orthogonal Pump Interaction,” J. Lightwave Technol. 33(12), 2633–2638 (2015).
    [Crossref]
  14. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994).
    [Crossref]
  15. A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
    [Crossref]
  16. A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
    [Crossref]
  17. N. Kaneda, T. Pfau, H. Zhang, J. Lee, Y.-K. Chen, C. J. Youn, Y. H. Kwon, E. S. Num, and S. Chandrasekhar, “Field Demonstration of 100-Gb/s Real-Time Coherent Optical OFDM Detection,” J. Lightwave Technol. 33(7) 1365–1372 (2015).
    [Crossref]
  18. A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
    [Crossref] [PubMed]
  19. M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14(4), 1395–1400 (2006).
    [Crossref]
  20. J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

2017 (1)

2016 (3)

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016).
[Crossref] [PubMed]

A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
[Crossref] [PubMed]

2015 (3)

2013 (1)

2012 (1)

2011 (3)

2010 (1)

L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
[Crossref]

2008 (1)

2006 (1)

1999 (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

1994 (1)

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994).
[Crossref]

Amin, A. A.

Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85.
[Crossref]

Angulo-Vinuesa, X.

Ba, D.

Bao, H.

Bao, X.

X. Bao and L. Chen, “Recent Progress in Brillouin Scattering Based Fiber Sensors,” Sensors (Basels) 11(12), 4152–4187 (2011).
[Crossref]

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

Bergman, A.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Boot, A. J.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994).
[Crossref]

Bremner, T. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

Brown, A. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

Chandrasekhar, S.

Chen, L.

X. Bao and L. Chen, “Recent Progress in Brillouin Scattering Based Fiber Sensors,” Sensors (Basels) 11(12), 4152–4187 (2011).
[Crossref]

Chen, Y.-K.

DeMerchant, M. D.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

Dominguez-Lopez, A.

Dong, Y.

Fang, J.

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), AT5C.3.
[Crossref]

J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

Feng, Y.

Gonzalez-Herraez, M.

Guo, N.

Herráez, M. G.

Jiang, T.

Jin, C.

Kaneda, N.

Kwon, Y. H.

Lee, J.

Li, H.

Li, J.

Li, Z.

Liang, H.

Loayssa, A.

López-Fernandino, F.

Lopez-Gil, A.

Lu, C.

Lu, Z.

Martin-Lopez, S.

Masoudi, A.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016).
[Crossref] [PubMed]

Motil, A.

Newson, T. P.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016).
[Crossref] [PubMed]

Num, E. S.

Peled, Y.

Pfau, T.

Sagues, M.

Shieh, W.

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[Crossref] [PubMed]

Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85.
[Crossref]

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), AT5C.3.
[Crossref]

J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

Song, K. Y.

Soto, M. A.

Tang, Y.

Thévenaz, L.

Tur, M.

Urricelqui, J.

van Deventer, M. O.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994).
[Crossref]

Voskoboinik, A.

Wang, B.

Wang, L.

Willner, A. W.

Xu, P.

D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
[Crossref]

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), AT5C.3.
[Crossref]

J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

Yang, Q.

Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85.
[Crossref]

Yaron, L.

Yilmaz, O. F.

Youn, C. J.

Yu, C.

Zhang, H.

Zhou, D.

Front. Optoelectron. China (1)

L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (9)

A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
[Crossref] [PubMed]

M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14(4), 1395–1400 (2006).
[Crossref]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[Crossref] [PubMed]

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
[Crossref] [PubMed]

D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
[Crossref]

Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
[Crossref] [PubMed]

J. Urricelqui, M. Sagues, and A. Loayssa, “BOTDA measurements tolerant to non-local effects by using a phase-modulated probe wave and RF demodulation,” Opt. Express 21(14), 17186–17194 (2013).
[Crossref] [PubMed]

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
[Crossref]

Opt. Laser Technol. (1)

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Proc. SPIE (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359 (1999).
[Crossref]

Rev. Sci. Instrum. (1)

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87, 011501 (2016).
[Crossref] [PubMed]

Sensors (Basels) (1)

X. Bao and L. Chen, “Recent Progress in Brillouin Scattering Based Fiber Sensors,” Sensors (Basels) 11(12), 4152–4187 (2011).
[Crossref]

Other (3)

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), AT5C.3.
[Crossref]

Q. Yang, A. A. Amin, and W. Shieh, “Optical OFDM Basics,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (SpringerNew York, New York, NY, 2011), pp. 43–85.
[Crossref]

J. Fang, W. Shieh, and P. Xu, “Single-shot Brillouin optical time domain analysis for distributed fiber sensing” in Proceedings of IEEE Sensors 2016 (IEEE, 2016), B4L-C, pp. 1–3.

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Figures (8)

Fig. 1
Fig. 1 Principle of single-shot BOTDA. (a) Subcarriers of OFDM signal in frequency domain. (b) Dual-polarized double-sideband OFDM probe. LSB: lower sideband, USB: upper sideband. (c) Pump and probe interaction and Brillouin spectrum extraction. FFT: fast Fourier transform, DSP: digital signal processing. (d) The Brillouin gain spectrum (BGS) and the Brillouin loss spectrum (BLS) after the stimulated Brillouin scattering process.
Fig. 2
Fig. 2 Experimental setup of single-shot BOTDA. ECL: external cavity laser, BS: beam splitter, AWG: arbitrary waveform generator, MSS: microwave synthesized sweeper, EOM: electro-optic modulator, EDFA: Erbium-doped fiber amplifier, DWDM: dense wavelength division multiplexer, PC: polarization controller, PBS: polarization beam splitter, BPD: balanced photo-detector, DSO: digital storage oscilloscope.
Fig. 3
Fig. 3 (a) Generation of the complex baseband OFDM signal. (b) The real part (I) and imaginary part (Q) of one generated OFDM frame in time domain. (c) The electric spectrum of the baseband OFDM probe. (d) The optical spectrum of the double-sideband OFDM probe with orthogonal sideband polarizations.
Fig. 4
Fig. 4 (a) Procedure of data processing. (b) Received time-domain signal. (c) The phase drift between the probe and LO in the SBS region. (d) Channel distortion of the OFDM subcarriers.
Fig. 5
Fig. 5 (a) Logarithmic gain of BGS vector Γ+(z) and BLS vector Γ(z) of a OFDM frame in the SBS region. (b) Logarithmic gain profile of ΔΓ(z). Inset is the data points which used for curve fitting and BFS estimation.
Fig. 6
Fig. 6 Reconstructed Brillouin spectrogram for (a) x-polarization (b) y-polarization and (c) combined dual-polarization. (d). Measured data and Lorentzian fitting curves of markers A, B and C in (c).
Fig. 7
Fig. 7 (a) Estimated BFS along the fiber with different water bath temperature. The inset figure shows the increment of BFS of the heated fiber segment. (b) BFS of the hotspot as a function of temperature. Blue line is the linear curve fitting.
Fig. 8
Fig. 8 (a) BFS of 25 times of measurements along the fiber when the water bath is set to 65°C. Blue line is the mean value for each segment. Dots are BFS data points. Red lines are the error bars. (b) Normalized probability density distribution of BFS deviation. Blue line denotes the Gaussian curve fitting of the probability density function (PDF).

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

s ( t ) = i = + k = 1 N c k i s k ( t i T s )
s k ( t ) = ( t ) e j 2 π v k t
( t ) = { 1 , ( 0 < t < T s ) 0 , ( t 0 , t > T s )
s ( t ) = k = 1 N c k e j 2 π v k t
E t ( t ) = A 0 k = 1 N c k e j 2 π ( v k + v 0 ) t
E ^ t ( v ) = A 0 k = 1 N c k δ [ 2 π ( v v k v 0 ) ]
E ^ DSB = a 0 A 0 k = 1 N c k δ [ 2 π ( v v k v 0 + v RF ) ] E ^ s + + a 0 A 0 k = 1 N c k δ [ 2 π ( v v k v 0 v RF ) ] E ^ s
E r + + h + ( t , z ) E s +
E r = h ( t , z ) E s
E ^ r + = H + ( v , z ) E ^ s +
E ^ r = H ( v , z ) E ^ s
H ± ( v , z ) = exp [ ± η ± g 0 Δ v B Δ v B + 2 j ( v v p ± v B ( z ) ) ]
η ± = 1 2 ( 1 + s 1 p s 1 s ± + s 2 p s 2 s ± s 3 p s 3 s ± )
R + = 1 2 γ E + E LO *
R = 1 2 γ E E LO *
Γ ± ( v , z ) = ± 2 η ± g 0 Δ v B 2 Δ v B 2 + 4 ( v ± v B ( z ) ) 2
G ( v , z ) = 2 g 0 Δ v B 2 Δ v B 2 + 4 ( v v RF + v B ( z ) ) 2
c k = exp ( j π k 2 / N ) , k = 1 , 2 , , N
PAPR = max [ | s ( t ) | 2 ] / E [ | s ( t ) | 2 ]

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