Abstract

We report a large-scale multi-channel fiber sensing network, where ultra-short FBGs (USFBGs) instead of conventional narrow-band ultra-weak FBGs are used as the sensors. In the time division multiplexing scheme of the network, each grating response is resolved as three adjacent discrete peaks. The central wavelengths of USFBGs are tracked with the differential detection, which is achieved by calculating the peak-to-peak ratio of two maximum peaks. Compared with previous large-scale hybrid multiplexing sensing networks (e.g., WDM/TDM) which typically have relatively low interrogation speed and very high complexity, the proposed system can achieve interrogation of all channel sensors through very fast and simple intensity measurements with a broad dynamic range. A proof-of-concept experiment with twenty USFBGs, at two wavelength channels, was performed and a fast static strain measurements were demonstrated, with a high average sensitivity of ~0.54dB/µƐ and wide dynamic range of over ~3000µƐ. The channel to channel switching time was 10ms and total network interrogation time was 50ms.

© 2017 Optical Society of America

1. Introduction

Identical weak fiber Bragg gratings (FBGs) based quasi-distributed sensors have wide range of applications in large civil construction, manufacturing industry, defense industry and so on. The use of weak gratings eliminates the capacity limitation caused by the limited optical source bandwidth, allowing for the multiplexing of a large number of sensors. In addition, weak gratings are also greatly facilitated by the well-established on-line grating writing technologies, through which hundreds or even thousands of identical gratings can be easily fabricated along a single optical fiber [1]. Currently, time-division multiplexing (TDM) is the most common method to multiplex identical weak gratings. It is also feasible to improve the capacity by combining two or three multiplexing methods together, such as TDM/WDM [2], and the multiplexing of thousands of gratings have been demonstrated through this concept [3]. The interrogation performance is another key factor for a large sensing network, which can directly determine the sensing performance [4]. An ideal interrogator in a sensing network should offer simultaneous detection of a large number of gratings, a wide operational range, and high-speed, reliable and accurate measurements. However, for traditional wavelength based interrogation methods using the active spectrum-detection devices, such as Fabry–Pérot filters and tunable lasers, the measurement typically suffers from a low response-rate, due to the inevitable mechanical wavelength scanning and spectrum peak (or notch) search processes. As an example, in [5], the network design distinguishes different FBGs via reconstructing the reflection spectrum of each FBG during the entire scanning period in time domain, but the total interrogation time for all the sensors is more than a minute.

In practice, the intensity-based interrogation technique is still of great interest because of its simplicity, potential cost-effectiveness, much faster and more robust measurement [6]. However, though they have been widely used in many single-point sensing applications, it is still challenging to implement them in distributed sensing applications. This is due to the fact that intensity-based interrogation, such as edge filter, generally have a higher requirement for the signal power and SNR. However, these parameters are typically low in TDM sensing network with weak gratings. More importantly, they generally suffer from a small wavelength operational range, for example, the dynamic range of edge filtering methods is low [7], and that of matched filtering approaches for conventional FBGs is usually less than 1nm [8]. This directly prevents their use in some hybrid multiplexing sensing networks (i.e. WDM/TDM), where the wavelength channels that need to be detected can usually cover a large wavelength band (usually tens of even hundreds of nm) [9].

Recently, we have demonstrated an intensity-based interrogation concept of large scale USFBGs TDM sensing network by using shifted optical Gaussian filters (SOGF) [10]. The SOGF-TDM distinguishes the sensors by different time delays and their interrogation is based on differential intensity output of two tunable optical Gaussian filters with shifted transmission spectra. Different from the actual return power of the FBGs which might be attenuated over the round-trip light path due to transmission losses and reflections of upstream FBGs, the differential power of each grating is intensity fluctuation invariant and has a self-referencing capability for FBGs Bragg wavelength shift detection in time domain.

However, the ASE broadband light source used in that setup, leads to some drawbacks such as high power consumption, and limits both the distances over which fiber sensors can be remotely driven and the number of sensors. The present work is aimed at the development of an intensity-based demodulation of USFBGs network by using the narrowband multi-wavelength channels of a laser source, which fully eliminates the shortcoming mentioned above and enhances the dynamic range of the system. This new approach offers a higher power within a narrow spectral width and thus improves SNR. In addition, it allows for a larger number of sensors of the same nominal Bragg wavelength, which are multiplexed by using TDM. The sensor number can be further increased using a combination of WDM and TDM by switching the laser lines to other channels of the network.

The proposed method inherits the important features of a conventional tunable laser interrogation technique, namely its high flexibility and natural insensitivity to intensity variations. While for traditional tunable laser based methods, it requires to sweep the whole spectrum range to obtain the center wavelength of FBG spectrum. In the proposed scheme, the interrogation is just implemented by three wavelength lines of a multi-line laser source, which leads to significant improvements of the efficiency and measurement speed. Our approach avoids the need for standard continuous sweeping through the entire spectrum, and all the associated limitations, such as ultra-high sampling requirements, and can reduce the cost of the interrogation system.

2. Sensing Concept

3.1 USFBG intensity-based wavelength demodulation

The ultra-short fiber Bragg grating (USFBG) is a type of weak FBGs with typical grating lengths reduced to only hundreds or even tens of microns. This short length can broaden the reflection spectrum significantly, typically several (or even tens of) nm, with a very symmetrical Gaussian shape, which could provide significant benefits in terms of the central wavelength tracking. According to Fig. 1, the FBG returned signal power of any discrete line of the laser on photodetector is proportional to the overlap integral of the spectrum density function of the laser source and the reflection spectrum of the fiber Bragg grating [11]:

 figure: Fig. 1

Fig. 1 USFBG spectrum and the three laser lines.

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I=0R(λλB)S(λλl)dλ

Where, S(λ-λl) and R(λ-λB), are the laser and the USFBG spectrum, respectively. Since the spectral width of the laser is much narrower than the USFBG bandwidth, by assuming that ‘S’ is a delta function and USFBG spectrum is ideal Gaussian, the total reflected power of the laser can be expressed as:

I=CSλlRmexp[(4ln2)(λlλBΔλB)2]

Where λl, Rm, λB and ΔλB are the laser line wavelength, FBG’s peak reflectivity, Bragg wavelength, and bandwidth of the sensing FBG, respectively. The constant coefficient, C, describes the power allocation coefficient, which depends on several factors, including the photoelectric conversion efficiency of the PD and propagation losses.

Thus, for a three line laser based USFBG demodulation scheme, there are three Gaussian functions as the power responses for different laser lines.

I1=C1Sλ1Rmexp[(4ln2)((λ1λB)/ΔλB)2]I2=C2Sλ1Rmexp[(4ln2)((λ1+ΔλλB)/ΔλB)2]I3=C3Sλ1Rmexp[(4ln2)((λ1+2ΔλλB)/ΔλB)2]

Where Δλ is the multiline laser wavelength spacing and Sλ1 = Sλ2 = Sλ3. So by subtracting the two Gaussian functions in the log-domain, the differential readout of the signal, as the sensing response, can be obtained as:

{P(λB)=log(I2(λB)I1(λB))if(I1I3)P(λB)=log(I2(λB)I3(λB))if(I1<I3)(E+FλB)

Where,E=C+4log(e)ln2ΔλB(Δλ2+2λ1Δλ), C = log(C2/C1 or C3) and F=8log(e)Δλln2ΔλB2 are constant terms for the given experimental setup. From Eq. (4), it is clear that the differential readout of USFBG have a linear response with respect to the Bragg wavelength. Different from the real returned power of the FBG which is vulnerable against intensity noises and transmission losses, it is intensity fluctuation invariant and has a self-referencing capability for the FBG’s Bragg wavelength shift detection. In addition, the slope, represented by F in Eq. (4), is proportional to the laser lines distance Δλ and inverse square of USFBG bandwidth ΔλB. increasing the laser line wavelength spacing or decreasing the FBG’s bandwidth, for a given implementation, will result in higher sensitivity. However, this operation would also lead to a decrease of the operational range, if the minimum optical power determined by a certain SNR requirement is taken into consideration. It is because when the laser line spacing increases, the shift range of the FBG wavelength in which both line signal power can be higher than the minimum level will decrease.

As shown in Fig. 2, it is clear that the measurement range highly depends on the grating bandwidth, and the use of USFBGs instead of conventional narrowband weak FBGs can result in a much wider measurement range.

 figure: Fig. 2

Fig. 2 Sensing sensitivity and measurement range for different Bragg grating bandwidths.

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This demodulation technique takes the merit of the cost-effective mass production of the USFBGs with the identical spectral. However, in some special cases, if the different spectral responses among gratings come from the fabrication process, the non-identically of the network sensors will result in different strain to intensity transfer function, which will affect sensing performance. Hence to have accurate sensing results it is necessary to fully characterizing USFBGs differential response in the calibration procedure. By implementing the calibration procedure in the sensing network, the interrogation of this kind of FBGs with non-identical responses, would still be possible.

2.2 USFBG sensing network interrogation

A schematic view of the sensing network setup is shown in Fig. 3, The sensing network contains n sensing wavelength channels with the 3Δλ central wavelength spacing, and each channel has m identical USFBGs. A LiNbO3 electro-optical amplitude modulator is used to generate short pulses from three narrowband tunable laser sources. The laser lines can be tuned to the sensing network wavelength channel as λn, λn + Δλ and λn + 2Δλ, where Δλ is selected to be same as the USFBG’s bandwidth to acquire the maximum measurement range. The time referenced light pulse train after amplification is then sent into the sensor network through an optic fiber circulator. After reflected from the multichannel sensors, they are routed to the passive periodical filtering device, named optical multi-bandpass filter (OMBFP). OMBPF acts as a comb filter with passbands matching all the sensing channel laser lines. Therefore, it can split the reflected laser lines into three photodetectors (PD). Furthermore, considering that well-performed comb filters with wide operational ranges over the C + L band are now readily available, tens of wavelength channels in the sensing network can potentially be covered in the proposed system, implying a great potential of capacity.

 figure: Fig. 3

Fig. 3 Basic structure of the proposed sensing setup. OMBFP: optical multi-bandpass filter, EOM: electro-optic modulator, PC: polarization controller, PD: photodetector.

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Compared with conventional wavelength tunable OTDR interrogation methods, using the three line laser source that can be switched into n wavelength channels of the sensing network, will be much faster and potentially capable of real-time monitoring. Here, the ‘multi-line laser–OMBPF’ scheme actually acts as a passive wavelength-to-intensity converter, which can simultaneously transform the wavelength shifts of all identical cascaded sensors to their reflection intensity changes without any time delay. In addition, this approach eliminates any active channel switching devices that are normally complex, expensive and time-consuming, and greatly increase the reliability and stability of the measurements.

3. Experimental results and discussion

We have conducted experimental studies based on the setup described in Fig. 3, three narrowband (5MHz), single chip MG-Y laser were combined to generate three line tunable laser source. MG-Y-Branch tunable laser is generally used solely for telecommunication applications. The wavelength channels of the MG-Y laser can be chosen based on the ITU grid. Its wavelength tuning is based on high speed electronics and the tuning range can cover the whole C + L band. The combined laser output was then modulated by a −60 dB excitation ratio cascade pulse modulator, consisting of two Mach–Zehnder interferometers. The modulation generated a time referenced probe pulse train (20ns, 500Hz), and the pulse train was then amplified by an EDFA before injected into the sensing network. The coherence length of the laser is long, so the leakage waves from other channels in a TDM system due to the limited extinction ratio of the optical pulse modulator may interfere with the signal wave and result in unwanted interferometric signals. To reduce the impact of this crosstalk on the wavelength detection accuracy, the high excitation ratio pulse modulator setup was used [12].

A polarization controller (PC) was placed in front of the EOM to enhance the modulation efficiency. Two sensing channels at the wavelengths of ~1550.80nm and ~1555nm were used, and ten identical cascaded USFBGs were used in each channel.

These USFBGs were fabricated with ultrashort lengths of ~500 µm. Their 3 dB bandwidth were ~1.4 nm, and reflectivity were 2-3%. The fiber separation between two neighboring gratings were ~5 m. The receiver section consisted of a 3-port OMBPF acting as a band splitter separating the laser pulses at the three wavelength lines. The OMBPF was implemented by 4 ports programmable optical filter (Waveshaper, 4000s). The filter transmission spectrum was set to have the rectangular interleaved passbands with the wavelengths of each port matching the three lines of the laser. The spectra of the OMBPF as well as the 3 output ports of the filter transmission function are shown in Fig. 4, three low noise DC-100MHz APDs and transimpedence amplifiers were used for detection, and a three channel ADC was connected to a computer for data processing. The required power level at the receiver is balanced with the goal of reducing any sources of interference while maintaining sufficient SNR at the receiver.

 figure: Fig. 4

Fig. 4 OMBPF three port periodic transmission function.

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In the experimental setup, different identical USFBGs in the same channel was resolved with different time delays of the probe pulse, and the central wavelength of each USFBG was obtained by using the differential output of two maximum APD signals. The laser lines were switched from [1550.6 nm, 1552 nm, 1553.4 nm] to the [1554.8 nm, 1556.2 nm, 1557.8 nm] to cover the two sensing channels by using the high speed electronic tuning, as shown at Fig. 5, the switching time of the MG-Y lasers was less than 10 msec.

 figure: Fig. 5

Fig. 5 Identical USFBGs spectrum and wavelength lines of laser for two channels.

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We applied axial strains at the first channel’s second sensor, USFBG12, was glued to a micrometric translation stage with a resolution of 2 µm. The integrated time domain detected signals of three APD under different strain are shown in Fig. 6, where we can find that the three APDs peak values corresponding to the tested USFBG12 change significantly, while the others keep relatively stable. Since the USFBGs reflectivity are low, the measurement of any sensing channels was integrated 10 times. Thus the sensing speed slowed down to 50Hz.

 figure: Fig. 6

Fig. 6 First channel time domain signals of three APDs under (a) zero strain, (b) ~500 µƐ, (c) ~750 µƐ, (d) ~1500 µƐ, (e) ~2250 µƐ and (f) ~3000 µƐ, (APD1, APD2 and ADP3 signals are shown in blue, red and green respectively).

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To characterize the sensing response, which is the relationship between the peak differential value and the strain, we performed a detailed analysis for the USFBG12. The center wavelength of the grating was tuned by gradually increasing the applied stain. At the same time, the corresponding peak height change of three lines of the laser was also recorded. The obtained differential measurement data, as well as the linear fitted curve is shown in Fig. 7.

 figure: Fig. 7

Fig. 7 (a) The evolution of the USFBG12 spectrum as the applied strain increases, (b) three APDs peak responses under different strains and (c) differential values [log(I2)-log(I1)] and [log(I2)-log(I3)] vs. strain, red and blue line, respectively.

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It is found that the strain sensitivity of USFBG12 is ~0.54dB/µƐ, and a relatively wide linear sensing range of >3000 µƐ was obtained. To verify the immunity of the proposed system to the transmission-link related intensity changes, we introduced bending losses between the third and the fourth grating, by coiling the fiber of that section around a cylinder, and compared the sensor responses. It is found that the peak level decreases after the bending. But the differential output is still unchanged after the bending.

3.1 Measurement resolution

Theoretically, during the measurement, with the stationary physical (temperature/strain) conditions, the wavelength shift should be zero. But the actual sensing signals are affected by the systematic errors, i.e., they suffer from the intensity noise of the laser and the APD. The intensity noise in the measurement setup includes the shot noise of the light, the electronic noise of the APD, and the quantization noise in the A/D conversion process. It mainly leads to some fluctuations in the APD output signal, which strongly depends on the USFBG’s reflectivity. These errors can be considered as the origin for the system resolution deterioration. In other words, the interrogation systematic error can be regarded as the system resolution, as shown in Eq. (5):

ΔP(λB)=log((I2(λB)+ΔI2)/(I1(λB)+ΔI1))

In order to investigate the strain sensing resolution, ΔP, the measurement was repeated 50 times under the same condition. There are some differences in the measurement of the USFBGs differential output, which are mainly caused by the different peak reflectivity and hence different signal-to-noise ratios of the USFBGs. Experimentally measured resolution of the two sensing channels USFBGs are less than ΔP = ± 0.017 dB/µƐ.

3.2 Multiplexibility

FBG’s multiplexing capacity is determined by several factors, including the fiber optic transmission loss, Rayleigh scattering, FBG excess losses, coupling ratios of up-stream sensors, the source input power and photodetector accuracy.

Multiple-reflection crosstalk

The proposed network, similar to other TDM weak FBGs sensing networks, will mainly suffer from spectral-shadowing crosstalk and multiple-reflection crosstalk [13]. Multiple-reflection crosstalk refers to the interference by the false signal, which undergoes multiple reflections between the upstream USFBGs and arrives at the detector at the same time with the real signal of the downstream FBGs. It is clear that as the sensor count of the system grows, the total number of reflected crosstalk pulses may become quite large.

The actual returned power of the kth USFBG can be considered as the sum of the returned power, Ik, from the kth USFBG:

Ik(λ)=λ(1R(λλB))2k2R(λλB)S(λλl)dλ=i=02(k1)C2(i1)i(1)iRmi+1λR(λλB)S(λλl)dλ

After substituting Eq. (2) into Eq. (7) the integral result of Eq. (7) can be given by:

Ik(λl)RmSλlexp[(4ln2)(λlλBΔλB)2]i=02(k1)C2(k1)i(1)iRmi+1/i+1

From Eq. (8), the intensity at the far-end USFBG, after passing many previously multiplexed USFBGs, not only depends upon its spectral function, but also upon the multiplexed attenuation factorA(k)=i=02(k1)C2(k1)i(1)iRmi+1/i+1. It can be found that the crosstalk level significantly increase with the grating reflectivity, which agrees with the previous studies for the ultra-weak FBGs. Therefore, we need to account for the multiple reflection crosstalk noise. Here if Rm is less than −38dB and the number of FBGs is < 1000, the negative influence of multi-reflection can be ignored.

The simulation shows that the crosstalk level remains unchanged as the grating FWHM, ΔλB, increases as shown in Fig. 8, this indicates that, in contrast to the tradeoff between the grating reflectivity and the multi-reflection crosstalk, there exists no dependence of the crosstalk on the grating bandwidth. Thus, one can conclude that when the same grating reflectivity, total sensor number, detector and light source are used, the network using USFBGs as sensors instead of traditional gratings has the same multi-reflection crosstalk level, highlighting the simple demodulation advantage of the USFBGs based sensing network.

 figure: Fig. 8

Fig. 8 Multi-reflection crosstalk level versus grating bandwidth

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Another crosstalk source of the sensing network is the spectral shadowing crosstalk, which is defined as the distortion of a downstream FBG’s spectrum due to light having to pass twice through an upstream FBG. Then the returned spectrum from that grating will not be a true representation of its spectrum, but will be the multiplication of the upstream grating and the interrogation spectrum. When the conventional wavelength-domain detections are used, the distorted relfection spectrum increases the uncertainty when determining the spectrum center, and thus introduces errors in the measurement. In our case, this spectral-distortion will be transferred to a response curve change of the interrogator, which then, if not be re-calibrated, affects the accuracy of the measurements.

To investigate how the USFBGs spectral distortion change the response curve of the interrogator, numerical studies were conducted. In the simulation, laser lines are switched to 1550.6 nm, 1552 nm and 1553.4 nm. The total identical USFBG number is 1000, each grating reflectivity is −38 dB, and the grating FWHM is 1.4 nm. The Bragg wavelengths of all the upstream gratings are randomly centered around 1550.8 nm with a range of 1.4 nm, as the real situation. Figure 9(a) shows the reflection spectra of the 1000th USFBG and the first one. Due to the spectral shadowing effect, the spectrum of the 1000th grating is wider, and the reflectivity appears lower.

 figure: Fig. 9

Fig. 9 (a) Spectrum distortion of the 1000th grating due to spectral shadowing effect. (b) Response curve change of the 1000th grating.

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Figure 9(b) compares the simulated response curves between these two gratings. The slope of the curve for the 1000th grating is slightly lower than the first one, and the linear range is reduced. It is due to the fact that the ratio of the laser line spacing to the 1000th grating FWHM is smaller . However, it can be found that the measuring curve still maintains a good linear response.

Power decay rate

Ideally, it is highly desirable to have the same reflected power from each of the FBG sensors to balance the performance of the FBGs within the array, but it is not possible for the identical FBGs based sensing network, due to the insertion loss of each FBG. Thus, to better balance the sensor performance, it is critical to acquire a low power decay rate of the returned signals from a sensor array [13]. Obviously, the power decay rate is strongly related to the FBG reflectivity, because it will directly affect the insertion loss of each FBG. From Eq. (8), we can find the reflected power changes with the sensor number, grating reflectivity, and bandwidth. One can clearly find that the decay rate becomes larger as higher reflectivity gratings are employed, but appears unchanged with different grating bandwidths. These results suggest that the use of broadband gratings will not change the power decay rate, and thus not increase the performance difference between the sensors within an array.

4. Conclusion

A multi-wavelength channel USFBG sensing network based on a triple-wavelength laser interrogation has been reported. The whole sensing system is implemented as a very fast and low cost intensity-based sensing network. In the time division multiplexing of the network, each grating will be resolved as three adjacent discrete peaks. The differential detection of each grating thus can be achieved by calculating the peak-to-peak ratio of two maximum peaks. At least tens of wavelength channels of 1000 identical USFBGs can be potentially multiplexed in our system with a high measuring performance by switching the MG-Y lasers wavelength matched to the OMBPF wavelength channels.

Twenty USFBGs in two wavelengths have been fabricated and used to prove the concept. The self-referencing capability of the distributed sensing has been verified and the application for the distributed strain measurement has been demonstrated. Relatively wide linear measurement range of near 3000 µƐ has been realized with the current ~1.4 nm FWHM USFBGs.

The remarkable features of high flexibility, self-referenced capability, wide measuring range, and fast and simple interrogation make the proposed system very attractive for various large-scale distributed sensing applications, especially for the applications requiring real-time monitoring over a large area, such as homeland security, wind power systems, and natural disaster warning.

5. Acknowledgment

This work is supported by the National Natural Science Foundation of China (No. 61675078) and the sub-Project of the Major Program of the National Natural Science Foundation of China (No. 61290315).

References

1. H. Guo, F. Liu, Y. Yuan, H. Yu, and M. Yang, “Ultra-weak FBG and its refractive index distribution in the drawing optical fiber,” Opt. Express 23(4), 4829–4838 (2015). [CrossRef]   [PubMed]  

2. X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012). [CrossRef]  

3. Z. Luo, H. Wen, H. Guo, and M. Yang, “A time- and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings,” Opt. Express 21(19), 22799–22807 (2013). [CrossRef]   [PubMed]  

4. A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015). [CrossRef]  

5. W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012). [CrossRef]  

6. P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003). [CrossRef]  

7. Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008). [CrossRef]  

8. L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013). [CrossRef]  

9. Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

10. J. Rohollahnejad, L. Xia, and R. Cheng, “Shifted Optical Gaussian Filters based Time Division Multiplexing of USFBGs Sensing Network,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTu5A.91. [CrossRef]  

11. W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000). [CrossRef]  

12. C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001). [CrossRef]  

13. P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000). [CrossRef]  

References

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  1. H. Guo, F. Liu, Y. Yuan, H. Yu, and M. Yang, “Ultra-weak FBG and its refractive index distribution in the drawing optical fiber,” Opt. Express 23(4), 4829–4838 (2015).
    [Crossref] [PubMed]
  2. X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
    [Crossref]
  3. Z. Luo, H. Wen, H. Guo, and M. Yang, “A time- and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings,” Opt. Express 21(19), 22799–22807 (2013).
    [Crossref] [PubMed]
  4. A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
    [Crossref]
  5. W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
    [Crossref]
  6. P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
    [Crossref]
  7. Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
    [Crossref]
  8. L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
    [Crossref]
  9. Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.
  10. J. Rohollahnejad, L. Xia, and R. Cheng, “Shifted Optical Gaussian Filters based Time Division Multiplexing of USFBGs Sensing Network,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTu5A.91.
    [Crossref]
  11. W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
    [Crossref]
  12. C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
    [Crossref]
  13. P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
    [Crossref]

2015 (2)

H. Guo, F. Liu, Y. Yuan, H. Yu, and M. Yang, “Ultra-weak FBG and its refractive index distribution in the drawing optical fiber,” Opt. Express 23(4), 4829–4838 (2015).
[Crossref] [PubMed]

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

2013 (2)

Z. Luo, H. Wen, H. Guo, and M. Yang, “A time- and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings,” Opt. Express 21(19), 22799–22807 (2013).
[Crossref] [PubMed]

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

2012 (2)

2008 (1)

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

2003 (1)

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

2001 (1)

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

2000 (2)

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Cerecedo-Núñez, H. H.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Chan, C. C.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

Chan, P. K.

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

Chen, P. K. C.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Delbeke, D.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Demokan, M. S.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

Dong, B.

W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Dumon, P.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Gong, J.

Guo, H.

Jin, W.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Li, X.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, B.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Liu, D.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, F.

Liu, H.

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, Q.

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Luo, B.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Luo, Z.

Miao, Y.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Pan, W.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Pathak, S.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Pickrell, G.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Qi, B.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Shillig, T. J.

Sun, Q.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Trita, A.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Van Thourhout, D.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Vermeiren, J.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Voet, E.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Wang, A.

W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Wang, D. Y.

Wang, P.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Wang, W.

Wen, H.

Wo, J.

Wu, Z.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Xu, H. G.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Yan, L.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Yang, M.

Yu, H.

Yuan, Y.

Zhang, M.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Zhang, P.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Zhang, W.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhang, Z.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Zhao, Q.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhou, H.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhou, Y.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

IEEE Photonics J. (2)

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

J. Lightwave Technol. (2)

Opt. Eng. (1)

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Opt. Express (2)

Proc. SPIE (1)

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

Sens. Actuators A Phys. (1)

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Other (2)

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

J. Rohollahnejad, L. Xia, and R. Cheng, “Shifted Optical Gaussian Filters based Time Division Multiplexing of USFBGs Sensing Network,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTu5A.91.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 USFBG spectrum and the three laser lines.
Fig. 2
Fig. 2 Sensing sensitivity and measurement range for different Bragg grating bandwidths.
Fig. 3
Fig. 3 Basic structure of the proposed sensing setup. OMBFP: optical multi-bandpass filter, EOM: electro-optic modulator, PC: polarization controller, PD: photodetector.
Fig. 4
Fig. 4 OMBPF three port periodic transmission function.
Fig. 5
Fig. 5 Identical USFBGs spectrum and wavelength lines of laser for two channels.
Fig. 6
Fig. 6 First channel time domain signals of three APDs under (a) zero strain, (b) ~500 µƐ, (c) ~750 µƐ, (d) ~1500 µƐ, (e) ~2250 µƐ and (f) ~3000 µƐ, (APD1, APD2 and ADP3 signals are shown in blue, red and green respectively).
Fig. 7
Fig. 7 (a) The evolution of the USFBG12 spectrum as the applied strain increases, (b) three APDs peak responses under different strains and (c) differential values [log(I2)-log(I1)] and [log(I2)-log(I3)] vs. strain, red and blue line, respectively.
Fig. 8
Fig. 8 Multi-reflection crosstalk level versus grating bandwidth
Fig. 9
Fig. 9 (a) Spectrum distortion of the 1000th grating due to spectral shadowing effect. (b) Response curve change of the 1000th grating.

Equations (7)

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

I= 0 R( λ λ B ) S( λ λ l )dλ
I=C S λ l R m exp[ ( 4ln2 ) ( λ l λ B Δ λ B ) 2 ]
I 1 = C 1 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 λ B ) / Δ λ B ) 2 ] I 2 = C 2 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 +Δλ λ B ) / Δ λ B ) 2 ] I 3 = C 3 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 +2Δλ λ B ) / Δ λ B ) 2 ]
{ P( λ B )=log( I 2 ( λ B ) I 1 ( λ B ) ) if( I 1 I 3 ) P( λ B )=log( I 2 ( λ B ) I 3 ( λ B ) ) if( I 1 < I 3 ) ( E+F λ B )
ΔP( λ B )=log( ( I 2 ( λ B )+Δ I 2 ) / ( I 1 ( λ B )+Δ I 1 ) )
I k (λ)= λ ( 1R( λ λ B ) ) 2k2 R( λ λ B )S( λ λ l )dλ = i=0 2(k1) C 2(i1) i (1) i R m i+1 λ R( λ λ B ) S( λ λ l )dλ
I k ( λ l ) R m S λ l exp[ ( 4ln2 ) ( λ l λ B Δ λ B ) 2 ] i=0 2(k1) C 2(k1) i (1) i R m i+1 / i+1

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