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Effect of spectrum broadening on photon-counting fiber Bragg grating sensing

ZhongHua Ou, XiaoDong Fan, LiXun Zhang, YunRu Fan, ChenZhi Yuan, LiXing You, Shuang Liu, Yong Liu, Guangcan Guo, and Qiang Zhou

Open Access Open Access

Abstract

In a photon-counting fiber Bragg grating (FBG) sensing system, a shorter probe pulse width reaches a higher spatial resolution, which inevitably causes a spectrum broadening according to the Fourier transform theory, thus affecting the sensitivity of the sensing system. In this work, we investigate the effect of spectrum broadening on a photon-counting FBG sensing system with a dual-wavelength differential detection method. A theoretical model is developed, and a proof-of-principle experimental demonstration is realized. Our results give a numerical relationship between the sensitivity and spatial resolution at the different spectral widths of FBG. In our experiment, for a commercial FBG with a spectral width of 0.6 nm, an optimal spatial resolution of 3 mm and a corresponding sensitivity of 2.03 nm−1 can be achieved.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Fiber Bragg grating (FBG) sensor is of great significance in fields of monitoring strain, temperature, and vibration [1,2]. To improve the performance of FBG sensors, many different interrogation techniques have been proposed, such as optical frequency domain reflectometry (OFDR) and random speckles [3,4]. By reducing the reflectivity of FBG, a large-scale temporally multiplexed FBG sensing system with identical central wavelength is experiencing rapid development. The FBG sensing system with high spatial resolution can improve the number of sensors per unit length and provide accurate spatial information for the field of shape sensing and source locating [5,6].

Recently, we have demonstrated a photon-counting FBG sensing system at 1.5 ${\mu }$m [7]. Due to the broad bandwidth and high sensitivity of single-photon detection, a photon-counting FBG sensing system with 5 cm spatial resolution and 0.5 pm wavelength resolution has been demonstrated. Photon-counting FBG sensing breaks through the trade-off between sensitivity and spatial resolution in the linear photo-detector, and this technique has a potential advantage to achieve a spatial resolution of millimeters. Moreover, such a system with a high spatial resolution also reaches a high sensitivity with the dual-wavelength differential detection (DWDD) method [7–10], which usually requires a probe light with a narrow spectral width <0.1 pm to increase the photons at a central wavelength of the probe light reflected by FBG. However, in the photon-counting FBG sensing with high spatial resolution, an ultra-short pulse is usually required, and the spectrum of a shorter probe pulse will be broadened according to the Fourier transform theory [11,12], which reduces the sensitivity of demodulating the central wavelength of FBG reflected spectrum. Therefore, high-precision FBG sensing in the time and spectrum domains cannot be performed simultaneously. Although the demodulation for the reflected spectrum of FBG by using a probe light with a narrow spectral width is a well-known method, the numerical and experimental investigations for the effect of spectrum broadening on photon-counting FBG sensing still lack.

In this work, we investigate the effect of spectrum broadening on photon-counting FBG sensing with the DWDD method. We develop a theoretical model and measure the performance of photon-counting FBG sensing systems at different spatial resolutions. Our results give a numerical relationship between the sensitivity and spatial resolution at the different spectral widths of FBG. In our experiment, for a commercial FBG with a spectral width of 0.6 nm, an optimal spatial resolution of 3 mm and a corresponding sensitivity of 2.03 nm$^{-1}$ can be achieved. Our works provide a theoretical and experimental direction for temporally multiplexed FBG sensing with high spatial resolution.

2. Theoretical analysis

In the DWDD method, the central wavelength of FBG reflected spectrum can be demodulated by calculating the difference between the reflected photons from FBG at ${\lambda _1}$ and ${\lambda _2}$ [7]:

$$D=\ln {C_1}-\ln {C_2},$$

The number of reflected photons for ${\lambda _i}(i=1,2)$ is calculated by the integration of reflected photons at all spectral range [9]:

$${C_i} \propto \int {\frac{{\lambda \cdot {S_i}(\lambda ) \cdot {S_{FBG}}(\lambda ) \cdot {n_{eff}}}}{{hc}}d\lambda },$$
where $h$ is Planck constant, $c$ is the speed of light in vacuum and $n_{eff}$ is the effective refractive index in optical fiber; ${S_i}(\lambda )$ is the spectrum of probe light; and ${S_{FBG}}(\lambda )$ is the reflection spectrum of FBG. To distinguish the FBG at different locations along a single fiber, the probe light must be a pulse light, which is assumed as a Gaussian pulse in the temporal domain, and the corresponding spectrum of probe light can also be written as a Gaussian function:
$${S_i}(\lambda ) = \exp \left[ { - 4\ln (2)\frac{{{{(\lambda - {\lambda _i})}^2}}}{{\Delta {\lambda ^2}}}} \right],$$
where ${\Delta \lambda }$ is the full width at half maximum (FWHM) of the spectrum of pulse light. According to the Fourier transform theory, an in-equation between the spectral width ${\Delta \lambda }$ and temporal pulse width ${\Delta \tau }$ can be expressed as [12]:
$$\Delta \tau \cdot \Delta \lambda \ge 0.441 \cdot {\lambda ^2}\cdot{{n_{eff}}}/c,$$
and thus, the spectrum of probe light will be broadened with the reduction of the pulse width. On the other hand, the DWDD method considers a Gaussian shape for the reflected spectrum of FBG, which can be written as [8]:
$${S_{FBG}}(\lambda ) = \exp \left[ { - 4\ln (2)\frac{{{{(\lambda - {\lambda _{FBG}})}^2}}}{{\Delta \lambda _{FBG}^2}}} \right],$$
where ${\Delta {\lambda _{FBG}}}$ is the FWHM of the reflected spectrum of FBG. By substituting Eqs. (3) and (5) into Eqs. (1) and (2), a linear function of ${\lambda _{FBG}}$ can be obtained:
$$D \propto 8\ln (2)\left[ {\frac{{\delta ({\lambda _c} - {\lambda _{FBG}})}}{{\Delta {\lambda ^2} + \Delta \lambda _{FBG}^2}}} \right],$$
here, ${{\lambda _c}=({\lambda _1}+{\lambda _2})/2}$ is the average wavelength, and ${\delta =({\lambda _1}-{\lambda _2})}$ is the wavelength difference between the ${\lambda _1}$ and ${\lambda _2}$. Therefore, by monitoring the signal of ${D}$, the central wavelength shift of FBG can be linearly tracked. The sensitivity of DWDD can be calculated by the derivatives of ${D}$ with respect to ${\lambda _B}$, and the upper limit of sensitivity can be further expressed according to Eq. (4):
$$s = \frac{{\partial D}}{{\partial {\lambda _{FBG}}}} \le 8\ln (2)\frac{\delta }{{{{(\frac{{0.441 \cdot {\lambda ^2}}}{{c \cdot \Delta \tau }})}^2} + \Delta \lambda _{FBG}^2}}.$$

Due to the different maximum sensitivity at each ${\Delta {\lambda _{FBG}}}$, the ${s}$ is normalized by using the maximal sensitivity, i.e., the sensitivity at ${\Delta \tau \to \infty }$, and ${s}$ can be further written as:

$${s^*} = \frac{s}{{{s_{\max }}}} \le \frac{{{{(c \cdot \Delta \tau \cdot \Delta {\lambda _{FBG}})}^2}}}{{{{(0.441 \cdot {\lambda ^2})}^2} + {{(c \cdot \Delta \tau \cdot \Delta {\lambda _{FBG}})}^2}}}.$$

Figure 1(a) shows the three-dimensional surface between the normalized sensitivity ${s^*}$, the pulse width of probe light ${\Delta \tau }$ and the reflected spectral width of FBG ${\Delta {\lambda _{FBG}}}$. Figure 1(b) is the contour line of ${s^*}$, in which the sensitivity is decreased with the reduction of ${\Delta {\lambda _{FBG}}}$ and ${\Delta \tau }$, and indicates that the ${\Delta {\lambda _{FBG}}}$ need to be increased if a high sensitivity and a high temporal resolution are required simultaneously. For example, for the FBG with a spectral width of 0.6 nm, the temporal resolution can reach 22 ps with a normalized sensitivity of 0.9; without the reduction of sensitivity, a better temporal resolution requires an FBG with a wider reflected spectrum, such as ultra-short FBG [8]. Figure 1(c) plots the curves between the normalized sensitivity and the pulse width ${\Delta \tau }$ at the different ${\Delta {\lambda _{FBG}}}$, and further indicates the degeneration of sensitivity with the reduction of the temporal resolution. Moreover, according to Eq. (6), the ratio of spectral width between the probe pulse and FBG (${{\Delta {\lambda }}}/{\Delta {\lambda _{FBG}}}$) influences the sensitivity of DWDD. A high ratio means that the measured sensitivity is the average response of the spectral envelope of the reflected probe light, which is always lower than the maximal sensitivity. Since the spectral width is inevitably broadened with the reduction of probe pulse width according to the Fourier transforms theory, a narrow probe pulse width causes a decrease in sensitivity. These numerical results not only give the constraint between the ${s^*}$, ${\Delta {\lambda _{FBG}}}$, and ${\Delta \tau }$, but also provide the theoretical direction for improving the performance of temporally multiplexed FBG sensing.

 figure: Fig. 1.

Fig. 1. Theoretical and simulated results. (a) Three-dimensional surface between the normalized sensitivity ${s^*}$, the pulse width ${\Delta \tau }$ and the spectral width of FBG ${\Delta {\lambda _{FBG}}}$. (b) Contour line of normalized sensitivity ${s^*}$. (c) Curves between the normalized sensitivity ${s^*}$ and the pulse width ${\Delta \tau }$ at the different ${\Delta {\lambda _{FBG}}}$.

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3. Experimental setup and results

In our proof-of-principle demonstration, we used a commercial FBG temperature sensor to investigate the spectrum-broadening effect of probe light on the photon-counting FBG sensing. The experimental diagram with the DWDD method is depicted in Fig. 2(a). A broadband coherent probe laser is generated from a mode-locked pulsed laser (Alnair Labs, PFL-200M) with a central wavelength of 1548.7 nm, a pulse width of 1 ps, a repetition rate of 20 MHz, and a 3dB spectral bandwidth of 3.5 nm. The output spectrum of the mode-locked pulsed laser is shown in Fig. 2(b). The probe light is divided into two parts by an optical coupler (OC, 90:10), one of them is received by an InGaAs photoelectric detector (Thorlabs, PDA015C/M) to generate a synchronizing electronic signal for a high-resolution time-to-digital converter (TDC, ID Quantique, ID900). Then, another probe light is injected in a tunable filter (EXFO, XTA-50) to generate a pulse light with different spectral width at the central wavelength of 1550.1 nm or 1550.2 nm in turn, so that the DWDD method can demodulate the central wavelength of FBG within a linear range. It is worth noting that the optical power difference of the laser in two wavelengths is corrected by using the normalization method [10]. In order to avoid the saturation of a single-photon detector, the mean photon number of the probe laser is attenuated to 0.003 per pulse by adjusting a variable optical attenuator (VOA).

 figure: Fig. 2.

Fig. 2. Experimental setup of high spatial resolution DWDD and operating results. (a) Schematic of the experimental setup. OC: optical coupler, PD: photoelectric detector, VOA: variable optical attenuator, PC: polarization controller, TC: temperature controller, SNSPD: superconducting nanowire single-photon detector, TDC: time-to-digital converter. (b) The output spectrum of a mode-locked pulsed laser. (c) Reflected spectrum of FBG. (d) Recorded histogram of reflected photons by TDC. Photon number is calculated by the sum of counts in the time windows of 300 ps.

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An FBG sensor with a central wavelength of 1550 nm, an FWHM of 0.6 nm, and a reflectivity of 5% is used to demonstrate, and its reflected spectrum is measured by an amplified spontaneous emission source, and an optical spectrum analyzer (Yokogawa, AQ6370D) with a resolution of 0.02 nm, as shown in Fig. 2(c). We use an SNSPD (Photon Technology Co., P-CS-6) to receive the optical signals with a high detection efficiency and a low noise at 1550 nm. Its time jitter is 140 ps, the detection efficiency is about 75% while the dark counts are 80 cps, corresponding to a noise equivalent power (NEP) of ${2.16 \times {10^{ - 18}}{\rm {\ W/}}\sqrt {Hz}}$ [13,14]. To obtain an SNSPD with a high polarization sensitivity, a polarization controller (PC) is used to control the polarization of reflected photons. Then, SNSPD converts an incident photon into a voltage signal which is recorded by TDC with a high time resolution of 13 ps. Figure 2(d) shows the recorded histogram of reflected photons when the acquisition time of TDC is set as 1 s, and the photon number is calculated by the sum of counts in the time windows of 300 ps.

To measure the sensitivity of DWDD, temperature sensing is demonstrated by placing the FBG sensor into a heating unit controlled by a temperature controller (Thorlabs, TED200C) with a temperature stability of 0.01 $^{\circ }\textrm {C}$. The temperature change of 1 $^{\circ }\textrm {C}$ leads to a wavelength change of around 10 pm for the conventional FBG temperature sensors [15]. The temperature is increased from 35 $^{\circ }\textrm {C}$ to 50 $^{\circ }\textrm {C}$ with an interval of 5 $^{\circ }\textrm {C}$, and thus the measurement range is in the FWHM of FBG. By setting the spectral width of probe light as 50 pm, 100 pm, 200 pm, 300 pm, 400 pm, and 500 pm, respectively, the results of temperature sensing with different spectral widths are obtained as shown in Fig. 3. The solid lines are the linear fitting curves obtained by the Monte Carlo method [16], in which 1000-time random sampling is performed around the measured data assumed to be Poisson distribution. By averaging the slope of the 1000 fitting curves, the sensitivity of each temperature sensing curve is fitted and is listed in Table 1, in which the R-square (R${^2}$) is also used to evaluate the linearity of fitting curves [17].

 figure: Fig. 3.

Fig. 3. Temperature sensing curves measured by DWDD with different pulse spectral widths. (a) ${\Delta {\lambda }}$ = 50 pm; (b) ${\Delta {\lambda }}$ = 100 pm; (c) ${\Delta {\lambda }}$ = 200 pm; (d) ${\Delta {\lambda }}$ = 300 pm; (e) ${\Delta {\lambda }}$ = 400 pm; (f) ${\Delta {\lambda }}$ = 500 pm. The error bar is calculated by the Monte Carlo method.

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Tables Icon

Table 1. Sensitivity and R${^2}$ of temperature sensing curves with different pulse spectral width ${\Delta \lambda }$.

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Furthermore, the experimental curve between the spectrum and pulse width is obtained to show the spectrum broadening, and the normalized temporal pulse data with different pulse spectral widths are presented in Fig. 4(a). Each temporal curve is fitted by the Gaussian function, and the FWHM of temporal pulse width can be estimated. In addition, the temporal pulse width directly obtained by TDC contains the time jitters of SNSPD and TDC, and the jitters from the laser source and its synchronization signal, their relationship can be expressed as a sum of squares [18]:

$$\Delta \tau _{Detected}^2 = \Delta \tau _{SNSPD}^2 + \Delta \tau _{TDC}^2 + \Delta \tau _{Sync}^2 + \Delta {\tau ^2}.$$

In our experiment, ${\Delta {\tau _{SNSPD}}}$ is 140 ps, ${\Delta {\tau _{TDC}}}$ is 5.7 ps, and ${\Delta {\tau _{Sync}}}$ is 1 ps. By this calibration process, ${\Delta {\tau }}$ can be obtained, and the Fourier transforms between spectrum width and pulse width is shown in Fig. 4(b).

 figure: Fig. 4.

Fig. 4. Experimental results. (a) Temporal pulse data with different spectral widths of the probe light is directly obtained by TDC and the solid line is the Gaussian fitting curves. (b) Experimental Fourier transform curve between spectrum width and pulse width. The green solid line is the theoretical curve, pink points are experimental data. (c) Experimental curves of the relationship between the pulse width and sensitivity for FBG with a spectral width of 0.6 nm. The error bar is the standard deviation calculated by 30 sets of data.

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Next, the relationship between the ${\Delta {\tau }}$ and sensitivity for an FBG reflected spectral width of 0.6 nm is shown in Fig. 4(c). Compared to the theoretical curve, the experimental sensitivity shows the same downward trend with the reduction of ${\Delta {\tau }}$. The approximate Gaussian spectrum of the probe laser and FBG causes deviations between the experimental data and the theoretical curve. When ${\Delta {\tau }}$ decreases from 100 ps to 10 ps, the sensitivity decreases from 2.45 nm${^{-1}}$ to 1.31 nm${^{-1}}$. To balance the temporal resolution and sensitivity, an optimal temporal resolution of 31 ps is found, which has reached a normalized sensitivity of 0.8 and corresponds to the spatial resolution of 3 mm and the sensitivity of 2.03 nm${^{-1}}$ [19]. Therefore, our proof-of-principle experiment has verified the effect of spectrum broadening on photon-counting FBG sensing. Although our theory and experiment are performed by the DWDD method, our methods can also analyze the optimal spatial resolution of other FBG methods with multi-wavelength demodulation for temporally multiplexed FBG sensing [20,21]. In addition, whether the FBGs have high or low reflectivity, the reflected spectral width of FBG for temporally multiplexed FBG sensing can also be realized with our method. Finally, the cost of SNSPD may constrain the application of photon-counting FBG sensors to cost-insensitive scenarios [22]. In the near future, a cost-effective photon-counting FBG sensor based on a silicon single-photon avalanche photodiode could be possible for practical applications.

4. Conclusion

In this work, we have investigated the effect of spectrum broadening on a photon-counting FBG sensing with a DWDD method. We develop a theoretical model and measure the performance variation of the photon-counting FBG sensing system by using probe light with different spectral widths. Our results give a numerical relationship between the sensitivity and spatial resolution at the different spectral widths of FBG. In our experiment, for a commercial FBG with a spectral width of 0.6 nm, an optimal spatial resolution of 3 mm and a corresponding sensitivity of 2.03 nm${^{-1}}$ can be achieved. Our works provide a theoretical and experimental direction for temporally multiplexed FBG sensing with high spatial resolution.

Funding

Sichuan Science and Technology Program (2021YFSY0062, 2021YFSY0063, 2021YFSY0064, 2021YFSY0065, 2021YFSY0066, 2022YFG0080, 2022YFG0084); Innovation Program for Quantum Science and Technology (2021ZD0301702); National Natural Science Foundation of China (61775025, 62005039, 91836102, U19A2076); National Key Research and Development Program of China (2018YFA0306102, 2018YFA0307400).

Acknowledgment

The authors thank Prof. Wei Zhang from Tsinghua University for providing us with the mode-locked femtosecond laser.

Disclosures

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. Theoretical and simulated results. (a) Three-dimensional surface between the normalized sensitivity ${s^*}$, the pulse width ${\Delta \tau }$ and the spectral width of FBG ${\Delta {\lambda _{FBG}}}$. (b) Contour line of normalized sensitivity ${s^*}$. (c) Curves between the normalized sensitivity ${s^*}$ and the pulse width ${\Delta \tau }$ at the different ${\Delta {\lambda _{FBG}}}$.

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Fig. 2.
Fig. 2. Experimental setup of high spatial resolution DWDD and operating results. (a) Schematic of the experimental setup. OC: optical coupler, PD: photoelectric detector, VOA: variable optical attenuator, PC: polarization controller, TC: temperature controller, SNSPD: superconducting nanowire single-photon detector, TDC: time-to-digital converter. (b) The output spectrum of a mode-locked pulsed laser. (c) Reflected spectrum of FBG. (d) Recorded histogram of reflected photons by TDC. Photon number is calculated by the sum of counts in the time windows of 300 ps.

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Fig. 3.
Fig. 3. Temperature sensing curves measured by DWDD with different pulse spectral widths. (a) ${\Delta {\lambda }}$ = 50 pm; (b) ${\Delta {\lambda }}$ = 100 pm; (c) ${\Delta {\lambda }}$ = 200 pm; (d) ${\Delta {\lambda }}$ = 300 pm; (e) ${\Delta {\lambda }}$ = 400 pm; (f) ${\Delta {\lambda }}$ = 500 pm. The error bar is calculated by the Monte Carlo method.

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Fig. 4.
Fig. 4. Experimental results. (a) Temporal pulse data with different spectral widths of the probe light is directly obtained by TDC and the solid line is the Gaussian fitting curves. (b) Experimental Fourier transform curve between spectrum width and pulse width. The green solid line is the theoretical curve, pink points are experimental data. (c) Experimental curves of the relationship between the pulse width and sensitivity for FBG with a spectral width of 0.6 nm. The error bar is the standard deviation calculated by 30 sets of data.

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Tables (1)

Tables Icon

Table 1. Sensitivity and R 2 of temperature sensing curves with different pulse spectral width Δ λ .

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Equations (9)

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

D = ln ⁡ C 1 − ln ⁡ C 2 ,
C i ∝ ∫ λ ⋅ S i ( λ ) ⋅ S F B G ( λ ) ⋅ n e f f h c d λ ,
S i ( λ ) = exp ⁡ [ − 4 ln ⁡ ( 2 ) ( λ − λ i ) 2 Δ λ 2 ] ,
Δ τ ⋅ Δ λ ≥ 0.441 ⋅ λ 2 ⋅ n e f f / c ,
S F B G ( λ ) = exp ⁡ [ − 4 ln ⁡ ( 2 ) ( λ − λ F B G ) 2 Δ λ F B G 2 ] ,
D ∝ 8 ln ⁡ ( 2 ) [ δ ( λ c − λ F B G ) Δ λ 2 + Δ λ F B G 2 ] ,
s = ∂ D ∂ λ F B G ≤ 8 ln ⁡ ( 2 ) δ ( 0.441 ⋅ λ 2 c ⋅ Δ τ ) 2 + Δ λ F B G 2 .
s ∗ = s s max ≤ ( c ⋅ Δ τ ⋅ Δ λ F B G ) 2 ( 0.441 ⋅ λ 2 ) 2 + ( c ⋅ Δ τ ⋅ Δ λ F B G ) 2 .
Δ τ D e t e c t e d 2 = Δ τ S N S P D 2 + Δ τ T D C 2 + Δ τ S y n c 2 + Δ τ 2 .
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