1. Introduction

Stimulated Brillouin scattering (SBS) is a fundamental nonlinear optical effect [1], which have been investigated extensively in the past [2]. This prominent fiber nonlinearity has been applied in numerous applications, such as band-pass microwave photonic filter (MPF) [3–5], distributed fiber-optic temperature and strain sensing [6,7], and high-resolution optical spectrum analyzer (OSA) [8–10]. SBS can be regarded as tunable narrow-band optical filter (TNOF) because of its inherent narrow bandwidth and flexible tunability [11]. The center wavelength of the SBS-TNOF can be tuned easily by applying external modulation to the pump [4], or using tunable laser source (TLS) as the pump [8]. The bandwidth and the shape of SBS-TNOF can also be flexibly reconfigured by tailoring the pump profile [12–15]. These merits make SBS-TNOF desirable to meet different requirements.

For SBS-TNOFs, the frequency response is one of the most important characteristics, which is closely related to the Brillouin gain spectrum (BGS). The basic SBS-TNOF frequency response is Lorentzian/Gaussian shape because of the natural BGS shape [11], and the typical bandwidth is a few tens of MHz for different fibers and different incident power conditions [16,17]. Because the effective BGS for a broadband pump can be obtained by convolving the natural Brillouin gain with the pump spectrum [1], SBS-based filters with rectangular-shaped and arbitrary-shaped frequency response have been proposed utilizing with variable-amplitude pump lines or frequency-sweeping pump [13,14]. While all these implementations pay attention to the filter frequency response, few of them have addressed the issue of the frequency response under continuously tuning condition, corresponding response model is not further analyzed either. Actually, the filter response will change and cause spectral distortion when the pump tuning speed is too fast in practical applications such as SBS-OSA. In order to maintain narrow-band response during tuning process, the frequency response of SBS-TNOF under continuously tuning condition need to be modeled and analyzed.

Additionally, in order to recover the distorted optical spectrum and extend the available sweep speed, we adopt deconvolution to eliminate distortion. Deconvolution algorithms have been extensively studied in deblurring signals and images for space telescopes [18], spectral resolution enhancement for Raman spectroscopy [19], and spatial resolution improving for distributed fiber-optic sensing [20,21]. Richardson–Lucy (RL) iterative algorithm provides a maximum likelihood estimation, which has been widely used in fore mentioned areas [22–24].

In this paper, we firstly propose modified expressions under wavelength-swept pump (i.e., continuously tuning) condition based on the nonlinear coupled-wave equations. Then the frequency response model of SBS-TNOF are derived and analyzed. Theoretical analysis shows that the broadening of SBS-TNOF response is induced by pump wavelength difference at different positions of the fiber, which can be equivalent to the convolution with broadband pump as well. Experimental results under different parameters demonstrate that the response profile of SBS-TNOF becomes asymmetrically broadened. And the bandwidth of SBS-TNOF depends on the combined effect of gain saturation and sweep induced response broadening. By adopting the widely used RL deconvolution method and utilizing SBS-TNOF response model as the point spread function (PSF), the distorted responses are effectively reconstructed and the sweep speed dependency is almost eliminated. Furthermore, commonly used on-off keying (OOK) signal is tested using the proposed reconstruction method to assess its performance in the SBS-OSA. The reconstructed results show that both the overall profile and the detail are significantly recovered. Quantitatively evaluation illustrates that the feasible sweep speed can be improved from ∼45 nm/s to over 95nm/s.

2. Theoretical analysis

2.1 Modification under the wavelength-swept pump condition

The process of SBS can be classically described as a nonlinear interaction between the pump and Stokes fields through an acoustic wave [1]. In a single mode fiber where Stokes wave counter-propagates with the pump wave, a set of two partial differential equations can be used to describe the SBS process. When the pump is swept to tune the center wavelength of the SBS-TNOF, only the wavelength changes, the intensity of the pump is independent with time. So that the sweep pump can be regarded as a quasi-continuous wave (CW) pump, and under steady-state conditions, the coupled equations could be expressed as follows [2]:

In SBS process, the backscattered Stokes wave interferes with the input pump wave and generates an acoustic wave through the effect of electrostriction when they counter propagate [2]. The scattered wave creates an SBS gain window, which is downshifted in frequency by the Brillouin shift, as shown in Fig. 1. When the pump is sufficiently strong, the scattered wave carries most of the pump power and only the spectral components falling within the SBS gain window can be amplified. Therefore, the SBS effect in fiber can be considered as an optical filter with an ultra-narrow bandwidth.

 

Fig. 1. Principle of SBS-TNOF response broadening under wavelength-swept pump condition. As the pump with time-variant frequency travels through the fiber, the frequency difference at different positions of the fiber introduces a mismatch of the BGS and results in the filter response broadening.

Download Full Size | PDF

When the pump is not swept, the pump wavelength keeps constant, and the filter response has the same shape as the BGS. The coupled Eqs. (1) and (2) can be directly used. However, for the case of continuously tuning using tunable laser, pump wavelength varies with time. Assuming the pump sweeps from low wavelength to high wavelength continuously and linearly, and use $\Delta f\textrm{ = }({ - c/{\lambda^2}} )\Delta \lambda$ to convert wavelength sweep to frequency sweep, the time-variant pump frequency can be expressed as:

Therefore, as the pump light travels through the fiber, there will be a frequency difference at different positions of the fiber as shown in Fig. 1. Because the center frequency of the BGS is determined by the pump frequency, this frequency difference directly introduces a mismatch of the BGS along the fiber and results in the filter response broadening.

The BGS mismatch Δf can be expressed as:

By combining Eqs. (3)–(5), g_B can be modified as:

Actually, the sweep induced mismatch can be regarded as a broadband pump, which seems to have a similar mechanism as the frequency-sweeping pump used in SBS-MPFs [13,14]. The frequency-sweeping pump in such filters can be considered as a broadband pump as long as the pump propagation time through the fiber is much longer than the duration of the sweeping cycle [13]. Short sweeping duration ensures the convolution of the entire broadband pump and BGS, and in order to realize that, their sweeping speed is up to several thousands of nm/s (e.g., 1 GHz/1 μs (8000 nm/s @1550 nm) [13] or 500 MHz/2.5 μs (1600 nm/s @1550 nm) [14]). In contrast, for SBS-TNOF, the BGS convolves with the pump that travel through the whole fiber, and response broadening in SBS-TNOF is a side effect of continuously tuning rather than achieving a wider bandwidth. And because of the narrow bandwidth, response broadening occurs easily. If we take n_eff, L and V_p as 1.5, 2 km and 2.5 THz/s (20 nm/s @ 1550 nm) respectively in Eq. (5), Δf (i.e., bandwidth of the broadband pump) will be 25 MHz (0.2 pm @ 1550 nm), which means that sweep speed of just several tens of nm/s will cause significant degradation.

2.2 Frequency response of SBS-TNOF

Based on the coupled equations and modified Brillouin gain expression, we are able to analyze frequency response of the SBS-TNOF under wavelength-swept condition. The output optical power of SBS-TNOF is acquired by integrating Eq. (2) over the whole spectrum and fiber length. Several kilometers of highly nonlinear fiber (HNLF) is used in SBS-TNOF to generate SBS effect [9], so the fiber attenuation on signal is negligible and the second term of Eq. (2) dominates. Thus, the total Stokes power at z=0 can be written as:

Equation (7) indicates that the Stokes power is time variant according to the change of pump frequency. And it can be explained as two convolutions: the BGS first convolves with the power distribution along the fiber to form the filter response, and then the response convolves with the input signal spectrum to obtain the output signal spectrum.

By using Eq. (4) to map wavelength to time, the frequency response of SBS-TNOF $H(f )$ under wavelength-swept pump condition can be succinctly expressed as:

And the output signal spectrum ${P_{s\_out}}(f )$ can be readily expressed as:

Thereby, if the pump is not swept or the sweep speed is low, the filter response has a narrow band Lorentzian shape without broadening, and it can be treated as an ideal pulse shape function. Equation (9) becomes ${P_{s\_out}}(f )\approx {P_s}({{f_s}} )$, which means the result is approximately equal to the input signal spectrum. However, under wavelength-swept condition, $H(f )$ is no longer ideal pulse shape, but broadened as shown in Fig. 1. And the measured spectrum will be distorted consequently.

On the other hand, when a narrow linewidth laser is used as the input signal, if its linewidth is much narrower than the BGS bandwidth, Eq. (9) becomes ${P_{s\_out}}(f )\approx H(f )$, which means the result will be equal to the response, as shown in Fig. 2. Linewidth of 100 kHz can be easily achieved with commercialized narrow linewidth laser [25]. And this technique has already been applied to the characterization of the SBS gain profile [26]. Therefore, we use this method to analyze the response broadening under wavelength-swept condition. In this method, as narrow linewidth laser signal passes through the moving SBS-TNOF, measured time-domain signal represents its response information. And the response is mapped from the time-domain signal to frequency domain by using Eq. (4), which also causes the response to be the reversal of BGS, as shown in Fig. 2. This reversal is consistent with the reflection in convolution when the process is explained as the convolution of the broadband pump and natural BGS.

 

Fig. 2. Principle of SBS-TNOF response measurement using narrow linewidth laser signal and wavelength-swept pump.

Download Full Size | PDF

2.3 System bandwidth consideration

In order to avoid signal distortion due to the electric filtering in other electronic part such as photo detector and data acquisition module, we estimate the bandwidth of the SBS-TNOF. For simplicity, the BGS broadening can be ignored because it doesn’t add up the high frequency component. And we obtain the normalized output signal with Eq. (7).

It is worth noting that the frequency response is obtained by time-frequency mapping and it is still a time-domain signal in essence. Using Fourier transform, we obtain:

And then we can obtain the FWHM of the output signal, which is also the -3dB bandwidth of SBS-TNOF.

Equation (12) indicates that the bandwidth of SBS-TNOF depends on the sweep speed and BGS bandwidth. We can use Eq. (12) to estimate the bandwidth. Δf_B is typically several tens of MHz, taking 25 MHz as an example, when V_p is up to 12.5 THz/s (100 nm/s at 1550 nm), the output signal bandwidth is still about 1.4 MHz, which can be easily satisfied.

2.4 Deconvolution method

According to Eq. (9), the spectral distortion is caused by convolution of original spectrum and distorted response. In order to obtain the original spectrum, the reverse operation of convolution—deconvolution—can be used to eliminate distortion. We adopt the widely used deconvolution method—Richardson–Lucy (RL) iterative algorithm to reconstruct spectrum. The spectrum degradation is modeled as [24]:

The iterative R–L algorithm can be expressed as:

The SBS-TNOF response can be used as the PSF in the R-L iteration. However, it can only be solved numerically. In order to obtain the analytical model, pump depletion is neglected [1]. Using ${P_P}(z )= {P_P}(0 ){e^{ - \alpha z}}$ in Eq. (8), PSF can be modeled as:

For the more general case of pump depletion, we can use experimental data of distorted response under various sweep speed to calibrate the PSF. Therefore, the reconstruction method can be concluded into three steps, that is PSF calibration, R-L deconvolution, and convolution of estimated spectrum and ideal PSF. The reason of the last convolution is that our purpose is to eliminate distortion rather than improve spectral resolution.

3. Experimental setup and results

3.1 Experimental setup

Figure 3 shows the experimental system for frequency response measurement of SBS-TNOF and subsequent deconvolution test. The setup consists of three parts: signal generation part, SBS-TNOF part and data acquisition and control part. Solid and dash lines represent optical and electrical links respectively.

 

Fig. 3. Experimental setup for SBS-TNOF frequency response measurement and deconvolution test under different wavelength sweep speed.

Download Full Size | PDF

In the signal generation part, a narrow linewidth laser with the linewidth of 100 kHz is generated using the laser source (SANTEC TSL-710). The center wavelength of the signal is set to 1550 nm during the experiment. The electro-optic modulator (EOM) and the RF signal are used to generate the 2.5 GHz non-return-to-zero (NRZ) OOK signal. The state of polarization (SOP) of signal is adjusted by polarization controller (PC 1) to match the polarization axis of the modulator.

In the SBS-TNOF part, a tunable laser source (LUNA Phoenix 1200) with the intrinsic linewidth of 100 kHz operates as the wavelength-swept pump source. The mode hop free tuning range of this laser is 1515∼1565 nm, and it is continuously tuned from 1549.5 nm to 1550.5 nm (i.e., 193.414 THz ± 125 GHz) in the experiment. An erbium doped fiber amplifier (EDFA) is utilized to stabilize and amplify the optical power. PC 2 is used to match the SOP of the signal and the pump. The HNLF with single peak BGS, bandwidth of ∼ 17 MHz, and peak Brillouin gain efficiency of ∼ 2.45 m⁻¹·W⁻¹ is used as the SBS medium [27]. Because both ends of the HNLF are fused with standard single mode fiber (SSMF) FC/APC connectors, the pump and signal power are calibrated by deducting the splice loss from HNLF to SSMF.

In the data acquisition and control part, A photo detector (PD) is used to detect the optical power of the Stokes wave. Data acquisition module (NI PXIe-5122) is used to collect the analog signal from PD and convert it to digital signal. Embedded controller (NI PXIe-8840) is used to control the TLS and EDFA and process the signal from data acquisition module to obtain the spectrum.

3.2 Experimental results of the SBS-TNOF response

The SBS-TNOF response is firstly measured using the experimental setup where only the narrow linewidth laser is used while PC1, EOM and RF signal are not connected. Figure 4 shows the experimental results of the response and corresponding -3dB bandwidth and selectivity. In Fig. 4(a) and (b), the pump power and signal power are set to 8 dBm and -20 dBm respectively. It can be observed that the response profiles become broader as sweep speed or fiber length increases. It is worth noting that the response broadening is asymmetric. The reason is twofold: the pump sweep in the direction towards lower frequency and the decreasing pump power distribution along the fiber. Therefore, higher frequency corresponds to lower gain in the overall BGS, and the measured response is obtained as illustrated in Fig. 2.

 

Fig. 4. Experimental results of (a, b) the SBS-TNOF response and (c, d) their -3dB bandwidth and selectivity under different sweep speed and fiber length. (a), (c) L=1000m; (b), (d) L=2000m.

Download Full Size | PDF

Then in Fig. 4(c) and (d), we change pump power and calculate -3dB bandwidth and selectivity of the SBS-TNOF response to further investigate the broadening phenomenon. It can be seen that the -3dB bandwidth shows an approximately linearity with the sweep speed when it is over 45 nm/s, especially in Fig. 4(d). By comparing Fig. 4(c) and (d), we can see that the -3dB bandwidth of 2 km fiber is nearly twice as the 1 km fiber under the same pump power and sweep speed. These regularities accord with Eq. (5). However, when pump power becomes higher, the -3dB bandwidth tends to increase in both low and high sweep speed while selectivity tends to decrease (see blue lines in Fig. 4(c) and (d)).

This phenomenon is caused by both gain saturation effect and sweep induced broadening effect. For low sweep speed situation (5 nm/s∼35 nm/s), the sweep induced broadening effect is not significant because the frequency mismatch Δf of BGS is small, the gain saturation effect is the main factor to decrease the selectivity and increase -3dB bandwidth. The SBS threshold of the HNLF is tested to prove this point. Method from ITU-T G.650.2 is applied, and the SBS threshold definition used here is the maximum point of the first derivative of the reflected power. For 2000 m HNLF, the SBS threshold is measured to be 8.89 dBm as shown in Fig. 5. The results show that the gain saturation is slight when pump power is 8 dBm, and becomes significant for pump power of 10 dBm.

 

Fig. 5. SBS threshold measurement for 2000m HNLF.

Download Full Size | PDF

And as the increase of sweep speed, the increasing mismatch reduces the gain and subsequently suppresses the gain saturation effect. So that -3dB bandwidth decreases and selectivity increases. When sweep speed becomes higher (>35 nm/s), Δf increases obviously. In this case, the gain saturation effect basically disappears while the sweep induced broadening effect dominates, so the -3dB bandwidth increases and selectivity decreases.

Briefly, lower sweep speed always means smaller -3dB bandwidth and higher selectivity for low pump power, while for higher pump power, there is an optimal sweep speed. This means that we can obtain ideal SBS-TNOF response and avoid distortion by selecting proper system parameters. But the limited parameters may restrain other performances (e.g., Brillouin gain). Therefore, we use deconvolution to recover the distortion. It will not just contribute to increase sweep speed but also help to bring a more flexible adjustment range for system parameters.

To apply deconvolution, we first calibrate the PSF using experimental data under various sweep speed. As shown in Fig. 6, scatter points represent the normalized experimental data chosen from Fig. 4(b), solid lines represent the least-square fit results using Eq. (15). Their consistency indicates that the PSF can be effectively calibrated by the model.

 

Fig. 6. PSF calibration under different sweep speed.

Download Full Size | PDF

After PSF calibration, the reconstruction procedure based on RL algorithm is carried out to recover the distorted responses. Figure 7 shows that the reconstructed SBS-TNOF responses under different sweep speed almost completely overlap within -150 MHz to 150 MHz. But at around ± 200 MHz, they deviate slightly and even form small fake signal (75 nm/s in Fig. 7(b)) owing to the noise amplification of RL deconvolution [19]. The -3dB bandwidths maintain at 30 MHz ∼ 40 MHz for different sweep speed, which means that the sweep speed dependency is almost eliminated. The selectivity also becomes almost constant under different sweep speed, except for blue dash line in Fig. 7(d). In this case, although the selectivities of first two points are not completely recovered due to the significant gain saturation effect, the reconstruction method still makes the improvement (e.g., from 13.39 dB to 15.95 dB @ 5 nm/s), and works fairly well when sweep speed is over 25 nm/s.

 

Fig. 7. Reconstruction results of (a, b) the SBS-TNOF response and (c, d) their -3dB bandwidth and selectivity under different sweep speed and fiber length. (a), (c) L=1000m; (b), (d) L=2000m.

Download Full Size | PDF

3.3 Application in SBS-OSA

The combination of the SBS-TNOF and data acquisition and control in the experimental setup can be regarded as an SBS-OSA. By scanning the pump wavelength, the spectral components of different wavelengths are extracted and the optical spectrum of the input signal is obtained. In order to further assess the performance of reconstruction method in the practical applications, we connect PC1, EOM and RF signal in the experimental setup to generate a 2.5GHz NRZ-OOK signal and measure its optical spectra. The pump power and the fiber length are set to 8 dBm and 2 km respectively. Figure 8 shows the spectra of OOK signal before and after reconstruction under different sweep speed. A simulated OOK signal generated by VPI TransmissionMaker 9.0 is used as the reference (gray dash line positioned at 0 nm/s). It can be seen that not only the main lobe of the signal is significantly recovered but the detail such as the sidelobes and the central spectral line (insets in Fig. 8) are also improved. The spectral resolution is recovered from 1.08 pm to 0.42 pm (138 MHz and 53 MHz @ 1550 nm) for the sweep speed of 95 nm/s. The resolution improvement is a little worse than that of Fig. 7 because the deconvolution iteration is stopped before unacceptable noise accumulation. Noise amplification has already appeared in Fig. 8 for high sweep speed.

 

Fig. 8. Spectra of 2.5GHz NRZ-OOK signal (a) before and (b) after reconstruction under different sweep speed.

Download Full Size | PDF

Correlation coefficient (CC) is used as the quantitatively performance evaluation to assess the merits of the reconstructed spectra [28]. It is defined as follows:

The test results of CC are calculated by comparing the spectra before and after reconstruction with simulated reference spectrum. From Fig. 9, we can see that the CC of OOK signal spectra after reconstruction always keep higher than those before reconstruction, and the recovery effect becomes better when the distortion is severer. The CC curve become flatter after reconstruction, which means that the deconvolution method reduces the sweep speed dependence of the spectra. We set 0.9 as the reference criteria to indicate that the tested spectra match well with the reference one and the spectra are not obviously distorted. It can be seen that before reconstruction the feasible sweep speed should be lower than ∼45 nm/s, and it can be enhanced to over 95 nm/s after reconstruction. The feasible sweep speed may be possibly further improved by optimizing the deconvolution process or applying more advanced algorithm to restrain the noise amplification. The phenomenon that CC increase firstly and then decrease is caused by the above-mentioned combined effect of gain saturation and signal broadening. A bit more obvious gain saturation effect can be observed here compared with Fig. 4. It is because of that the OOK spectrum is wider and meanwhile the CC evaluates the overall similarity.

 

Fig. 9. CC of the OOK signal before (black solid line) and after reconstruction (red dash line) under different sweep speed.

Download Full Size | PDF

4. Conclusion

In summary, we modify the coupled-wave equations under wavelength-swept pump condition, derive and analyze the frequency response and model of SBS-TNOF. Experimental results are in good agreement with the theoretical analysis, and verify that the broadening of SBS-TNOF response is induced by pump wavelength difference at different positions of the fiber which can also be explained as the broadband pump. The bandwidth of SBS-TNOF depends on the combined effect of gain saturation and sweep induced signal broadening. Utilizing PSF model based on the SBS-TNOF response, the distorted responses are recovered by adopting the widely used RL deconvolution method. The -3dB bandwidths maintain nearly constant for different sweep speed after reconstruction, shows that the sweep speed dependency is almost eliminated. The commonly used OOK signal is tested using the proposed reconstruction method to further verify its performance applied in the SBS-OSA. The reconstructed results show that both the overall profile and the detail are significantly recovered. Quantitatively evaluation illustrates that the feasible sweep speed (CC>0.9) can be improved from ∼45 nm/s to over 95nm/s.