Signal-to-Noise Ratio of Brillouin Grating Measurement with Micrometer-Resolution Optical Low Coherence Reflectometry

Signal-dependent speckle-like noise was the dominant noise in a Brillouin grating measurement with micrometer-resolution optical low coherence reflectometry (OLCR). The noise was produced by the interaction of a Stokes signal with beat noise caused by a leaked pump light via square-law detection. The resultant signal-to-noise ratio (SNR) was calculated and found to be proportional to the square root of the dynamic range (DR) defined by the ratio of the Stokes signal magnitude to the variance of the beat noise. The calculation showed that even when we achieved a DR of 20 dB on a logarithmic scale, the SNR value was only 7 on a linear scale and the detected signal tended to fluctuate over ±14% with respect to the mean level. We achieved an SNR of 24 by attenuating the pump light power entering the balanced mixer by 55 dB, and this success enabled us to measure the Brillouin spectrum distributions of mated fiber connectors and a 3-dB fused fiber coupler with a micrometer resolution as examples of OLCR diagnosis.


Introduction
The Brillouin-enhanced four-wave mixing induced by counter-propagating pump lights and one probe light produces a backward Stokes light in a waveguide under test [1], which is assumed to be the reflection of the probe light by an acoustic wave or a dynamic Brillouin grating excited in the waveguide by the two pump lights. Optical time-domain reflectometry (OTDR) has been used to detect the reflection distribution, or Brillouin grating distribution, while changing the frequency difference between the pump lights to obtain the Brillouin spectrum distribution in an optical fiber [2][3][4][5][6]. A micrometer-scale spatial resolution is indispensable for diagnosing miniaturized optical circuits and modules with the same four-wave mixing technique. Since the spatial resolution of the OTDR is determined by the temporal width of the employed optical pulses, we should launch a picosecond optical pulse into an optical waveguide under test and detect return pulses without deformation by using a high-speed optical detector. However, such an attempt to increase the spatial resolution often results in increasing the electrical noise level due to the ultra-wide detection bandwidth over 10 GHz, and thus degrading the signal-to-noise ratio (SNR). To our knowledge, therefore, there have been no reports on the Brillouin grating measurement at a micrometer-scale spatial resolution with the OTDR method. To overcome the degradation of the SNR, we have proposed Brillouin grating-based optical low coherence reflectometry (OLCR) [7], which detects Stokes light by utilizing its interference with local oscillator (LO) light at a detection bandwidth less than 100 kHz. Since the center frequency of a Brillouin spectrum changes with strain, OLCR is expected to provide us with useful information The conventional OLCR part consisted of main and auxiliary interferometers. The former was a fiber-optic Mach-Zehnder interferometer with optical fiber couplers (CP1 and CP2), which was designed to detect the reflection whose optical frequency was down-converted by the acoustic wave induced in the DUT. We replaced the LiNbO3 intensity modulator that we employed earlier with a LiNbO3 phase modulator (PM3) to increase the signal level. We drove the phase modulators PM1 and PM3 at f0 = 145 kHz and f1 = 150 kHz, respectively, with sawtooth voltage waveforms so that the carrier frequency of the Stokes signal was fc = f1 − f0 = 5 kHz. The photocurrent output from the balanced mixer was converted to a voltage with a transimpedance amplifier (TIA). The latter auxiliary interferometer was a fiber-optic Mach-Zehnder interferometer with optical fiber couplers (CP4 and CP5) and a 1 × 2 wavelength-division multiplexer (WDM), which shared a bulk-optic variable delay line with the former. We used a DFB laser diode operating at λ0 = 1316.077 nm as the light source, and the beat signal, which was produced during the translation of the linear stage installed in the variable delay line, enabled us to change the grid of the interferogram from equal time to equal path length increments.
In the optical fiber loop, the output from the pump light source was divided into two with an optical fiber coupler (CP3) for use as two counter-propagating pump lights. We describe the fixed frequency of the laser output as ωp in the figure. To up-convert the frequency of the pump light by the same frequency Ω as the down-conversion frequency of the probe light, we applied phase modulation at Ω to the pump light with a LiNbO3 phase modulator (PM2) and extracted the up-converted light component with another narrow-band filter. The resultant pump light at ωp + Ω was amplified with another erbium-doped fiber amplifier (EDFA), combined with the probe light at the polarization beam splitter PBS1, and launched into the DUT after passing through the optical circulator (CL1). The other laser output was launched into the DUT from the opposite direction for use as a pump light at ωp after passing through the optical circulator (CL2). The optical path lengths of the counter-propagating pump light waves were adjusted so that they were equal at the joint between the mated fiber connectors.
We used a configuration consisting of a pair of polarization controllers (PC1 and PC2) and a polarization beam splitter (PBS2) to block the pump light at ωp from entering the balanced mixer. We incorporated a fiber-optic polarizer (polarizer #2) in front of the PC1 to ensure that the pump light was linearly polarized with a negligibly small orthogonal component. By precisely adjusting the state of polarization (SOP) of the pump light with PC2, we were able to attenuate the pump light The conventional OLCR part consisted of main and auxiliary interferometers. The former was a fiber-optic Mach-Zehnder interferometer with optical fiber couplers (CP1 and CP2), which was designed to detect the reflection whose optical frequency was down-converted by the acoustic wave induced in the DUT. We replaced the LiNbO 3 intensity modulator that we employed earlier with a LiNbO 3 phase modulator (PM3) to increase the signal level. We drove the phase modulators PM1 and PM3 at f 0 = 145 kHz and f 1 = 150 kHz, respectively, with sawtooth voltage waveforms so that the carrier frequency of the Stokes signal was f c = f 1 − f 0 = 5 kHz. The photocurrent output from the balanced mixer was converted to a voltage with a transimpedance amplifier (TIA). The latter auxiliary interferometer was a fiber-optic Mach-Zehnder interferometer with optical fiber couplers (CP4 and CP5) and a 1 × 2 wavelength-division multiplexer (WDM), which shared a bulk-optic variable delay line with the former. We used a DFB laser diode operating at λ 0 = 1316.077 nm as the light source, and the beat signal, which was produced during the translation of the linear stage installed in the variable delay line, enabled us to change the grid of the interferogram from equal time to equal path length increments.
In the optical fiber loop, the output from the pump light source was divided into two with an optical fiber coupler (CP3) for use as two counter-propagating pump lights. We describe the fixed frequency of the laser output as ω p in the figure. To up-convert the frequency of the pump light by the same frequency Ω as the down-conversion frequency of the probe light, we applied phase modulation at Ω to the pump light with a LiNbO 3 phase modulator (PM2) and extracted the up-converted light component with another narrow-band filter. The resultant pump light at ω p + Ω was amplified with another erbium-doped fiber amplifier (EDFA), combined with the probe light at the polarization beam splitter PBS1, and launched into the DUT after passing through the optical circulator (CL1). The other laser output was launched into the DUT from the opposite direction for use as a pump light at ω p after passing through the optical circulator (CL2). The optical path lengths of the counter-propagating pump light waves were adjusted so that they were equal at the joint between the mated fiber connectors.
We used a configuration consisting of a pair of polarization controllers (PC1 and PC2) and a polarization beam splitter (PBS2) to block the pump light at ω p from entering the balanced mixer. We incorporated a fiber-optic polarizer (polarizer #2) in front of the PC1 to ensure that the pump Sensors 2020, 20, 936 4 of 21 light was linearly polarized with a negligibly small orthogonal component. By precisely adjusting the state of polarization (SOP) of the pump light with PC2, we were able to attenuate the pump light power to around 55 dB. While maintaining the attenuation and fixing the stage at a particular position, we detected the Stokes signal with an FFT signal analyzer and achieved a dynamic range of 37 dB as shown in the inset of Figure 1. The range was 14 dB greater than that in our previous paper, where the resolution and band shape of the analyzer were the same.

Dispersive Fourier Spectroscopy (DFS)
We introduced dispersive Fourier spectroscopy (DFS) [10,11] to numerically cancel the residual dispersion in the main interferometer and thus achieve the highest possible spatial resolution of 30 µm, which was dictated by the spectrum of the low coherence light output. We passed the balanced mixer output through an antialiasing filter with a cutoff frequency of 20 kHz. We set the controller to translate the linear stage at a speed of 400 µm/s and simultaneously oversampled the signal from the filter at a rate of 40k samples/s together with the reference signal from the auxiliary interferometer. The analytic signal of the Stokes signal was demodulated numerically by filtering the signal with a digital rectangular band-pass filter centered at 5 kHz and with a width of 630 Hz, digitally mixing the band-passed signal with in-phase and quadrature waveforms at 5 kHz, and low-pass filtering the result.
In principle the absolute square of the analytic signal as a function of the time delay provided us with a Brillouin grating reflectogram. Before straightforwardly deriving the reflectogram from the demodulated signal, we changed the grid of the interferogram from equal time to equal path length increments, calculated the Fourier inverse transform, and removed the residual phase term of the transform, which was the origin of the degradation of the spatial resolution. We then calculated the Fourier transform to reconstitute the analytic signal after removing the noise that was distributed outside the range from 1527 to 1576 nm where the spectrum of the low coherence light was finite. Although the nominal spatial resolution of the constructed OLCR system was 30 µm, the best way to determine the actual spatial resolution was to measure the response of the OLCR system against a localized Brillouin grating whose length was assumed to be much shorter than the spatial resolution. However, it would be very difficult to generate such an ultimately narrow Brillouin grating artificially. Therefore, to check the spatial resolution beforehand, we measured the Stokes signal distribution (or Brillouin grating distribution) around the joint of angled physical contact fiber connectors that were mated with each other while changing the down-conversion frequency and found the narrowest Brillouin gratings whose full widths at half-maxima were 81 and 84 µm, at 10.47 and 10.5 GHz, respectively, as shown in Figure 2. The result shows that the actual spatial resolution was 81 µm or higher.
Sensors 2020, 20, x 4 of 21 power to around 55 dB. While maintaining the attenuation and fixing the stage at a particular position, we detected the Stokes signal with an FFT signal analyzer and achieved a dynamic range of 37 dB as shown in the inset of Figure 1. The range was 14 dB greater than that in our previous paper, where the resolution and band shape of the analyzer were the same.

Dispersive Fourier Spectroscopy (DFS)
We introduced dispersive Fourier spectroscopy (DFS) [10,11] to numerically cancel the residual dispersion in the main interferometer and thus achieve the highest possible spatial resolution of 30 μm, which was dictated by the spectrum of the low coherence light output. We passed the balanced mixer output through an antialiasing filter with a cutoff frequency of 20 kHz. We set the controller to translate the linear stage at a speed of 400 μm/s and simultaneously oversampled the signal from the filter at a rate of 40k samples/s together with the reference signal from the auxiliary interferometer. The analytic signal of the Stokes signal was demodulated numerically by filtering the signal with a digital rectangular band-pass filter centered at 5 kHz and with a width of 630 Hz, digitally mixing the band-passed signal with in-phase and quadrature waveforms at 5 kHz, and low-pass filtering the result.
In principle the absolute square of the analytic signal as a function of the time delay provided us with a Brillouin grating reflectogram. Before straightforwardly deriving the reflectogram from the demodulated signal, we changed the grid of the interferogram from equal time to equal path length increments, calculated the Fourier inverse transform, and removed the residual phase term of the transform, which was the origin of the degradation of the spatial resolution. We then calculated the Fourier transform to reconstitute the analytic signal after removing the noise that was distributed outside the range from 1527 to 1576 nm where the spectrum of the low coherence light was finite. Although the nominal spatial resolution of the constructed OLCR system was 30 μm, the best way to determine the actual spatial resolution was to measure the response of the OLCR system against a localized Brillouin grating whose length was assumed to be much shorter than the spatial resolution. However, it would be very difficult to generate such an ultimately narrow Brillouin grating artificially. Therefore, to check the spatial resolution beforehand, we measured the Stokes signal distribution (or Brillouin grating distribution) around the joint of angled physical contact fiber connectors that were mated with each other while changing the down-conversion frequency and found the narrowest Brillouin gratings whose full widths at half-maxima were 81 and 84 μm, at 10.47 and 10.5 GHz, respectively, as shown in Figure 2. The result shows that the actual spatial resolution was 81 μm or higher.

Calculations
We denote the electric fields of the LO, probe, pump light waves at ω p + Ω and ω p , as E LO + c.c., E pr + c.c., E p1 + c.c. and E p2 + c.c., respectively, where c.c. denotes complex conjugate. In this paper E LO , E pr , E p1 and E p2 are referred to as the analytic signals of the respective electric fields. In Figure 3, the probe light E pr passes through the circulator CL1, enters the DUT, and is reflected and down-converted by the Brillouin grating. The reflected light is referred to as the Stokes light that enters the balanced mixer after passing through the CL1, where the analytic signal is denoted as E s . In addition to the Stokes light, there are three kinds of light waves ∆E pr , ∆E p1 and ∆E p2 , which enter the balanced mixer and generate beat noise. That is, due to the finite directivity of CL1, the probe light and the pump light at ω p + Ω, are transmitted straight through it into the balanced mixer without entering the DUT, whereas they are Fresnel reflected at the endfaces of the DUT, return to CL1 and are combined with the respective transmitted light components. We denote the combined light wave that originates from the probe light as ∆E pr and from the pump light as ∆E p1 . Although the pump light at ω p is attenuated by the polarization beam splitter PBS2, a small amount is transmitted through PBS2 and enters the balanced mixer. We denote the analytic signal of the pump light component entering the balanced mixer as ∆E p2 .

Calculations
We denote the electric fields of the LO, probe, pump light waves at ωp + Ω and ωp, as ELO + c.c., Epr + c.c., Ep1 + c.c. and Ep2 + c.c., respectively, where c.c. denotes complex conjugate. In this paper ELO, Epr, Ep1 and Ep2 are referred to as the analytic signals of the respective electric fields. In Figure 3, the probe light Epr passes through the circulator CL1, enters the DUT, and is reflected and down-converted by the Brillouin grating. The reflected light is referred to as the Stokes light that enters the balanced mixer after passing through the CL1, where the analytic signal is denoted as Es.
In addition to the Stokes light, there are three kinds of light waves ΔEpr, ΔEp1 and ΔEp2, which enter the balanced mixer and generate beat noise. That is, due to the finite directivity of CL1, the probe light and the pump light at ωp + Ω, are transmitted straight through it into the balanced mixer without entering the DUT, whereas they are Fresnel reflected at the endfaces of the DUT, return to CL1 and are combined with the respective transmitted light components. We denote the combined light wave that originates from the probe light as ΔEpr and from the pump light as ΔEp1. Although the pump light at ωp is attenuated by the polarization beam splitter PBS2, a small amount is transmitted through PBS2 and enters the balanced mixer. We denote the analytic signal of the pump light component entering the balanced mixer as ΔEp2. Figure 3. Propagation of the light waves ΔEpr, ΔEp1 and ΔEp2, which were generated from the probe light Epr, pump light Ep1 at ωp + Ω and Ep2 at ωp, respectively, and entered the balanced mixer to produce beat noise. ILO and Ipr are the mean photocurrents produced if the respective LO and probe lights would be detected with a photodiode. Pp1 and Pp2 are the light powers of the pump lights at ωp + Ω and ωp, respectively. CL1: optical circulator, TIA: transimpedance amplifier. A schematic of the square-law detection and the relation between the constituent spectral densities are also shown. (a) shape of the low-pass filter whose width is VBW. The center frequency and the bandwidth of the band-pass filter are fc and RBW, respectively. The constituent spectral density functions denoted as Equations (A1)-(A3) are represented by (b), (c) and (d), respectively.
Since the electric field Es + ΔEpr + ΔEp1 + ΔEp2 + c.c. is superimposed on that of the LO light by the 3-dB fiber coupler CP2, the instantaneous photocurrent from the balanced mixer is represented by: where i = √−1 and α is a proportionality coefficient that converts the square value of the electric field to the photocurrent. The first term on the right-hand side of Equation (1) is the desired Stokes signal, whereas the second to fourth terms generate beat noise. Since ELO(t) is no longer correlated with ΔEpr(t) and ΔEp1(t) does not interfere with ΔEp2(t) because of their 10-GHz frequency difference, we assume the three kinds of noises to be statistically independent. Since a Brillouin spectrum at a given location is obtained by measuring the mean square value of the Stokes signal as a function of the frequency difference between the pump lights, the voltage Figure 3. Propagation of the light waves ∆E pr , ∆E p1 and ∆E p2 , which were generated from the probe light E pr , pump light E p1 at ω p + Ω and E p2 at ω p , respectively, and entered the balanced mixer to produce beat noise. I LO and I pr are the mean photocurrents produced if the respective LO and probe lights would be detected with a photodiode. P p1 and P p2 are the light powers of the pump lights at ω p + Ω and ω p , respectively. CL1: optical circulator, TIA: transimpedance amplifier. A schematic of the square-law detection and the relation between the constituent spectral densities are also shown. (a) shape of the low-pass filter whose width is VBW. The center frequency and the bandwidth of the band-pass filter are f c and RBW, respectively. The constituent spectral density functions denoted as Equations (A1)-(A3) are represented by (b), (c) and (d), respectively.
Since the electric field E s + ∆E pr + ∆E p1 + ∆E p2 + c.c. is superimposed on that of the LO light by the 3-dB fiber coupler CP2, the instantaneous photocurrent from the balanced mixer is represented by: where i = √ −1 and α is a proportionality coefficient that converts the square value of the electric field to the photocurrent. The first term on the right-hand side of Equation (1) is the desired Stokes signal, whereas the second to fourth terms generate beat noise. Since E LO (t) is no longer correlated with ∆E pr (t) and ∆E p1 (t) does not interfere with ∆E p2 (t) because of their 10-GHz frequency difference, we assume the three kinds of noises to be statistically independent.
Since a Brillouin spectrum at a given location is obtained by measuring the mean square value of the Stokes signal as a function of the frequency difference between the pump lights, the voltage output from the TIA should be launched into the square-law detection system installed in the RF Sensors 2020, 20, 936 6 of 21 spectrum analyzer, which is shown enclosed by the dotted square in Figure 3. The detection system usually consists of a band-pass filter whose center frequency is set at the carrier frequency f c = 5 kHz and whose bandwidth determines the variance of the beat noise, a square-law device to convert the signal to its squared value, and a low-pass filter to extract the slowly-varying component. Since the output of a conventional RF spectrum analyzer is scaled into a logarithmic magnitude in units of dBV or dBm, we converted the measured logarithmic values to the corresponding squared currents on a linear scale by using the gain of the TIA and compared them with the theoretical results. Hereafter, we calculate the magnitude of the Stokes signal and the power spectral density of the noise in units of A 2 and A 2 /Hz, respectively.

Light Power Dependence of Stokes Signal
Since the coherent time of the broadband light from the OLCR source was much shorter than the response time of the detection system, the Stokes signal is obtained by performing statistical averaging over the first term on the right-hand side of Equation (1) as follows [12]: when the delay induced by the variable delay line is τ. < > denotes the statistical average [13], G(ω 1 + Ω) is the light spectrum of the broadband light whose center frequency is ω 1 , and r(τ) is the slowly-varying envelope of the acoustic wave. γ e and ρ 0 are the electrostrictive constant and mean density of the waveguide material of the DUT, respectively. n is the refractive index of the waveguide, ∆ϕ is a constant and ω c is the angular frequency, which is equal to 2πf c where f c =5 kHz. When deriving Equation (2) we assume that F(t) is the slowly-varying envelope of the electric field of the broadband light, and κ 1 and κ 2 are constants where the envelopes of the LO and probe lights are denoted as κ 1 F(t) and κ 2 F(t), respectively. Then the mean square values of the LO and probe lights are 2|κ 1 | 2 <|F(t)| 2 > and 2|κ 2 | 2 <|F(t)| 2 >, respectively, and the individual mean photocurrents, which would be measured if the individual lights were detected with a photodetector, are given by I LO = 2|α|·|κ 1 | 2 <|F(t)| 2 > and I pr = 2|α|·|κ 2 | 2 <|F(t)| 2 >. Similarly, I p2 denotes the photocurrent corresponding to the pump light at ω p . V(τ) takes a finite value when τ is included in the range |τ|<τ coh , where τ coh is the coherence time of the light source. When r(τ) changes with τ so slowly that r(τ ) is considered to be a constant within the range of |τ-τ |<τ coh , Equation (2) can be simplified to: where: is an effective width of the broadband light. Since the center of the band-pass filter is set at the carrier frequency of the Stokes signal, it can pass through the filter, and the output from the low-pass filter is denoted as: Sensors 2020, 20, 936 7 of 21 V(τ) is defined by Equation (3) by using the light spectrum G(ω 1 + Ω), which is a function of the RF angular frequency ω. In the following subsection, we discuss the power spectral density of the beat noise as a function of the effective bandwidth. In Equation (5) we replace G(ω 1 + Ω) with a new function G(ν), which is a function of the light frequency ν.
The slowly-varying envelope r(τ) of the acoustic wave in a steady state is [1]: where ε 0 is the free-space permittivity, q is the wavenumber of the acoustic wave, Ω B is the Brillouin frequency shift as a function of τ, and Γ B is the Brillouin linewidth. We introduced a constant Q 1 in Equation (7) to take account of the cross-sectional distributions of the acoustic and pump light waves in the waveguide of the DUT [14]. Since A p1 and A p2 are the envelopes of the pump light waves, the mean square values of the electric fields are 2|A p1 | 2 and 2|A p2 | 2 , and therefore, the total mean powers that propagate in the DUT are P p1 = 2Q 2 |A p1 | 2 and P p2 = 2Q 2 |A p2 | 2 , respectively, where Q 2 is a constant. By substituting Equation (7) into Equation (6), we finally obtain: The Stokes signal is proportional to the product of the mean powers of these four light waves. Since the spatial resolution increases with δν LO×p , Equation (8) means that the Stokes signal decreases approximately with the square of the resolution.

Current Noise Spectral Density
First, we derive the current noise spectral density of the second noise term ∆I(t) = 2iαE LO (t)∆E pr * (t) + c.c. in Equation (1) by calculating its autocorrelation function. The whole calculation process is described in detail in [15]. By assuming that the electric fields obey a Gaussian random process [16], we can simplify their fourth-order moment as: where G LO (ν) and G pr (ν) are light spectra of the LO and probe lights, respectively. Since f is a low frequency around the detection frequency at 5 kHz, and ν is the light frequency around 200 THz, we can make a good approximation of G pr (ν + f ) ≈ G pr (ν) in Equation (9). Considering that the mean photocurrents of the LO and probe lights are I LO = 2|α|<|E LO (t)| 2 > and ∆I pr = 2|α|<|E pr (t)| 2 >, respectively, the current noise spectral density is obtained as: Since G pr (ν) has the same shape as G LO (ν) and shifts by only 10 GHz, which is much less than the light frequency ν, we can assume G pr (ν) to be proportional to G LO (ν) in Equation (10), and we obtain the final result of the spectral density for the beat between the LO and probe light as: Sensors 2020, 20, 936 where δν LO×pr is another effective width of the broadband light defined by [17] as: The current noise spectral density of the third noise term 2iαE LO (t)∆E p1 * (t) + c.c. in Equation (1) is obtained by replacing ∆I pr with ∆I p1 and G pr (ν) with G p (ν) in Equation (10), where ∆I p1 and G p (ν) are the mean photocurrent and spectrum of the pump light component ∆E p1 . Since the origin of ∆E p1 is the coherent light from the laser source whose bandwidth is 100 kHz, G p1 (ν) is approximated as a delta function δ(ν-ν p1 ) with ν p1 = (ω p + Ω)/2π and the resultant width of Equation (12) becomes the effective width defined by Equation (5). Therefore, we obtain: Similarly, the current noise spectral density of the fourth term in Equation (1) is: Then the total current noise spectral density including the shot noise effect is: where e is the elementary charge. In Equation (15) we assume that the intensity noise of the broadband LO light is reduced to a negligible level by using balanced detection and the shot noise caused by the weak Stokes light is also negligible. It should be noted that the derived spectral density is the variance of the total noise observed when the bandwidth of the band-pass filter is unity. The last two terms in Equation (15) are characteristic of an OLCR system designed for Brillouin grating measurement.
The OLCR system was originally developed to measure the Rayleigh backscattering distribution in a silica-based waveguide. The Rayleigh backscatter signal was so weak that we defined the SNR of the Rayleigh backscattering measurement as the ratio of the mean level of the fluctuated Rayleigh backscatter signals to the current noise spectral density when estimating the minimum-detectable reflectivity at a unit detection bandwidth [18]. Instead, in this paper the ratio of the mean Stokes signal level to the variance is referred to as the dynamic range (DR): because the range would be observed on the display of an RF spectrum analyzer if the output of the TIA were connected to it and the spectrum around the carrier frequency of 5 kHz were measured, as shown by the inset of Figure 1.

Relationship between DR and SNR
The output from the band-pass filter is denoted as I s (t) + N(t), where I s (t) is a sine wave of the form Acos(2πf c t + θ) with a constant A as described by Equation (4), and N(t) is the fluctuating term of the total noise in Equation (1) whose variance is <N 2 (t)> = σ 2 . Then, the output from the square-law device is Y(t) = {I s (t) + N(t)} 2 = I s 2 (t) + 2I s (t)N(t) + N 2 (t) whose slowly-varying component can pass the low-pass filter. The second term, 2I s (t)N(t) in Y(t), fluctuates greatly as the signal I s (t) increases, and this is the main origin of the speckle-like noise we observed with OLCR. The signal fluctuation is Sensors 2020, 20, 936 9 of 21 expressed by the standard deviation σ Y of the output from the low-pass filter, where the variance σ Y 2 is written as the general form: From the definition of I s (t) = Acos(2πf c t + θ), the signal from the low-pass filter output is <I s 2 (t)> = A 2 /2. Assuming that I s (t) and N(t) are statistically independent, we have <I s (t)N(t)> = 0, and so <Y(t)> in Equation (17) is written as: Once the first term <Y 2 (t)> in Equation (17) is calculated for the low-pass filter output, it is certain that the SNR of the Stokes signal measurement is defined by: The actual form of <Y 2 (t)> is derived in [19] where the band-pass filter has a Gaussian window. In our experiment, however, we used three types of windows to measure the magnitude of the signal and noise. One was a Gaussian window, which was installed in an analog RF spectrum analyzer, the second was the flat-top window of the FFT signal analyzer, which we used to measure the spectrum shown by the inset of Figure 1 and the current noise spectral density, and the third was the rectangular (or uniform) window that we employed in the DFS. Therefore, we calculated <Y 2 (t)> at an arbitrary window function H(f ) according to the reference as shown in the Appendix A of this paper, and we found that: with any window function as long as the variance of the noise from the band-pass filter is σ 2 . By substituting equations (18) and (20) into Equation (17), we have: and the SNR defined by Equation (19) is into Equation (16), we have DR = (A 2 /2)/σ 2 and so the SNR is written with the DR as: When DR>>1, as is usually the case, we have SNR≈ √ (DR/2) meaning that the SNR is proportional to the square root of the DR which increases in proportion to the product of the mean powers of the four light waves. On the other hand, the DR and thus the SNR degrade as the power of the pump light entering the balanced mixer increases, where the variance of the generated beat noise is proportional to the product of the mean photocurrents of the LO light and the leaked pump light. The square root dependence of the SNR on the DR means that even when we achieve a DR of 20 dB on a logarithmic scale, the SNR value is only 7 on a linear scale, which indicates that the detected signal tends to fluctuate over ±14% with respect to the mean level.

Power Dependence of the Stokes Signal
We measured the Stokes signal at a distance of 2 cm from the joint of the mated fiber connectors with the FFT signal analyzer. We set the center frequency at 5 kHz and chose a flat-top window since there were small fluctuations in the carrier frequency caused by perturbations experienced by the main interferometer. The down-conversion frequency was 10.77 GHz. To measure the magnitude of the signal accurately, we averaged ten spectra, which we acquired with repetitive frequency scans and determined the peak in the mean spectrum by employing a least square fitting procedure. We converted the magnitude of the signal scaled in dBV units to pA 2 units by using the gain of the TIA. The maximum powers of the pump lights at ω p + Ω and ω p were P p1 = 200 mW and P p2 = 12 mW, respectively. We increased the output power from the low coherence light source and measured the signal as a function of the photocurrent I LO of the LO light as shown in Figure 4a,b. It is noted that when the power of the probe light was 114 mW, the photocurrent was I LO = 47 µA. Figure 4a shows the measurement result at pump powers of P p1 = 50, 100 and 200 mW while P p2 was fixed at 12 mW, and Figure 4b shows the result at pump powers of P p2 = 3, 6 and 12 mW while P p1 was fixed at 200 mW, where logarithmic scales were used on both the horizontal and vertical axes. Since the variation of each data point was 10% with respect to the mean value, we did not add error bars to the log-log graphs. At each I LO value, the signal increased twofold when the pump power of either P p1 or P p2 was doubled, and this dependence means that the signal was proportional to the product of the individual pump light powers, or P p1 × P p2.
Sensors 2020, 20, x 10 of 21 vertical axes. Since the variation of each data point was 10% with respect to the mean value, we did not add error bars to the log-log graphs. At each ILO value, the signal increased twofold when the pump power of either Pp1 or Pp2 was doubled, and this dependence means that the signal was proportional to the product of the individual pump light powers, or Pp1 × Pp2. The signal dependence on ILO for every pair of (Pp1, Pp2) agreed with the square function of ILO, which is shown by the solid line, and this dependence means that the signal was proportional to ILO 2 . As described in Section 3.1, the photocurrents of the LO and probe lights are defined by ILO = 2|α|·|κ1| 2 <|F(t)| 2 > and Ipr = 2|α|·|κ2| 2 <|F(t)| 2 >, respectively. Since 2|α|<|F(t)| 2 > is the photocurrent corresponding to the output power from the low coherence light source, both ILO and Ipr changed in proportion to the output power and so we have the relation ILO 2 = |κ1/κ2| 2 ILO × Ipr. Thus the square dependence of the signal that we observed in the figures means that the signal was proportional to ILO × Ipr. Combining the two dependences leads to the conclusion that the signal changed in proportion to ILO × Ipr × Pp1 × Pp2, which agreed with the theoretical result provided by Equation (8), and therefore we confirmed that OLCR detected the exact Stokes signal produced by the Brillouin-enhanced four-wave mixing. It is noted that all the solid lines in the figures were plotted by calculating the magnitude CsILOIprPp1Pp2 in units of pA 2 , where Cs= 4.29 × 10 −5 , and the units of (ILO, Ipr) and (Pp1, Pp2) were restricted to μA and mW, respectively. Since Cs is the coefficient taken at δνLO×p = 2.05 THz and Ω ≈ ΩB in Equation (8), the coefficient should be Csʹ = Cs × (δνLO×p/δνʹLO×p) 2 × (ΓBΩB) 2 /{(ΩB 2 -Ω 2 ) 2 + (ΓBΩ) 2 } at arbitrary values of δνʹLO×p and Ω.

Current Noise Spectral Density
The three kinds of light waves entered the balanced mixer to produce beat noise as shown schematically in Figure 3. The current noise spectral density was measured as follows. The vertical scale of the FFT signal analyzer was set at the power spectral density (PSD) in units of V 2 /Hz. We converted the PSD values measured with the analyzer at 5 kHz to the corresponding squared currents by using the gain of the TIA. To show the effect of the beat noise caused by the leaked pump light, we blocked the probe light and the pump light at ωp + Ω from entering the DUT by disconnecting the optical path between PBS1 and CL1. The power of the pump light at ωp launched into the DUT was 12 mW, and we increased the photocurrent ΔIpr of the leaked pump light from 80  The signal dependence on I LO for every pair of (P p1 , P p2 ) agreed with the square function of I LO , which is shown by the solid line, and this dependence means that the signal was proportional to I LO 2 . As described in Section 3.1, the photocurrents of the LO and probe lights are defined by I LO = 2|α|·|κ 1 | 2 <|F(t)| 2 > and I pr = 2|α|·|κ 2 | 2 <|F(t)| 2 >, respectively. Since 2|α|<|F(t)| 2 > is the photocurrent corresponding to the output power from the low coherence light source, both I LO and I pr changed in proportion to the output power and so we have the relation I LO 2 = |κ 1 /κ 2 | 2 I LO × I pr . Thus the square dependence of the signal that we observed in the figures means that the signal was proportional to I LO × I pr . Combining the two dependences leads to the conclusion that the signal changed in proportion to I LO × I pr × P p1 × P p2 , which agreed with the theoretical result provided by Equation (8), and therefore we confirmed that OLCR detected the exact Stokes signal produced by the Brillouin-enhanced four-wave mixing. It is noted that all the solid lines in the figures were plotted by calculating the magnitude C s I LO I pr P p1 P p2 in units of pA 2 , where C s = 4.29 × 10 −5 , and the units of (I LO , I pr ) and (P p1 , P p2 ) were restricted to µA and mW, respectively. Since C s is the coefficient taken at δν LO×p = 2.05 THz and Ω ≈ Ω B in Equation (8)

Current Noise Spectral Density
The three kinds of light waves entered the balanced mixer to produce beat noise as shown schematically in Figure 3. The current noise spectral density was measured as follows. The vertical scale of the FFT signal analyzer was set at the power spectral density (PSD) in units of V 2 /Hz.
We converted the PSD values measured with the analyzer at 5 kHz to the corresponding squared currents by using the gain of the TIA. To show the effect of the beat noise caused by the leaked pump light, we blocked the probe light and the pump light at ω p + Ω from entering the DUT by disconnecting the optical path between PBS1 and CL1. The power of the pump light at ω p launched into the DUT was 12 mW, and we increased the photocurrent ∆I pr of the leaked pump light from 80 to 320 to 1280 nA by finely adjusting its SOP with PC2. We changed the output power of the low coherence light source and measured the current noise spectral density at 5 kHz as a function of the photocurrent I LO as shown in Figure 5. It should be noted that the photocurrent of ∆I pr = 80 nA was measured when the pump light power was attenuated by 55 dB with PBS2. At every photocurrent value ΔIP2, the measured spectral density as a function of ILO agreed with the theoretical curve shown by the solid line of σ 2 = 2e(ILO + ΔIp2) + σ 2 LO×P2, which was obtained by letting ΔIpr = 0 and ΔIp1 = 0 in Equation (15). To calculate the values of σ 2 LO×P2 according to Equation (14), we estimated the effective width defined by Equation (5) at δνLO×p = 2.05 THz beforehand from the measured spectrum of the low coherence light output. The shot noise limited spectral density was observed by completely blocking the pump light from entering the DUT as shown by the open circles in the figure, revealing that the inherent intensity noise of the low coherence light was suppressed by the balanced detection. As the pump light power increased, the beat noise overcame the shot noise and the former is shown to be a dominant noise that degrades the DR and SNR of the Brillouin grating measurement.
Finally, we launched all the lights into the interferometer while the serrodyne modulation to the phase modulators (PM1 and PM3) was switched off to prevent the generation of a Stokes signal at 5 kHz. We measured the spectral density at 5 kHz as a function of ILO by increasing the power of the low coherence light output, as shown in Figure 6, where we changed ILO within the range of practical interest. The pump light powers Pp1 and Pp2 were 200 and 12 mW, respectively. Since both ΔIp1 and ΔIp2 were launched into the mixer, we increased the total photocurrent ΔIp = ΔIp1 + ΔIp2 from 80 to 160 to 320 nA by making the fine adjustment of the SOP of the pump lights with the PC2. From Equation (15) the total noise was expressed by σ 2 = 2e(ILO + ΔIpr + ΔIp) + σ 2 LO×pr + σ 2 LO×p, with which we calculated the noise change as shown by the solid lines. Here σ 2 LO×pr was evaluated from Equation (11) with the measured reflection of -62 dB and δνLO×pr = 2.88 THz, whereas σ 2 LO×p was evaluated from 4ILOΔIp/δνLO×p with δνLO×p = 2.05 THz. The noise σ 2 LO×p overcame the lower bound of the shot noise + beat noise σ 2 LO×pr, which is the main noise of the conventional OLCR system, and so the total noise increased linearly with ILO. The agreement between the measured and calculated values shows that the noise of the Brillouin-grating-based OLCR is expressed by Equation (15). As long as ΔIp≥80 nA, we could maintain a given value of ΔIp for 20 min under our laboratory Figure 5. Current noise spectral density as a function of photocurrent I LO measured when the power of the pump light at ω p that entered the balanced mixer was changed so that the photocurrent was ∆I p2 = 80, 320 or 1280 nA. Both the probe light and the pump light at ω p + Ω were blocked. The photocurrent was ∆I p2 = 80 nA when the pump light was attenuated by 55 dB. The solid lines were plotted by calculating σ 2 = 2e(I LO + ∆I p2 ) + σ 2 LO×P2 at different ∆I p2 values.
At every photocurrent value ∆I P2 , the measured spectral density as a function of I LO agreed with the theoretical curve shown by the solid line of σ 2 = 2e(I LO + ∆I p2 ) + σ 2 LO×P2 , which was obtained by letting ∆I pr = 0 and ∆I p1 = 0 in Equation (15). To calculate the values of σ 2 LO×P2 according to Equation (14), we estimated the effective width defined by Equation (5) at δν LO×p = 2.05 THz beforehand from the measured spectrum of the low coherence light output. The shot noise limited spectral density was observed by completely blocking the pump light from entering the DUT as shown by the open circles in the figure, revealing that the inherent intensity noise of the low coherence light was suppressed by the balanced detection. As the pump light power increased, the beat noise overcame the shot noise and the former is shown to be a dominant noise that degrades the DR and SNR of the Brillouin grating measurement.
Finally, we launched all the lights into the interferometer while the serrodyne modulation to the phase modulators (PM1 and PM3) was switched off to prevent the generation of a Stokes signal at 5 kHz. We measured the spectral density at 5 kHz as a function of I LO by increasing the power of the low coherence light output, as shown in Figure 6, where we changed I LO within the range of practical interest. The pump light powers P p1 and P p2 were 200 and 12 mW, respectively. Since both ∆I p1 and ∆I p2 were launched into the mixer, we increased the total photocurrent ∆I p = ∆I p1 + ∆I p2 from 80 to 160 to 320 nA by making the fine adjustment of the SOP of the pump lights with the PC2. From Equation (15) the total noise was expressed by σ 2 = 2e(I LO + ∆I pr + ∆I p ) + σ 2 LO×pr + σ 2 LO×p , with which we calculated the noise change as shown by the solid lines. Here σ 2 LO×pr was evaluated from Equation (11) with the measured reflection of -62 dB and δν LO×pr = 2.88 THz, whereas σ 2 LO×p was evaluated from 4I LO ∆I p /δν LO×p with δν LO×p = 2.05 THz. The noise σ 2 LO×p overcame the lower bound of the shot noise + beat noise σ 2 LO×pr , which is the main noise of the conventional OLCR system, and so the total noise increased linearly with I LO . The agreement between the measured and calculated values shows that the noise of the Brillouin-grating-based OLCR is expressed by Equation (15). As long as ∆I p ≥80 nA, we could maintain a given value of ∆I p for 20 min under our laboratory conditions. We could temporarily reduce ∆I p to a minimal level of the order of 10 nA so that the noise level approached the lower bound, as shown by the open circles in Figure 6.
Sensors 2020, 20, x 12 of 21 Figure 6. Current noise spectral density as a function of photocurrent ILO measured when all the lights were launched and the total power of both pump lights that entered the balanced mixer was changed so that their total photocurrent was ΔIp = 80, 160 or 320 nA. The solid lines were plotted by calculating σ 2 = 2e(ILO + ΔIpr + ΔIp) + σ 2 LO×pr + σ 2 LO×p at different ΔIp values.

Relationship between DR and SNR
We fixed the pump light powers Pp1 and Pp2 at 200 and 12 mW, respectively. The output power of the low coherence light was 114 mW so that the photocurrent was ILO = 47 μA. The down-conversion frequency was fixed at 10.77 GHz to measure the Stokes signal from the same position at a distance of 2 cm from the joint of the mated fiber connectors. To check the validity of the relationship between the DR and SNR, which we derived as Equation (22), we had to measure the time change of the low-pass filter output under a condition where the bandwidth VBW of the low-pass filter was at least twice the RBW of the band-pass filter, as described in Appendix A. To meet this condition, we used an analog-type RF spectrum analyzer where we set the center frequency, span, VBW and RBW at 5 kHz, 0 Hz (or a zero span), 100 Hz, and 30 Hz, respectively. The spectrum analyzer had a video output port from which the square-law detection output was streamed.
We reduced the photocurrent ΔIp gradually to reduce the noise, for every value of which we acquired the time changes of the Stokes signal and noise for ten seconds by switching the serrodyne modulation to both phase modulators on and off. Figure 7a shows one example of the acquired waveforms, where the noise (lower trace) was so high that it was close to the Stokes signal (upper trace), which also fluctuated with time. We calculated the standard deviation and mean value of the fluctuating Stokes signal from the acquired waveform, which corresponded to σY in Equation (17), and <Y(t)> = A 2 /2 + σ 2 in Equation (18). Figure 6. Current noise spectral density as a function of photocurrent I LO measured when all the lights were launched and the total power of both pump lights that entered the balanced mixer was changed so that their total photocurrent was ∆I p = 80, 160 or 320 nA. The solid lines were plotted by calculating σ 2 = 2e(I LO + ∆I pr + ∆I p ) + σ 2 LO×pr + σ 2 LO×p at different ∆I p values.

Relationship between DR and SNR
We fixed the pump light powers P p1 and P p2 at 200 and 12 mW, respectively. The output power of the low coherence light was 114 mW so that the photocurrent was I LO = 47 µA. The down-conversion frequency was fixed at 10.77 GHz to measure the Stokes signal from the same position at a distance of 2 cm from the joint of the mated fiber connectors. To check the validity of the relationship between the DR and SNR, which we derived as Equation (22), we had to measure the time change of the low-pass filter output under a condition where the bandwidth VBW of the low-pass filter was at least twice the RBW of the band-pass filter, as described in Appendix A. To meet this condition, we used an analog-type RF spectrum analyzer where we set the center frequency, span, VBW and RBW at 5 kHz, 0 Hz (or a zero span), 100 Hz, and 30 Hz, respectively. The spectrum analyzer had a video output port from which the square-law detection output was streamed.
We reduced the photocurrent ∆I p gradually to reduce the noise, for every value of which we acquired the time changes of the Stokes signal and noise for ten seconds by switching the serrodyne modulation to both phase modulators on and off. Figure 7a shows one example of the acquired waveforms, where the noise (lower trace) was so high that it was close to the Stokes signal (upper trace), which also fluctuated with time. We calculated the standard deviation and mean value of the fluctuating Stokes signal from the acquired waveform, which corresponded to σ Y in Equation (17), and <Y(t)> = A 2 /2 + σ 2 in Equation (18).
After calculating the variance σ 2 from the time change of the noise we reduced the photocurrent ∆I p gradually to reduce the noise, for every value of which we acquired the time changes of the Stokes signal and noise for ten seconds by switching the serrodyne modulation to both phase modulators on and off. Figure 7a shows one example of the acquired waveforms, where the noise (lower trace) was so high that it was close to the Stokes signal (upper trace), which also fluctuated with time. We calculated the standard deviation and mean value of the fluctuating Stokes signal from the acquired waveform, which corresponded to σ Y in Equation (17), and <Y(t)> = A 2 /2 + σ 2 in Equation (18). After calculating the variance σ 2 from the time change of the noise waveform, we derived the Stokes signal A 2 /2 by substituting σ 2 from the value <Y(t)>. From this calculation, we estimated the DR from (A 2 /2)/σ 2 and the SNR from (A 2 /2)/σ Y of Equation (19) to be 12.5 dB on a logarithmic scale and 3.05 on a linear scale, Sensors 2020, 20, 936 13 of 21 respectively. As we reduced the photocurrent ∆I p by adjusting the SOP of the pump lights which entered the balanced mixer, the fluctuations of both the Stokes signal and the noise decreased to DR = 28 dB and SNR = 18.1, respectively, as shown in Figure 7b. We estimated both the DR and SNR values at different photocurrent values of ∆I p and plotted their relation in Figure 8 together with the calculation according to Equation (22). The agreement between the measured and calculated values shows that the fluctuations in the Stokes signal originated from the interaction between the Stokes signal and the noise via square-law detection.
down-conversion frequency was fixed at 10.77 GHz to measure the Stokes signal from the same position at a distance of 2 cm from the joint of the mated fiber connectors. To check the validity of the relationship between the DR and SNR, which we derived as Equation (22), we had to measure the time change of the low-pass filter output under a condition where the bandwidth VBW of the low-pass filter was at least twice the RBW of the band-pass filter, as described in Appendix A. To meet this condition, we used an analog-type RF spectrum analyzer where we set the center frequency, span, VBW and RBW at 5 kHz, 0 Hz (or a zero span), 100 Hz, and 30 Hz, respectively. The spectrum analyzer had a video output port from which the square-law detection output was streamed.
We reduced the photocurrent ΔIp gradually to reduce the noise, for every value of which we acquired the time changes of the Stokes signal and noise for ten seconds by switching the serrodyne modulation to both phase modulators on and off. Figure 7a shows one example of the acquired waveforms, where the noise (lower trace) was so high that it was close to the Stokes signal (upper trace), which also fluctuated with time. We calculated the standard deviation and mean value of the fluctuating Stokes signal from the acquired waveform, which corresponded to σY in Equation (17), and <Y(t)> = A 2 /2 + σ 2 in Equation (18).  After calculating the variance σ 2 from the time change of the noise we reduced the photocurrent ΔIp gradually to reduce the noise, for every value of which we acquired the time changes of the Stokes signal and noise for ten seconds by switching the serrodyne modulation to both phase modulators on and off. Figure 7a shows one example of the acquired waveforms, where the noise (lower trace) was so high that it was close to the Stokes signal (upper trace), which also fluctuated with time. We calculated the standard deviation and mean value of the fluctuating Stokes signal from the acquired waveform, which corresponded to σY in Equation (17), and <Y(t)> = A 2 /2 + σ 2 in Equation (18). After calculating the variance σ 2 from the time change of the noise waveform, we derived the Stokes signal A 2 /2 by substituting σ 2 from the value <Y(t)>. From this calculation, we estimated the DR from (A 2 /2)/σ 2 and the SNR from (A 2 /2)/σY of Equation (19)

Brillouin Grating Reflectogram Revealed with DFS
The light powers of the two pump lights and probe light were the same as those in Section 4.3, and the down-conversion frequency was also 10.77 GHz. The photocurrent ΔIp of the leaked pump light was adjusted at 80 nA. The reflectogram around the joint of the angle-polished fiber connectors, which we obtained with the DFS, is shown in Figure 9a. We calculated the mean level and standard deviation of the fluctuating signal ranging from −2.5 to −1 cm and estimated the SNR by the ratio of the former to the latter to be 24. The achieved SNR was 6 times higher than that in our previous paper [8]. After performing ten repetitive measurements for 4 min, we had a smoothed reflectogram as shown in Figure 9b, where the SNR was improved to 65, which was the same as that we previously obtained with 200 repetitive measurements.

Brillouin Grating Reflectogram Revealed with DFS
The light powers of the two pump lights and probe light were the same as those in Section 4.3, and the down-conversion frequency was also 10.77 GHz. The photocurrent ∆I p of the leaked pump light was adjusted at 80 nA. The reflectogram around the joint of the angle-polished fiber connectors, which we obtained with the DFS, is shown in Figure 9a. We calculated the mean level and standard deviation of the fluctuating signal ranging from −2.5 to −1 cm and estimated the SNR by the ratio of the former to the latter to be 24. The achieved SNR was 6 times higher than that in our previous paper [8]. After performing ten repetitive measurements for 4 min, we had a smoothed reflectogram as shown in Figure 9b, where the SNR was improved to 65, which was the same as that we previously obtained with 200 repetitive measurements. We increased the photocurrent ΔIp from 80 to 160 to 320 nA, and at each value we acquired ten reflectograms from the mated fiber connectors and plotted the mean value and distribution of their SNRs along the fiber from -2.5 to -1 cm with the open circles and error bars in Figure 10. The horizontal axis is the attenuation expressed by ΔIp/Ip2 on a dB scale. While calculating a reflectogram with the DFS, we employed two kinds of rectangular windows for band-pass filtering. We employed a 630-Hz-wide window when we demodulated the signal, and a window ranging from 1527 to 1576 nm in the spectral region. It was not clear a priori which bandwidth should be the resolution bandwidth RBW of the square-law detection, which was included in the expression of the DR as <Is 2 (t)>/(σ 2 × RBW) = CsILOIprPp1Pp2/(σ 2 × RBW), where Cs = 4.29 × 10 −5 and σ 2 is denoted by Equation (15). The theoretical SNR at the former bandwidth RBW = 630 Hz as a function of the attenuation was too low to fit the measured values, as shown in the figure. By changing the RBW until the theoretical curve fitted the measured values, we found that the actual RBW was 25 Hz. If the linear stage had translated at a constant speed of 400 μm, the beat frequency produced by the translation would distribute from 508 to 524 Hz depending on the frequency of the individual spectral components. That is, the narrowest bandwidth that we could apply to the interferogram without deformation was RBW = 16 Hz, and the resultant SNR is also plotted in the figure. Although the stage did not translate at a constant speed, we concluded that the combination of changing the grid to the equal path increments and the band-pass filtering in the spectral region enabled us to reduce the noise level close to the minimum level.  We increased the photocurrent ∆I p from 80 to 160 to 320 nA, and at each value we acquired ten reflectograms from the mated fiber connectors and plotted the mean value and distribution of their SNRs along the fiber from -2.5 to -1 cm with the open circles and error bars in Figure 10. The horizontal axis is the attenuation expressed by ∆I p /I p2 on a dB scale. While calculating a reflectogram with the DFS, we employed two kinds of rectangular windows for band-pass filtering. We employed a 630-Hz-wide window when we demodulated the signal, and a window ranging from 1527 to 1576 nm in the spectral region. It was not clear a priori which bandwidth should be the resolution bandwidth RBW of the square-law detection, which was included in the expression of the DR as <I s 2 (t)>/(σ 2 × RBW) = C s I LO I pr P p1 P p2 /(σ 2 × RBW), where C s = 4.29 × 10 −5 and σ 2 is denoted by Equation (15). The theoretical SNR at the former bandwidth RBW = 630 Hz as a function of the attenuation was too low to fit the measured values, as shown in the figure. By changing the RBW until the theoretical curve fitted the measured values, we found that the actual RBW was 25 Hz. If the linear stage had translated at a constant speed of 400 µm, the beat frequency produced by the translation would distribute from 508 to 524 Hz depending on the frequency of the individual spectral components. That is, the narrowest bandwidth that we could apply to the interferogram without deformation was RBW = 16 Hz, and the resultant SNR is also plotted in the figure. Although the stage did not translate at a constant speed, we concluded that the combination of changing the grid to the equal path increments and the band-pass filtering in the spectral region enabled us to reduce the noise level close to the minimum level.
(a) (b) Figure 9. (a) Brillouin grating reflectogram around the joint of the mated fiber connectors, which was derived from one scan of the linear stage. (b) Mean reflectogram derived by averaging 10 reflectograms. The reflectograms were derived by processing the acquired interferograms with the DFS.
We increased the photocurrent ΔIp from 80 to 160 to 320 nA, and at each value we acquired ten reflectograms from the mated fiber connectors and plotted the mean value and distribution of their SNRs along the fiber from -2.5 to -1 cm with the open circles and error bars in Figure 10. The horizontal axis is the attenuation expressed by ΔIp/Ip2 on a dB scale. While calculating a reflectogram with the DFS, we employed two kinds of rectangular windows for band-pass filtering. We employed a 630-Hz-wide window when we demodulated the signal, and a window ranging from 1527 to 1576 nm in the spectral region. It was not clear a priori which bandwidth should be the resolution bandwidth RBW of the square-law detection, which was included in the expression of the DR as <Is 2 (t)>/(σ 2 × RBW) = CsILOIprPp1Pp2/(σ 2 × RBW), where Cs = 4.29 × 10 −5 and σ 2 is denoted by Equation (15). The theoretical SNR at the former bandwidth RBW = 630 Hz as a function of the attenuation was too low to fit the measured values, as shown in the figure. By changing the RBW until the theoretical curve fitted the measured values, we found that the actual RBW was 25 Hz. If the linear stage had translated at a constant speed of 400 μm, the beat frequency produced by the translation would distribute from 508 to 524 Hz depending on the frequency of the individual spectral components. That is, the narrowest bandwidth that we could apply to the interferogram without deformation was RBW = 16 Hz, and the resultant SNR is also plotted in the figure. Although the stage did not translate at a constant speed, we concluded that the combination of changing the grid to the equal path increments and the band-pass filtering in the spectral region enabled us to reduce the noise level close to the minimum level.

Examples of OLCR Diagnosis
The light powers of the two pump lights and probe light were the same as those in Section 4.3. Since we succeeded in reducing the measurement time required for obtaining one Brillouin grating reflectogram with a high SNR to 4 min, we were able to obtain the reflectograms from the mated fiber connectors as a function of the down-conversion frequency, which changed from 10.4 to 10.85 GHz in steps of 10 MHz, as shown in Figure 11. Throughout the measurements, every 20 min we made a fine adjustment of PC2 to maintain the leaked pump light at ∆I p = 80 nA. Figure 11a shows the reflectograms we obtained around the joint of the glass cores in the rear and front connectors. Since these connectors were spring-loaded and mated, the angle-polished fiber endfaces were pressed together and stress applied to the glass core at a distance of up to 0.5 mm from the joint moved that part of the Brillouin grating to a different frequency region. In the front connector, the part of the fiber from its endface to the base of the glass core, which was 9.8 mm long, passed through a small hole in the ferrule and was fixed to it with adhesive. The rest of the fiber had a primary coating that was fixed to a metal flange with adhesive, as shown in the inset of Figure 11b. The Brillouin grating was generated at 10.76 GHz in the glass core at a distance of 0.5 mm from the joint, moved dynamically in a positive direction along the fiber, suddenly disappeared at the base of the core, which was located at a distance of 9.8 mm, and appeared again at 10.805 GHz to move dynamically through the primary-coated glass core. This propagation means that the primary coating of the fiber fixed in the metal flange relieved the glass core of the stress that was applied in the ferrule by 9.1 × 10 −4 , where we assumed that the strain coefficient was 4.93 MHz/10 −4 at 1.55 µm [20].

Examples of OLCR Diagnosis
The light powers of the two pump lights and probe light were the same as those in Section 4.3. Since we succeeded in reducing the measurement time required for obtaining one Brillouin grating reflectogram with a high SNR to 4 min, we were able to obtain the reflectograms from the mated fiber connectors as a function of the down-conversion frequency, which changed from 10.4 to 10.85 GHz in steps of 10 MHz, as shown in Figure 11. Throughout the measurements, every 20 min we made a fine adjustment of PC2 to maintain the leaked pump light at ΔIp = 80 nA. Figure 11a shows the reflectograms we obtained around the joint of the glass cores in the rear and front connectors. Since these connectors were spring-loaded and mated, the angle-polished fiber endfaces were pressed together and stress applied to the glass core at a distance of up to 0.5 mm from the joint moved that part of the Brillouin grating to a different frequency region. In the front connector, the part of the fiber from its endface to the base of the glass core, which was 9.8 mm long, passed through a small hole in the ferrule and was fixed to it with adhesive. The rest of the fiber had a primary coating that was fixed to a metal flange with adhesive, as shown in the inset of Figure 11b. The Brillouin grating was generated at 10.76 GHz in the glass core at a distance of 0.5 mm from the joint, moved dynamically in a positive direction along the fiber, suddenly disappeared at the base of the core, which was located at a distance of 9.8 mm, and appeared again at 10.805 GHz to move dynamically through the primary-coated glass core. This propagation means that the primary coating of the fiber fixed in the metal flange relieved the glass core of the stress that was applied in the ferrule by 9.1 × 10 −4 , where we assumed that the strain coefficient was 4.93 MHz/10 −4 at 1.55 μm [20]. We changed the down-conversion frequency from 10.35 to 10.875 GHz more precisely in steps of 5 MHz and measured the reflectograms around joint of the mated connectors. Then we plotted the Stokes signal as a function of the RF frequency at equidistant spaces along the fiber to display the Brillouin spectrum distribution as shown in Figure 12. The center frequency of each spectrum provided us with the Brillouin frequency shift along the fiber as shown in Figure 13, from which we could observe the change in the strain along the fiber with respect to that at point A by rescaling the vertical axis in the figure. The spectra at the particular positions A, B, C, and D in the front connector are shown by the inset in Figure 13. The spectrum, which we acquired at a certain point with OLCR, is the weighted average of the Brillouin spectra distributed along the fiber within a distance of around 0.1 mm from the point. Therefore, when the stress applied to the glass core increased rapidly within a short distance of 0.5 mm as shown in Figure 13, Brillouin spectra with We changed the down-conversion frequency from 10.35 to 10.875 GHz more precisely in steps of 5 MHz and measured the reflectograms around joint of the mated connectors. Then we plotted the Stokes signal as a function of the RF frequency at equidistant spaces along the fiber to display the Brillouin spectrum distribution as shown in Figure 12. The center frequency of each spectrum provided us with the Brillouin frequency shift along the fiber as shown in Figure 13, from which we could observe the change in the strain along the fiber with respect to that at point A by rescaling the vertical axis in the figure. The spectra at the particular positions A, B, C, and D in the front connector are shown by the inset in Figure 13. The spectrum, which we acquired at a certain point with OLCR, is the weighted average of the Brillouin spectra distributed along the fiber within a distance of around 0.1 mm from the point. Therefore, when the stress applied to the glass core increased rapidly within a short distance of 0.5 mm as shown in Figure 13, Brillouin spectra with different frequency shifts contributed to the weighted averaging, resulting in the spectral broadening and deformation shown in the inset.   As another example of the OLCR diagnosis, we tested a 3-dB fused biconically tapered single-mode fiber coupler [21] operating at 1.55 μm, as shown in Figure 14. Since the coupler was molded in a package, we could not obtain any information regarding the coupler parameters such as the length of the heated section and the taper size. We assume that the pump light at ωp was launched from an output port of the coupler, and that the pump light at ωp + Ω and the probe light were launched from an input port. We chose origin C of the position coordinate z as the point about which the reflectogram profiles appeared to be symmetrical, and A and B denote the points at z = ±0.8 mm. Since outside the line segment AB the Brillouin gratings were generated at the same down-conversion frequency of 10.805 GHz as in the stress-released fiber (as shown in Figure 11b), we considered the outside to consist of the input and output ports of the constituent fibers, whereas the inside was the fused region where the two lowest supermodes were coupling. The reflectograms observed at a 1.5 cm position D were cleaved so sharply that the fiber part was considered to be fixed on an invar plate with adhesive. broadening and deformation shown in the inset.  As another example of the OLCR diagnosis, we tested a 3-dB fused biconically tapered single-mode fiber coupler [21] operating at 1.55 μm, as shown in Figure 14. Since the coupler was molded in a package, we could not obtain any information regarding the coupler parameters such as the length of the heated section and the taper size. We assume that the pump light at ωp was launched from an output port of the coupler, and that the pump light at ωp + Ω and the probe light were launched from an input port. We chose origin C of the position coordinate z as the point about which the reflectogram profiles appeared to be symmetrical, and A and B denote the points at z = ±0.8 mm. Since outside the line segment AB the Brillouin gratings were generated at the same down-conversion frequency of 10.805 GHz as in the stress-released fiber (as shown in Figure 11b), we considered the outside to consist of the input and output ports of the constituent fibers, whereas the inside was the fused region where the two lowest supermodes were coupling. The reflectograms observed at a 1.5 cm position D were cleaved so sharply that the fiber part was As another example of the OLCR diagnosis, we tested a 3-dB fused biconically tapered single-mode fiber coupler [21] operating at 1.55 µm, as shown in Figure 14. Since the coupler was molded in a package, we could not obtain any information regarding the coupler parameters such as the length of the heated section and the taper size. We assume that the pump light at ω p was launched from an output port of the coupler, and that the pump light at ω p + Ω and the probe light were launched from an input port. We chose origin C of the position coordinate z as the point about which the reflectogram profiles appeared to be symmetrical, and A and B denote the points at z = ±0.8 mm. Since outside the line segment AB the Brillouin gratings were generated at the same down-conversion frequency of 10.805 GHz as in the stress-released fiber (as shown in Figure 11b), we considered the outside to consist of the input and output ports of the constituent fibers, whereas the inside was the fused region where the two lowest supermodes were coupling. The reflectograms observed at a 1.5 cm position D were cleaved so sharply that the fiber part was considered to be fixed on an invar plate with adhesive. The power of the pump light, which was launched from the output port, was attenuated by 3 dB to reach the input port, where the generated Stokes light power should decrease by 3 dB compared with that from the fiber connector. On the other hand, the powers of the pump and probe lights, which were launched from the input port, were both attenuated by 3 dB to reach the output port, where the generated Stokes light power should decrease by 6 dB. After passing through the fused region, the Stokes light power was attenuated by 3 dB to reach the input port, resulting in atotal attenuation of 9 dB. From this estimation, therefore, the Stokes signal observed at the output port was considered to be 6 dB lower than that at the input port, which is consistent with the difference between the signal levels of the input and output ports.
The reflectograms that we observed in the fused region were so complicated that we performed the reflectogram measurements again by translating the stage over a shorter span of ± 4mm, changing the RF frequency more precisely in steps of 5 MHz, and doubling the number of repetitive measurements to increase the SNR. The resultant Brillouin spectrum distribution around the center is shown in Figure 15. That is, we found that the complicated appearance of the reflectogram distribution arose from the fact that the Brillouin frequency shift changed along the fiber symmetrically with respect to the center from 11.05 to 11.2 GHz, as shown by curve (i). Since the Brillouin frequency shift is closely related to the interaction of the supermodes with the acoustic waves, the frequency shift as a function of the position will provide us with useful information about the longitudinal taper profile, which is an important step when modeling the coupler in the calculation. The power of the pump light, which was launched from the output port, was attenuated by 3 dB to reach the input port, where the generated Stokes light power should decrease by 3 dB compared with that from the fiber connector. On the other hand, the powers of the pump and probe lights, which were launched from the input port, were both attenuated by 3 dB to reach the output port, where the generated Stokes light power should decrease by 6 dB. After passing through the fused region, the Stokes light power was attenuated by 3 dB to reach the input port, resulting in atotal attenuation of 9 dB. From this estimation, therefore, the Stokes signal observed at the output port was considered to be 6 dB lower than that at the input port, which is consistent with the difference between the signal levels of the input and output ports.
The reflectograms that we observed in the fused region were so complicated that we performed the reflectogram measurements again by translating the stage over a shorter span of ± 4mm, changing the RF frequency more precisely in steps of 5 MHz, and doubling the number of repetitive measurements to increase the SNR. The resultant Brillouin spectrum distribution around the center is shown in Figure 15. That is, we found that the complicated appearance of the reflectogram distribution arose from the fact that the Brillouin frequency shift changed along the fiber symmetrically with respect to the center from 11.05 to 11.2 GHz, as shown by curve (i). Since the Brillouin frequency shift is closely related to the interaction of the supermodes with the acoustic waves, the frequency shift as a function of the position will provide us with useful information about the longitudinal taper profile, which is an important step when modeling the coupler in the calculation.
Together with the main distribution (i), we observed another small signal distribution shown by (ii) whose Brillouin frequency shift changed gradually from 11.15 to 11.2 GHz. The fused region consisted of the coalesced tapered region and a uniform-waist region. On the assumption that the fused part became a core and the surrounding air was cladding in the latter region, there is a possibility that higher order modes were excited there and propagated along the tapered fiber in the former region. Together with the main distribution (i), we observed another small signal distribution shown by (ii) whose Brillouin frequency shift changed gradually from 11.15 to 11.2 GHz. The fused region consisted of the coalesced tapered region and a uniform-waist region. On the assumption that the fused part became a core and the surrounding air was cladding in the latter region, there is a possibility that higher order modes were excited there and propagated along the tapered fiber in the former region.

Conclusions
We have shown theoretically and experimentally that speckle-like noise was generated during Brillouin grating measurements with micrometer-resolution OLCR by the interaction of the Stokes signal with the beat noise via square-law detection. To reduce the noise we had no choice but to reduce the noise generated by the beat between the LO light and a pump light entering the balanced mixer. This was achieved by using a fiber-optic polarizer and making a fine adjustment to the SOP of the polarizer output. We achieved the SNR of 24 for one Brillouin grating reflectogram and that of 65 by averaging ten individual reflectograms. The achievement of such a high SNR enabled us to acquire a reflectogram as a function of the down-conversion frequency, which provided us with the Brillouin spectrum distributions in mated fiber connectors and a 3-dB fused fiber coupler.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The autocorrelation function of the output Y(t) from the square-law device is: where: Figure 15. Distribution of Brillouin spectrum around the fused region of the fiber coupler. The main distribution is shown by (ii), which was associated with the small signal distribution shown by (ii).

Conclusions
We have shown theoretically and experimentally that speckle-like noise was generated during Brillouin grating measurements with micrometer-resolution OLCR by the interaction of the Stokes signal with the beat noise via square-law detection. To reduce the noise we had no choice but to reduce the noise generated by the beat between the LO light and a pump light entering the balanced mixer. This was achieved by using a fiber-optic polarizer and making a fine adjustment to the SOP of the polarizer output. We achieved the SNR of 24 for one Brillouin grating reflectogram and that of 65 by averaging ten individual reflectograms. The achievement of such a high SNR enabled us to acquire a reflectogram as a function of the down-conversion frequency, which provided us with the Brillouin spectrum distributions in mated fiber connectors and a 3-dB fused fiber coupler.

Conflicts of Interest:
The authors declare no conflict of interest.
The first to third terms are generated by the coupling of the signal with itself, that of the signal with the noise, and that of the noise with itself, respectively [19]. Since the Stokes signal is a sinusoidal waveform, we let I(t) = Acos(2πf c t + θ). Here we introduce the spectral density function S N (f ) of