Sensing is a key technology for acquiring various types of information, serving as a core component of modern science and technology. It is widely applied in fields such as the Internet of Things, artificial intelligence, automation, and other fields.1 With advancements in optics and photonics technology,2^–4 optical fiber sensing technology has been widely applied due to its outstanding advantages, such as high sensitivity, immunity to electromagnetic interference, high integration, and exceptional precision. It has become particularly indispensable in fields demanding high precision and reliability. However, in many extreme application environments, the measured signals often exhibit characteristics such as weak energy, wide frequency range, susceptibility to external interference, and transmission attenuation. These challenges necessitate sensing systems with enhanced precision and signal-to-noise ratios (SNRs).

Enhanced detection precision has been achieved through advancements in both software and hardware. On the software side, various denoising algorithms have been developed.5^–9 Wavelet decomposition has been applied to the detection signal, leading to improvements in the SNR of acoustic recovery.10 On the hardware side, the adoption of polarization-maintaining fiber loop mirrors has demonstrated a notable enhancement in SNR.11

Traditional optical sensors primarily rely on mechanical, thermal, or electrical effects to reflect the true value of the quantity to be measured. The sensing parameters in these effects, whether utilizing light intensity,12 phases,13^,14 polarization,15^,16 or frequency17 for sensing, are all real parameters. In fact, the electric field of light is inherently a complex value. The real and imaginary parts of an optical signal are interconnected through the Kramers–Kronig relationship.18 Neglecting this intrinsic relationship between the real and imaginary parts of optical signals has limited further improvements in sensing precision.

The intensity and phase of light can reflect the variations in the real and imaginary parts of the electromagnetic field of the light wave. In this paper, we introduce a method for simultaneous analysis of optical phase and intensity using Bayesian estimation, aiming to enhance the precision of optical sensing. Bayesian estimation is a statistical approach that refines parameter estimates by integrating prior probabilities with observed data.19^,20 It effectively reveals the characteristics of data and offers advantages such as simplicity, low computational cost, and online processing capability. Through Bayesian estimation, the algorithm proposed in this paper achieves statistically optimal processing of measured information, by considering the interaction between the real and imaginary parts of optical signals. Moreover, after applying a self-Bayesian estimation algorithm, other denoising algorithms or hardware optimizations can still be utilized in the sensing system, enhancing measurement precision further. The effectiveness of this self-Bayesian estimation was validated in the experiment. The experimental results show that, without altering the experimental setup of the sensing system, the algorithm proposed in this paper exhibits excellent denoising performance over a broad frequency range from 200 to 500 kHz. The noise variance is significantly reduced from 1.86×10−4 and 1.66×10−4 to 8.53×10−5, and the SNR is increased by 1.67 dB.

To simultaneously obtain the intensity and phase of the sensing light, a fiber Bragg grating (FBG) is employed in this paper to measure vibration signals. The sensing signals are demodulated through I/Q quadrature demodulation in the heterodyne interference system.21^,22 As illustrated in Fig. 1, a narrow linewidth laser (NLL) emits a laser beam, which is divided at the fiber optical coupler 1 (OC1) into reference and signal lights. The reference light is modulated to incorporate an additional frequency shift via an acousto-optic modulator (AOM). Meanwhile, the signal light is directed to the FBG sensor through a circulator. The light reflected by the FBG, having undergone intensity and phase modulation due to external signals, re-enters the system through another port of the circulator and proceeds to fiber OC2. At OC2, the reference and reflected signal lights interfere, creating a beat frequency signal. This signal is then converted into a voltage signal by a photodetector and transmitted to a computer for subsequent demodulation.

The beat frequency signal generated at OC2 can be expressed by Is=E12+(E2δ)2+2E1E2δcos[ωt+φ(t)+φS(t)],(1)where E1 and E2δ represent the amplitudes of the electric field in the two optical paths, respectively. δ denotes the reflectance of FBG, ω is the additional frequency shift introduced by the AOM, φ(t) corresponds to the phase modulation of the reflected light from the FBG, and φs(t) is the phase noise induced by ambient disturbances.

In I/Q quadrature demodulation, the interference signal is separately multiplied by sine and cosine signals of angular frequency ω, I0(t)={E12+(E2δ)2+2E1E2δcos[ωt+φ(t)]+φS(t)}·cos(ωt),(2)Q0(t)={E12+(E2δ)2+2E1E2δcos[ωt+φ(t)]+φS(t)}·sin(ωt).(3)

After the carrier signal passes through the low-pass filter, the resulting orthogonal components are obtained, I(t)=E1E2δcos[φ(t)],(4)Q(t)=E1E2δsin[φ(t)].(5)

The phase and the intensity are obtained in Eqs. (6) and (7), φ(t)=−arctanQ(t)I(t),(6)I1(t)=2E1E2δ=2I(t)2+Q(t)2.(7)

The intensity and phase of the beat signals include both the measured signals and Gaussian noise, which are expressed in Eqs. (8) and (9), I1(t)=I1true+I1noise,(8)φ(t)=φtrue+φnoise.(9)

The electric field of light is inherently complex with the real and imaginary parts satisfying the Kramers–Kronig relationship.18 However, extracting the information to be measured from the light is still treating the optical signal as a real function. Such treatment discards the relationship between the real and imaginary parts of a complex number, which will inevitably limit the improvement of sensing precision.

A self-Bayesian estimation approach is introduced in this paper, which utilizes the intensity demodulation and phase demodulation results as the observed values, and establishes the state equation based on the initial frequency corresponding to the peak frequency of the spectrum. The core principle of self-Bayesian estimation involves creating a linear stochastic system and a discrete-time state space model, integrating state prediction and observation update equations, which can be expressed as xk=F·xk−1+wk−1,(10)yk=H·xk+vk,(11)where xk∈Rn represents the state parameter of the system at the current time, F∈Rn×n represents the state transition matrix of the system, yk∈Rm represents the external observation value at the current time, H∈Rm×n represents the external observation matrix, wk and vk represent the state prediction noise vector and observation noise vector, respectively. The mean value of wk and vk is both 0. The covariance matrix is the Gaussian distribution of Q and R, that is, P(w):N(0,Q),P(v):N(0,R), Q and R can be expressed as $$\begin{gathered} Q=E[wwT], \\ R=E[vvT]. \end{gathered}$$(12)

Considering that at time k, the prior value x^k− obtained by the state prediction equation and the observation value x^k,mea obtained by the observation equation can be expressed as $$\begin{gathered} x^k−=F·x^k−1, \\ x^k,mea=H−·yk. \end{gathered}$$(13)The estimated value x^k at time k can be obtained by data fusion of the predicted prior value x^k− and the observed value x^k,mea, x^k=x^k−+G(x^k,mea−x^k−).(14)We make the transformation G=KkH and substitute Eq. (14) to get x^k=x^k−+Kk(yk−H·x^k−).(15)

The error function ek is introduced, ek=xk−x^k.(16)ek satisfies a Gaussian distribution with a mean of 0 and a covariance matrix of P=E[eeT], i.e., P(e):N(0,P). Substituting Eqs. (11) and (15) into Eq. (16) yields $$\begin{gathered} ek=xk−[x^k−+Kk(yk−H·x^k−)] \\ =xk−x^k−−KkHxk−Kkvk+KkHx^k− \\ =(I−KkH)(xk−x^k−)−Kkvk. \end{gathered}$$(17)

A prior error function ek− is introduced as ek−=xk−x^k−.(18)Then, Pk, the covariance matrix at time k, can be expressed as $$\begin{gathered} Pk=E[ekekT] \\ =E{[(I−KkH)(xk−x^k−)−Kkvk][(I−KkH)(xk−x^k−)−Kkvk]T} \\ =E{[(I−KkH)ek−−Kkvk][(I−KkH)ek−−Kkvk]T} \\ =E[(I−KkH)ek−ek−T(I−KkH)T−(I−KkH)ek−vkTKkT \\ −Kkvkek−T(I−KkH)T+KkvkvkTKkT]. \end{gathered}$$(19)

As the noise satisfies a Gaussian distribution with a mean of 0, there is E[ek−]=0, so Eq. (19) can be simplified as Pk=(I−KkH)E[ek−ek−T](I−KkH)T+KkE[vkvkT]KkT.(20)

Let the distribution of the prior error function ek− be Pk−=E[ek−ek−T] and substitute Eq. (20) to obtain $$\begin{gathered} Pk=(I−KkH)Pk−(I−KkH)T+KkRKkT \\ =(Pk−−KkHPk−)(I−KkH)T+KkRKkT \\ =Pk−−KkHPk−−Pk−HTKkT+KkHPk−HTKkT+KkRKkT. \end{gathered}$$(21)

When x^k→xk, a trace tr(Pk) of Pk is minimized, tr(Pk)=tr(Pk−)−2tr(KkHPk−)+tr(KkHPk−HTKkT)+tr(KkRKkT).(22)

Let ∂tr(Pk)∂Kk=0. It can be obtained that −2(HPk−)T+2KkHPk−HT+2KkR=0.(23)By sorting out Eq. (23), we can get Kk=Pk−HT(HPk−HT+R)−1.(24)Substituting Eq. (24) into Eq. (21) yields Pk=(I−KkH)Pk−.(25)We update Pk− through iteration and substitute Eqs. (10) and (13) into Eq. (18) to obtain $$\begin{gathered} ek−=Fxk−1+wk−1−Fx^k−1 \\ =Fek−1+wk−1. \end{gathered}$$(26)Substituting Eq. (26) into Pk−=E[ek−ek−T] yields $$\begin{gathered} Pk−=E[(Fek−1+wk−1)(Fek−1+wk−1)T] \\ =E(Fek−1ek−1TFT+Fek−1wk−1T+wk−1ek−1TFT+wk−1wk−1T). \end{gathered}$$(27)As ek−1 and wk−1 are independent, Eq. (27) can be simplified as $$\begin{gathered} Pk−=FE(ek−1ek−1T)FT+E(wk−1wk−1T) \\ =FPk−1FT+Q. \end{gathered}$$(28)

Based on the above theory, for the intensity and phase obtained in demodulation, the noise can be removed by the following steps:

1. Step I:Convert intensity demodulation results to phase quantities.

The light-intensity spectrum and the phase spectrum (Fig. 2) of FBG can be simulated using the transfer matrix method and measured by a wavelength-tunable laser in the experiment. For a particular wavelength, the intensity obtained through demodulation can be converted to the phase by transfer matrix of FBG according to the correspondence between the intensity and the phase of the reflected light shown in Fig. 2. In the experiment, the intensity and phase of the FBG reflection light can be obtained, respectively, according to Eqs. (6) and (7) for a particular wavelength.

The laser wavelength is then fixed at the 3-dB point for sensing demodulation. The light intensity obtained by Eq. (7) is transformed into the phase through the intensity-phase correspondence. Both the phase obtained through this transformation and the phase obtained from Eq. (6) serve as observed values for self-Bayesian estimation. The inset in Fig. 2 illustrates the working interval, and the transformation between intensity and phase can be approximated as a linear relationship, represented as φ1(t)=kI1(t).

1. Step II:Establish the state prediction equation and observation update equation.

The signal x(t) is obtained by performing a Taylor expansion of time at x(t−1), $$\begin{gathered} x(t)=x(t−1)+x′(t−1)·dt+x′(t−1)·dt2/2!+o[x′(t−1)] \\ x′(t)=x′(t−1)+x′(t−1)·dt+o[x′(t−1)]. \end{gathered}$$(29)If the signal is an ideal cosine function expressed as x=acos(ωst), where a is the signal amplitude and ωs is the signal frequency, it can be obtained by substituting the ideal signal into the sorting {x(t)≈(1−ωs2dt2/2)·x(t−1)+dt·x′(t−1)x′(t)≈(−ωs2dt)·x(t−1)+x′(t−1).(30)

The signal is sampled in the time domain, generating a discrete vector. The equations of state and observation are then established as $$\begin{gathered} xk=F·xk−1+wk−1, \\ yk=H·xk+vk, \end{gathered}$$(31)where xk=[x(k)x′(k)], F=[1−ωs2dt2/2dt−ωs2dt1], yk=[φ1(k)φ(k)], and H=[1010].

1. Step III:Initialize state and covariance.

During state initialization, the sensor-provided initial demodulation value is typically used to estimate the system’s initial state x^0. For covariance initialization, the initial covariance matrix Q is configured based on the correlation between the system’s characteristics and prior knowledge. A more accurate initial state estimate allows for a smaller covariance matrix, indicating high confidence in the initial state. Conversely, if the initial state estimate is imprecise or prior knowledge is unavailable, the covariance matrix is set larger to indicate greater uncertainty.

1. Step IV:Update the state prediction equation according to Eqs. (13) and (28) $$\begin{gathered} x^k−=F·x^k−1, \\ Pk−=FPk−1FT+Q. \end{gathered}$$(32)
2. Step V:Update the observation equation according to Eqs. (15), (24), and (25) {Kk=Pk−HT(HPk−HT+R)−1x^k=x^k−+Kk(yk−H·x^k−)Pk=(I−KkH)Pk−.(33)
3. Step VI:Complete the update by repeating steps IV–V; the flow chart of self-Bayesian estimation is shown in Fig. 3.

To demonstrate the self-Bayesian estimation numerically, the original signal is set as a standard sine curve with a frequency of 200 kHz. In the simulation, the sampling rate is 625 MS/s, and the sampling number is 1×104. Two Gaussian noises with a variance of 1×10−3, uncorrelated with each other, are, respectively, superimposed on the standard sine wave, generating two observed signals. The simulation results are shown in Fig. 4. In Step III of Sec. 2.2, the initialization of the state prediction noise vector variance Q is often obtained based on experience. To study the impact of the initial value of variance on the demodulation results and the convergence of the algorithm, the state prediction noise vector variance Q is set in a wide range (from 1×10−4 to 1000×10−4). The theoretical posteriori variance [Pk in Eq. (33)] and the actual variance of the filtered signal [x^k in Eq. (33)] are shown in Fig. 4(a). The results show that when the initial value of Q is small, Pk has been decreasing, but the variance of x^k increases significantly due to the increase of the prediction weight. When the initial value of Q is large, the prediction information will be wasted, resulting in the increase of Pk and the variance of x^k. Therefore, to achieve the optimal performance, the initial value of the variance Q of the state prediction noise vector should be set to 12×10−4, at which time the variance of x^k reaches the minimum value, and Pk is consistent with the variance of x^k. Figure 4(b) shows the iterative process of theoretical variance. It can be seen that after five iterations, the variance almost reaches the theoretical result, demonstrating the rapid convergence of the self-Bayesian algorithm. Figure 4(c) shows the filtered signal and the noise in the time domain. In Fig. 4(d), it can be seen that the noise level is significantly reduced after filtering, and the variance of the actual calculation is 3.8×10−4, which is significantly reduced compared with the variance of the observed signal of 1×10−3.

Based on the aforementioned self-Bayesian performance simulations, further simulations of the experimental system shown in Fig. 1 are conducted to validate the performance of this method in sensing. The laser frequency is set to 1550.05 nm, the linewidth is set to 100 Hz, the frequency shift in the AOM is set to 80 MHz, the Bragg wavelength of FBG is set to 1550 nm, the length of the grating area is set to 5 mm, the effective refractive index is set to 1.468, and the refractive index modulation amplitude is set to 0.00012. The reflected light signal is simulated using the transfer matrix method for the FBG. The frequency of the vibration signals ranges from 200 to 500 kHz. Based on the simulation parameters set above, the reflection intensity spectrum and reflection phase spectrum around the laser frequency are shown in the inset panels in Fig. 2.

Figure 5 shows the demodulation results under different signal frequencies and vibration amplitudes. Figure 5(a) compares the variances of phase demodulation, intensity transformation, and self-Bayesian estimation. It is clear that for any given frequency and vibration amplitude, the variances from self-Bayesian estimation are consistently lower than those from phase demodulation or intensity transformation. Corresponding SNR values are shown in Fig. 5(b). Compared with direct phase demodulation or intensity transformation, the SNR achieved through self-Bayesian estimation is significantly enhanced. These results indicate that, compared with phase demodulation and intensity transformation, the self-Bayesian estimation method can more effectively reduce variance and improve the SNR.

The experimental setup illustrated in Fig. 6 is built up to validate the self-Bayesian estimation. The light source is a narrow linewidth laser (E15 manufactured by NKT Photonics, Birkerød, Denmark). The light source has a linewidth of less than 100 Hz and a central wavelength of 1550.050 nm. An optical isolator is employed to prevent backreflection interference. The laser beam is split into two beams at the fiber OC1: one beam enters the optical power meter (OPM) for intensity monitoring, whereas the other is further divided into reference light and signal light via the fiber OC2. An additional frequency shift of 80 MHz is introduced by an AOM (T-M080-0.4C2J-3-F2S manufactured by Gooch & Housego, Ilminster, United Kingdom). The beat signal is converted into a voltage signal by a balanced photodetector (BPD) and then sent to the computer for demodulation by a data acquisition card (DAQ, M4i.2221-x8 manufactured by Spectrum Instrumentation, Großhansdorf, Germany). The sampling rate of the acquisition card used in the experiment is 625 MS/s.

An FBG sensor is attached to the center of an aluminum plate adjacent to a piezoelectric ceramic (PZT) ultrasonic transducer. The PZT is driven by an arbitrary waveform generator, which produces a sinusoidal signal at 200 kHz with a peak-to-peak voltage of 80 V. This sinusoidal vibration induces periodic modulation in both the intensity and phase of the reflected light from the FBG. The intensity and phase are determined by Eqs. (6) and (7).

In self-Bayesian estimation, the noise from phase demodulation and intensity transformation should be Gaussian-distributed and uncorrelated. Figure 7 shows the noise distribution of the two demodulation results: Fig. 7(a) shows the noise scatter diagram, whereas Fig. 7(b) provides the cross-correlation plot. At a significance level of p<0.001, the noise correlation coefficient r is 0.0079, indicating the correlation is negligible. Here, p represents the significance test result of the correlation coefficient, which is calculated by the t-distribution based on the value of the t statistic and the degree of freedom n−2.

Figure 7(c) displays the two-dimensional probability density distribution, revealing that both noises align with a Gaussian distribution. The mean values of the two types of noise are 3.36×10−6 and 4.10×10−6, and the variances are 1.86×10−4 and 1.66×10−4, respectively. The result in Fig. 7 shows that the noise satisfies the Gaussian distribution with a mean of ∼0.

Furthermore, the intensity noise and phase noise are uncorrelated. In practical optical sensing systems, the main sources of noise are thermal noise and shot noise, both of which belong to Gaussian white noise. When the number of sampling points is sufficiently large, noise tends to follow a Gaussian distribution due to the central limit theorem. The signal fluctuations in the real and imaginary parts of the optical signal caused by thermal noise and shot noise are usually uncorrelated or have very low correlation. As a result, the self-Bayesian estimation proposed in this paper will be an optimum estimator, which is expected to be widely applied in optical sensing.

Figure 8(a) shows the results of phase demodulation, intensity transformation, and self-Bayesian estimation of the 200 kHz vibration signal in the time domain. Figure 8(b) depicts the fluctuation differences between the sensing results in Fig. 8(a) and the input sinusoidal signal. It is obvious that the error in self-Bayesian estimation is the smallest. Figure 8(c) displays the probability density distribution of the three types of noise, demonstrating a narrower distribution range for the self-Bayesian estimation results. The calculated variance for self-Bayesian estimation is 8.53×10−5, which is smaller compared with those from phase demodulation and intensity transformation [Fig. 7(c)].

Furthermore, Figs. 9(a)–9(c) illustrate the power spectral density curves of the three different methods. The SNR of phase demodulation is 37.55 dB, the intensity transformation SNR stands at 35.68 dB, and self-Bayesian estimation achieves an SNR of 39.22 dB. These results demonstrate that the sensing precision is enhanced by the self-Bayesian estimation method without altering the experimental equipment or structure.

Compared with phase demodulation, the SNR of the self-Bayesian estimation algorithm is improved by 1.67 dB, which is equivalent to a 47% increase in SNR. The reason for the improvement in SNR is that the self-Bayesian estimation algorithm takes into account the correlation between the real and imaginary parts of the optical signal. This is rather a significant improvement for statistical algorithms.

To validate the algorithm’s general applicability, 100 data groups were collected. Figures 10(a) and 10(b) show the variance and SNR of phase demodulation, intensity transformation, and self-Bayesian estimation. The results indicate that the variances are reduced by the self-Bayesian estimation. The SNR is also improved except for three of the 100 groups. Figures 10(c) and 10(d) show the density curves and confidence intervals of the measured variance and SNR. The mean variances are 1.81×10−4, 1.52×10−4, and 8.62×10−5, whereas the mean SNRs are 37.73, 36.46, and 39.49 dB, respectively. These results confirm the stability and broad applicability of the self-Bayesian estimation method.

To investigate the relationship between self-Bayesian estimation and vibration frequency, the signal frequency generated by the AWG was varied from 200 to 500 kHz. Each data set was collected and recorded 4 times to assess results across different frequencies. Figures 11(a) and 11(b) show the SNR and variance at different frequencies. The PZT’s response is frequency-dependent, causing fluctuations in SNR and variance. Despite these fluctuations, the results demonstrate that self-Bayesian estimation effectively reduces variance and enhances SNR.

To examine the algorithm’s linear response, the peak-to-peak voltage of the AWG-driven signal was adjusted from 20 to 200 V in steps of 20 V. As shown in Figs. 12(a) and 12(b), the variances of the results obtained after demodulation using three different methods, respectively, remain relatively unchanged for voltages below 160 V, whereas SNR increases with voltage. However, at 180 and 200 V, the variance rises sharply and SNR fluctuates. Figure 12(c) shows that the demodulated amplitude exhibits a linear relationship with the increase in voltage below 160 V. When the input voltage increases to 180 V, the vibration signal shifts the FBG reflection peak out of the linear region, resulting in nonlinear amplitude variations and the generation of multiple harmonics, which significantly increase the variance of the demodulation results and reduce the effectiveness of the algorithm.

In conclusion, a self-Bayesian estimation method is proposed in this paper to enhance the precision of optical sensing by utilizing the relationship between the real and imaginary parts of optical signals. This method applies the Bayesian theorem to combine prior probability with detected data, updating the posterior probability of optical sensing results to improve sensing precision. Experimental results demonstrate that this method enhances the SNR of the measured signal by 1.67 dB. This approach effectively integrates intensity and phase information in the optical field, offering new insights and methods for improving the performance of various optical sensors.

Biographies of the authors are not available.