Theoretical Investigation of Ultraprecise Position Determination by Spatially Modulated Low-Coherence Interference Fringe in Optical Fiber Sensing

A position determination method for low-coherence interference fringe in optical fiber sensing is proposed. This method can be used to measure displacement or length. An amplitude ratio curve is created from spatially modulated interference fringes. There is a one-to-one correspondence relationship between the peaks of the amplitude ratio curve and the extremes of the low-coherence interference fringe. The peak positions of the amplitude ratio curve only rely on the Fabry–Pérot (F-P) cavity length. In addition, the peaks of the amplitude ratio curve are very sharp, which can be easily identified. In the calculation of the amplitude ratio curve, the low-coherence interference fringe is spatially modulated by Pockels effects. The impacts of the modulated voltage on the sensing precision are analytically and numerically discussed. Compared with the envelop detection method, the position determination precision of low-coherence interference fringe is enhanced about 30 times by creating an amplitude ratio curve. This method can be widely employed in the sensing of all kinds of parameters that can be converted to displacement or length.

SECTION 1. Introduction

Optical fiber sensing has been studied extensively over the past 30 years due to its outstanding advantages: small dimension, light weight, electromagnetic immunity, embedability, and so on [1]–​ [3]. Displacement is a basic and important physical parameter. Many other kinds of parameters can be detected by converting them to the displacement, such as temperature or pressure [4]. Displacement can be measured by different optical methods, including time-of-flight distant measurement method, optical triangulation method, interferometry method, photoelectromotive force, etc. [5]. The precision of displacement sensing is limited by $2\pi$ ambiguities. Low-coherence interferometry, which is also known as white light interferometry, is an effective non-contact and non-destructive method in measuring Fabry–Pérot (F-P) cavity length [6]. The $2\pi$ ambiguities can be overcome by low-coherence interferometry method. In spatial scanning low-coherence interferometry, the displacement or position can be measured by using the optical wedge to realize the scanning of optical path difference (OPD) with no mechanical movement involved [7]. Thus, this configuration is more stable and reliable in harsh applications. The position determination precision of spatial scanning low-coherence interferometry can be improved by a curve fitting algorithm, such as the centroid algorithm, quadratic polynomial fitting, envelop method, and so on [8]–​ [14]. However, the precision is still constrained by such adverse effects as fringe ripples due to the reflections in the optical path, fluctuation of the laser power, imperfection of optical wedge, and non-ideal reflection mirror of F-P cavity. These adverse effects can distort the fringes and make a precise position determination difficult. One of the solutions to solve these problems is to narrow down the full width half maximum (FWHM) of peaks in low-coherence interference fringe, making the position of fringe more obvious to be located. Increasing the spectral width of light source will be helpful, however, that method will require the spectral windows of all the optical components are wide enough, which is difficult to be satisfied. Recently, the FWHM of resonance peak was narrowed significantly in the Whisper-Gallery-Mode resonator [15], [16] by a modulated signal in frequency domain, which made this scheme approaching feasible.

In this paper, we proposed an ultraprecise displacement measurement method by locating the low-coherence interference fringe precisely from a created amplitude ratio curve. The peaks of amplitude ratio curve have a one-to-one correspondence with the extremes of the low-coherence interference fringe but are much sharper to identify. The amplitude ratio curve is created from spatially modulated low-coherence interference fringes and only sensitive to the length of F-P cavity. To modulate the low-coherence interference fringe, modulated voltage is imposed on Strontium-Barium Niobate (SBN) birefringence wedge. The precision of peak positions determination can be improved further by inverse proportional curve fitting. Based on our numerical simulation, the precision of position determination can be enhanced to 5 nm from 0.16 $\mu\text{m}$ in conventional while light interferometry. The impacts of the modulated voltage amplitude and the signal-to-noise ratio will be analyzed in this paper.

SECTION 2. Theory

The scheme diagram of the spatially modulated low-coherence interferometer is similar to the configuration demonstrated by Dändliker et al. in 1992 [7], as shown in Fig. 1(a). The displacement or the position can be measured by scanning OPD with optical wedge. Mechanical movement is avoided in our scheme, making the system more stable, compact, and simple. A broad band light from a light-emitting diode (LED) is injected into a 2 $\times$ 2 coupler. An LED is a cheaper laser source comparing with the super luminescent diode. The fiber is commercial multimode fiber, whose core/cladding diameter is 62.5/125 $\mu\text{m}$. The coupler is a fiber beam splitter with 50% splitting ratio. The sensing unit in our scheme is an F-P cavity, as shown in Fig. 1(b), which consists of a silicon membrane and a glass substrate. Light sent to the optical fiber F-P sensor will be either reflected by the interface between the glass and air or reflected by the interface between air and silicon. The second output of coupler is terminated to ensure that any reflection from the end of that fiber is not reflected to the readout. The reflection of interfaces in the F-P cavity is about 2%, which can be adjusted by coating suitable film. The reflected signal of the F-P cavity can be approximately seen as two-beam interference. The light is reflected back to the coupler and sent to the demodulation module, which composes a cylindrical lens, a polarizer, a birefringent optical wedge, and an analyzer. The light output from fiber is focused by the cylindrical lens to increase the signal-to-noise ratio. The OPD between ordinary light and extraordinary light can be converted into a spatial distribution by the birefringent effect of the wedge. Low-coherence interference fringe is generated when the OPD caused by the F-P cavity is compensated or equalized with the OPD caused by the thickness of the birefringent wedge. In our scheme, the optical axis directions of polarizer and the analyzer are parallel with each other. The optical axis direction of the optical wedge is vertical. The angle between optical axis direction of birefringent optical wedge and the axis of polarizer (analyzer) is $\pi/4$. The intensities of ordinary and extraordinary light are almost the same, and the low-coherent interference fringe can achieve the highest resolution. Finally, the interference fringe is projected onto a linear charge-coupled device (CCD), and is further processed in computer. The changing of F-P cavity length, which is the displacement to be measured, can be obtained by the movement of low-coherence interference fringe on CCD.

Fig. 1.

(a) Scheme diagram of the spatially modulated low-coherence interferometer. (b) Structure of the F-P cavity sensor.

Besides length or displacement, other parameters which can be converted to displacement also can be measured by this method. For example, if an external pressure is imposed on the silicon membrane in Fig. 1(b), the position of silicon membrane and the cavity length will be affected by external pressure. As a result, this scheme can also be used as a pressure sensor. The response of the F-P cavity length can quickly satisfy the demand of high speed sensing.

To modulate the low-coherence interference fringe, the birefringence wedge is made of SBN crystal [17]–​ [19], which has birefringent Pockels effects. The refractive index and the OPD between ordinary and extraordinary light in SBN wedge can be modulated by a voltage, which obeys \begin{align*}\Delta \left(\frac{1}{n^{2}}\right)_{ij} =& \sum_{k} r_{ijk}{V_{k}}\tag{1}\\ \text{OPD} =& ({n_{31}}-{n_{33}}) \cdot L\tag{2}\end{align*}where $n$ is the refractive index of SBN, $r_{ijk}$ is the linear electro-optic coefficient tensor, $V_{k}$ is the voltage, and $L$ is optical path in the wedge. When the voltage is 0, the power distribution of low-coherence interference fringe on CCD camera is expressed by \begin{equation*}P(x)=\gamma \exp \left\{\left[-\alpha (x-{x_{0}})\right]^{2}\right\}\cos \left[\beta (x-{x_{0}})\right]\tag{3}\end{equation*}where $\alpha, \beta, \mbox{and}\ \gamma$ are the parameters related to the optical path, $x$ is the position on CCD, and $x_{0}$ is the center position of low-coherence interference fringe on CCD where the OPD between ordinary and extraordinary light in optical wedge equals to two times of the length of F-P cavity [20]. When using a modulated voltage $V=A \cdot \sin (2\pi\ ft)$ with amplitude $A$ and frequency $f$, the center position of the low-coherence interference fringe will move from $x_{0}$ to $x_{0}+\Delta x\cdot\sin(2\pi\ ft)$. The low-coherent interference fringe swings around the center position. The low-coherence interference fringe will be rewritten as \begin{equation*}P(x,t)=\gamma \exp \left\{\left[-\alpha \left(x-{x_{0}}-\Delta x \cdot \sin(2 \pi ft)\right)\right]^{2}\right\} \cdot \cos \left[\beta \left(x-{x_{0}}-\Delta x \cdot \sin(2\pi ft)\right)\right].\tag{4}\end{equation*}

$P(x, t)$ has a fundamental frequency component $P(f)$ and a second harmonic frequency component $P(2f)$, which satisfy \begin{align*}P(f) &\propto \Delta x \cdot P^{\prime}(x)\tag{5}\\ P(2f) &\propto \Delta {x^{2}}\cdot P^{\prime\prime}(x).\tag{6}\end{align*}

$N$ frames of low-coherence interference fringe are sampled in a period of modulated voltage with equal time interval. The time interval of sampling is $1/Nf$. $P(f)$ and $P(2f)$ can be derived by discrete Fourier transform from the sampled low-coherence interference fringes.

As an intrinsic feature of all curves with extreme, the function $P(t)$ mainly contains $P(2f)$ component in the area around the extreme, while $P(t)$ is mainly made up by $P(f)$ in the linear area. The fundamental frequency component $P(f)$ vanishes from the modulated signal $P(x,t)$ on CCD camera, while the second order frequency component $P(2f)$ arrives its maximum value on the very point of extreme of low-coherence interference. Theoretically speaking, on the extreme point of the low-coherence interference fringe, $P^{\prime}(x)$ is 0, and the amplitude ratio $P(2f)/P(f)$ should be infinity. The amplitude ratio between $P(2f)$ and $P(f)$ is calculated for every pixel on CCD \begin{equation*}\frac{P(2f)}{P(f)}=\frac{{\Delta {x^{2}}\cdot P^{\prime\prime}(x)}}{{\Delta x \cdot P^{\prime}(x)}}\propto \frac{{P^{\prime\prime}(x)}}{{P^{\prime}(x)}}.\tag{7}\end{equation*}

There is a one-to-one correspondence relationship in position between the peaks of the amplitude ratio curve and the extremes of low-coherent interference fringe, but the FWHM of the peaks of amplitude ratio curve is much narrower. When the optical power on the CCD is affected by the factors other than the cavity length, such as the fluctuation of the LED source power or imperfection in optical wedge, both $P(2f)$ and $P(f)$ change in the same way, and the amplitude ratio does not change. The extremes and the position of low-coherent interference fringe can be located ultra-precisely by the amplitude ratio curve.

We employ a curve-fitting algorithm to improve the precision of peak position determination. The low-coherence interference fringe around extreme can be approximated as a parabolic curve by omitting the higher order components in Taylor expansion. This parabolic curve will shift periodically along the $x$-axis on CCD accompanying with the modulated voltage, which can be expressed by \begin{equation*}P(x,t)=p-q \left(x-\left({x_\mathrm{ext}} + \Delta x \cdot \sin (2\pi ft)\right)\right)^{2}\tag{8}\end{equation*}where $p$ and $q$ are parameters related to the shape of low-coherence interference fringe, and $x_{\mathrm{ext}}$ is the center position of the selected extreme without modulated voltage. Based on (7) and (8), a pair of inverse proportional curves can be derived around the selected extreme \begin{equation*}\frac{{P(2f)}}{{P(f)}}\propto \frac{{\pm \Delta x}}{{x-{x_\mathrm{ext}}}}\equiv \frac{{\pm w}}{{x-{x_\mathrm{ext}}}}\tag{9}\end{equation*}where $w$ is a parameter related to the modulated voltage amplitude $A$. This parameter determines the peak width. The dimension of $w$ is length. Every peak of the amplitude ratio curve is made up by a pair of inverse proportional curves. One of them is on the left of the peak, while the other one is on the right. In the inverse proportional curve fitting, a pair of “ $w$” ( $w_\mathrm{left}$ and $w_\mathrm{right}$) can be obtained. $w=(w_\mathrm{right} + w_\mathrm{left})/2$ is defined to be the peak width of amplitude ratio curve in the discussion below to make the results more concise. Therefore, the peak width can be found exactly without complete fitting to the whole curve.

It can be seen from (9) that, for every peak on amplitude ratio curve, both the peak position and width can be obtained precisely by Taylor's theorem and curve fitting algorithm. With these precise positions of extremes, the shift of every extreme on low-coherence interference fringe can be calculated by comparing with the reference position of fringe. The average extremes shifting and the precise length changing of F-P cavity can be found out in sequence.

SECTION 3. Results and Discussion

To demonstrate the amplitude ratio curve method we proposed, numerical simulation is carried out for a particular configuration of the spatially modulated low-coherence interferometry. Equations (1)–​ (9) are applied to our model in MATLAB. In the simulation, the optical axis of SBN is parallel with the modulated electric field, the thickness of the wedge is 1 mm, the Electro-optic coefficient $r_{13}=47\ \text{pm/V}$, $r_{33}=235\ \text{pm/V}$ [17], and the length of FP cavity is assumed to be 30 $\mu\text{m}$. The output spectrum of LED source is in Gaussian shape distribution around $5\times 10^{14}\ \text{Hz}$. The FWHM of the Gaussian shape is 80 THz. This source can cover the waveband from 550 nm to 650 nm. This waveband is divided into 1000 parts in the simulation. The power of light source is normalized because the low-coherence interference fringe only related to the relative electric field power of every frequency component. The fluctuation of the light source will reduce the precision of the low-coherence interference fringe position determination. The amplitude of modulated voltage is assumed to be 8 V, five frames of low-coherence interference fringe are sampled in every period of modulated voltage, and the angle of optical wedge is 2°.

The amplitude ratio of every pixel on CCD is shown in Fig. 2 by a series of red dots. In our simulation, every dot on this curve covers 7 $\mu\text{m}$ space on CCD. For a comparison, the original low-coherence interference fringe without modulated voltage is also plotted on this figure by blue dashed curve. It is obvious that the extremes of the fringe and the peaks of amplitude ratio curve are a one-to-one correspondence. Theoretically speaking, the value of amplitude ratio at the extreme position on low-coherence interference fringe should be infinity. However, limited by the resolution of CCD, the peak values of amplitude ratio curve fall in the range between 5 and 25, as shown in Fig. 2(a).

Fig. 2.

(a) One-to-one correspondence relationship between peaks of amplitude ratio curve and extremes of low-coherence interference fringe. Blue dashed line is the low-coherence interference fringe without modulated voltage. Red solid curve is the amplitude ratio curve. (b) Comparison between the FWHM in original low-coherence interference fringe and amplitude ratio curve for an arbitrary selected peak. The FWHM is reduced to less than 3%.

In Fig. 2(b), an arbitrary extreme of low-coherence interference fringe (the extreme around 57.55 mm) in Fig. 2(a) is selected and zoomed in to show the results in detail. The space between adjacent dots is reduced from 7 $\mu\text{m}$ to 1 $\mu\text{m}$ in simulation to make the curve smooth. The extreme position of the low-coherence interference fringes can be extracted precisely from the amplitude ratio curve. The peak position determination precision is defined as the FWHM of the peak. The FWHM of the amplitude ratio curve is reduced from 0.3 mm to less than 0.01 mm, which means the precision of extreme position of low-coherence interference fringe found from amplitude ratio curve can be enhanced for about 30 times. If the peak width of amplitude ratio curve is about 0.01 mm on CCD, the precision of cavity length determination will be 5 nm correspondingly. This precision will be 0.16 $\mu\text{m}$ if we calculate from the low-coherence interference fringe.

Modulated voltages with different amplitudes are imposed on the SBN wedge. The results of amplitude ratio curves are shown in Fig. 3. There are two reasons for the problem that the curve in Fig. 3(a) is discontinuity. On the one hand, limited by the resolution of CCD camera (every pixel of CCD covers 7 $\mu\text{m}$), the singular points of amplitude ratio curve are abandoned because the amplitude ratio on the point which $P^{\prime}(x)$ is zero cannot be detected exactly. On the other hand, every point on Fig. 3(a) covers 7 $\mu\text{m}$ on CCD, while every point covers 1 $\mu\text{m}$ in Fig. 2(b). The curves from the bottom to top correspond to modulated voltages with amplitude increasing from 2 V to 20 V. As shown in this figure, higher modulated voltage can make the peak wider on the amplitude ratio curve. The red circles are the original data from the simulation, and the blue lines are the inverse proportional curve fitting results. By curve fitting, the peak width and position of amplitude ratio curve can be obtained. The results are shown in Fig. 3(b) and (c). In Fig. 3(b), the peak width increases with amplitude of modulated voltage linearly when the modulated voltage is < 20 V This result is consistent with the results shown in Fig. 3(a).

Fig. 3(c) shows the relationship between the peak position deviation and the amplitude of modulated voltage. The peak position obtained with 2 V modulated voltage is assumed to be “standard position.” The “peak position deviation” shown in Fig. 3(c) means the deviation between standard position and peak position obtained from amplitude ratio curve method under a given modulated voltage. The peak position on low-coherence interference fringe can be found by our scheme for every particular modulated voltage. Compared with the peak width $(\sim\!\!\mu\text{m})$, the impact of the modulated voltage amplitude on the peak position (∼nm) should be negligible. However, the curve shown in Fig. 3(c) accelerated declines with the increasing of modulated voltage amplitude, which means the deviation will be prominent if the amplitude of modulated voltage is too huge. This phenomenon is caused by the error of Taylor expansion in deriving (9). The original curve on low-coherence interference fringe is assumed to be a parabolic curve, but that assumption is only valid for the region near extreme. With the increasing of modulated voltage amplitude, the extreme of low-coherence interference fringe will shift far away from the center position (extreme position without modulated voltage). The higher order terms in Taylor expansion cannot be omitted, and the amplitude ratio curve will no longer be a perfect inverse proportional curve.

Fig. 3.

Impacts of modulated voltage amplitude on the amplitude ratio curve. (a) Amplitude ratio curves for the arbitrary selected peak with modulated voltage amplitudes increases from 2 V to 20 V. (b) Relationship between the modulated voltage amplitude and the peak width of amplitude ratio curves. (c) Deviation of peak positions of amplitude ratio curve with the increasing of modulated voltage amplitude. The $y$-axis is the difference between the standard position and the peak position value under a given modulated voltage amplitude.

There should be a tradeoff for the amplitude of modulated voltage. If the voltage is too small, the modulation effect of the modulated voltage on the low-coherence interference fringe will be so weak that the peaks of amplitude ratio curve are too narrow compared with the volume of pixel on CCD and maybe submerged in the noise. Measuring the amplitude ratio curve precisely will be a challenge for the resolution of CCD. However, if the amplitude of modulated voltage is too large, the extreme of low-coherence interference fringe will be severely dragged to the area where cannot be assumed to be a parabolic curve. The precision of curve fitting will also be reduced. Therefore, the most suitable modulate voltage amplitude should be in a certain range. Based on the numerical simulation results, this range is approximately between 1 V and 8 V, which will be tested in experiment in the next stage.

According to the Nyquist-Shannon sampling theorem and Eq. (7), sampling only four frames per period is enough to calculate the amplitude ratio curve. However, only the fundamental frequency and the second harmonic frequency components are obtained if four frames are sampled per period and the entire higher frequency information is neglected. Fortunately, a relative good result can be obtained if five frames are sampled per period according to our numerical investigation, which means the sensing rate could be as high as 2 kHz, if a high speed CCD camera with reaction time in the order of 100 $\mu\text{s}$ is employed in practice.

Finally, the impact of signal-to-noise ratio is analyzed by a comparison of the coefficient of variation in measurement between the amplitude ratio curve method and envelop detection method [11]. The coefficient of variation is the ratio of the standard deviation between the actual length and the measured one relative to the actual length induced in the simulation. The length of F-P cavity is compressed from 30 $\mu\text{m}$ to 25 $\mu\text{m}$ in our simulation. Accordingly, the center position of low-coherence interference fringe is shifted roughly from the initial position 57.4 mm to 47.7 mm on CCD. Comparing the amplitude ratio curves under these two cases, the cavity length changing can be obtained from the average peak shifting. Random noises are superposed on the low-coherence interference fringes, making a precise sensing difficult. The signal-to-noise ratio of the random noise is set from 10 dB to 30 dB. The noise is mainly caused by the fluctuation of the LED source power. The SNR can be as high as 30 dB [7]. Every peak of amplitude ratio curve is fitted by inverse proportional function. To select a better fitting result, the coefficient of determination is considered in every inverse proportional fitting process, and the poor fitting results are excluded. The average shifting of the selected extremes from the initial positions and the F-P cavity length changing can be obtained in sequence. This process is repeated for 100 times under every given signal-to-noise ratio. The standard deviation and the average of cavity length changing can be obtained. Since the coefficient of variation will go to a certain value with the increasing of time, we believe the result we calculated from 100 times repetition can represent the coefficient of variation in a long time measurement unless the SNR is changed.

The red dots in Fig. 4 are coefficient of variation of the cavity length changing as a function of signal to noise ratio. In this simulation, the volume of CCD pixel is assumed to be 7 $\mu\text{m}$, and 5 frames are sampled in every period. For a comparison, results from envelope detection method, which is also repeated for 100 times, are plotted in the same figure with blue circles. The dashed lines are the linear curve fitting results of the discrete simulated results. It is obvious that the sensing precision of F-P cavity length is improved for about 30 times by employing the amplitude ratio curve method, which is inconsistent with the results shown in Fig. 2.

Fig. 4.

Coefficient of variation for cavity length changing is measured with different signal-to-noise ratio. The red dots stand for the results obtained from the amplitude ratio curve method. The blue circles are the results from the envelop method, for a comparison.

SECTION 4. Conclusion

In summary, a position demodulation algorithm for low-coherence interference fringes with high precision is proposed based on spatially modulated low-coherence interference fringe. The low-coherence interference fringe is modulated by Pockels effects. Amplitude ratio curve is created by the organic combination of a series of low-coherence interference fringes to enhance the precision of fringes position location. To further increase the sensing precision, inverse proportional curve fitting method is demonstrated both numerically and analytically for calculating the peak position and width in the amplitude ratio curve. The most suitable modulated voltage amplitude is determined by the requirements on resolution of CCD camera and the shape of low-coherence interference fringe. The sampling rate of low-coherence interference fringes can be reduced down to four frames per period, which is helpful in improving the demodulation speed. The coefficient of variation for cavity length changing under different signal-to-noise ratio is analyzed, and the numerical investigation result shows that the precision of the length sensing can be enhanced for about 30 times compared with the envelop method. This algorithm can be extended to the high precision measurement of all parameters if they can be converted to the displacement.