Influence of fiber bending on wavelength demodulation of fiber-optic Fabry-Perot interferometric sensors

Guigen Liu; Qiwen Sheng; Weilin Hou; Ming Han

doi:10.1364/OE.24.026732

1. Introduction

One well-known advantage of the optical fiber sensors (OFSs) is the capability for remote sensing [1], which makes full use of the ultra-low transmission losses of optical fibers. Minimal optical loss is obtained when the optical fiber is laid straight. However, dynamic or static bending is often unavoidable during the deployment and operation of the sensor systems in the field. For example, environmental perturbations such as vibrations may cause dynamic bending to the fiber and, for some cases, bending fiber is necessary to accommodate limited installation space. It is well-known that fiber bending causes additional loss to the optical signals. As a result, fiber bending could be a serious concern for intensity demodulated OFSs. Bending losses are also known to be wavelength dependent [2–7]. Therefore, fiber bending can also influence the responses of the wavelength-demodulated OFSs. To date, although there are plenty of OFSs that are based on bending losses or are used to measure fiber bending [8–17], little work has been reported to quantify the bending-induced wavelength shift (BIWS) of fringe peaks and valleys for wavelength-demodulated OFSs.

This paper investigates the BIWS for fiber-optic sensors based on Fabry-Perot interferometers (FPIs) [18]. The sensor used as an example here is a low-finesse FPI formed by a silicon pillar attached to a cleaved fiber end face, which was developed recently for high-resolution profiling of the thermal structures in the ocean for oceanographic applications [19]. A partially explicit expression for the BIWSs of fringe peaks and valleys is derived for this low-finesse FPI. This model is then used to quantify the influence of bending (including the bending radius and length) on the fringe peaks and valleys. In addition, parameters of the FPI itself, such as fringe visibility and optical path length of the cavity, are also discovered to have significant effect on the BIWSs. Finally, experimental results are provided to verify the theoretical predictions.

2. Theoretical investigation

2.1 Qualitative illustration

The mechanisms responsible for the bending loss of the lead-in single mode fiber (SMF) are schematically shown in Fig. 1(a). Losses from the bending section include two parts [3, 20]: one is the transition loss arising from the mismatch between the modal fields of the straight and curved fiber sections and the other is the radiation loss along the curved fiber section. Since the bending loss is wavelength-dependent, the reflection spectrum from the FPI is modulated by an extra envelope, which is shown later in Fig. 2(c). Assuming the reflection modulation envelope induced by bending is $B (λ)$ and the higher-order modes (cladding modes) generated from bending are lost before reaching the FPI, the total reflectance $R (λ)$ from the sensor head can be expressed as

R (λ) = B (λ) F (λ),

where

F (λ)

is the reflection spectrum of the FPI that is independent of fiber bending. The peak and valley wavelengths are obtained at the positions where the derivative of

R (λ)

is zero, i.e.,

\frac{d R (λ)}{d λ} = \frac{d B (λ)}{d λ} F (λ) + \frac{d F (λ)}{d λ} B (λ) = 0.

Rearrangement of Eq. (2) leads to

Fig. 1 (a) Schematic representation of the radiation losses introduced by bending to the lead-in single-mode fiber of the sensor. (b) Reflection spectrum of a 200-µm-thick silicon FPI without disturbance from bending. (c) Schematic showing some parameters to model the sensor head.

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Fig. 2 Theoretical (a) transition loss coefficient and (b) pure bend loss coefficient as a function of wavelength with different bending radii. (c) The according reflection spectra with different bending radii. In (c), both transition loss and pure bend loss are included, the pink solid and void dots denote, respectively, the peak and valley wavelengths that are calculated in Fig. 3(a).

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\frac{d F (λ)}{d λ} = - \frac{d B (λ)}{d λ} \frac{F (λ)}{B (λ)} .

From Eq. (3), without bending, i.e., $B (λ) =$ 1, we have $\frac{d F (λ)}{d λ} = 0$ , which corresponds to the peak or valley wavelengths of the unperturbed FPI, as shown in Fig. 1(b). With bending losses, derivative of $B (λ)$ can take negative, zero, or positive values, depending on the shape of $B (λ)$ . If $\frac{d B (λ)}{d λ} > 0$ , the peak or valley wavelengths are obtained at a region where $\frac{d F (λ)}{d λ} < 0$ , indicating that the peak wavelengths shift slightly toward longer wavelength while the valley wavelengths shift slightly toward shorter wavelength, as shown in Fig. 1(b). On the other hand, if $\frac{d B (λ)}{d λ} < 0$ , the peak or valley wavelengths are obtained at a region where $\frac{d F (λ)}{d λ} > 0$ , suggesting a blueshift of peak wavelengths and a redshift of valley wavelengths. If $\frac{d F (λ)}{d λ} = \frac{d B (λ)}{d λ} = 0$ , no wavelength shift is induced by the fiber bending.

2.2 General quantitative formulism

To quantitatively evaluate the influence of bending, the reflection modulation envelope through the bending section, $B (λ)$ , along with the reflection spectrum from the FPI, $F (λ)$ , are evaluated. According to [20], the optical transmission through the transition point between the straight and curved fiber can be calculated from

T_{T r a n} = \exp (- α_{1}) = \exp [- \frac{1}{16} {(\frac{a}{R})}^{2} \frac{V^{4}}{{(\ln V)}^{3}} \frac{n_{1}^{4}}{{(n_{1}^{2} - n_{2}^{2})}^{2}}],

where

α_{1}

is the loss coefficient due to the transition,

a

is the core radius,

R

is the bending radius,

n_{1}

is core refractive index (RI),

n_{2}

is the cladding RI,

V = k a \sqrt{n_{1}^{2} - n_{2}^{2}}

is the normalized frequency of the fiber with

k = 2 π / λ

being the wavenumber of the light in vacuum. Denotations of some of the parameters are also shown in Figs. 1(a) and 1(c). The transmission through a pure bending section of the fiber with length L can be given as

T_{B e n d} = \exp (- α_{2} L),

where

α_{2}

is the radiation loss coefficient with the unit of

m^{- 1}

. According to [6],

α_{2}

can be calculated from

α_{2} = \frac{2 (k^{2} n_{1}^{2} - β_{0}^{2})}{β_{0} V^{2} K_{1}^{2} (a γ)} \int_{0}^{\infty} \frac{\exp [- a {(γ^{2} + ζ^{2})}^{1 / 2}]}{{(γ^{2} + ζ^{2})}^{1 / 2}} \frac{A i [X_{2} (0, ζ)]}{B i [X_{2} (a, ζ)]} \frac{x_{2}^{1 / 2} x_{3}^{1 / 3}}{x_{2} \cos^{2} θ (ζ) + x_{3} \sin^{2} θ (ζ)} d ζ,

where

β_{0}

is the propagation constant of the straight optical fiber,

K_{1}

is the modified Bessel function,

A_{i}

and

B_{i}

are the Airy functions, expressions for

X_{q} (y, ζ)

,

x_{q}

and

θ (ζ)

are given as follows

{\begin{cases} X_{q} (y, ζ) = {(\frac{R}{2 k^{2} n_{q}^{2}})}^{2 / 3} [β_{0}^{2} + ζ^{2} - k^{2} n_{q}^{2} (1 + \frac{2 y}{R})], q = 2, 3 \\ x_{q} = - X_{q} (b, ζ) {(\frac{2 k^{2} n_{q}^{2}}{R})}^{2 / 3}, q = 2, 3 \\ θ (ζ) = \frac{2}{3} {[- X_{2} (b, ζ)]}^{3 / 2} + \frac{π}{4} \end{cases},

where

n_{3}

and

b

are the RI of the plastic coating and radius of the cladding, respectively.

Because the optical signals travel through the bending section and the two transitions twice, the reflection modulation envelope, $B (λ)$ , is written as

B (λ) = {(T_{T r a n})}^{4} {(T_{B e n d})}^{2} = \exp (- 4 α_{1} - 2 α_{2} L) .

Reflectance

F (λ)

from the silicon FPI (

n_{s i} > n_{1}, n_{s i} > n_{e}

) is given by [21]

F (λ) = \frac{R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ})}{1 + R_{1} R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ})},

where

R_{1}

and

R_{2}

are the reflectances of the two sides of the silicon FPI. After substituting Eqs. (4)-(9) into Eq. (1), the peak and valley wavelengths can be numerically found. By calculating the wavelength difference for the cases with and without bending, the BIWSs are obtained.

2.3 Simplified expressions for low-finesse FPI

For low-finesse FPIs where $R_{1} ≪ 1$ and $R_{2} ≪ 1$ , such as the sensor based on a silicon FPI used as an example in this paper, Eq. (9) can be simplified to

F (λ) = R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ}) .

The wavelength of the

N^{t h}

peak (

λ_{P N}

) or valley (

λ_{V N}

) in the absence of bending satisfies

{\frac{d F (λ)}{d λ} |}_{λ_{P N,} λ_{V N}} = {- \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{λ^{2}} \sin (\frac{4 π n_{s i} d}{λ}) |}_{λ_{P N,} λ_{V N}} = 0,

In the presence of bending, the BIWSs for the

N^{t h}

peak (

Δ λ_{P N}

) or valley (

Δ λ_{V N}

) can be derived from Eqs. (3), (8), (10), and Eq. (11) as

Δ λ_{P N} = - (4 {\frac{\partial α_{1}}{\partial λ} |}_{{\bar{λ}}_{P N}} + 2 L {\frac{\partial α_{2}}{\partial λ} |}_{{\bar{λ}}_{P N}}) (\frac{1}{V_{b}} + 1) \frac{λ_{P N}^{4}}{{(4 π n_{s i} d)}^{2}} .

Δ λ_{V N} = (4 {\frac{\partial α_{1}}{\partial λ} |}_{{\bar{λ}}_{V N}} + 2 L {\frac{\partial α_{2}}{\partial λ} |}_{{\bar{λ}}_{V N}}) (\frac{1}{V_{b}} - 1) \frac{λ_{V N}^{4}}{{(4 π n_{s i} d)}^{2}} .

where

V_{b}

is visibility of the FPI,

{\bar{λ}}_{P N}

and

{\bar{λ}}_{V N}

are, respectively, the peak and valley wavelengths in the presence of bending. Detailed derivations of Eq. (12) can be found in the appendix. Although Eq. (12) is only partially explicit, it clearly demonstrates that the BIWS is not only dependent on the bending radius (implicit in the derivative of loss coefficients) and the bending length (L) of the fiber, but also on the visibility (

V_{b}

) and the optical path length (

n_{s i} d

) of the FPI.

There are a few interesting observations from Eq. (12) for fringe peaks and fringe valleys that are subject to the same spectral modulation slope by bending. Because the visibility of the FPI follows $0 < V_{b} \leq = 1$ , Eq. (12) suggests that the BIWS of a fringe peak is opposite to that of a fringe valley, which is consistent with the qualitative analysis in Section 2.1. The fringe peak experiences larger wavelength shifts than the valley does because the factor $(\frac{1}{V_{b}} + 1)$ for $Δ λ_{P N}$ is larger than that $(\frac{1}{V_{b}} - 1)$ for $Δ λ_{V N}$ . In particular, the bending does not introduce any shift to the valley wavelength for FPIs with a visibility $V_{b} = 1$ . In this case, the reflection at the valley reaches 0 and the valley wavelength is not affected by the spectral modulation. The explanation on the dependence of BIWS on the visibility and optical length of the FPI is straightforward: a FPI with higher visibility or longer optical path length produces narrower peaks and valleys that improve the accuracy of finding the fringe peak and valley wavelengths.

2.4 Theoretical results

Because Eq. (12) is only semi-explicit, it is impossible to calculate directly BIWSs from it. However, it’s easy to numerically find the peak or valley positions using Eq. (1). Thus, all the following calculations are done from Eq. (1). To help interpret the calculated results, comparison between the calculated results and Eq. (12) is provided. In the following calculations, the following parameters for the lead-in fiber and the FPI were used [7]: $n_{1}$ = 1.4504, $n_{2}$ = 1.4447, $n_{3}$ = 1.4786, $n_{s i}$ = 3.4, $n_{e}$ = 1, 2a = 8.3 µm, 2b = 125 µm.

We started from calculating the bending loss coefficients and the reflection spectrum modulated by the bend losses. The results are shown in Fig. 2. It is seen from Figs. 2(a) and (b) that most of the bend losses are attributed to the pure bending section instead of the transition section, especially for small bend radius. The reflection spectra of the FPI at different bending radii are shown in Fig. 2(c). When the bend radius is larger than 15 mm, there is no visible envelope modulation. As bend radius is reduced to 7.5 mm, the spectrum is modulated by a monotonic slope with larger loss at longer wavelength. If the bend radius is further reduced to 5 mm, in addition to the dramatic drop in the overall reflectance, fluctuation in the modulation envelope appears. As explained both theoretically and experimentally in [3–7], the back reflection of the radiated rays from the cladding-coating layer interface causes the reflection modulation envelope which is especially pronounced when the bend radius is small.

BIWSs of the two neighboring peak and valley around 1550 nm, denoted respectively by the solid and void pink dots in Fig. 2(c), are calculated and compared in Fig. 3(a). Due to the slow modulation slope by bending, the slopes are similar at most of the two neighboring peak and valley wavelengths and thus the BIWSs for most fringes and valleys are opposite. When the envelope slope is negative (i.e., $B^{'} (λ) > 0$ ), as shown for the case of R = 7.5 mm in the middle panel in Fig. 2(c), the peak wavelength experiences blueshifts while the valley wavelength shows redshifts. On the contrary, when the envelope slope is positive, as shown in the bottom spectrum of Fig. 2(c) for the case of R = 5 mm, the peak wavelength experiences redshifts while the valley wavelength experiences blueshifts.

Fig. 3 BIWS as functions of (a) bending radius, (b) length of bending section, (c) visibility of the FPI, and (d) bending radius for different cavity lengths of the silicon FPI.

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The BIWS for both the fringe peaks and fringe valleys are linear functions of the bending length (in unit of turns), as shown in Fig. 3(b), which agrees with Eq. (12). Dependence of the BWISs on the visibility of the silicon FPI is shown in Fig. 3(c). The visibility was varied by changing $n_{e}$ from 1 to 1.4504 (i.e., $n_{1}$ ). From Fig. 3(c), as visibility increases, the BIWSs of both the peak and the valley decrease. The BIWS of the valley wavelength approaches 0 when the visibility is close to 1, agreeing with the prediction from Eq. (12b). The optical path length of the FPI also affects the BIWS. As shown in Fig. 3(d), the BIWS for a 10 µm thick FPI is more than two orders of magnitude higher than that of a 200 µm thick FPI. For example, with the bending radius of ~10.5 mm, the wavelength shift (−290.2 pm) for the 10-µm-thick FPI is around 392 times larger than the shift (−0.74 pm) for the 200-µm-thick FPI, again, in agreement with Eq. (12).

3. Experimental demonstration

Experiments were carried out to verify the theoretical analysis. Firstly, the influence of modulation slope on peak and valley wavelengths was visualized by fiber-bending-and-releasing experiments. Secondly, dependence of BIWS on bending length and FPI parameters (visibility and cavity length) was tested.

3.1 Experimental setup and method

The experimental setup is schematically shown in Fig. 4. White light from a superluminescent LED (SLED) source (DL-BP1-1501A, DenseLight Semiconductors) was guided to the sensor head via a circulator, the reflected signal was analyzed by a high-speed spectrometer (I-MON 512 OEM with the developer’s kit, Ibsen) which was logged into a computer. The SLED has a 3dB spectral width of >70 nm centered around 1550 nm. The high-speed spectrometer has an array of 512 detectors to read the diffracted spectrum, and a nominal spectral resolution of <0.5 pm can be obtained after curve fitting for the peak and valley wavelengths. In our experiments, the peak or valley wavelengths were fitted over 7 data points centered around the local maxima or minima using a cosine function. More details on the data processing can be found in our previous work [18]. In the meanwhile, the high-speed spectrometer possesses a maximum spectrum frame rate of 3 kHz. This ultra-high-speed data reading guarantees the acquisition of spectrum without distortion during the bending experiments. Fiber bending was introduced by wrapping a section of the lead-in SMF on a mandrel mounted on a translation stage. The sensor head was fabricated using the protocol detailed in [18]. Inset in Fig. 4 is a microscope image of the sensor head with a 100-µm-diameter, 200-µm-long silicon pillar glued on the cleaved end of a SMF. During the experiments, the sensor was placed in an enclosed environment with stable temperature to eliminate the interference from temperature variation.

Fig. 4 Experimental setup. Inset shows a 200-µm-long, 100-µm-diameter silicon cylinder mounted on the end face of a SMF, and the box on the right shows top view of the bending angle.

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Using the above setup, the fiber-bending-and-releasing experiments were performed to study the effect of the bending radius on the BWISs. The steps are outlined as follows: (1) the SMF was very loosely placed around the mandrel with unnoticeable loss to the spectrum; (2) the translation stage was moved away from the fixed stage to reduce the bending radius and stopped at a distance for about 10 s; (3) the translation stage was moved further away from the fixed stage until the bending radius reached its smallest value when the fiber was tightly pressed on the mandrel surface (half-turn bending). This status of half-turn bending maintained for about 10 s; (4) the fiber was released to an intermediate bending radius and was kept for another 10 s; (5) finally, the fiber was fully released from the mandrel to the original state.

3.2 Experimental results and discussion

We first examined the fiber-bending-induced envelope modulation to the reflection spectrum during a bending-and-releasing experiment on a 13.5 mm diameter mandrel. The results are shown in Fig. 5(a). Without bending, the normalized spectrum (top panel) showed very flat reflection fringes. As bending was introduced, the overall intensity began to drop and slight envelope modulation appeared, as illustrated by the middle panel. The bottom panel is the spectrum when the bending radius reaches its smallest value (half turn bending). The spectrum showed very pronounced envelope modulation in addition to the severe reduction in the overall intensity.

Fig. 5 Normalized reflection spectra (a) and relative wavelength and peak intensity versus time (b) during the bending-and-releasing experiment on a mandrel diameter of 13.5 mm. The spectrum at time t1, t2 and t3 in (a) were captured at the initial, intermediate, and ultimate bending moments, respectively, which are located by the green arrows in (b). The red, black and pink circles in (a) highlight the peak-valley pair 1, 2 and 3, respectively, which were tracked during the bending process and are shown in (b).

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We then tracked the shifts of three pairs of adjacent peak and valley wavelengths during the above fiber-bending-and-releasing experiment. Three peak-valley pairs [the red, black and pink circles in Fig. 5(a)] were chosen this way so that they were subject to different modulation slopes by bending. Thus, the direction of wavelength shift can be easily related to the sign of the modulation slope. The BIWSs and peak intensities for the three peak-valley pairs during the fiber-bending-and-releasing process are shown in Fig. 5(b). The top, middle and bottom panels correspond, respectively, to the results for Pair 1, 2, and 3. To associate the spectra with the three different times, t1, t2 and t3 are denoted by the three green arrows in Fig. 5(b), and the spectra in the top, middle and bottom panels of Fig. 5(a) were captured at t1, t2 and t3, respectively. It can be seen from Fig. 5(b) that, in general, the BIWS increased as the bending radius decreased (the level of bending can be inferred from the peak intensity shown by the cyan curves). When the fiber was kept at the maximum bending (t3), the peak wavelength of Pair 2 showed blueshifts while the valley wavelength showed redshifts, which is in accordance with the theoretical results in Section 2.4 since Pair 2 was subject to a negative modulation slope. The case for Pair 3 was reversed because of the positive modulation slope. When the modulation slope was small, which was the case for Pair 1 at time t3, the BIWS was much smaller compared to those of Pairs 2 and 3. Also note that both the peak and valley of Pair 3 experienced small blueshifts at time t3 instead of opposite shifts, which is believed to be due to the opposite slopes they underwent around the 0-transition point of the modulation slope. Furthermore, in the experiments the absolute value of the BIWS of the valley is close to that of the peak, which is different from the theoretical expectations. Considering that the interferometric spectrum in the experiments [Fig. 5(a)] is much more complex than that of theoretical calculations [Fig. 2(c)], two adjacent peak and valley could undergo very different modulation slopes that leads to similar absolute values of BIWSs. By taking advantage of the comparable amplitudes of BIWSs for the peak and valley, a mean wavelength [the red curves in Fig. 5(b)] can be used to partially suppress the BIWS, which is illustrated by the close-up view in the inset of Fig. 5(b).

Lastly, dependence of BIWS on the length of the bending section, and the visibility and optical path length of the FPI was verified by the experimental results shown in Fig. 6. To improve the repeatability of the experiments, plastic coating of the bending section of the lead-in SMF was stripped off and coated with a thin layer of index matching gel to remove the interference from the reflected light at the cladding-coating boundary. The fiber was removed from the fixed stage and bent manually along the mandrel surface with the desired bending angle, then the fiber was released abruptly to the straight status or large radius with negligible bending loss, driven by the rigidity of the silica fiber itself. This swift transition from bending to releasing status minimizes the affects from the possible temperature drift. Due to the high frame reading rate of the high-speed spectrometer (set at 100 frames/second in this case), this fast transition was readily captured without distorting the spectrum. The BIWS as a function of the bending angle (equivalently the bending length L in Eq. (12)) is shown in Fig. 6(a), which shows a reasonably linear dependence. As visibility increases, the BIWS decreases, which is proven by the peak wavelength shifts of −8.7, −3.9 and −2.2 pm for the FPIs with a visibility of 0.45, 0.66 and 0.79, respectively, as shown in Fig. 6(b). Visibility of the silicon FPI was highly dependent on the alignment during fabrication in our lab, and three of them with different visibilities were picked out to carry out the experiments. In Fig. 6(c), results suggest that BIWSs of −257 and −2.17 pm were obtained respectively for the silicon FPIs with cavity length of 10 and 200 µm. The ratio of 118 ( = 257/2.17) is lower than the theoretical value of 400 ( = (200/10)²), which is attributed to the different fringe visibilities of the two FPIs that are not considered here. Despite these discrepancies, the experimental results shown in Fig. 6 support the theoretical predictions reasonably well.

Fig. 6 (a) BIWS versus bending angle. Angle of 180 degrees means a half-turn bend. Relative peak wavelength during the bending-and-releasing experiments for the silicon FPIs with different (b) visibilities of 0.45, 0.66, 0.79 and (c) cavity lengths of 10 and 200 µm. In these experiments, diameter of the mandrel was 21 mm.

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4. Conclusion

We have investigated both theoretically and experimentally the BIWS on wavelength demodulation of fiber-optic FPI sensors. The underlying mechanism of BIWS lies in the wavelength-dependent bending losses introduced to the reflection spectrum of the FPI. A partially explicit expression of BIWS has been derived for the low-finesse FPIs. The following conclusions are theoretically drawn:

1) A larger bending-induced modulation slope, which usually associates with a smaller bending radius, gives birth to a larger BIWS. The direction of the wavelength shift is dependent on the sign of the modulation slope. The peak wavelength blueshifts and redshifts for a negative and positive modulation slope, respectively. However, the case for the valley wavelength is opposite, i.e., the valley wavelength redshifts and blueshifts for a negative and positive modulation slope, respectively. For the same modulation slope, the absolute value of BIWS for the valley is smaller than that of the peak. Furthermore, BIWS is a linearly proportional to the length of bending section.

2) In addition to the bending itself, BIWS is also determined by the visibility and optical path length of the FPI. Results suggest that the BIWS is proportional to the reciprocals of visibility and square of the optical path length of the FPI. For FPIs with a high visibility close to 1, the valley wavelengths are immune to the bending losses, which is due to the fact that the valley intensity is close to zero and thus is maintained zero regardless of the bending losses (or modulation slopes). Intuitive explanation to the dependence of BIWS on the visibility and optical cavity length of the FPI is that a higher visibility and a longer optical length lead to narrower peaks and valleys whose wavelengths are more resistant to the bending-induced spectral modulation. However, longer cavity length is usually associated with reduced spatial resolution of the FPI sensor.

Experiments have been performed on the silicon FPI which was recently developed by us for profiling fast and high resolution temperature variations in water, and the results in general support the theoretical predictions very well. In one experiment, for a half-turn bend with a bending diameter of 21 mm, the BIWSs of −257 and −2.17 pm were experimentally obtained for the silicon FPIs with a silicon cavity length of 10 and 200 µm, respectively.

Appendix

Derivative of Eq. (12) is given below. From Eq. (8), the derivative of the total bending transmission $B (λ)$ is given by

\frac{d B (λ)}{d λ} = - (4 \frac{d α_{1}}{d λ} + 2 L \frac{d α_{2}}{d λ}) \exp (- 4 α_{1} - 2 α_{2} L) = - (4 \frac{d α_{1}}{d λ} + 2 L \frac{d α_{2}}{d λ}) B (λ) .

Using Eqs. (3), (10), (11) and (13), the equation that governs the peak (

{\bar{λ}}_{P N}

) or valley (

{\bar{λ}}_{V N}

) wavelengths in the presence of bending is given as

{- \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{λ^{2}} \sin (\frac{4 π n_{s i} d}{λ}) |}_{{\bar{λ}}_{P N}, {\bar{λ}}_{V N}} = {(4 \frac{d α_{1}}{d λ} + 2 L \frac{d α_{2}}{d λ}) [R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ})] |}_{{\bar{λ}}_{P N}, {\bar{λ}}_{V N}}

The peak and valley wavelength can be written as

{\bar{λ}}_{P N} = λ_{P N} + Δ λ_{P N}

and

{\bar{λ}}_{V N} = λ_{V N} + Δ λ_{V N}

, Then Eq. (14) can be expressed as

- \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{{(λ_{P N} + Δ λ_{P N})}^{2}} \sin (\frac{4 π n_{s i} d}{λ_{P N} + Δ λ_{P N}}) = ({4 \frac{d α_{1}}{d λ} |}_{{\bar{λ}}_{P N}} + {2 L \frac{d α_{2}}{d λ} |}_{{\bar{λ}}_{P N}}) [R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ_{P N} + Δ λ_{P N}})]

- \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{{(λ_{V N} + Δ λ_{V N})}^{2}} \sin (\frac{4 π n_{s i} d}{λ_{V N} + Δ λ_{V N}}) = ({4 \frac{d α_{1}}{d λ} |}_{{\bar{λ}}_{V N}} + {2 L \frac{d α_{2}}{d λ} |}_{{\bar{λ}}_{V N}}) [R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos (2 π \frac{2 n_{s i} d}{λ_{V N} + Δ λ_{V N}})]

Because

| Δ λ_{P N} | ≪ λ_{P N}

and

| Δ λ_{V N} | ≪ λ_{V N}

, Eq. (15) can be simplified as

\begin{array}{l} - \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{λ_{P N}^{2}} \sin [4 π n_{s i} d (\frac{1}{λ_{P N}} - \frac{Δ λ_{P N}}{λ_{P N}^{2}})] = \\ ({4 \frac{d α_{1}}{d λ} |}_{{\bar{λ}}_{P N}} + {2 L \frac{d α_{2}}{d λ} |}_{{\bar{λ}}_{P N}}) {R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos [4 π n_{s i} d (\frac{1}{λ_{P N}} - \frac{Δ λ_{P N}}{λ_{P N}^{2}})]}, \end{array}

\begin{array}{l} - \frac{8 π n_{s i} d \sqrt{R_{1} R_{2}}}{λ_{V N}^{2}} \sin [4 π n_{s i} d (\frac{1}{λ_{V N}} - \frac{Δ λ_{V N}}{λ_{V N}^{2}})] = \\ ({4 \frac{d α_{1}}{d λ} |}_{{\bar{λ}}_{V N}} + {2 L \frac{d α_{2}}{d λ} |}_{{\bar{λ}}_{V N}}) {R_{1} + R_{2} - 2 \sqrt{R_{1} R_{2}} \cos [4 π n_{s i} d (\frac{1}{λ_{V N}} - \frac{Δ λ_{V N}}{λ_{V N}^{2}})]} . \end{array}

At the peak and valley wavelength positions, Eq. (11) holds and Eq. (10) reaches its maxima and minima, respectively, at.,

{\begin{cases} \sin (\frac{4 π n_{s i} d}{λ_{P N}}) = \sin (\frac{4 π n_{s i} d}{λ_{V N}}) = 0 \\ \cos (\frac{4 π n_{s i} d}{λ_{P N}}) = - 1, \cos (\frac{4 π n_{s i} d}{λ_{V N}}) = 1 \end{cases} .

Combining Eq. (17) and these approximations,

\sin (x) = x

and

\cos (x) = 1

when

| x |

is close to zero, after some trigonometric and alphabetic manipulations the following expression for BIWS is achieved

Δ λ_{P N} = - (4 {\frac{\partial α_{1}}{\partial λ} |}_{{\bar{λ}}_{P N}} + 2 L {\frac{\partial α_{2}}{\partial λ} |}_{{\bar{λ}}_{P N}}) (\frac{R_{1} + R_{2}}{2 \sqrt{R_{1} R_{2}}} + 1) \frac{λ_{P N}^{4}}{{(4 π n_{s i} d)}^{2}},

Δ λ_{V N} = (4 {\frac{\partial α_{1}}{\partial λ} |}_{{\bar{λ}}_{V N}} + 2 L {\frac{\partial α_{2}}{\partial λ} |}_{{\bar{λ}}_{V N}}) (\frac{R_{1} + R_{2}}{2 \sqrt{R_{1} R_{2}}} - 1) \frac{λ_{V N}^{4}}{{(4 π n_{s i} d)}^{2}} .

Taking the following definition of visibility for the FPI

V_{b} = \frac{F_{\max} - F_{\min}}{F_{\max} + F_{\min}} = \frac{2 \sqrt{R_{1} R_{2}}}{R_{1} + R_{2}},

Equation (12) is obtained by inserting Eq. (19) into Eq. (18).

Funding

U.S. Naval Research Laboratory (N0017315P3755); U.S. Office of Naval Research (N000141410139, N000141410456).

References and links

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Influence of fiber bending on wavelength demodulation of fiber-optic Fabry-Perot interferometric sensors

Abstract

1. Introduction

2. Theoretical investigation

2.1 Qualitative illustration

2.2 General quantitative formulism

2.3 Simplified expressions for low-finesse FPI

2.4 Theoretical results

3. Experimental demonstration

3.1 Experimental setup and method

3.2 Experimental results and discussion

4. Conclusion

Appendix

Funding

References and links

Cited By

Figures (6)

Equations (23)

Optics Express