Thermal and mechanical properties of tapered single mode fiber measured by OFDR and its application for high-sensitivity force measurement

Xiaozhen Wang; Wenhai Li; Liang Chen; Xiaoyi Bao

doi:10.1364/OE.20.014779

1. Introduction

Tapered fiber has attracted much attention over the last two decades due to many interesting applications, such as supercontinuum generation [1], fiber sensors [2–4] and fiber devices [5]. In order to excite high order modes, the diameter of fiber is reduced to a few micro-meters using heating and stretching technique [6]. During tapering process, stress is introduced to the taper and causes the inhomogeneous refractive index distribution. This will degrade the optical quality of taper for its applications. To the best of our knowledge, the thermal and mechanical properties, particularly their variations at different locations along tapered fiber, have not been studied.

Fiber tapers with small diameters become interesting also for the force measurement. When a section of fiber taper is stretched by an axial force F, the strain applied on the taper scales inversely with the cross-sectional area or the square of the diameter of the taper as long as Young’s modulus is considered independent of the diameter [7]. Therefore, a high force measurement has been achieved using Fiber Bragg gratings (FBGs) inscribed in the waist of a tapered fiber [7]. However, for tapers with a diameter below:16μm, the variation of the effective refractive index with diameter should be taken into account [7], so that it becomes hard to fabricate FBG in tapered single-mode fibers (SMFs) with smaller waist diameters. As reported in this paper, a high-sensitivity force sensor can be achieved by measuring the refractive index change, thus the force applied to the fiber taper. Here we realize the force sensor by measuring Rayleigh backscattering spectral change by using high-sensitivity, high-resolution Optical Frequency-domain Reflectometry (OFDR) technique.

Rayleigh backscatter in optical fiber is caused by random fluctuations in the refractive index profile along the fiber length, and it varies randomly along the fiber length due to inhomogeneity of the fiber. Changes in the refractive index profile caused by an external stimulus, such as temperature or strain, will induce spectral shift being measured by OFDR. Recent publications have demonstrated that SMF and polarization-maintaining fiber (PMF) can be used as Rayleigh scattering-based temperature or strain sensor by using high-resolution, high-sensitivity OFDR, and the use of PMF makes it possible to discriminate the temperature and strain through data processing schemes including complex Fast Fourier Transfer (FFT), autocorrelation and cross-correlation, etc [8, 9]. In the case of PMF, the birefringence of the fiber, which is refractive index difference between the fast and slow modes, can be induced by spectral shifts through autocorrelation calculation, while the refractive index change of the fast (or slow) mode can be induced by using cross-correlation. These two parameters are temperature and strain dependent, which makes it possible to assess these parameters simultaneously. In the case of fiber tapers, the autocorrelation wavelength shift indicates the refractive index difference between the fundamental mode and high order modes through mode coupling process; In the uniform region of taper, the cross-correlation wavelength shift indicates the refractive index change of the fundamental mode as the fundamental mode becomes the dominating one [10].

In this report, we represent our study on fiber taper's thermal and mechanical properties by using an OFDR scheme, and a highly sensitive force sensor is proposed based on the fiber taper. In Section 2, the fabrication of the gentle taper and the principle of OFDR measurement based on our previous work [10] are described. The wavelength shift is calculated through cross-correlation of Rayleigh scatter spectra from a selected section of fiber before and after the temperature or force is applied to the fiber. As described in section 3.1, the non-annealed taper has its inhomogeneous thermal property and the annealed taper has homogeneous temperature coefficient, similar to that of SMF (~0.01nm/°C). In section 3.2, the mechanical property of the taper and its variation along the taper length are studied and measured with a spatial resolution of 3.85mm. It is demonstrated that the force sensitivity is improved by decreasing the waist diameter of the uniform segment along the taper, e.g., when the waist diameter is reduced to ~6μm, a force sensitivity of 620.83nm/N with a force resolution of 6.35μN is achieved, which is about 500 times higher than that of the SMF. It is noted that fiber taper can be used as a temperature-independent force sensor due to huge difference between its force and temperature sensitivities.

2. Taper fabrication and principle

Heating and stretching technique [6] is introduced to fabricate the gentle tapers. A section of SMF is fixed on two linear translation stages with submicron precision and the middle jacket-off fiber is hanging. A small region of the fiber is heated by a scanning hydrogen flame. After a pre-heating process, the fiber is stretched slowly while the scanning range of the flame is increasing. The pulling speed of the translation stages, the scanning speed and the scanning range of the flame are accurately controlled by a PC. The slow stretching technique leads to the taper with gentle slopes in the down-taper and up-taper regions and a uniform region with reduced diameter waist.

In OFDR, the interference Rayleigh backscatter signal of the fiber taper is monitored and the complex Fourier transform is carried out to obtain OTDR-like (Optical Time-domain Reflectometry) curves along the taper length. A polarization-independent signal is obtained by calculating the vector sum of the P- and S-components of the detected signals. During the uniform range of the gentle taper, LP₀₁ mode is a dominating mode, which contains most of energy, with refractive index n and its corresponding phase velocity $V = c / n$ inside the fiber, where c is the speed of light in vacuum. The OTDR-like signal along the uniform taper length is expressed as [10],

S (\bar{z}) = \frac{V}{2 Ω} κ (\bar{z})

where

\bar{z} = (ω V) / (2 Ω)

and Ω is a constant sweep rate of tunable laser,

κ (\bar{z})

is directly related to local Rayleigh backscatter strength. An appropriate window with a width of Δz is applied at the position

z^{'}

along the OTDR-like curve and the spectrum response is obtained through inverse Fourier transform data processing, as follows:

I_{Δ z} (ω, z^{'}) = | \sqrt{2 π} \int_{z^{'}}^{z^{'} + Δ z} e^{i ω z} S (z) d z | = \frac{V}{2 Ω} | Κ (ω, z^{'}, Δ z) |

with

Κ (ω, z^{'}, Δ z) = \sqrt{2 π} \int_{z^{'}}^{z^{'} + Δ z} e^{i ω z} κ (z) d z

.

A reference measurement is taken in the initial state. Then a sensing measurement of the same taper is taken after the temperature or force is changed. Therefore, one can measure the spectral shift in the cross-correlation functional plot of $I_{Δ z} (ω, z^{'})$ , which gives the amount of changed effective refractive index of LP₀₁ mode. Because the refractive index is determined by the change of temperature or force, the thermal and mechanical properties of the taper can be investigated by measuring the wavelength shift through cross-correlation calculation of the spectra. Here, the sensitivity is defined as the ratio of wavelength shift of cross-correlation calculation and the change of temperature or force, which is given by $ζ_{T} = Δ λ_{c . c .} / Δ T$ and $ζ_{F} = Δ λ_{c . c .} / Δ F$ , respectively.

When a section of fiber is stretched, it is subjected to a deformation proportional to the amplitude of the force within elastic range of the fiber. Within this elastic range, the mechanical deformation is reversible. Here, the applied force F is expressed as

F = E A \frac{Δ x}{x}

where for fiber without coating, Young’s modulus E = 7.18 × 10¹⁰N/m² [11], A is the cross-sectional area of the fiber,

x

is the initial length of the fiber and

Δ x

is the length change of relative deformation imposed by the force F. As shown in Fig. 1 , we have

Δ x = Δ x_{T a p e r U} + Δ x_{S M F} + \sum_{z}^{} Δ x_{T a p e r L} (z)

x = x_{T a p e r U} + x_{S M F} + \sum_{z}^{} x_{T a p e r L} (z)

where

Δ x_{S M F}

,

Δ x_{T a p e r U}

, and

Δ x_{T a p e r L} (z)

are the length change of SMF, the uniform segment in taper, and a sub-segment in the up- and down-segment in taper, respectively.

x_{S M F}

,

x_{T a p e r U}

, and

x_{T a p e r L} (z)

are the initial length of jacket-off SMF, the uniform segment in taper, and a sub-segment in the up- and down-segment in taper, respectively.

Δ x

and

x

are the total length change and initial length of the whole fiber, respectively. The length changes of each segment satisfy the following relations,

\frac{Δ x_{S M F}}{Δ x_{T a p e r U}} = \frac{F x_{S M F}}{E A_{S M F}} \frac{E A_{T a p e r U}}{F x_{T a p e r U}} = \frac{x_{S M F}}{x_{T a p e r U}} {(\frac{d_{T a p e r U}}{d_{S M F}})}^{2}

\frac{Δ x_{T a p e r L} (z)}{Δ x_{T a p e r U}} = \frac{F x_{T a p e r L} (z)}{E A_{T a p e r L} (z)} \frac{E A_{T a p e r U}}{F x_{T a p e r U}} = \frac{x_{T a p e r L} (z)}{x_{T a p e r U}} {(\frac{d_{T a p e r U}}{d_{T a p e r L} (z)})}^{2}

where

d_{S M F}

,

d_{T a p e r U}

, and

d_{T a p e r L} (z)

are the diameter of SMF, the uniform segment in taper, and a sub-segment in the up- and down-segment in taper, respectively. Young’s modulus is assumed to be constant when the fiber is tapered. Therefore, we have

Δ x_{T a p e r U} = \frac{Δ x}{γ}

where

γ

is a normalization factor and given by

γ = 1 + \frac{x_{S M F}}{x_{T a p e r U}} {(\frac{d_{T a p e r U}}{d_{S M F}})}^{2} + \sum_{z} \frac{x_{T a p e r L} (z)}{x_{T a p e r U}} {(\frac{d_{T a p e r U}}{d_{T a p e r L} (z)})}^{2}

Inserting Eq. (8) into Eq. (3), the force is calculated by

F = E A_{T a p e r U} \frac{Δ x_{T a p e r U}}{x_{T a p e r U}} = \frac{π E d_{T a p e r U}^{2} Δ x}{4 γ x_{T a p e r U}}

Thus, the force is calculated through the parameters of taper and the stretched length. Since the strain applied on the fiber is given by

ε = F / (E A)

, the strain applied on the uniform segment of taper is higher than that of the SMF. In theory, the ratio of the strains is given by

\frac{ε_{T a p e r U}}{ε_{S M F}} = \frac{A_{S M F}}{A_{T a p e r U}} = {(\frac{d_{S M F}}{d_{T a p e r U}})}^{2}

The applied strain scales inversely with the cross-sectional area or the square of the diameter of the fiber. As the applied strain increases, the change of refractive index increases and then measured wavelength shift in cross-correlation is getting larger. Therefore, the uniform range of taper with smaller waist diameter has a higher force sensitivity

ζ_{F}

.

Fig. 1 Sketch image of a gentle fiber taper. $x_{S M F}$ , $x_{T a p e r U}$ , and $x_{T a p e r L} (z)$ are the initial length of jacket-off SMF, the uniform segment in taper, and a sub-segment in the up- and down-segments in taper, respectively. $x_{S M F} = x_{S M F}_{1} + x_{S M F}_{2}$ where $x_{S M F}_{1}$ and $x_{S M F}_{2}$ are initial lengths of left and right side jacket-off SMF, respectively.

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3. Experimental Setup and Results

The schematic diagram to measure Rayleigh backscatter as a function of fiber length is shown in Fig. 2 . It consists of a tunable laser source (TLS) with a sweep rate of 40nm/s, two polarization controllers (PC₁ and PC₂), a measurement interferometer, an auxiliary interferometer with 64m differential delay, and a polarization beam diversity receiver. A wide sweep range of tunable laser source (1510-1570nm) produces high spatial resolution of 13μm. The auxiliary Mach-Zehnder interferometer (MZI) provides interference fringes, which are used to trigger data acquisition and help to mitigate the tuning errors of the laser. The polarization beam diversity receiver includes a polarization beam splitter (PBS), two photo-detectors (PD₁ and PD₂) and a high-speed data acquisition card (DAQ). The interference fringes are digitized as a function of laser frequency and stored in memory. By scanning the segment window, all fiber length is mapped.

Fig. 2 Schematic setup of OFDR system. TLS: tunable light source; C₁: 99:1 coupler; C₂, C₃, C₄, C₅: 50:50 couplers; PC: polarization controller; PBS: polarization beam splitter; OPD: optical path difference; PD: photo-detector; DAQ: data acquisition.

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With the help of a high-magnification microscope, the cladding diameter of taper A is measured point by point, as shown in Fig. 3(a) . Figure 3(a) also shows the reflected spectrum of Rayleigh backscatter as a function of distance along the tapered SMF. The spectrum has been smoothed using a low-pass filter with an effective spatial resolution of ~1.7mm. The amplitude fluctuation of Rayleigh backscatter as a function of fiber distance shows the energy re-distribution among different modes continuously along the taper length [10].

Fig. 3 (a) Backscatter signal versus distance along the taper (left axis) and the cladding diameter of the taper (right axis). (b) Cross-correlation calculation of the spectra of the segment in the uniform taper range of non-annealed taper A (the blue window in (a)) with increasing temperature from 25°C to 38°C.

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3.1 Thermal properties

To investigate the thermal properties of a taper, taper A was put into the oven. A reference measurement was taken when the taper was at the initial temperature of 25 °C. As shown in Fig. 3(a), we choose a window at position 4.37m with a width of ~3.85mm, including 300 data points, along the OTDR-like curve. The spectrum response is obtained through inverse Fourier transform data processing. Then a sensing measurement of the same taper was taken after the temperature was changed. The cross-correlation of these two spectra reveals a central peak shift due to changed effective refractive index n, which represents the changed temperature. Figure 3(b) shows the left-shifted central peak in cross-correlation of the spectra of the non-annealed taper A, which indicates the temperature-induced refractive index change when the temperature is increased from 25°C to 38°C.

Figure 4 compares the thermal property of the non-annealed taper, annealed taper and SMF. From Fig. 4(a), we observe that the non-annealed taper has the inhomogeneous thermal property. When the temperature increases, the wavelength shift of cross-correlation calculation is negative and decreases to ~-0.9nm at the temperature of ~40°C and then increases as the temperature increases. When the temperature decreases from 110°C to 30°C, the wavelength shift is a linear function of the temperature, with a negative slope. Then the taper is put in a closed oven at a constant temperature T = 120°. After 24 hours, the taper is slowly cooled to room temperature. Figure 4(b) indicates that the annealed taper has the same linear temperature dependence when the temperature increases or decreases. Comparing Figs. 4(b) and 4(c), the temperature sensitivity of the annealed taper is similar to that of SMF. After annealing, the inhomogeneous thermal property has been reduced significantly. Apparently, the residual stress from the taper process has been released by annealing treatment.

Fig. 4 Wavelength shift vs. temperature for (a) non-annealed taper, (b) annealed taper, and (c) SMF, respectively. The left and right axes show the temperature increasing case and the temperature decreasing case, respectively. The samples are the experimental data and the linear curves are the fitting results.

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Now we discuss the system resolution, including the spectral resolution and the spatial resolution. A larger window in OTDR-like curve includes more data points and gives better spectral resolution, but lower spatial resolution. There is a trade-off between spectral resolution and spatial resolution due to tuning range of the TLS and the sampling rate of DAQ. The OFDR measurement is performed 100 times at the room temperature and the standard deviation of cross-correlation calculation versus spatial resolution of OFDR system is plotted as a function of spatial resolution of OFDR system, as shown in Fig. 5(a) . The standard deviation of the wavelength shift of cross-correlation measurement is as high as ~17pm when the spatial resolution is ~1mm. The standard deviation of wavelength shift is decreased to ~3.94pm as the spatial resolution is ~3.85mm. There is a trade-off between spatial resolution and standard deviation of the OFDR system. As an example, Fig. 5(b) displays the standard deviation of cross-correlation calculation of ~3.94pm when the spatial resolution is set at ~3.85mm, including 300 data points.

Fig. 5 (a) The standard deviation of wavelength shift of cross-correlation calculation as a function of the spatial resolution. (b) The wavelength shifts of cross-correlation calculation over 100 measurements at the room temperature, when the spatial resolution is ~3.85mm.

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3.2 Mechanical properties

A section of the jacket-off fiber with a length of ~19cm was stretched by using two translation stages at room temperature. A reference measurement was taken when the taper was in the initial state. Then the force measurement was taken when the fiber was stretched with a step of Δx. A ~3.85mm window with 300 data points is applied along the OTDR-like curve and the spectrum response is obtained through inverse Fourier transform data processing. As discussed in previous subsection, the residual stress leads to inhomogeneous change of the refractive index. Tapers A, B and C are fabricated with different parameters and then all of these tapers are annealed. Taper A has a length of $x_{A} =10 .4cm$ and a uniform segment with diameter of $d_{T a p e r U_{A}} = 12 .2 μ m$ and length of $x_{T a p e r U_{A}} =4 .9cm$ . The parameters for taper B and C are $x_{B} =12 .6cm$ , $d_{T a p e r U_{B}} =7 .8 μ m$ , $x_{T a p e r U_{B}} =4 .7cm$ , $x_{C} =13 .7cm$ , $d_{T a p e r U_{C}} =6 μ m$ and $x_{T a p e r U_{C}} = 5.5 c m$ . Using Eq. (9), we have $γ_{A} = 1.176$ , $γ_{B} = 1.205$ , and $γ_{C} = 1.127$ . Using Eq. (11), in theory, the uniform ranges of taper A, B and C should have a ~105, ~257 and ~434 times more strain than that of SMF.

We take taper A as an illustration. Taper A was stretched with a step of 125μm. Using Eq. (10), the corresponding force step is 0.0182N. Compared with a reference measurement, the sensing measurement is taken with an additional force of 0.0182N. Cross-correlation of the spectra reveals the wavelength shift between two conditions. As shown in Fig. 6(a) , each wavelength shift data is calculated within a selected ~3.85mm window along the OTDR-like curve. In the first 15 states, the wavelength shift of cross-correlation in the uniform range of taper shows the refractive index change of LP₀₁ mode. The wavelength shift between states 15 and 16 shows negative values. It means that the taper starts to be deformed in transverse direction in state 16, with a 0.2912N-force applied. The energy in original dominating mode LP₀₁ is redistributed to higher order mode; hence correlation between different modes gives negative values, while correlation results of the same mode show positive correlation. It is because higher order mode has lower refractive index [10]. Then the cross-correlation wavelength shifts in states 16-19 become positive again because they show the refractive index change of the same high order mode, which has most energy in these states. The energy coupling between high order modes happens again in states 19 and 20, with negative cross-correlation wavelength shifts. Figure 6(b) indicates that the wavelength shift is not a linear function to the force when the force is larger than ~0.2N, which is the elastic limit of the taper. The wavelength shift-force sensitivities for different modes are different. Linear fitting in separated ranges find that LP₀₁ mode has a force sensitivity of 124.27nm/N, high order modes have sensitivity of 48.39nm/N and 18.15nm/N respectively. Here we only test the tapers with reversible mechanical perturbations, and consider the force sensitivity of LP₀₁ mode. Reference [12] showed the experimental result of $Δ λ_{c . c .} / λ = 0.7314 Δ ε$ , where $Δ ε$ is the change strain in the fiber, which results in a wavelength shift-strain sensitivity of $ζ_{ε} = Δ λ_{c . c .} / Δ ε = 0.7314 λ$ = 1.13pm/με, here λ = 1540nm. We transform it to the wavelength shift-force sensitivity by using

ζ_{F} = \frac{Δ λ_{c . c .}}{Δ F} = \frac{Δ λ_{c . c .}}{Δ ε} \frac{Δ ε}{Δ F} = \frac{ζ_{ε}}{E A} = \frac{4 ζ_{ε}}{π E d_{S M F}^{2}}

and the force sensitivity of SMF is 1.25nm/N. In our experiment, as shown in Fig. 6(b), SMF has a force sensitivity of 1.29nm/N, which is similar to the reference value. Compared with SMF, the taper is ~96 times more sensitive, which is equivalent to 124.27nm/N. The optimization ratio is comparable to the theoretical strain ratio between taper A and SMF (~105). Figure 6(c) displays distributed force sensitivity along the whole taper fiber. There is small force sensitivity fluctuation along the uniform segment of taper A. Fluctuation of flame temperature during tapering process results in inhomogeneity of geometric structure and mechanical strength of the taper. Hence, the inhomogeneity of geometric structure causes the inhomogeneous refractive index distribution in the taper.

Fig. 6 (a) Wavelength shifts of cross-correlation calculation along the taper length (left axis) and the cladding diameter of the gently taper A (right axis). Taper A: Length of taper: x_A = 10.4cm; Uniform segment of taper: $d_{T a p e r U_{A}}$ = 12.2μm, $x_{T a p e r U_{A}}$ = 4.9cm. (b) Wavelength shifts as a function of force for the uniform segment of taper A (left axis) and SMF (right axis). The samples are the experimental data and the curves are the linear fitting results. (c) The force sensitivities along the fiber (left axis) and the cladding diameter of the gently taper A (right axis).

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Taper B was tested with a stretched step of 125μm. The total length of the stretched fiber is ~19cm. The cross-correlation of the spectra reveals the wavelength shift between two states with an additional force of 7.574 × 10⁻³N applied. Figures 7(a) and 7(b) show that an increased force leads to a positive cross-correlation wavelength shift value and the wavelength shift obeys a linear relationship with the applied force. There is no new spatial mode generation. Figure 7(c) displays force sensitivity along the fiber. The uniform range of taper B obtains a force sensitivity of 339.89nm/N, which is ~279 times higher than that of SMF, over a force range of 0-0.07Ν. The optimization ratio is close to the theoretical strain ratio between taper B and SMF (~257).

Fig. 7 (a) Wavelength shift of cross-correlation calculation along the fiber length (left axis) and the cladding diameter of the gentle taper B (right axis). Taper B: Length of taper: x_B = 12.6cm; Uniform segment of taper: $d_{T a p e r U_{B}}$ = 7.8μm, $x_{T a p e r U_{B}}$ = 4.7cm. (b) Wavelength shift as a function of force for SMF (left axis) and uniform segment of taper B (right axis). The samples are the experimental data and the curves are the linear fitting results. (c) The force sensitivity along the fiber (left axis) and the cladding diameter of taper B (right axis).

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Now we reduce the force to the level of 1.456 × 10⁻⁴N. A section of fiber (~19cm) was stretched by a step of 1μm, which corresponds to a step force of 1.456 × 10⁻⁴N. The measurement is performed with zero force applied to the fiber to produce a reference and then again after a force has been induced. A ~3.85mm window is applied at the location of 4.38m along taper A and as shown in Fig. 8(a) , the increasing force results in a right-shifted main peak in the cross-correlation plot. Figure 8(b) shows the wavelength shift of cross-correlation calculation along the whole taper length. In the taper region, compared with the zero force applied state, cross-correlation of the spectra reveals that the wavelength shift increases as the force applied on the taper increases. From Fig. 8(c), we observe that the taper with smaller diameter has higher force sensitivity. The force sensitivities of tapers A, B, and C are $ζ_{F_A}$ = 132.31nm/N, $ζ_{F_B}$ = 424.74nm/N and $ζ_{F_C}$ = 620.83nm/N, respectively. The resolution of the force measurement is related to the wavelength resolution of cross-correlation calculation in OFDR scheme. As mentioned in section 3.1, the standard deviation is optimized to ~3.94pm as the spatial resolution is ~3.85mm. Thus, the corresponding force resolutions are 29.78μN, 9.28μN and 6.35μN for tapers A, B and C, respectively. Figure 8(d) displays the force sensitivity along the uniform segments of these tapers. The force sensitivities fluctuate slightly along the whole uniform segment.

Fig. 8 (a) Cross-correlation calculation of the spectra of the segment in the uniform taper range between the zero force applied state and three increasing force applied states. (b) Wavelength shifts of cross-correlation calculation along the taper length as the force increases (left axis), and the cladding diameter of taper A (right axis). (c) Wavelength shifts as a function of force in the case of tapers A, B and C. The samples are the experimental data and the curves are the linear fitting results. (d) The force sensitivities along the uniform segments of these tapers.

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4. Conclusions

In conclusion, we present the thermal and mechanical properties of tapered SMF by using high-resolution, high-sensitivity OFDR technique. The cross-correlation spectral shifts induced by thermal or force in the uniform region of fiber taper indicate the refractive index change of the dominating fundamental mode. The annealed fiber taper has similar homogeneous temperature coefficient to that of SMF. It was demonstrated experimentally that a fiber taper with a reduced waist diameter obtains high wavelength-force sensitivity.

Acknowledgments

X. Wang is supported by the China Scholar Council. This research is supported by the Canadian Institute for Photonic Innovations (CIPI) — a Networks of Centres of Excellence program and the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grants.

References and links

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Thermal and mechanical properties of tapered single mode fiber measured by OFDR and its application for high-sensitivity force measurement

Abstract

1. Introduction

2. Taper fabrication and principle

3. Experimental Setup and Results

3.1 Thermal properties

3.2 Mechanical properties

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (12)

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