We present a simple technique to experimentally determine the optical-path length change with temperature for optical single-mode fibers. Standard single-mode fibers act as natural low-finesse cavities, with the Fresnel reflection of the straight cleaved surfaces being ∼3%, for the laser light coupled to them. By measuring the intensity variations due to interference of light reflected from the fiber front and end surfaces, while ramping the ambient temperature, the thermal sensitivity of the optical-path length of the fiber can be derived. Light was generated by a narrow linewidth, low drift laser. With our fairly short test fibers, we found that it was possible to reach a relative precision of the temperature sensitivity, compared to a reference fiber, on the 0.4%–2% scale and an absolute precision of 2%–5%, with the potential to improve both by an order of magnitude. The results for single-acrylate, dual-acrylate, and copper- and aluminum-coated fibers are presented. Values are compared with analytic models and results from a finite element method simulation. With the aid of these measurements, a simple fiber-interferometer, which is insensitive to thermal drifts, could be constructed.

## I. INTRODUCTION

Good knowledge of the thermal sensitivity of single-mode optical fibers can be important for precision optics, where signal variations originating from unintentional changes in optical-path length can be limiting. Examples of this include laser stabilization based on fiber cavities, fiber interferometers, fiber sensing, and also more broadly in phase-sensitive measurements that use optical fibers.^{1,2} Temperature sensing with optical fibers also profits from an increased temperature sensitivity.^{3} As a result, knowledge of the optical-path length change of single-mode fibers with temperature can be an important parameter for experimental design. This parameter is, however, not commonly specified by manufacturers. A simple method to characterize optical fibers regarding their thermal sensitivity is thus of interest.

We present a simple method to determine the thermal sensitivity of optical fibers, which reaches a relative precision between two fibers of 0.4%–2% and an absolute precision of 2%–5%. The precision of this method has the potential to be improved by more than an order of magnitude by reducing the error on the physical length of the fibers from cm to mm or by the use of longer fibers. The method is ultimately limited by the absorption of the fiber becoming significantly larger than the reflectivity of the fiber edges and the homogeneity of the temperature distribution along the fiber. To our knowledge, measurements in the configuration presented here are carried out for the first time.

Methods to characterize the thermal sensitivity of optical fibers usually rely on optical-phase measurements. One common method is fringe counting using a Mach–Zehnder interferometer.^{4–7} An alternate method is based on the use of Fiber Bragg gratings connecting to their use as temperature sensors^{8,9} and, more recently, wavelength-sweeping interferometry.^{10} Another method, presented by Li *et al.*, used very short Fabry–Pérot interferometers (FPIs) made of tellurite and germanite fibers spliced onto a fiber interferometer to measure the thermo-optic coefficient of those glasses. Producing and splicing these glass fibers is challenging.^{11} Here, we can take advantage of meter long fibers that are used as FPIs directly allowing for a particularly simple setup. One application of our method could be the construction of a simple fiber-interferometer, which is insensitive to thermal drifts. Fiber interferometers already find application as a cheap and simple method to narrow the laser linewidth significantly.^{12} However, environmental disturbances such as thermal fluctuations impose limitations on simple setups, and significant effort regarding isolation^{13} or a more sophisticated setup has to be designed to overcome these limitations.^{14,15}

## II. PRINCIPLE

While Fabry–Pérot interferometers are commonly used for frequency-sensitive measurements that record reflection (or transmission) as a function of frequency, it is also possible to express the reflection as a function of varying optical-path length keeping the frequency constant. The response of the FPI is recorded while the path length is scanned by ramping the ambient temperature. The reflected signal from the low finesse FPI is almost sinusoidal. Intuitively, it is readily understood that a phase change of Δ*φ* = 2*π* corresponds to a single pass optical-path length change of the optical fiber of *λ*/2*n*, where *λ* denotes the wavelength in vacuum and *n* is the refractive index. We can therefore easily derive the relation between phase change with temperature and optical-path length change with temperature *T* as follows:

Here, *L* denotes the length of the optical fiber. As the phase can directly be derived from the reflected intensity measurements of the FPI, this gives a very straight forward way to measure the thermal sensitivity of the optical-path change *d*(*nL*)/*dT*.

A more rigorous derivation is given in the following. The reflectivity of a FPI as a function of optical-path length *R*(*nL*) is given as

where *τ*_{max} denotes the maximum transmittance, *ν* is the frequency, and *c* is the speed of light in vacuum. The finesse $F$ is given as $F\u2248\pi |R1R2|41\u2212|R1R2|$, where *R*_{1} and *R*_{2} are the Fresnel reflection at the fiber ends due to the change in refractive index at the glass–air boundary. Here, *R*_{1} = *R*_{2} = 0.032, assuming a refractive index of fused silica glass of 1.44. This yields a finesse of 0.58. To derive the phase relation, we can make the Taylor approximation 1/(1 + *x*) ≈ 1 − *x* and use the trigonometric identity sin^{2}(Θ) = (1 − cos(2Θ))/2 to rewrite the expression for reflectivity as

where we set *τ*_{max} to 1. Using *ν*/*c* = 1/*λ*, the relation of optical phase and optical path can then be written as

## III. EXPERIMENTAL REALIZATION

To demonstrate this method, the thermal sensitivity of four single-mode silica fibers with different coatings were measured, referred to as fiber 1 in the following: single-acrylate coated, aluminum-coated, copper-coated, and double-acrylate coated. Each measurement is taken with a reference fiber (in the following referred to as fiber 2). The aluminum coated fiber was a 1 m sample of type AIS4.3/125/175A from Fiberguide Industries. The copper coated fiber tested was a 3 m single-mode fiber by IVG fibers with the product number Cu600, the single acrylate coated fiber was a 630HP fiber by Thorlabs, and the dual acrylate fiber was a F-SV fiber by Newport.

The free-spectral range of the fibers is on the scale of 10 MHz–100 MHz. The length of the fiber is inversely proportional to the free-spectral range determining the frequency at which fringes appear. The light source used in these experiments is a frequency-stabilized dye laser with a sub-kHz linewidth and with an expected drift of less than 1 kHz during the measurements. Since the linewidth was a factor of >10^{5} smaller than the free-spectral range of the optical fiber, any effects on the measurements from shifts in laser frequency can be neglected. In general, a linewidth significantly smaller than the free-spectral range of the optical fibers should be sufficient.

In order to record the reflections from the fibers, they were optically coupled and placed into a simple, home-built, heat-box. Part of the beam was picked-off and recorded to enable compensation for any laser power intensity fluctuations. The setup is depicted in Fig. 1. As can be seen in the figure, light reflected from fiber 2 could potentially also be detected on photodiode (PD) 1, which would have resulted in crosstalk. In order to rule out crosstalk from fiber 2 incident on PD 1, the Fourier spectrum, where signals from the fibers were at significantly different frequencies, was checked. No significant crosstalk was observed. We suspect that the fibers might have been cleaved at a slight angle, thus spatially separating the path of back reflection from incident light. In configurations where back reflections from fiber 2 on PD 1 prove to be a challenge, an optical isolator could be used.

The heat-box consisted of a card-board box with armaflex insulation and a heat plate, which is composed of evenly spread heating resistors in thermal contact with an aluminum plate. Applying a constant voltage across the resistors resulted in a temperature ramp with respect to time, as depicted in Fig. 2(a). One might believe that a linear increase in temperature with time is necessary. However, experiments showed that the data analysis is robust enough to handle moderate fluctuations from a linear increase. The temperature was recorded with a Lakeshore 201 digital thermometer with a Lakeshore DT670 silicon diode temperature sensor.

The fibers studied here are, due to their high thermal sensitivity, much better temperature sensors than a common temperature sensor. Therefore, all fibers were measured with a single acrylate fiber as a reference fiber. Both fibers are wrapped around the same spool. As a result, the reference fiber acts as a sensor that experiences a very similar ambient temperature distribution as the sample fiber. Although efforts were made to keep the temperature distribution homogeneous in the heat-box, our construction allows inhomogeneities to impact both fibers in the same way, which should partly equalize any remaining differences. The fiber was loosely wound on the spool, as not to be stretched and affected by its thermal expansion. The spool was made of cardboard. The polarization was adjusted before experiments with a polarizing beamsplitter followed by a *λ*/2 waveplate for maximum contrast for the fringes. Throughout measurements, the amplitude of the fringes varied somewhat, which could be a result of birefringence. The envelope of the fringes typically only had a few oscillations and should therefore not affect the result in any significant way.

In order to analyze the recorded data and deduct phase information, the analytic signal was retrieved. The data are first corrected for intensity fluctuations by dividing by the intensity recorded with the reference detector. Next, the data are normalized by subtracting the average intensity value denoted. The result is Δ*I*. To deduct phase information and filter out noise, the data are Fourier transformed, and the relevant frequency spectrum filtered out with a Supergaussian filter. By only applying the filter to the positive frequency components, before applying the inverse Fourier transform, you get a complex representation of your data in time. The complex data are then unwrapped to gain phase information. To make sure no essential information was lost, the real part of the transformed and filtered data is checked against the raw experimental data, as seen in Figs. 2(b) and 2(c). In principal, this modified Hilbert transform also makes it possible to filter out crosstalk wherever it becomes problematic (provided the peaks are sufficiently separated). The ratio of the thermal sensitivities of the two fibers was then obtained by dividing the phase change in both fibers with respect to time. A snippet from a typical measurement is presented in Fig. 2 for visualization.

The absolute thermal sensitivity of the optical-path length of the single acrylate coated reference fiber was obtained by interpolating the obtained time data with temperature and deriving *d*(*nL*)/*dT* from *dφ*/*dT*. Representative graphs obtained from this analysis are depicted in Figs. 3(a) and 3(b). For higher accuracy on the absolute value, a statistical mean was calculated from all measurements carried out.

## IV. EXPERIMENTAL RESULTS

To test the reliability of the method, a measurement with two single acrylate fibers with comparable lengths was carried out (see Table I). Deviation from the expected value of 1 is within the uncertainty. Generally, we differentiate between two different errors for these measurements. First, a systematic error results from the uncertainty of the fiber length. Second, there is the statistical error given as the standard deviation of the mean of the ratio recorded in one measurement. For the initial measurement with two single acrylate fibers, the systematic error is on the percent scale with 0.013, while the statistical error is on the per mill scale with 0.006. This reflects values typically obtained for these two errors throughout experiments.

Fiber . | Measured thermal . | . | . |
---|---|---|---|

coating . | sensitivity ratio . | $1LdnLdT$ (K^{−1})
. | L (m)
. |

Single acrylate | 1.003 ± 0.014 | (9.5 ± 0.2) · 10^{−6} | (1.18 ± 0.01)/(1.15 ± 0.01) |

Aluminum | 1.877 ± 0.007 | (17.9 ± 0.8) · 10^{−6} | (1.25 ± 0.01)/(1.15 ± 0.01) |

Copper | 1.533 ± 0.015 | (14.6 ± 0.7) · 10^{−6} | (2.85 ± 0.02)/(3.05 ± 0.02) |

Dual acrylate | 1.14 ± 0.02 | (10.9 ± 0.5) · 10^{−6} | (12.90 ± 0.02)/(4.90 ± 0.02) |

Fiber . | Measured thermal . | . | . |
---|---|---|---|

coating . | sensitivity ratio . | $1LdnLdT$ (K^{−1})
. | L (m)
. |

Single acrylate | 1.003 ± 0.014 | (9.5 ± 0.2) · 10^{−6} | (1.18 ± 0.01)/(1.15 ± 0.01) |

Aluminum | 1.877 ± 0.007 | (17.9 ± 0.8) · 10^{−6} | (1.25 ± 0.01)/(1.15 ± 0.01) |

Copper | 1.533 ± 0.015 | (14.6 ± 0.7) · 10^{−6} | (2.85 ± 0.02)/(3.05 ± 0.02) |

Dual acrylate | 1.14 ± 0.02 | (10.9 ± 0.5) · 10^{−6} | (12.90 ± 0.02)/(4.90 ± 0.02) |

Throughout the course of experiments, we found a faster, although slightly nonlinear, temperature ramping process to be more advantageous than a slower linear process. While the latter allows for more narrow peaks in the Fourier spectrum, the shift of the peak away from low frequency noise and the DC peak observed was crucial for a good signal-to-noise ratio (SNR). As discussed previously, while the method does allow for the derivation of absolute values, it was confirmed that their error is significantly higher than the error on relative measurements, which had errors an order of magnitude lower (see Table I).

The aluminum, copper, and dual acrylate coated fibers were measured with single acrylate fibers of similar length as the reference fiber. For each fiber pairing, three to four measurement runs were evaluated. Lengths of the fibers as well as the results of the thermal sensitivity ratio are given in Table I. The error of the thermal sensitivity ratio is derived from the statistical error of the average of multiple experimental runs and was checked to be in general agreement with propagated statistical and systematic errors on single experimental runs. Absolute values were calculated by averaging the absolute thermal sensitivity of the optical-path length change of the reference single acrylate fiber from all measurements and multiplying with the experimentally determined ratio.

Note that the temperature coefficient of a material is approximated as to be constant throughout this paper, even though it weakly depends on temperature. However, this dependence could not be experimentally resolved in our measurements and was thus neglected.

As the limiting systematic error of this type of measurement scales directly with the relative error of the lengths of the fibers, one way to achieve higher precision is to reduce this systematic error from the cm scale, used in this work, to the mm scale. This should be possible with commercially available tools. Alternatively, one can use longer fibers. This has the additional advantage of producing more fringes at a higher frequency, which would likely also reduce the statistical error.

Ultimately, the maximum suitable length of the fibers is limited by absorption in the fiber. For longer fibers, this could be a significant effect at the working wavelength of 606 nm used for this work and would reduce the modulation depth by half for a fiber length of 330 m. At telecom wavelength, where the attenuation is much lower, longer fibers could be used. However, for longer fibers, the free-spectral range decreases, which would also require lasers with narrower linewidths. Furthermore, homogeneous heating will be more challenging for larger fiber spools, which can result in experimental difficulties. With these considerations in mind, we still judge that it is possible to improve the systematic length error by 1–3 orders of magnitude and further improve on the statistical error.

Note that due to varying properties of single-mode fibers, e.g., varying coating thickness, the numerical results presented here may differ slightly from the results obtained for different production runs.

## V. COMPARISON WITH THEORY AND SIMULATIONS

In this paper, optical fibers with different coatings were studied. Due to the mismatch in the coefficient of thermal expansion of the fiber coating and the fiber cladding and core, the fiber experiences thermal stress upon variations in temperature. As a consequence, in addition to the thermo-optical effect *∂n*/*∂T*, the fiber exhibits a different thermal behavior expressed in an altered change in the physical length of the fiber with temperature and an additional change in refractive index due to the stress. This photo-elastic effect on an optical fiber can be expressed as

where *ε*_{z} and *ε*_{r} denote the axial and radial strain, respectively, and *P*_{11} and *P*_{12} represent the photo-elastic coefficients of silica glass. The approximation made holds for a fiber with a silica core and a temperature range sufficiently close to room temperature.^{4} Induced radial and axial stress can be calculated based on the elastic modulus *E*, Poisson’s ratio *ρ*, coefficient of thermal expansion (CTE), *α* as well as the cladding and coating diameter. An analytic expression for calculating the axial and radial thermal stress *σ*_{z} and *σ*_{r} in coated fibers was first derived by Lagakos *et al.*^{5} and has been refined for metal-coated fibers by Shiue *et al.*^{16} The formulas used here were taken from the latter. Thermal stress and strain are related via the elastic modulus.

For pure aluminum and copper, the elastic modulus and Poisson’s ratio are well-studied quantities and can be obtained from the literature.^{17,18} The copper fiber also has an inner carbon layer. Values for the properties of carbon are taken from the work of Shiue *et al.*^{19} Note that the copper coating is actually only 99% pure copper. Mechanical properties of single and dual acrylate are not specified by the manufacturers and are not well-known although these coatings are very common. Few papers have been published on this matter, and they have deviating results. Values stated and used here are taken from the work by Olson *et al.*^{20} who carried out a more extensive study. Mitra *et al.* reported values on the same order of magnitude.^{21} The inner coating diameter of the dual acrylate fiber is also taken from the work of Olson *et al.* An overview of parameters used for the analytical calculations, as well as the results obtained by Finite Element Method (FEM) simulations, is given in Table II.

Material . | E (GPa)
. | ρ
. | α (K^{−1})
. | Thickness r (μm)
. |
---|---|---|---|---|

Silica glass (cladding) | 72.5 | 0.155 | 0.5 · 10^{−6} | 125 |

Aluminum coating | 69 | 0.345 | 23 · 10^{−6} | 164 |

Copper coating | 117 | 0.330 | 17 · 10^{−6} | 165 |

Acrylate single coating | 0.047 | 0.49 | 100 · 10^{−6} | 245 |

Dual acrylate inner coating | 0.0042 | 0.44 | 100 · 10^{−6} | 190 |

Dual acrylate outer coating | 0.85 | 0.44 | 100 · 10^{−6} | 245 |

Carbon inner coating | 45 | 0.3 | 22 · 10^{−6} | 30 nm |

Material . | E (GPa)
. | ρ
. | α (K^{−1})
. | Thickness r (μm)
. |
---|---|---|---|---|

Silica glass (cladding) | 72.5 | 0.155 | 0.5 · 10^{−6} | 125 |

Aluminum coating | 69 | 0.345 | 23 · 10^{−6} | 164 |

Copper coating | 117 | 0.330 | 17 · 10^{−6} | 165 |

Acrylate single coating | 0.047 | 0.49 | 100 · 10^{−6} | 245 |

Dual acrylate inner coating | 0.0042 | 0.44 | 100 · 10^{−6} | 190 |

Dual acrylate outer coating | 0.85 | 0.44 | 100 · 10^{−6} | 245 |

Carbon inner coating | 45 | 0.3 | 22 · 10^{−6} | 30 nm |

Errors on the analytical results calculated according to the work of Shiue *et al.* are derived from uncertainties on the coating thickness provided by the manufacturer. For the aluminum fiber, this uncertainty produced an error; the fiber was therefore control measured under a microscope, yielding a coating diameter of (164 ± 5) *µ*m. The values for axial and radial stress based on the analytic expressions are compared with the results from FEM simulations carried out with COMSOL Multiphysics to validate these results. Since the problem of thermal stress is satisfactorily solved with a classical approach, COMSOL is a suitable choice for modeling all key parameters. Radial and axial stress are analyzed in two separate simulations. The simulation for radial stress *σ*_{r} is based on an infinitesimally short fiber piece implemented as a 2D simulation and solved with the COMSOL thermal stress module. The axial stress *σ*_{z} simulation is set up as a radial symmetric 2D simulation. As FEM simulations are more challenging for objects that are much larger in one dimension, the length of the fiber piece in the simulation was chosen just long enough so that boundary effects did not affect the result. We found that a length of few mm was sufficient. Again, we made use of the COMSOL thermal stress module. Typical COMSOL simulations for the aluminum coated fiber are presented in Fig. 4. The underlying geometry of the simulation is indicated with black lines, and thermal expansion is exaggerated. The value of the thermal stress is indicated by color, and the direction is indicated with white arrows. Note that core and cladding are modeled as one. This is a sufficient approximation as the relevant material constants are very similar. From the computed radial and axial stress, we once again calculated a value for the total optical path change with temperature 1/*L d*(*nL*)/*dT* according to Eq. (6), adding the thermo-optic coefficient and photo-elastic coefficients.

Figure 5 compares the experimental, analytical, and simulated results of the absolute thermal sensitivity, as given in Eq. (6), of the four studied fibers measured and calculated in this paper. For the single acrylate fiber, the dual acrylate fiber, and the aluminum coated fiber, the results derived with the different methods are in good agreement validating the experimental results. However, a clear deviation is observed between experimental results and analytical and simulated results for the copper coated fiber. Cladding and coating diameters provided by the manufacturer were confirmed with a microscope. Multiple measurements of the coating thickness along the fiber dismissed variations along the fiber as possible explanation. Possible explanations for the deviation are that calculations are based on the assumption of a fiber coated with pure copper, while, as stated previously, it is actually a copper alloy. Even alloys with a high percentage of purity can behave substantially different. Furthermore, due to the carbon layer being extremely thin, FEM simulations were challenging and might not reflect the effects of this layer correctly. Finally, as these measurements were carried out last, the quality of the heat-box due to usage had degraded, resulting in a worse SNR for these measurements, as indicated by a larger statistical error on single measurements and the mean.

## VI. CONCLUSION AND OUTLOOK

We have presented a simple experimental method for measuring the thermal sensitivity of single-mode fibers and presented the results for four silica fibers: standard single acrylate, dual acrylate, and copper and aluminum coating, respectively. We have reached a precision ranging from 0.4% to 2% on ratios of the thermal sensitivity of the optical-path length change measured with the single acrylate fiber as a reference. We expect that this precision can be improved by at least one order of magnitude by reducing the dominating systematic error on the length of the fibers. In the experiments presented here, we were limited to a cm precision. We further expect that the statistical error resulting from noise, which was typically under 1% in these experiments, can be improved by the use of longer fibers, proper thermal insulation, and repeated measurements. One possible application that the authors had in mind carrying out this work is the construction of a fiber interferometer with an optical-path difference that is insensitive to thermal fluctuations.^{22} This can be achieved by choosing appropriate fibers and a length ratio for the two arms reflecting the ratio of the thermal sensitivity of the optical-path length change of the fibers used.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

This research was supported by the by the Swedish Research Council (Grant Nos. 2016-05121, 2015-03989, and 2016-04375), the Knut and Alice Wallenberg Foundation (Grant No. KAW 2016.0081) and the Wallenberg Center for Quantum Technology (WACQT) (Grant No. KAW 2017.0449), and Smarter Electronics Systems through Sweden’s Innovation Agency (Vinnova, Grant No. DNR 2019-02110). The authors further would like to thank Aylin Ahadi and Solveig Melin for helpful discussion.