Light inside Fabry–Perot interferometers (FPIs) travels many times between the two cavity mirrors, enabling long path lengths in a compact form. For example, vacuum-based FPI cavities with a finesse in excess of 100,000 are used in metrology [1] to reduce the noise of lasers. Such cavities are typically 10–50 cm long, enabling an “equivalent” light path length of up to almost 150 km [2]. However, due to various technical reasons (i.e., mirror alignment accuracy and mirror quality requirements that increase rapidly with finesse), most FPIs based on light propagation between two mirrors in free space have a finesse of 100–1000, with associated effective light path lengths of up to hundreds of meters. An increase in the cavity round trip time, reduction of the mirror alignment sensitivity, and less weight/volume can be achieved in fiber-based FPIs, where the mirrors are deposited directly on the end-facets of a single-mode fiber (SMF). A finesse in excess of 2000 has been demonstrated with fiber lengths of 10 m, enabling an equivalent path length of 20 km [3].

To achieve the highly stable operation required in applications such as laser stabilization [2], the FPIs’ sensitivity to temperature must be managed. In free-space FPIs, this is addressed, e.g., by fixing the distance between mirrors with materials with ultra-low or even zero-crossing coefficient of thermal expansion (CTE) such as silica glass [4], Invar, or ultra-low expansion (e.g., ULE) glass. For SMF-based FPIs, the thermal sensitivity is relatively large, as light propagates through the glass core, which has a relatively strong thermo-optic coefficient. This can be significantly reduced by using a hollow-core fiber (HCF) instead of an SMF, as light propagates almost entirely through the empty core which is surrounded by a glass microstructure responsible for light guidance [5]. HCFs with attenuation levels close to that of an SMF in the low-loss C-band (down to 0.22 dB/km, with attenuation even lower than for an SMF at shorter wavelengths, e.g., below 1000 nm) were recently demonstrated [6]. Free-space coupled HCF–FPIs were demonstrated over >10 m lengths with finesse values >1000, corresponding to effective path lengths of >10 km [7]. A 22-m-long alignment-free (after fabrication) HCF–FPI with a finesse of over 120 and with the light coupled directly from an SMF has also been demonstrated [8], providing an effective path length of >2.5 km.

Although HCF–FPIs already have a thermal sensitivity approximately 15 times lower than SMF–FPIs of the same optical length [8], it is desirable to reduce it even further. As the thermal sensitivity of the HCF is mostly given by the CTE of silica glass [5], reducing the thermal sensitivity of HCF–FPIs requires a reduction in the thermally induced expansion of the HCF. Published approaches include operating at very low temperatures where the CTE of silica glass crosses zero [9,10]. This was shown to occur at −71°C in an uncoated HCF [9] and was predicted to occur at even lower temperatures for a coated HCF [10] since the standard acrylate fiber coating becomes very stiff at low temperatures, significantly altering the thermal expansion of the coated HCF. Another option is to use an HCF with open ends, where changes in the optical phase due to HCF elongation can be compensated by refractive index changes associated with pressure-driven ingress/egress of air to/from the central hole [11]. At atmospheric pressure, this allows for zero thermal sensitivity to be achieved close to 110°C. However, for many applications, operation at or close to room temperature is desired.

Here we elaborate on our preliminary work [12] in which we suggested a method to reduce the thermal sensitivity of an HCF–FPI at room temperature down to potentially zero. Our method is based on tightly winding the HCF–FPI on a spool made of a material with zero or slightly negative CTE (such as Zerodur from Schott [13] or ULE glass from Corning [14]). As we show through simulations, although zero thermal sensitivity can, in principle, be achieved already with a zero-CTE spool, for a coated HCF, a slightly negative CTE of the spool is required. In our proof-of-principle demonstration, we reduce the HCF–FPI thermal sensitivity by a factor of three even with a spool with a small positive CTE, demonstrating a fiber FPI approximately 60 times less sensitive than a fiber FPI based on an SMF. Although we demonstrate this on an FPI, the same approach can be used to reduce the thermal sensitivity of other interferometers based on an HCF or HCF-based delay lines.

Our technique involves initial pre-stretching of the HCF at room temperature and fixing its stretched length (we explain later how we do this). We can visualize this by thinking of the length of the HCF as a string on a guitar. When we heat the HCF/string up, the tension is reduced (due to temperature-induced elongation of the HCF/string, which then requires less force to keep it in the original elongated state). The string then plays at a lower pitch whilst maintaining its length. This keeps the HCF/string straight (under tension) as long as the heat-induced elongation is smaller than the original pre-stretching. The associated glass refractive index change through the stress-optic effect has negligible influence on the light guided through the HCF thanks to its propagation mostly through the air-filled core rather than the silica glass.

To fix the stretched HCF length, we coil the HCF under tension on a spool made of a very low or zero CTE material, as shown in Fig. 1(a). We use Zerodur from Schott in our experiment, but there are several other materials with similar properties, e.g., ULE glass from Corning, ZERØ from Nippon, or IC-ZX from Shinhokoku. With increasing temperature, the spool maintains its size, forcing the HCF to maintain its length as well. The heating would have elongated a freely coiled HCF, but as it is under a stretch, it only reduces the tensile force, meaning the length is fully dictated by the spool size. In this way, the CTE of the HCF should be identical to the CTE of the spool. Silica glass, from which HCF is made, has CTE = 3 × 10⁻⁷/K [15], meaning that a 100 K temperature change would change its length by 0.003%. Fibers can be stretched significantly more (usually proof-tested at 0.5 or 1%), making our technique applicable over a very broad temperature range.

 

Fig. 1. (a) Schematic of the spooled HCF and (b) the 2D axisymmetric model in COMSOL Multiphysics (not to scale as the spool is significantly larger than the HCF).

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In practice, there are several aspects that may reduce the effectiveness of this method. The most important is the coating that has very different mechanical and thermal properties to the glass and which will serve as a “cushion” between the spool and the glass fiber. To understand these effects, we performed simulation with Comsol Multiphysics.

We used a 2D axisymmetric model with the geometry shown in Fig. 1(b). We neglect the mechanical properties of the HCF microstructure, approximating the HCF as a capillary. We consider it to be coated with a single acrylate coating. A contact boundary is applied between the HCF’s outer surface and the coil. A body load in the direction of the spool center is applied to the HCF to simulate the effect of the stretching tension. For a spool radius R, the relationship between the body load F­_L and the stretching tension F_T is

First, we studied the influence of the HCF coating thickness. We considered a spool with R = 75 mm and coating thickness of either 0 µm (bare HCF), 10 µm (thinly coated HCF presented in [15] and used in our experiments), and 50 µm (typical thickness used, e.g., in [10]). As shown in Fig. 2, the uncoated and uncoiled HCF is expected to have a thermal sensitivity of 0.3 ppm/K, corresponding to the silica glass CTE. When coiled on the CTE = 0 spool, its thermal sensitivity is almost eliminated, in line with our expectations. For the uncoiled thinly coated HCF, the coating slightly increases the HCF thermal sensitivity due to the significantly larger CTE of the coating relative to the silica glass. Thanks to the relatively low coating material stiffness (low Young’s modulus) and small thickness, its effect on the overall HCF thermal sensitivity is very small, increasing the HCF thermal sensitivity by only 0.05 ppm/K. When coiled on the CTE = 0 spool, the HCF thermal sensitivity is strongly reduced, but it does not reach zero like for the uncoated HCF. However, the seven-fold reduction predicted is still significant. For the standard-thickness coated HCF, the thermal sensitivity of the uncoiled HCF is relatively strongly influenced by the coating, increasing it by ∼1.5 times. When coiled on the CTE = 0 spool, the thermal sensitivity is reduced as in the previous two cases, but only by a factor of two.

 

Fig. 2. Thermal sensitivity calculated for free-coiled HCFs (yellow) and spool-coiled HCFs (green) with different coating thickness (bare, 0 µm; thinly coated, 10 µm; and standard-coated, 50 µm).

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A bare fiber is impractical (fragile) and a standard-coated HCF (50-µm coating) does not show significant improvement when coiled on the CTE = 0 spool. Thus, with further simulations (and subsequent experimental demonstration), we focus on the thinly coated HCFs.

First, we analyze how the thermal sensitivity of the spool-coiled HCF changes with the HCF pre-stretching. By varying the pre-stretch force between 0 and 2 N, we found that the HCF thermal sensitivity is virtually insensitive to the pre-stretching. In the following simulations, we set this parameter to 0.5 N.

Further simulations revealed rather non-intuitive behaviors. The thermal sensitivity of the spool-coiled HCF did not change significantly with the HCF silica thickness, while for an uncoiled HCF, thicker silica glass reduced the influence of the coating on the thermal sensitivity, as shown in Fig. 3.

 

Fig. 3. Thermal sensitivity of spool-coiled HCF with respect to the fiber diameter and its comparison to free-coiled HCFs.

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In addition, we found that the Young’s modulus of the coating had almost no influence on the thermal sensitivity of the spool-coiled HCF, while its CTE did, as shown in Fig. 4. This is very different to the uncoiled HCFs, where the CTE × Young’s modulus product influences the HCF thermal sensitivity (Eq. (2) in [15]). This suggests that the CTE = 0 spool-coiled HCF should have a coating with a small CTE, irrespective of its Young’s modulus. A candidate for such a coating is polyimide [17] used as a fiber coating for high-temperature applications. In particular, Novastrat905 polyimide from NeXolve has a CTE close to 0 ppm/K.

 

Fig. 4. Thermal sensitivity of spool-coiled HCF with respect to coating CTE.

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Finally, we studied how the coiled HCF thermal sensitivity changes with the CTE of the spool material and the spool radius. Figure 5 shows the results for a CTE between −0.1 and 0 ppm/K, which is within the CTE range specified by the Zerodur or ULE manufacturers (<±0.1 ppm/K [13,14]). As expected, a lower spool material CTE means a lower thermal sensitivity of the coiled HCF. We observe that the coiled HCF thermal sensitivity increases for smaller coil diameters. Our simulations confirmed that this is due to the HCF thermal expansion in its cross section, which increases the HCF center circumference with temperature. The influence of this, however, reduces as the spool diameter increases, becoming negligible for an infinitely large spool. The data in Fig. 5 suggest a design rule for thermally insensitive coiled HCFs. First, we should target spools with a CTE between −0.1 and −0.05 ppm/K. Once the spool CTE is known, the spool diameter can be set to ensure a zero-sensitivity coiled HCF (e.g., in Fig. 5, R = 44 mm gives zero HCF thermal sensitivity for a spool with CTE = −0.1 ppm, as does R = 82 mm for CTE = −0.05 ppm). Further tuning can be achieved by drilling a hole in the center of the spool, as this reduces its stiffness. This is shown in Fig. 6, where a coiled HCF thermal sensitivity of −0.02 ppm/K for a homogeneous Zerodur spool with CTE = −0.05 ppm/K and R = 55 mm is tuned to zero sensitivity when a hole of 46 mm is drilled in the middle of the spool.

 

Fig. 5. Thermal sensitivity of coiled HCF with respect to the radius at different spool CTE values.

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Fig. 6. Thermal sensitivity of coiled HCF on spool with CTE of −0.05 ppm/K with respect to the central hole radius for spool with R = 55 mm.

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To demonstrate our method experimentally, we used a Zerodur cylinder with R = 75 mm that was salvaged from an old mirror, and a thinly coated HCF (10 µm coating thickness) with an outer diameter of 206 µm [15]. Two HCF samples were prepared: one coiled on Zerodur (4.9-m long, fixed on the Zerodur spool with Capton tape), and one freely coiled (4.4-m long). The HCF samples were each spliced with two mode-field adapters and SMF pigtails, as described in [18]. At the HCF end-facets, there is 4% Fresnel reflection, as light travels from the solid glass core to the hollow core, forming the FPI.

The characterization setup is shown in Fig. 7. Both HCF samples were put into a thermal chamber. The probe laser wavelength λ was locked to a carrier envelope offset stabilized optical frequency comb to avoid any interference fringe instability due to any drift in laser wavelength. The signal was then split in two to enable probing of both interferometers at the same time. The HCF–FPI signals were observed in reflection (retrieved via two circulators), which gives better interference contrast in a low-finesse FPI than the transmitted signal.

 

Fig. 7. Thermal sensitivity measurement setup using the free-coiled and zerodur-coiled HCF–FPIs.

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The thermal chamber temperature was changed between 25 and 55°C. Figure 8 shows how the reflection changed as a function of temperature for both HCF–FPIs. The reflected signal amplitude most likely changes because the polarization state inside the HCF–FPIs changes with temperature. Each extrema corresponds to a phase change of π/2 and the thermal sensitivity can then be calculated by counting these extrema (N_extrema) observed over a temperature change of ΔT:

 

Fig. 8. Example measument of Zerodur-coiled HCF–FPI and free-coiled HCF–FPI.

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Fig. 9. Measured and simulated thermal sensitivities of Zerodur-coiled and free-coiled HCF-FPI. Simulations consider a Zerodur CTE of 0.07 ppm/K.

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Our experimental results show an improvement of the Zerodur-coiled HCF–FPI compared with the freely coiled embodiment by a factor of three. However, we expect improvements by up to seven times when using a coil with CTE = 0 and a fully insensitive HCF–FPI when selecting Zerodur with CTE = -0.06 ppm/K, which is within its specifications. Zerodur with a selected specific CTE can be purchased from the manufacturer.

In conclusion, we analyzed a new method of achieving HCF-based FPIs with low and potentially even zero senstivity to temperature. Due to the HCF coating, zero thermal sensitivity requires a spool made of a material with negative CTE. Thinly coated HCFs require a spool with only very small negative CTE (−0.1 to −0.05 ppm/K), which is available commercially (e.g., selected Zerodur or ULE materials). The HCF–FPI thermal sensitivity can be slightly tuned by controlling the spool and its central hole diameters, offering avenues to fine-tune the thermal sensitivity close to zero.

In our experimental demonstration, we achieved a reduction of the HCF–FPI thermal sensitivity by a factor of three, most probably limited by the CTE of the available spool. Even though we plan to further reduce its thermal sensitivity by using optimized spools, the achieved thermal sensitivity of 0.13 ppm/K is three times lower than the lowest value so far achieved at room temperature for any fiber FPI. Finally, we emphasize that the proposed method can be used in any optical fiber interferometer configuration.