1. Introduction

In the past seven years several hundred articles have been published that describe the use of simple in-fiber interferometers to determine the change of the refractive index of liquids. Most interferometers are based on either the Michelson or Mach-Zehnder configuration where one of the interferometer arms is the core of the fiber waveguide and the other arm consists of the cladding material which propagates one or more cladding modes [1–3]. In all in-fiber interferometers a coupler splits the incoming core mode into two (groups of) modes. In the Mach-Zehnder configuration a second coupler then combines these two co-propagating modes into a single mode, whereas in the Michelson configuration a retroreflector doubles the interaction length and the recombination occurs at the same coupler. In either case, the accumulated phase shift in each arm depends on the length of the arm and the effective refractive index of the propagated mode(s),

It is straightforward to derive the sensitivity with which an interferometer can measure the refractive index of a sample, n_s. The sensitivity of the phase shift measurement of the sample’s refractive index depends on the extent with which either one of the two modes can interact with the sample medium. We define an sensitivity coefficient f_i such that

Equally important for the practical use of an interferometer is the “figure of merit (FoM)”, i.e. the ratio of the sensitivity and the full-width-at-half-maximum (FWHM) of the spectral feature, δλ. In optical resonators the linewidth, δλ, of a cavity resonance is used and in interferometers it would be most practical to use the width of an interference fringe, δϕ = π/2 which results with Eq. (3) in

In this article we follow the route to maximal sensitivity and largest figure of merit, i.e. we attempt to increase the effective interaction length by maximizing the overlap of the mode volume with the sample. This may be done most efficiently in a hollow-core photonic crystal fiber (HC-PCF), where the core mode propagates through the sample-filled hollow core of a microstructured fiber [7, 8]. The many cladding modes in HC-PCF propagate in the patterned silica-walled microholes surrounding the hollow core, but their phase indices are so different from the core modes (and from each other) that an interferogram is not analyzable. We therefore co-propagate two low-lying core modes with different phase indices and measure their interference. In this case the two values of f₁ and f₂ are both near unity, but unfortunately they are quite similar so that the overall sensitivity coefficient f = f₁ - f₂ = 0.0078 and is far from the maximal value of f = 1. Nevertheless, our simple design is able to outperform nearly all existing in-fiber interferometers and measure the refractive index of several microliter-sized gas samples with an accuracy of better than 1 ppm.

To demonstrate the capabilities of the setup we measure the refractive indices of several gases as a function of their pressure. Even polarizable gases do not exhibit a refractive index change of more than Δn_s/P = 10⁻³/bar thereby requiring great sensitivity of the interferometer setup. In particular, we measure the refractive indices of several gases at pressures from 0.07 bar to about 0.6 bar. In the near-infrared the refractive index of some molecular gases is a complicated function of wavelength, due to absorption features of overtones and combination bands and their corresponding dispersion behavior. Additionally, for some molecular gases we observe absorption phenomena in our wavelength window. Below we demonstrate that it is possible to detect refractive index variations as small as 10⁻⁷ refractive index units. These miniscule changes would be buried in thermal fluctuations for denser samples such as liquids, but in dry gases they remain detectable due to the gases’ lower thermal sensitivity of their refractive indices.

We first quantify the different terms that govern the interferometric phase of Eq. (1). This requires an explicit calculation of the eigenmodes to obtain the indices of the co-propagating core modes responsible for the interferogram and their dependence on the pressure and refractive index of the sample gas. We then describe the experimental setup and report on the results of the measurements.

2. Theory

To understand the phase shifts in the interferogram in response to gas pressure, P, and temperature, T, changes we consider the interdependence of all relevant variables in Eq. (1) above

The refractive indices of the propagated modes $n_1^{eff}$ and $n_2^{eff}$ are a function of pressure, temperature and wavelength, and the length of the PCF is also pressure- and temperature-dependent. Assuming only small and linear dependencies Eq. (1) can be written in partial derivatives

The pressure-induced length expansion was obtained using$\Delta L=LF/\left(A_{\text{fiber}}E\right)$, where L = 0.346 m is the length of the fiber and A_fiber = 3.8 × 10⁻⁸ m² is the surface area of the glass and acrylate coating. The force applied to the fiber by the gas is calculated using$F=\Delta P{A}_{\text{hole}}$ where A_hole = 3.4 × 10⁻⁹ m² is the cross sectional area of all holes in the PCF, i.e. the area of the center hole with diameter 10 µm and of 294 holes with a diameter of 3.8 µm. The Young modulus of the fiber E = 16.56 ± 0.39 GPa was previously determined for a fiber coated by a polymer jacket [9]. The contribution of the pressure-induced expansion of the PCF is ΔL/ΔP = 1.9 × 10⁻¹² m Pa⁻¹ and is negligible compared to the δn_i/δP terms.

The thermal expansion of the fiber, ∂L/∂T = α_LL = 2.1 × 10⁻⁵ m K⁻¹, was obtained from L = 0.346 m and α_L = 6.0 × 10⁻⁵ K⁻¹. The value α_L is the cross-section-weighted average of the linear thermal expansion coefficients of the hollow-core fiber material, α_silica = 5.5 × 10⁻⁷ K⁻¹ [10, 11] and its 50 μm acrylate coating, approximated with α_PMMA = 7.6 × 10⁻⁵ K⁻¹ [11, 12]. With Δn^eff = 0.0096 (see section 5) we determine ∂L/∂T (n₁^eff-n₂^eff) = 2.0 × 10⁻⁷ m K⁻¹. This value almost offsets the contribution from the difference of the thermoptic coefficients, (∂n₁^eff/∂T-∂n₂^eff/∂T)L = −2.04 × 10⁻⁷ m K⁻¹, which was calculated using the eigenmode solver with the thermooptic coefficients of Corning 7980 glass (see section 3).

The dispersion of gas and fiber material ∂n/∂λ could be neglected, since the phase shifts were obtained in wavelength windows that were as narrow as 100 pm. The term ∂n/∂P corresponds to the pressure dependence on the modes’ effective refractive index. Changing the pressure of the gas inside the PCF has a strong effect of the sample’s index, n_s, inside the holes. This index can be related to temperature and pressure as well as the gases’ polarizability, α_s, through the Lorentz-Lorenz equation [13]

3. Modelling of the eigenmodes

To model the modes within the PCF an eigenmode solver analysis program (MODE Solutions, Lumerical) was used. The PCF (HC-1550-02, NKT Photonics) was modelled as a patterned structure made from Corning 7980 glass at 23 °C having a bulk index of 1.445034 [14]. In the model the central core has a radius of 4.8 µm and the surrounding holes have radii of 1.7 µm and a pitch of 3.8 µm, corresponding to the values given by the manufacturer of the HC-PCF. The calculated five dominant core modes near 1532 nm are shown in Fig. 1 along with our scanning electron microscope image of the PCF (Quanta 250, FEI SEM). According to Aghaie et al. these modes are best labelled as “quasi-TEM_m,n modes”, but for convenience we will simply refer to them as TEM_m,n modes [15, 16].

 

Fig. 1 The top left image of a PCF (HC-1550 NKT) was obtained using a scanning electron microscope. The other images show five of the core modes simulated using the eigenmode solver analysis. The effective indices were obtained setting the hole interiors at vacuum, the bend radius to R = 30 mm, and the temperature to T = 23°C. The two polarization states of the core TEM₀₀ mode give different effective indices.

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The TEM₀₁ mode and the TEM₁₀ are not degenerate, since the fiber’s hole pattern does not have C₄ symmetry. Also, when the PCF is bent in the horizontal plane the TEM₀₁ mode is expected to be more affected than the TEM₁₀ mode. Even the TEM₀₀ mode has two polarization components that are not degenerate. We denote the horizontally polarized mode as TEM₀₀ and the vertically polarized mode as TEM₀₀*.

To compare with our experimental studies at different gas pressures, mode calculations were performed using different indices of the medium inside the holes in the range of 1.0000 to 1.0006 in steps of 10⁻⁴. The medium inside the holes was modelled as a gas with a polarizability α_s = 6.57 × 10⁻⁴⁰ J⁻¹C²m² (similar to ClF₂CH) with a temperature dependence according to Eq. (15), while the glass portion of the PCF was modelled as Corning 7980 glass with respective Sellmeier coefficients [14].

We identified the modes responsible for the interference patterns from their accumulated phase difference and their experimentally measured Δn^eff ≈0.010 (see section 5, below). It is apparent that the core modes cannot interfere with any of the cladding modes, since then the respective value for Δn^eff would be much larger.

We further assume that only modes carrying light of the same polarization can interfere, i.e. that the interferograms arise from interference of either the horizontally polarized TEM₀₀:TEM₀₁ mode pair or the vertically polarized TEM₀₀*:TEM₁₀ mode pair. The TEM₀₀:TEM₁₁ mode pair was experimentally observed but was not further investigated.

The phase index associated with these modes was calculated as a function of material index inside the holes and the PCF bend radius (Fig. 2). It is found that the phase indices of the TEM₀₀ (polarized in the bend plane) and the equivalent TEM₀₀* mode (polarized normal to bend plane) change approximately linearly with increasing refractive index of the medium inside the holes, but do not vary greatly when the fiber is bent (Fig. 2). Similarly, the TEM₁₀ mode index increases linearly with the index of the sample gas, but is largely insensitive to the bend radius. On the other hand, the index of the TEM₀₁ mode, which is polarized in the bend plane, strongly depends on the bend radius as the fiber is bent in the polarization plane. When the empty PCF is bent to 30 mm the TEM₀₁ mode index increases from n₂^eff = 0.98351 to 0.98361.

 

Fig. 2 (a) The calculated phase indices, n^eff, of the four lowest lying core modes of Fig. 1 as the sample’s index in the holes, n_s, was increased. The hollow data points are the effective index with a bend radius of 30 mm and the solid data points represent the effective index of the modes in a straight fiber. For the TEM₀₀*, TEM₀₀, and TEM₁₀ modes the phase index is nearly independent on the bend radius. (b) The phase indices, n^eff, as a function of bending radius calculated as in (a) but for n_s = 1.0.

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More importantly, the difference of the horizontally polarized TEM₀₀ and TEM₀₁ mode indices, Δn^eff = 0.0095 (Fig. 3(a)), is consistent with the experimentally obtained index difference of Δn^eff = 0.010 associated with the 1.43 nm⁻¹ peak in the Fourier transform of the interferogram (Fig. 3(a) and Figs. 5 and 6 below). It is not impossible that the vertically polarized mode pair, TEM₀₀*:TEM₁₀, is contributing to the interference pattern giving Δn^eff = 0.0088, but we have not seen experimental evidence for this effect.

 

Fig. 3 (a) The difference in phase index responsible for the interference spectrum, Δn^eff, is shown for the two modes polarized in the bending plane Δn^eff = n^eff(TEM₀₀)- n^eff(TEM₀₁) in blue and black circles and the two out-of-plane polarized modes Δn^eff = n^eff(TEM₀₀*)- n^eff(TEM₁₀) in red and green squares. (b) The sensitivity dn^eff/dn_s for the core modes as the bend radius is increased is calculated from the first derivative of Fig. 3(a). The dashed lines are meant to guide the eye. The short blue arrow corresponds to a sensitivity of the interference measurement f = Δdn^eff/dn_s = dΔn^eff/dn_s = 0.008.

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Changing the index of the material inside the holes from n_s = 1.0 to 1.0006, allows us to estimate the sensitivity of measurement depending on the pair of beating modes that is involved as well as the bend radius. Figure 3(b) shows the dependence of the sensitivity dn^eff/dn_s on the bending radius. It is apparent that the modes that have no nodes along the bend axis are hardly affected by bending the waveguide, whereas the TEM₀₁ mode that has a nodal plane along the axis of the bend is strongly affected. Especially for the TEM₀₁ mode (and at very small bend radii the TEM₁₀ mode) the sensitivities depend strongly on the bend radius. At the experimental bend radius of approximately 34 mm the TEM₀₀:TEM₀₁ mode pair shows a sensitivity of the interference measurement f = Δdn^eff/dn_s = 0.008 (see also Eq. (5)).

4. Experimental setup

The experimental setup is shown in Fig. 4. The experiments were carried out using a tuneable laser source (AQ4320D, ANDO, linewidth <1MHz and typically 200 kHz), which was coupled to a single mode fiber (SMF). The hollow-core photonic crystal fiber (HC-PCF) (L = 346 mm, HC-1550-02, NKT Photonics) was coupled to the SMF using ceramic ferrules with a small gap of a few µm to allow gas to vent in and out of the PCF. The output from the PCF was coupled into a second SMF and detected using an InGaAs amplified detector (PDA 10CS, Thorlabs) set to 10 dB. The setup required that the fiber was bent with a bend radius of about 30-35 mm.

 

Fig. 4 Experimental setup. Laser light is coupled into a single mode fibre (SMF). Using gas-permeable ferrules the light is then coupled into the hollow-core photonic crystal fiber (length 346 mm). The photodetector signal is recorded through a lock-in amplifier.

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The laser was chopped with a frequency of 300 kHz and was scanned from 1527 nm to 1537 nm with a step increment of 1 pm (about 12 MHz). At each wavelength, 15 data points were collected and averaged. The detector output was sampled by a lock-in amplifier (Mode 5202, Princeton Applied Research) that was synchronized to the laser. The lock-in amplitude was recorded using a digital data acquisition card (USB-201, Measurement Computing). While the laser was polarized in the bending plane, polarization was not actively measured or controlled.

We also performed experiments similar to those described below at nine different temperatures in a 4 K range. In these experiments we were not able to discern any thermal effect on the interferograms that was significantly above the noise level of the measurements and will therefore not report on the results, here.

5. Interferometric measurements

The in-fiber PCF interferometer of Fig. 4 has been filled with nine different gases having different polarizabilities (Table 1); for each gas interferograms were obtained at nine different pressures between 0.07 bar and 0.6 bar.To remove any residual gas the system was flushed with the new gas three times before each measurement. The system was then filled with the gas to a pressure of up to approximately 0.6 bar and allowed to settle for 20 min to ensure that the gas completely diffused along the photonic crystal fiber before measurements were taken. The volume of the PCF holes is only A_holeL = 1.2 μL. The temperature was kept at 296 ± 1 K.

Table 1. Polarizabilities of the gases investigated in this paper with references. For He, Ar, N₂, and C₂H₂ the polarizability was calculated using known dispersion and refractive indices at 1532 nm. The static average electric dipole polarizabilities for the ground state of all gases are also shown.

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Figure 5 shows normalized spectra collected for all nine gases as the pressure within the fiber was decreased. The spectra show interference patterns that are caused by the TEM₀₀ core modes of the PCF beating with several higher-order core modes. The panels in Fig. 5 are arranged in order of increasing gas polarizabilities (Table 1) and accordingly show an increase in phase shift with pressure. Ammonia and acetylene are the only two gases with obvious absorption features. As expected the transmission spectra show a linear increase of the absorption with increasing gas pressure.

 

Fig. 5 Normalized intensities of all nine gases in the order of increasing polarizability, from top left to bottom right corners.

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To quantify the pressure-dependent phase shift in the interferograms, we perform a fast Fourier transform (FFT). The delta peak was reduced by subtracting the constant background and multiplying the spectra with a cosine function, where cos(0) is placed at the start wavelength and cos(π/2) at the end wavelength. The data was padded with 20,000 extra zeros to increase the Fourier transform resolution. Figure 6 shows the resulting Fourier transform of chlorodifluoromethane and acetylene interferograms recorded at three different pressures. The peaks at 1.43 nm⁻¹ and 3.1 nm⁻¹ in the chlorodifluoromethane and acetylene interferograms correspond to two mode pairs, i.e. TEM₀₀:TEM₀₁ (Fig. 3(a)) and probably TEM₀₀:TEM₁₁. The corresponding wavelength difference can either be obtained by fitting the interference fringes of Fig. 5 using Eq. (2) or from the reciprocal values of the peaks in the Fourier transform and Eq. (3). Using Eq. (12) both methods give the effective phase index differences of Δn^eff = 0.010 and 0.021, respectively. The other peaks in Fig. 6(b) arise from harmonics and combinations in the interferogram and are multiples of the beat frequencies of 1.43, 3.1 and 1.6 nm⁻¹. Minor peaks appear to be beats of higher order core modes in the fiber. In contrast to these interference fringes the amplitude of the peak in the acetylene interferograms at 1.6 nm⁻¹ increases with pressure and corresponds to the ν₁ + ν₃ combination absorption band with a lineto-line separation of ~0.63 nm. It is apparent in the interferograms as well as in their Fourier transform that the phase of this latter peak does not change with pressure.

 

Fig. 6 Spatial frequencies obtained using a fast Fourier transform of (a) the chlorodifluoromethane and (b) the acetylene spectra. The black, red, and blue lines represent the gases at low, medium and high pressure. The inset shows the peak located at 1.43 nm⁻¹, which corresponds to the interference pattern separated by 0.69 nm as shown in Fig. 5. Acetylene has a structured absorbance band at 1527-1537 nm and this is apparent in the Fourier transform by a concentration dependent peak at 1.60 nm⁻¹.

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To investigate how the interferograms shift as a function of gas pressure, the phase of the FFT was extracted using Eq. (16) and unwrapped to remove the steps caused by the discontinuity of the tangent function:

For the prominent 1.43 nm⁻¹ peak the unwrapped phases are shown in Fig. 7(a) as a function of pressure. All nine gases show a linear dependence of the interferometric phase on the gas pressure.

 

Fig. 7 (a) Phase as a function of pressure for each of the nine gases. The slope increases with the polarizability of the gas. (b) The dependence of the effective index difference between the beating modes and the index of the gas in the holes. Δn^eff was calculated using Eq. (3) from the phase at the centre wavelength of 1532 nm. The index of the gas in the holes n_s was obtained using Eq. (15) from the pressure and polarizability (Table 1) of each gas.

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For small changes of the gas index, n_s the Lorentz-Lorenz Eq. (15) predicts an approximately linear dependence on pressure. Accordingly, the interferometric phase changes from Eq. (6) depend linearly on gas pressure and on the polarizability.

The measured sensitivities may be compared to previously reported polarizabilities of the nine gases (Table 1). Figure 7(b) uses the experimentally obtained phase shift from the Fourier analysis and Eq. (3) to obtain Δn^eff. The index of the gas inside the holes, n_s, was calculated from the experimental pressure, the previously reported polarizabilities and the Lorentz-Lorenz Eq. (15). We expect all data to fall on the same straight line. Its slope corresponds to the sensitivity coefficient, f, and is expected to be between 0.05 and zero depending on the exact fiber bend radius (see section 3). The slopes in Fig. 7(b) vary in the range of f = 0.022 ± 0.003 for helium to 0.0052 ± 0.0003 for acetylene with an average value near f = 0.0078 for all nine gases. For comparison, if one assumes a phase shift corresponding to an “effective bend radius” of 33.5 mm, the calculated sensitivity f = 0.008 (Fig. 3(b)) is very close to that experimentally observed.

The variation of the observed sensitivity coefficients, f, may point to inadequacies of the reported polarizabilities in predicting the refractive index at 1532 nm. In addition, absorption features in acetylene and ammonia cause strong wavelength-dependent index changes [28] and for most gases the polarizability was obtained from an average value in the infrared region of the spectrum. It is also possible that the variation of the sensitivity coefficients observed for the different gases points towards our difficulty to control the polarization states in our experiment given that the fiber is bent and may also be under torsion.

The smallest detectable refractive index change at the 1σ level can be calculated from the calibration curves following [29] and is shown for 4 typical gases to be 0.5-1.0 × 10⁻⁶ (Fig. 8). This corresponds to a minimal detectable phase change of δΔϕ ≈0.1 rad. The sensitivity coefficients of the measurement which depends on the sensitivity of the phase indices of the beating modes towards the refractive index of the sample was calculated as f = 0.008 for a bend radius of 33.5 mm and was experimentally measured to be f = 0.0078.

 

Fig. 8 Interferometric phase measured for helium, neon, nitrogen and ClF₂CH at different pressures (from Fig. 7(a)). The corresponding Δn^eff was calculated using Eq. (6) from the phase at the centre wavelength of 1532 nm. The index of the gas in the holes n_s was obtained from the pressure and polarizability (Table 1) of each gas from the pressure using Eq. (15). The minimal detectable gas refractive index change δn_s = 0.5-1.0 × 10⁻⁶ is indicated by horizontal arrows and corresponds to the width of the 1σ−confidence interval (95%) [29].

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6. Discussion and concluding remarks

Figure 9 shows a calibration curve where the instrument-specific sensitivity coefficient f = 0.0078 (the average f value for all 9 gases) was used to correlate the pressure-corrected interferometric phase against previously measured gas polarizabilities (Table 1). Such a calibration curve may be used to determine the refractive index of an unknown gas at known pressure.

 

Fig. 9 Experimental calibration to determine the polarizability from the phase-pressure dependence. The black squares and red circles represent the static average electric dipole polarizabilities for the ground state and the true polarizability at 1532 nm calculated from dispersion, respectively. The line is a linear fit of the data with a slope and intercept of 6.00 ± 0.84 × 10⁻⁴¹ J⁻¹C²m²bar and −0.4 ± 4.9 × 10⁻⁴¹ J⁻¹C²m², respectively.

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These sensitivity coefficients f permit the calculation of the figure of merit, FoM = 4fL/λ = 7050, calculated from Eq. (11) and with f = 0.0078. This figure of merit implies that an interference fringe shifts by dΔϕ = π/2 when the sample’s index changes by dn_s = 1/7050 = 1.4 × 10⁻⁴. In more conventional terms the sensitivity of an attenuation minimum to the gases index change is 1233 nm/RIU. While these are respectable results by any measure, they are not optimal. The sensitivity, f, and accordingly the FoM and detection limit could all be improved, if the two modes that are beating were not both core modes, i.e. if they were not both propagating to a similar extent through the sample gas. Note that the sensitivity factor f = 0.78% is far below the optimal value of 100% expected for two modes in which one travels exclusively through the sample and the other one travels through material that is not affected by the sample medium. Such a setup would require a balanced in-fiber Mach-Zehnder interferometer consisting of one mode traveling entirely through the hollow core, which also contains the sample gas, and beating against a single second mode that propagates through, for example, a solid glass core contained in the same fiber. To our knowledge such a fiber does not exist, yet.

One might propose that a hollow-core PCF operating at longer wavelength, when it supports only a single core mode might act as such a Mach-Zehnder-interferometer. Here the second mode would be a cladding mode. However, it would be difficult to set up such an instrument as it would require filling the hole with an inert gas of low index (ideally helium) sealing the holes and then launching a single cladding mode of well-defined phase index together with the core mode. Alternatively, one can devise an unbalanced MZ-interferometer similar to the Young interferometer described by Shavrin et al. [28]. Here the two modes traveled through approximately identical lengths of HC-PCF and regular SMF and were beating when they recombined. The differential response to temperature and birefringence would likely make it difficult to increase the experimental sensitivity, however.

Almost all previous variants of in-fiber interferometers are designed to measure the refractive indices of liquid samples, n_s [24, 30–32]. Typically, one of the interfering modes is the core mode of a single mode fiber with effective (phase) index $n_1^{eff}$ which does not interact with the sample (the reference arm). In the sensing arm one propagates either a single cladding mode or the superposition of many similar cladding modes, with effective index $n_2^{eff}$, that may interact with the sample through the evanescent field. The interaction may be enhanced by, e.g., the use of tapers [33] or field access blocks [1–3, 34]. Usually, the sensitivity is reported as an interference wavelength shift with sample index. Typical values are dλ/dn_s ≈25 nm/RIU for 38 and 62 mm tapers [3], 200 nm/RIU for a 12 mm taper [33], and 1600 nm/RIU for a 0.66 mm taper [35]. The sensitivity coefficients may then be estimated with Eq. (9) and are between f = 7 × 10⁻⁵, 5 × 10⁻⁴ and 4 × 10⁻², showing that it is difficult to obtain strong interaction of the propagated modes in the sensing arm with the sample liquid.

Our setup is quick to construct and simple to use. The interpretation of the interferograms requires no more than a phase analysis using Fourier transforms. The setup is uniquely suited to measure the refractive index of small gas volumes of about a microliter (1.2 μL in our case), and could be adapted to measurements of liquid indices. It is nevertheless limited to relative measurements of refractive index, i.e. the interferometric phase at a given partial pressure has to be compared to either the phase at vacuum, or the phase with a different partial pressure of the same gas, or to that of a reference gas at the same pressure. This is a common limitation of interferometric measurements, however. We note that the sensitivity might simply be increased by careful control of the bending radius and polarization as Fig. 3(b) indicates. Both values may be improved by decreasing the bending radius beyond 30 mm and launching light that is polarized in the bending plane. Trivially, it is also possible to enhance the sensitivity by simply increasing the length of the fiber interferometer.