*5.3. MPD PTS in ARHCF*

In the majority, the multimode characteristic of the ARHCFs is not desired in the sensor configurations where the intermodal interference induces a significant increase in the baseline noise in the sensor. However, with a specific sensor design and signal retrieval method, the few-moded behavior of the ARHCF can be very beneficial, especially in combination with the PTS.

Zhao et al. proposed in [25] a novel approach to the PTS technique, the so-called modephase-difference photothermal spectroscopy (MPD-PTS), which measures the PT-induced differential phase change of the LP01 and LP11 modes of the *probe* beam propagating through the gas-filled ARHCF core, which can be expressed by [25]:

$$\delta\varphi = \Lambda\varphi\_{01} - \Lambda\varphi\_{11} = \mathbf{k}^\*(\mathfrak{n}\_\nu \mathbf{f}) \left( 1 - \mathbf{e}^{-\alpha(\lambda\_{\text{pump}}) \cdot \mathbf{CL}} \right) \mathbf{P}\_{\text{pump}} \approx \mathbf{k}^\*(\mathfrak{n}\_\nu \mathbf{f}) \mathbf{a}(\lambda\_{\text{pump}}) \mathbf{CLP}\_{\text{pump}}.\tag{11}$$

where Δϕ<sup>01</sup> and Δϕ<sup>11</sup> are the phase modulation for LP01 and LP11 modes of the *probe* light, respectively, k\* is the differential phase modulation coefficient, which is a function of the fractional *pump* power in the LP01 mode—η and sinewave modulation frequency—f of the *pump* light, α(λpump) is the absorption coefficient for the relative concentration of 100%, C corresponds to the target gas concentration, L is the ARHCF length and Ppump defines the average *pump* beam power over the fiber length. In principle, the *pump* light transmitted in the gas-filled ARHCF core in the LP01 and LP11 mode heats the gas sample in a way corresponding to the intensity distribution of each mode as shown in Figure 11a,b. This results in the periodically varying RI modulation following the temperature change trend over the entire core length as presented in Figure 11c. This leads to the coherent mixing of the LP01 and LP11 *pump* modes with the spatial period corresponding to their beat length [25]. If the *probe* beam is simultaneously coupled to the fundamental mode and the higher-order mode, it experiences the RI modulation induced by the corresponding *pump* light modes. This results in the different phase modulation of each *probe* mode, which can be detected in a sensor configuration based on an in-line dual-mode interferometer as presented in Figure 11d. Here, the hollow-core fiber can act as an absorption cell, which can be filled with the gas analyte through the gaps between its input/output end facets and the SMFs placed at both ends or via a laser-drilled microchannel along its length. The *pump* and *probe* beams were coupled into the ARHCF using the butt coupling approach from SMF, which provided excitation of the LP01 and LP11 modes. The interferometer cavity was implemented in the setup shown in Figure 10e, which enabled C2H2 sensing based on the 2f signal readout. The gas molecules were *pumped* at 1532.83 nm using a DFB laser and the induced RI modulation was *probed* with an ECDL operating at 1550 nm. The ARHCF forming absorption cell was designed to operate in the ~1.5 μm wavelength band. The length of the fiber was 4.67 m with an air core diameter of 28 μm. The WDM couplers provided perfect separation of the *probe* beam from the *pump* light before directing it to the photodetector and subsequently to a lock-in amplifier for 2f signal demodulation. The authors have demonstrated that the MPD-PTS sensor can reach an MDL at the level of 15 parts-per-trillion by volume (pptv) for 100 s integration time with an NEA of 1.6 × <sup>10</sup>−<sup>11</sup> cm<sup>−</sup>1. Furthermore, the long-term stability tests of the developed sensor confirmed its excellent robustness, greater in comparison to the ARHCF-aided sensors utilizing the MZI setup.

**Figure 11.** Principle of the MPD-PTS in ARHCF for gas sensing. (**a**) Intensity profiles of the *pump* LP01 *ψ*2 01 and LP11 *ψ*2 11 modes in the gas-filled ARHCF. (**b**) *Pump* intensity change within the modal beat length. (**c**) The temperature change, which induces the RI modulation in the gas-filled ARHCF core while the gas molecules are excited by the *pump* light. It can be seen that the *probe* modes LP01 Ψ2 01 and LP11 Ψ2 11 are experiencing different RI changes. (**d**) A schematic of the in-line dual-mode interferometer. (**e**) Experimental setup of the MPD-PTS sensor utilizing an ARHCF as a C2H2 absorption cell. HCF—hollow-core fiber, SMF—single-mode fiber, Ib—modal beat length, WDM—wavelength division multiplexer, PC—polarization controller, ECDL—external cavity diode laser, TF—tunable optical filter, EDFA—erbium-doped fiber amplifier, LIA—lock-in amplifier, DAQ—data acquisition card Reprinted from [25] with permission from Springer Nature.

The calculations performed by Zhao et al. in [22] have indicated that the differential phase modulation in an MPD-PTS gas sensor reaches its maximum if the *pump* light power is coupled into the LP01 mode. However, at the same time, the *probe* light must be transmitted within the LP01 and LP11 modes simultaneously so the induced RI modulation by the heated inside ARHCF gas molecules can introduce the phase difference between these modes, which can be subsequently analyzed. Excitation of the LP01 *pump* mode only and two *probe* modes at the same time is not trivial and almost impossible to realize by offsetting the SMFs with respect to the ARHCF core as it was reported by the Authors in their previous work [25]. According to this, the MPD-PTS gas sensor performance did not reach its maximum. To address this problem the Authors inscribed a long period grating (LPG) in a negative curvature – hollow core fiber (NC-HCF) forming an absorption cell as shown in Figure 12. The LPG enabled the excitation of the LP01 and LP11 modes at the *probe* wavelength maintaining *pump* light transmission in the LP01 mode only. This modification of the sensor resulted in the maximization of the differential phase modulation induced by the photothermal effect and reduced the complexity of the sensor setup. The approach was tested in an experimental configuration of an MPD-PTS detector targeting C2H2 at 1532.83 nm in an 85 cm long NC-HCF with a core size of ~30 μm. The induced RI modulation was *probed* at 1620 nm. The setup of the sensor and measurement procedure was similar to the described above. The sensor reached an MDL of 600 pptv at 100 s integration time with an NEA of 6.3 × <sup>10</sup>−<sup>10</sup> cm−1, maintaining a less complex configuration than reported in [25].

**Figure 12.** Schematic of the absorption cell formed by an NC-HCF with LPG inscribed in it. The absorption cell forms an in-line dual-mode interferometer. The *probe* and *pump* beams were delivered to and outcoupled from the NC-HCF using the butt-coupling approach with SMFs. Reprinted with permission from [22] © The Optical Society.

Zhao et al. reported in [53] an MPD-PTS CH4 sensor utilizing an ARHCF with an inscribed LPG. The CH4 molecules were *pumped* at ~1653.7 nm with the aid of a DFB laser, which was additionally amplified using a fiber-based Raman amplifier to maximize the photothermal effect. The *probe* light at 1550 nm was delivered from a fiber laser equipped with a wavelength stabilization unit. The absorption cell was realized based on a 2.4 m long ARHCF with a core size of ~30 μm, which was characterized by an attenuation of 0.16 dB/m and 0.25 dB/m at the *probe* and *pump* wavelengths, respectively. The sensor setup was built in a configuration similar to the one shown in Figure 11e and the spectroscopic signal retrieval was based on the 2f signal demodulation. In the proposed configuration, a part of the ARHCF forming the absorption cell was placed inside a column oven to investigate the influence of the temperature change on the induced differential phase modulation of the *probe* beam. The performed experiments indicated that the operating point of the dualmode interferometer (the quadrature point of the interference fringe) is highly sensitive to the temperature deviations, hence a proper compensation of the temperature drift is necessary to improve the sensor stability. It was possible to minimize the photothermal signal variations from ~9.4% to ~2.1% using a linear temperature compensation scheme and obtain an MDL of approximately 4.3 ppbv for 100 s integration time. The NEA coefficient reached the level of 1.6 × <sup>10</sup>−<sup>9</sup> cm<sup>−</sup>1.
