High Sensitivity Cryogenic Temperature Sensors Based on Arc-Induced Long-Period Fiber Gratings

In this paper, we investigated the evolution of the dispersion curves of long-period fiber gratings (LPFGs) from room temperature down to 0 K. We considered gratings arc-induced in the SMF28 fiber and in two B/Ge co-doped fibers. Computer simulations were performed based on previously published experimental data. We found that the dispersion curves belonging to the lowest-order cladding modes are the most affected by the temperature changes, but those changes are minute when considering cladding modes with dispersion turning points (DTP) in the telecommunication windows. The temperature sensitivity is higher for gratings inscribed in the B/Ge co-doped fibers near DTP and the optimum grating period can be chosen at room temperature. A temperature sensitivity as high as −850 pm/K can be obtained in the 100–200 K temperature range, while a value of −170 pm/K is reachable at 20 K.


Introduction
Systems operating at cryogenic temperatures are becoming increasingly important in the energy sector, transportation, medicine technology and many other fields (high energy physics, military, aerospace, etc.) [1][2][3]. Some applications also use superconducting magnets and high magnetic fields requiring electromagnetic shield and heat transfer control of their monitoring devices [4][5][6]. The standard equipment used in these environments may exhibit critical drawbacks [7,8] and, therefore, researchers have been looking for the intrinsic advantages of using optical fiber sensors [9,10]. Recently, different approaches have been tested ranging from distributed sensing, based on Raman, Rayleigh and Brillouin backscattering, to the use of interferometric and other wavelength-selective devices [11][12][13]. Nevertheless, fiber Bragg gratings (FBGs) have been the most explored, despite showing low-temperature sensitivity below 100 K [14,15]. For this reason, several techniques have been applied in order to enhance their thermal sensitivity, such as the deposition of metal or sol-gel coatings on the fiber cladding or by embedding or bonding them to substrates with very different thermal expansion coefficients [16][17][18][19]. When compared to conventional FBGs, the fabrication of modified FBGs is time consuming and the temperature sensitivity improvement is limited [20]. On the other hand, it is known that long-period fiber gratings (LPFGs) possess higher temperature sensitivity, typically an order of magnitude higher for temperatures above room temperature [21]. However, research involving the use of LPFGs at cryogenic temperatures is very scarce [22][23][24], possibly due to two main factors: they need access to both sides of the fiber, and they are also very sensitive to bending. In 2015 [25], we solved the problem by using a phase-shift LPFG (PS-LPFG) working in reflection placed inside a polyamide capillary with a 1 mm inside diameter. The results obtained were considerably better than for other fiber-grating sensors, but still limited at temperatures close to 0 K. In 2016 [26], we proved that it is possible to fabricate LPFGs with grating periods as short as 148 µm using the electric arc discharge technique. This allows the fabrication of LPFGs in the dispersion turning points, the region where they exhibit the highest sensitivity to changes in physical parameters such as temperature. This paper investigates, for the first time, to the best of our knowledge, the temperature dependence of the dispersion curves in order to find the optimum feasible grating period that allows the increase of the temperature sensitivity at cryogenic temperatures. For that goal, we performed computer simulations based on the experimental data that we obtained for PS-LPFGs inscribed in the SMF-28 Corning fiber and also on the B/Ge co-doped PS1250/1500 Fibercore fiber.

Long-Period Fiber Grating Structure
LPFGs are periodic structures with grating periods ranging from 100 µm up to 1 mm. Their transmission spectra exhibit a number of dips at specific wavelengths that satisfy the resonance condition, corresponding to coupling of the core mode to different copropagating cladding modes [27]. In this paper, gratings were arc-induced in a standard single-mode telecommunication GeO 2 -doped silica fiber, the SMF-28 fiber from Corning, and in two batches of a photosensitive fiber from Fibercore, the PS 1250/1500 GeO 2 -B 2 O 3 co-doped silica fiber. The first, with a core radius of 4.3 µm and a cut-off wavelength of 1.31 µm [28], results in a refractive index difference of~5 × 10 −3 . The two Fibercore fibers have the following parameters: B/Ge#1 (NA = 0.13, MFD = 9.6 µm, λ cut-off = 1.15 µm) and B/Ge#2 (NA = 0.14, MFD = 8.9 µm, λ cut-off = 1.24 µm), leading to a core radius of~3.4 µm and a refractive index difference of~6 × 10 −3 and~7 × 10 −3 , respectively.
The gratings were arc-induced [29] in those fibers using two different high-voltage power supplies: a homemade one [26] (Figure 1) that enables the writing of LPFGs in the dispersion turning points (DTP), and a commercial fusion splice machine (BICC AFS130) [30] used typically to fabricate gratings above 400 µm. The gratings used in 2015 for cryogenic temperature measurements [25] were produced in the SMF-28 fiber and in the B/Ge #1 fiber using the fabrication parameters: grating period Λ, electric current I, number of arc discharges N, arc duration ∆t and pulling weight w, presented in Table 1. temperature [21]. However, research involving the use of LPFGs at cryogenic temperatures is very scarce [22][23][24], possibly due to two main factors: they need access to both sides of the fiber, and they are also very sensitive to bending. In 2015 [25], we solved the problem by using a phase-shift LPFG (PS-LPFG) working in reflection placed inside a polyamide capillary with a 1 mm inside diameter. The results obtained were considerably better than for other fiber-grating sensors, but still limited at temperatures close to 0 K. In 2016 [26], we proved that it is possible to fabricate LPFGs with grating periods as short as 148 μm using the electric arc discharge technique. This allows the fabrication of LPFGs in the dispersion turning points, the region where they exhibit the highest sensitivity to changes in physical parameters such as temperature. This paper investigates, for the first time, to the best of our knowledge, the temperature dependence of the dispersion curves in order to find the optimum feasible grating period that allows the increase of the temperature sensitivity at cryogenic temperatures. For that goal, we performed computer simulations based on the experimental data that we obtained for PS-LPFGs inscribed in the SMF-28 Corning fiber and also on the B/Ge co-doped PS1250/1500 Fibercore fiber.

Long-Period Fiber Grating Structure
LPFGs are periodic structures with grating periods ranging from 100 μm up to 1 mm. Their transmission spectra exhibit a number of dips at specific wavelengths that satisfy the resonance condition, corresponding to coupling of the core mode to different copropagating cladding modes [27]. In this paper, gratings were arc-induced in a standard single-mode telecommunication GeO2-doped silica fiber, the SMF-28 fiber from Corning, and in two batches of a photosensitive fiber from Fibercore, the PS 1250/1500 GeO2-B2O3 co-doped silica fiber. The first, with a core radius of 4.3 μm and a cut-off wavelength of 1.31 μm [28], results in a refractive index difference of ~5 × 10 −3 . The two Fibercore fibers have the following parameters: B/Ge#1 (NA = 0.13, MFD = 9.6 μm, λcut-off = 1.15 μm) and B/Ge#2 (NA = 0.14, MFD = 8.9 μm, λcut-off = 1.24 μm), leading to a core radius of ~3.4 μm and a refractive index difference of ~6 × 10 −3 and ~7 × 10 −3 , respectively.
The gratings were arc-induced [29] in those fibers using two different high-voltage power supplies: a homemade one [26] (Figure 1) that enables the writing of LPFGs in the dispersion turning points (DTP), and a commercial fusion splice machine (BICC AFS130) [30] used typically to fabricate gratings above 400 μm. The gratings used in 2015 for cryogenic temperature measurements [25] were produced in the SMF-28 fiber and in the B/Ge #1 fiber using the fabrication parameters: grating period Λ, electric current I, number of arc discharges N, arc duration Δt and pulling weight w, presented in Table 1.   On the other hand, to achieve gratings in the DTP, we used the homemade high-voltage power supply and the fabrication parameters presented in Table 2 [26]. It should be stressed that under our fabrication conditions, coupling occurs to LP 1j cladding modes in the SMF-28 fiber and to LP 0j cladding modes in the B/Ge co-doped fibers [31][32][33]. Moreover, depending on the fabrication conditions, a geometric modulation of the fiber cross section may occur, which is particularly noticed for high pulling weight [34]. It is also accompanied with the creation of new stresses in the fiber [35]. On the other hand, for low pulling weight, the annealing of intrinsic stresses/viscoelastic stresses is observed, which in turn leads to an increase of the cladding refractive index of the order of 1 × 10 −4 [36]. This value is compatible with the fact that, for the same grating period, the resonances of arc-induced gratings in the SMF-28 fiber are shorter by~50 nm (depending on the cladding modes) than the ones belonging to mechanically induced gratings [37]. A small decrease of the core refractive index may also occur [38,39]. Furthermore, the dimensions of the arc-discharge, its electric current and duration will impact the length of the region affected by the arc. Thus, the duty cycle of the index modulation will decrease as the grating period increases. Therefore, the effects of the arc-discharge need to be considered in the computer simulations.

Refractive Index of the Fiber Structure
For calculating the spectra of a LPFG, we need to define the refractive indices of the fiber structure: first, the core and cladding refractive indices for the step index fiber or the continuous distribution of the refractive index along the radial direction for fibers with gradient profiles; second, the modulation of the refractive index along the fiber that creates the grating. In contrast to a fiber Bragg grating, precise knowledge of the difference between the core and cladding refractive indices is required due to the fact that the resonance condition for the LPFG involves a difference between the propagation constants of the core (β co ) and cladding modes (β cl ): It may be difficult to know the refractive indices, since their difference is determined by the concentration of various dopants having different wavelength dispersion. The material dispersion of doped silica is also important, because the LPFG spectrum covers a broad range of wavelengths.
Thus, we should construct the dependence of the refractive index of silica on four parameters: wavelength, germanium concentration, boron concentration, and temperature. The ranges are 0.9-1.7 µm for wavelength, 0-10% for germanium concentration, 0-20% for boron oxide concentration, and 0-300 K for temperature. This dependence is a fourdimensional function. There is limited information on this function in the literature, which provides different slices for some parameters to be fixed. For example, the dispersion of germanium-and boron/germanium-doped silica is known only at room temperature [40,41], but the thermo-optic coefficient is only measured for pure silica [42], etc.
In general, as far as fused silica is concerned, the Sellmeier equation [43] is used, which empirically relates the refractive indices as a function of wavelength: In this equation, the Sellmeier coefficients (D s,j and λ s,j ) found in [43] describe the refractive index of pure fused silica at room temperature. The fiber core doped with various dopants has a modified refractive index. In this paper, we consider germaniumand boron/germanium-doped fibers. For these types of doping, we can use Sellmeier's coefficients found by Fleming [40,41]: In Equation (3), we used corrected coefficients for boron oxide that were obtained from D B,j = D G,j + (D B,j − D G,j )/0.133, and λ B,j = λ G,j + (λ B,j − λ G,j )/0.133, since the initial coefficients D B,j were obtained for boron-doped silica with a concentration of 13.3%. So, if we calculate the refractive index of silica with a boron oxide concentration of 13.3%, we obtain exactly the refractive index measured. The numerical values of the coefficients are given in Table 3. The indices S, G and B in the coefficients stand for silica, germanium and boron. We assume that the changes in the refractive index induced by temperature are small; therefore, we can employ an approximation: where T 0 is the room temperature, η = dn/dT is the thermo-optic coefficient of silica, and c Ge , c B are the concentrations of germanium and boron dopants. The thermo-optic coefficient can be, in principle, wavelength dependent; however, experimental measurements of the dispersion of pure silica as a function of temperature presented in [42] have shown that this dependence is weak and can be neglected (measurements show that the value is 8.2 × 10 −6 in the third telecommunication window and the relative difference is within 2% from 1.0 µm up to 1.7 µm).
In order to determine the thermo-optic coefficient of pure silica, we have used Sellmeier's equation with temperature-dependent coefficients to generate the wavelengthdependent refractive indices valid for wavelengths between 0.4 µm and 2.6 µm and for the temperature range from 30 K up to 310 K [42]. To assess the results, we calculated the Sellmeier's coefficients for room temperature (295 K) and we compared the refractive index values with the ones obtained by Fleming [44] for quenched SiO 2 fibers, and the difference is better than 6 × 10 −5 for the wavelength range from 1.0 µm up to 1.7 µm. Note that the values achieved are slightly higher (<4 × 10 −4 ) than those obtained using the coefficients presented by Malitson [43]. After validation, we fitted the refractive index as a function of temperature by a 6th order polynomial for a wavelength of 1.55 µm: where the temperature is measured in Kelvin. There is no linear term in this sum, because it is known that the thermo-optic coefficient goes to zero as the temperature approaches 0 K [45][46][47]. The curve has a parabola shape at 0 K. This is experimentally corroborated by the fact that gratings are essentially insensitive to temperature changes near 0 K [15,23,25,48]. Thus, we extrapolated the refractive indices down to 0 K and the results show that the equation obtained by Leviton and Frey may be applied to lower temperatures. Through derivatives, we obtained the thermo-optic coefficient: ficients presented by Malitson [43]. After validation, we fitted function of temperature by a 6th order polynomial for a wavele

40
where the temperature is measured in Kelvin. There is no linear it is known that the thermo-optic coefficient goes to zero as the t K [45][46][47]. The curve has a parabola shape at 0 K. This is experi the fact that gratings are essentially insensitive to tempera [15,23,25,48]. Thus, we extrapolated the refractive indices dow show that the equation obtained by Leviton and Frey may be a tures.
Through derivatives, we obtained the thermo-optic coeffici The problem of calculating the LPFG spectra is that the wav resonances with temperature is determined by the difference be coefficients of pure and doped silica of the cladding and the cor reconstruct this dependence from cryogenic experiments with duced in germanium and boron/germanium fibers [49]. Howeve temperatures studied in that work above 77 K, but important p fibers used and also to the gratings inscribed are missing and, t the results would be insufficient.
It is well known that, at room temperature, the thermal exp neglected in comparison to the thermo-optic coefficients. Howe the thermo-optic coefficients and the thermal expansion coefficie of magnitude, and the latter may even become negative [45,50] coefficient depends on the fiber composition and also on its th depends on its fictive temperature, which is different for the core For the SMF-28 fiber, a germanium-doped fiber, by knowi additive model [52][53][54] to determine the concentration of GeO2 to GeO2 glass, the physical parameters used are from ref. [55]. After meier's equation for a binary glass with coefficients obtained b GeO2 glass and the ones obtained in [42] for pure silica glass. Th firmed by applying the same model to the calculus of the refracti based glasses under different concentration values. The problem of calculating the LPFG spectra is that the wavelength shift of the LPFG resonances with temperature is determined by the difference between the thermo-optic coefficients of pure and doped silica of the cladding and the core, respectively. We may reconstruct this dependence from cryogenic experiments with fiber Bragg gratings induced in germanium and boron/germanium fibers [49]. However, not only is the range of temperatures studied in that work above 77 K, but important parameters related to the fibers used and also to the gratings inscribed are missing and, therefore, the accuracy of the results would be insufficient.
It is well known that, at room temperature, the thermal expansion coefficient can be neglected in comparison to the thermo-optic coefficients. However, at low temperatures the thermo-optic coefficients and the thermal expansion coefficients are of the same order of magnitude, and the latter may even become negative [45,50]. The thermal expansion coefficient depends on the fiber composition and also on its thermal history, that is, it depends on its fictive temperature, which is different for the core and cladding region [51].
For the SMF-28 fiber, a germanium-doped fiber, by knowing the NA, we used the additive model [52][53][54] to determine the concentration of GeO 2 to be~3.2 mol%. For pure GeO 2 glass, the physical parameters used are from ref. [55]. Afterwards, we used the Sellmeier's equation for a binary glass with coefficients obtained by Fleming [40] for pure GeO 2 glass and the ones obtained in [42] for pure silica glass. The results were also confirmed by applying the same model to the calculus of the refractive index of binary silica-based glasses under different concentration values.
In this work, we used the results on wavelength shifts for LPFGs in germaniumand boron/germanium-doped fibers [25] to obtain the difference between the thermooptic coefficients of pure and doped silica. Figure 3 shows the refractive index difference between the doped and pure silica as a function of temperature for SMF-28 (Corning) and B/Ge-doped fiber (Fibercore). Following the same reasoning as for the refractive index dependence, there is no linear term in the polynomial expansion on temperature: where the parameters have the following values for the two types of fibers: One more factor that influences the refractive index change of an o temperature is thermal expansion. For the temperature dependence of th sion coefficient of pure silica, inside their validity intervals, the expressio et al. [50] yields a value of 5.0 × 10 −7 K −1 at room temperature. Using the [56], we can estimate the thermal expansion coefficient for the core of leading to a value of 8.9 × 10 −7 K −1 .
The core and the cladding have different thermal expansion coefficie deformed inhomogeneously with temperature. The solution of the probl it is too cumbersome, its effect is small, and the thermal expansion coef silica are unknown. Moreover, when we use the refractive index differen ure 3, we already take the thermal expansion effect on the refractive ind Therefore, we do not specifically consider thermal expansion in our calc tive index. The temperature behavior of FBGs and LPFGs can be used in estimate the thermal expansion coefficient of the fiber core and cladding

Dispersion Curves of LPFGs
We start our analysis of LPFGs by calculating the dispersion curve B/Ge fibers at room temperature. The structures of both fibers are assume index profile with several concentric silica layers containing different do tive index of each layer is calculated using the procedure described in the During the fabrication of LPFGs by arc discharges, the structure of is changed: the cladding and core diameters are reduced and their refra changed due to the modification of the internal stress distribution. There some of the fiber parameters to obtain the best fitting of simulated disp the experimentally measured curves.
In order to follow the procedure regarding the simulations, Figure 4 SMF-28: One more factor that influences the refractive index change of an optical fiber with temperature is thermal expansion. For the temperature dependence of the thermal expansion coefficient of pure silica, inside their validity intervals, the expression given by Okaji et al. [50] yields a value of 5.0 × 10 −7 K −1 at room temperature. Using the additive model [56], we can estimate the thermal expansion coefficient for the core of the SMF-28 fiber leading to a value of 8.9 × 10 −7 K −1 .
The core and the cladding have different thermal expansion coefficients, and they are deformed inhomogeneously with temperature. The solution of the problem is known, but it is too cumbersome, its effect is small, and the thermal expansion coefficients of doped silica are unknown. Moreover, when we use the refractive index difference shown in Figure 3, we already take the thermal expansion effect on the refractive index into account. Therefore, we do not specifically consider thermal expansion in our calculation of refractive index. The temperature behavior of FBGs and LPFGs can be used in future studies to estimate the thermal expansion coefficient of the fiber core and cladding.

Dispersion Curves of LPFGs
We start our analysis of LPFGs by calculating the dispersion curves of SMF-28 and B/Ge fibers at room temperature. The structures of both fibers are assumed to have a step-index profile with several concentric silica layers containing different doping. The refractive index of each layer is calculated using the procedure described in the previous section.
During the fabrication of LPFGs by arc discharges, the structure of the pristine fiber is changed: the cladding and core diameters are reduced and their refractive indices are changed due to the modification of the internal stress distribution. Therefore, we adjusted some of the fiber parameters to obtain the best fitting of simulated dispersion curves to the experimentally measured curves. The following parameters for the SMF-28 fiber have been found to produce fit to the experimental data: co 4.3 r = μm, cl 62.5 r = μm, and Ge 3.2% c = . The mental data points were taken from previous studies: the blue dots (s1) belong to g fabricated using the AC high-voltage power supply [26] and the red triangles (s2 the BICC AFS130 fusion splicer [30]. The simulation results are shown in Figure 5 lines. The resonance wavelengths of the LPFGs are shown depending on the gra riod for LP1j cladding modes.  The following parameters for the SMF-28 fiber have been found to produce the best fit to the experimental data: r co = 4.3 µm, r cl = 62.5 µm, and c Ge = 3.2%. The experimental data points were taken from previous studies: the blue dots (s1) belong to gratings fabricated using the AC high-voltage power supply [26] and the red triangles (s2) using the BICC AFS130 fusion splicer [30]. The simulation results are shown in Figure 5 by solid lines. The resonance wavelengths of the LPFGs are shown depending on the grating period for LP 1j cladding modes.
In general, the resonance wavelengths increase with the grating period, while the slope of the curve grows with the mode number. For mode numbers greater than 10, the curve becomes close to a vertical line with some curvature resulting in a two-valued function of wavelength on the period. The apex of the curve is the so-called "turning point". The first turning point appears for an LPFG with a period of 225 µm at 1.46 µm for the LP 1,10 mode. The next modes have turning points at lower wavelengths.
The LP 11 mode is a special case. The dispersion curve of this mode bends in the opposite direction for a period of 470 µm at 1.05 µm. This happens because the fiber becomes a two-mode waveguide, and the LP 11 mode is transferred to the core. When we move to shorter wavelengths, the period of the electromagnetic field becomes shorter and one more period can fit in the core. The number of modes in a waveguide is determined by the V number: V = 2πr co N A/λ. Here, NA is the numerical aperture, and r co is the core radius. Single-mode propagation is obtained when V < 2.4. With decreasing wavelength, V becomes greater than 2.4, and the fiber can guide two modes. Thus, the LP 11 mode is transferred to the core at 1.05 µm. In fact, all cladding modes have the same behavior, but at much shorter periods and wavelengths, when they become core modes. In general, the resonance wavelengths increase with the gratin slope of the curve grows with the mode number. For mode numbers curve becomes close to a vertical line with some curvature resulting in tion of wavelength on the period. The apex of the curve is the so-cal The first turning point appears for an LPFG with a period of 225 μm LP1,10 mode. The next modes have turning points at lower wavelengt The LP11 mode is a special case. The dispersion curve of this mo posite direction for a period of 470 μm at 1.05 μm. This happens becau a two-mode waveguide, and the LP11 mode is transferred to the core shorter wavelengths, the period of the electromagnetic field becom more period can fit in the core. The number of modes in a wavegui the V number: Here, NA is the numerical aper core radius. Single-mode propagation is obtained when V < 2.4. Wi Boron-germanium-doped fiber is another type of fiber that is used for the inscription of LPFGs. Arc-induced gratings in this fiber produce prevailing symmetric perturbations in the core and excite the LP 0j cladding modes [31]. These modes have somewhat different dispersion curves due to other mode symmetry and significantly different parameters of the B/Ge fiber itself. The dispersion curves for LP 0j cladding modes of two B/Ge fibers are demonstrated in Figure 6: (a) B/Ge#1 and (b) B/Ge#2. The following parameters for the fiber were used to obtain the best fit of the experimental data: B/Ge#1-r co = 3.8 µm, r cl = 64.6 µm, c Ge = 8.37%, c B = 18.9%; B/Ge#2-r co = 2.9 µm, r cl = 62.5 µm, c Ge = 9.9%, and c B = 18.9%. The experimental data points were taken from previous studies [26]. The dispersion curves for the B/Ge#1 fiber lie lower than for the B/Ge#2 fiber, and resonances of modes LP 0j with j ≥ 6 were experimentally measured, while modes with j starting from 1 are seen for the B/Ge#2 fiber. Three modes with j = 11, 12, 13 have dispersion turning points in the presented range for B/Ge#1 fiber at wavelengths 1.54, 1.46, and 1.38 µm, respectively, and one dispersion turning point for the mode with j = 12 for the B/Ge#2 fiber at 1.54 µm.

Sensitivity and Resolution of LPFGs
For optical fiber sensors based on LPFGs, it is important to determine the mai rameters that describe the performance such as resolution, sensitivity, and figure of m which is defined as the ratio between sensitivity and FWHM [57]. These parameter pend on the grating length, grating period, wavelength, mode number, and ot Among other methods, working around the dispersion turning point is often used crease the sensitivity of the grating sensor [58]. To evaluate temperature sensitivity figure of merit (FOM) of LPFGs, here, we use a theoretical approach along with a sim tion of gratings with particular parameters.
The temperature sensitivity of an LPFG is defined as the ratio between the w length shift of a grating resonance in the transmission spectrum upon a temperatu crease of 1 degree. The transmission coefficient of an LPFG for one of the resonance be written in the following form [59]: Here, κ is the coupling constant, and L is the length of the grating. The detu parameter δ is defined as the detuning from the center of resonance:

Sensitivity and Resolution of LPFGs
For optical fiber sensors based on LPFGs, it is important to determine the main parameters that describe the performance such as resolution, sensitivity, and figure of merit, which is defined as the ratio between sensitivity and FWHM [57]. These parameters depend on the grating length, grating period, wavelength, mode number, and others. Among other methods, working around the dispersion turning point is often used to increase the sensitivity of the grating sensor [58]. To evaluate temperature sensitivity and figure of merit (FOM) of LPFGs, here, we use a theoretical approach along with a simulation of gratings with particular parameters.
The temperature sensitivity of an LPFG is defined as the ratio between the wavelength shift of a grating resonance in the transmission spectrum upon a temperature increase of 1 degree. The transmission coefficient of an LPFG for one of the resonances can be written in the following form [59]: Here, κ is the coupling constant, and L is the length of the grating. The detuning parameter δ is defined as the detuning from the center of resonance: The maximum loss is observed in the center of the resonance β co (λ r ) − β cl (λ r ) = 2π/Λ for κL = π/2. Therefore, the first minimum is for the detuning parameter δ = δ s = √ 3π/2L. The corresponding wavelength of the first minimum is separated from the center by ∆λ s : λ s = λ r + ∆λ s . Assuming that the wavelength separation is small, we can obtain its relation with the detuning parameter at the first minimum: and reversely If we define the resonance width as the wavelength span between two minima, then it is equal to 2∆λ s (Figure 7). The derivative dΛ/dλ is the slope of the dispersion curve Λ(λ), which depends on the wavelength and the mode number.  Let us assume that there is a change in the effective refractive index for the cladding mode due to some changes in the refractive index of the fiber n eff + ∆n eff . There is a corresponding change in the propagation constant of the mode β cl = 2π(n eff + ∆n eff )/λ. This would result in a wavelength shift of the resonance position due to an addition in the detuning parameter δ n = π∆n/λ r . It can be approximated as follows: and reversely From (11) and (13), we can obtain the ratio between the wavelength shift and the resonance width, which is the figure of merit: As one can see, this ratio is independent of the slope of the dispersion curve dΛ/dλ. Therefore, the bandwidth is proportional to sensitivity at a fixed wavelength, and the figure of merit cannot be improved by moving close to the dispersion turning point, where dλ/dΛ has higher values. It follows from (14) that the figure of merit increases with grating length and decreases with wavelength.
However, it should be stressed that the minimum temperature change detectable at cryogenic temperatures improves if the LPFG works close to DTP, since the sensitivity can increase by an order of magnitude (due to the higher slope of the dispersion curves when compared to the one of a 540 µm grating) and the overall resolution that takes into account fluctuations in the optical power source, the minimum resolution of the OSA, and the fact that the peak detection through the fitting of a Gaussian/Lorentzian curve is not particularly impacted by a slightly wider resonance [57,58].
In order to illustrate how sensitivity and figure of merit depend on the grating period and wavelength, we demonstrate it in Figure 8 with the spectrum of LPFGs in the B/Ge#1 fiber as a function of the grating period. Each vertical line represents a spectrum in color form. One can see that the width of the spectral resonances slowly grows with mode number. The amplitudes decrease with wavelengths. The thickness of each curve increases slightly with wavelength; however, the spectral width, which is measured along the wavelength axis, increases strongly with the angle of the curve slope and is inversely proportional to its cosine. So, at the turning point, the curves have slopes that are close to vertical yielding high sensitivities, but their spectral widths are very large, so that the figure of merit is not improved.
ors 2022, 22, x FOR PEER REVIEW 12 of increase by an order of magnitude (due to the higher slope of the dispersion curves wh compared to the one of a 540 μm grating) and the overall resolution that takes into accou fluctuations in the optical power source, the minimum resolution of the OSA, and the f that the peak detection through the fitting of a Gaussian/Lorentzian curve is not parti larly impacted by a slightly wider resonance [57,58]. In order to illustrate how sensitivity and figure of merit depend on the grating peri and wavelength, we demonstrate it in Figure 8 with the spectrum of LPFGs in the B/Ge fiber as a function of the grating period. Each vertical line represents a spectrum in co form. One can see that the width of the spectral resonances slowly grows with mode nu ber. The amplitudes decrease with wavelengths. The thickness of each curve increa slightly with wavelength; however, the spectral width, which is measured along wavelength axis, increases strongly with the angle of the curve slope and is inversely p portional to its cosine. So, at the turning point, the curves have slopes that are close vertical yielding high sensitivities, but their spectral widths are very large, so that the f ure of merit is not improved.

Temperature Sensitivity of LPFGs
We use the temperature dependences of fiber refractive indices (4) to calculate h the dispersion curves are changed, the shifts of the resonance wavelengths, and the figu of merit for different cladding modes. Figure 9 demonstrates the dispersion curves SMF-28 fiber for temperatures changing between 0 and 300 K for cladding mode numb = 1 … 13. The dispersion curves belonging to the lower order cladding modes are most affected by the temperature change, but those changes occur for the smallest slop of the dispersion curves. Much higher slopes are observed for cladding modes with 10. . .13 near the DTP.
The resonance wavelengths are blue-shifted with decreasing temperature, if range before the dispersion turning point is considered. The dependence of the wa length shift on temperature is shown in Figure 10 for several cladding modes. We can that the shift is generally larger for higher-order modes due to their higher slopes of d

Temperature Sensitivity of LPFGs
We use the temperature dependences of fiber refractive indices (4) to calculate how the dispersion curves are changed, the shifts of the resonance wavelengths, and the figure of merit for different cladding modes. Figure 9 demonstrates the dispersion curves for SMF-28 fiber for temperatures changing between 0 and 300 K for cladding mode numbers j = 1 . . . 13. The dispersion curves belonging to the lower order cladding modes are the most affected by the temperature change, but those changes occur for the smallest slopes of the dispersion curves. Much higher slopes are observed for cladding modes with j = 10...13 near the DTP.

Wavelength Shift (nm)
Grating Period (μm) 1 Figure 10. Absolute value of the wavelength shift for LP1j modes as a function of grating period for the temperature change from 300 K to 0 K.
As we discussed above, high sensitivity, as a rule, is compromised by low resolution, and the figure of merit defined as the relation between the wavelength shift and the resonance width is a more appropriate value to calculate. In Figure 11, we show the figure of merit for different cladding modes of the SMF-28 fiber as a function of wavelength for a grating with a length of 20 mm and temperature change 10 K T Δ = − . We note that the grating period is not fixed in this figure; rather, it is adjusted to obtain a mode resonance at a certain wavelength. In general, the absolute value of the figure of merit decreases with the mode number and wavelength. A special case is the LP11 mode, which behaves nonmonotonically and has an extremum near 1.4 μm. So, in terms of the figure of merit, it is preferable to use lower-order modes at shorter wavelengths. This is somewhat contrary to what is required for high sensitivity. The resonance wavelengths are blue-shifted with decreasing temperature, if the range before the dispersion turning point is considered. The dependence of the wavelength shift on temperature is shown in Figure 10 for several cladding modes. We can see that the shift is generally larger for higher-order modes due to their higher slopes of dispersion curves (except for LP 11 mode). The shift is highest when the DPT is approached.  Figure 10. Absolute value of the wavelength shift for LP1j modes as a function of grating period for the temperature change from 300 K to 0 K.
As we discussed above, high sensitivity, as a rule, is compromised by low resolution, and the figure of merit defined as the relation between the wavelength shift and the resonance width is a more appropriate value to calculate. In Figure 11, we show the figure of merit for different cladding modes of the SMF-28 fiber as a function of wavelength for a grating with a length of 20 mm and temperature change 10 K T Δ = − . We note that the grating period is not fixed in this figure; rather, it is adjusted to obtain a mode resonance at a certain wavelength. In general, the absolute value of the figure of merit decreases with the mode number and wavelength. A special case is the LP11 mode, which behaves nonmonotonically and has an extremum near 1.4 μm. So, in terms of the figure of merit, it is preferable to use lower-order modes at shorter wavelengths. This is somewhat contrary to what is required for high sensitivity. Figure 10. Absolute value of the wavelength shift for LP 1j modes as a function of grating period for the temperature change from 300 K to 0 K. As we discussed above, high sensitivity, as a rule, is compromised by low resolution, and the figure of merit defined as the relation between the wavelength shift and the resonance width is a more appropriate value to calculate. In Figure 11, we show the figure of merit for different cladding modes of the SMF-28 fiber as a function of wavelength for a grating with a length of 20 mm and temperature change ∆T = −10 K. We note that the grating period is not fixed in this figure; rather, it is adjusted to obtain a mode resonance at a certain wavelength. In general, the absolute value of the figure of merit decreases with the mode number and wavelength. A special case is the LP 11 mode, which behaves non-monotonically and has an extremum near 1.4 µm. So, in terms of the figure of merit, it is preferable to use lower-order modes at shorter wavelengths. This is somewhat contrary to what is required for high sensitivity. Let us consider the behavior of the B/Ge fiber gratings at cryogenic temperatures. Figure 12 depicts series of dispersion curves for 13 modes at temperatures from 300 K to 0 K with a step of 30 K for the B/Ge#1 fiber. The resonance wavelengths are red-shifted with decreasing temperature, if the range before the dispersion turning point is considered. The curves become more condensed when approaching 0 K. The dependence of the wavelength shift on temperature is shown in Figure 13 for the cladding modes with = 1. . .13, when the resonance is at 1.3 μm (corresponding to different periods). We can see that the shift is larger for higher-order modes, it monotonically grows with decreasing temperature and the sensitivity tends to zero at 0 K. These curves follow the same trend as the difference between the core-cladding refractive index difference in Figure 3. The maximum sensitivity is in the temperature range 100-200 K and is as high as −850 pm/K for cladding mode with 13 j = . The temperature sensitivity is −170 pm/K at 20 K for the same mode and tends to zero at 0 K, since the thermo-optic coefficient vanishes at this temperature. These values are considerably higher than those obtained in our previous work using phase-shifted LPFGs ( Figure 14). Table 4 summarizes the typical values of sensitivity obtained in the temperature range of 0-300 K using fiber gratings: Let us consider the behavior of the B/Ge fiber gratings at cryogenic temperatures. Figure 12 depicts series of dispersion curves for 13 modes at temperatures from 300 K to 0 K with a step of 30 K for the B/Ge#1 fiber. The resonance wavelengths are red-shifted with decreasing temperature, if the range before the dispersion turning point is considered. The curves become more condensed when approaching 0 K. Let us consider the behavior of the B/Ge fiber gratings at cryogenic temperatures. Figure 12 depicts series of dispersion curves for 13 modes at temperatures from 300 K to 0 K with a step of 30 K for the B/Ge#1 fiber. The resonance wavelengths are red-shifted with decreasing temperature, if the range before the dispersion turning point is considered. The curves become more condensed when approaching 0 K. The dependence of the wavelength shift on temperature is shown in Figure 13 for the cladding modes with = 1. . .13, when the resonance is at 1.3 μm (corresponding to different periods). We can see that the shift is larger for higher-order modes, it monotonically grows with decreasing temperature and the sensitivity tends to zero at 0 K. These curves follow the same trend as the difference between the core-cladding refractive index difference in Figure 3. The maximum sensitivity is in the temperature range 100-200 K and is as high as −850 pm/K for cladding mode with 13 j = . The temperature sensitivity is −170 pm/K at 20 K for the same mode and tends to zero at 0 K, since the thermo-optic coefficient vanishes at this temperature. These values are considerably higher than those obtained in our previous work using phase-shifted LPFGs (Figure 14). Table 4 summarizes the typical values of sensitivity obtained in the temperature range of 0-300 K using fiber gratings: The dependence of the wavelength shift on temperature is shown in Figure 13 for the cladding modes with j = 1...13, when the resonance is at 1.3 µm (corresponding to different periods). We can see that the shift is larger for higher-order modes, it monotonically grows with decreasing temperature and the sensitivity tends to zero at 0 K. These curves follow the same trend as the difference between the core-cladding refractive index difference in Figure 3. The maximum sensitivity is in the temperature range 100-200 K and is as high as −850 pm/K for cladding mode with j = 13. The temperature sensitivity is −170 pm/K at 20 K for the same mode and tends to zero at 0 K, since the thermo-optic coefficient vanishes at this temperature. These values are considerably higher than those obtained in our previous work using phase-shifted LPFGs ( Figure 14). Table 4 summarizes the typical values of sensitivity obtained in the temperature range of 0-300 K using fiber gratings: coated and uncoated FBGs, LPFGs and PS-LPFGs. As it can be observed, the temperature sensitivity of LPFGs is larger than that obtained for FBGs, even the coated ones, and this work reveals that the sensitivity can duplicate using LPFGs in the dispersion turning points. coated and uncoated FBGs, LPFGs and PS-LPFGs. As it can be observed, the tempera sensitivity of LPFGs is larger than that obtained for FBGs, even the coated ones, and work reveals that the sensitivity can duplicate using LPFGs in the dispersion tur points.   work reveals that the sensitivity can duplicate using LPFGs in the dispersion tur points.     [25] The dependence of the figure of merit for a 20 mm grating in the B/Ge#1 fiber is demonstrated in Figure 15 for modes LP 11 -LP 1,13 and ∆T = 10 K. The curves for different modes almost coincide and decrease monotonically with wavelength. Thus, in terms of the figure of merit, it is preferable to use shorter wavelengths independently of the mode number. The period should be chosen so that a resonance at a certain wavelength is obtained. As we can see by comparing this with Figure 10, the figure of merit for the B/Ge fiber is one order of magnitude higher than for the standard fiber. The figure of merit is not improved near the DTP, because the high sensitivity is compensated by a wider resonance width.
The dependence of the figure of merit for a 20 mm grating in the B/Ge#1 fib demonstrated in Figure 15 for modes LP11-LP1,13 and 10 K T Δ = . The curves for diff modes almost coincide and decrease monotonically with wavelength. Thus, in term the figure of merit, it is preferable to use shorter wavelengths independently of the m number. The period should be chosen so that a resonance at a certain wavelength i tained. As we can see by comparing this with Figure 10, the figure of merit for the fiber is one order of magnitude higher than for the standard fiber. The figure of me not improved near the DTP, because the high sensitivity is compensated by a wider nance width.

Conclusions
We studied the evolution of the dispersion curves of long-period fiber gratings room temperature down to 0 K. We considered gratings arc-induced in the SMF-28 and in two B/Ge co-doped fibers. Computer simulations were performed based on p ously published experimental data.
We demonstrated that the dispersion curves belonging to the lower-order clad modes are the most affected by the temperature change, but the shift is generally l for higher-order modes due to their higher slopes of dispersion curves. The shift is la when the dispersion turning point is approached. The shift of dispersion curves fo B/Ge fibers is one order of magnitude higher than for the SMF-28 fiber.
We have shown that two parameters important for the implementation of a tem ature sensor are the sensitivity (wavelength shift per unit of temperature change) an figure of merit (wavelength shift related to resonance width). A temperature sensi as high as −850 pm/K can be obtained in the 100-200 K temperature range. The tem ture sensitivity is −170 pm/K at 20 K and tends to zero at 0 K due to the vanishing the optic coefficient. The figure of merit is higher at shorter wavelengths and is indepen of the mode number.
In future work, we plan to perform cryogenic temperature measurements on L with grating periods shorter than 200 μm, inscribed in B/Ge#1 fiber, and we also inte investigate the second resonance of the LP11 mode at shorter wavelengths and deter its turning point.

Conclusions
We studied the evolution of the dispersion curves of long-period fiber gratings from room temperature down to 0 K. We considered gratings arc-induced in the SMF-28 fiber and in two B/Ge co-doped fibers. Computer simulations were performed based on previously published experimental data.
We demonstrated that the dispersion curves belonging to the lower-order cladding modes are the most affected by the temperature change, but the shift is generally larger for higher-order modes due to their higher slopes of dispersion curves. The shift is largest when the dispersion turning point is approached. The shift of dispersion curves for the B/Ge fibers is one order of magnitude higher than for the SMF-28 fiber.
We have shown that two parameters important for the implementation of a temperature sensor are the sensitivity (wavelength shift per unit of temperature change) and the figure of merit (wavelength shift related to resonance width). A temperature sensitivity as high as −850 pm/K can be obtained in the 100-200 K temperature range. The temperature sensitivity is −170 pm/K at 20 K and tends to zero at 0 K due to the vanishing thermo-optic coefficient. The figure of merit is higher at shorter wavelengths and is independent of the mode number.
In future work, we plan to perform cryogenic temperature measurements on LPFGs with grating periods shorter than 200 µm, inscribed in B/Ge#1 fiber, and we also intend to investigate the second resonance of the LP 11 mode at shorter wavelengths and determine its turning point.  Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: The data segments can be obtained by contacting the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.