1. Introduction

Remote real-time chemical analysis of various substances is in demand in industries, medicine and ecology. As liquids, gaseous and solids have strong absorption bands of their vibrational spectra in the wavelength range of 3 – 10 $\mu m$, the middle infrared (mid-IR) fiber-based evanescent wave spectroscopy (FEWS) is considered as a prospective tool for collecting and delivering information about the “fingerprints” of molecules. This is possible because radiation propagating in a single-index optical fiber is partially extending outside the core and can be absorbed. Functionality of the FEWS sensors has already been demonstrated mostly in laboratories in spectrometer-based experiments with various liquids [1–4]. For FEWS probes, single-index multimode chalcogenide fibers having small optical losses in the mid-IR were mainly used as straight segments [1,2] or as U-shaped and loop bends [3,4]. With the current advanced developments of supercontinuum fiber sources in the mid-IR [5–7], creation of the fiber-based chemical sensors for the real-time distant monitoring of various processes becomes a real challenge. For the FEWS sensors, a computer aided design of sensing elements based on a theoretical analysis of electromagnetic waves in chalcogenide fibers is an urgent task.

In our recent work [2], we have used the theoretical approach based on electromagnetic theory of optical fibers [8] for design of the FEWS sensors. Selective excitation of higher-order evanescent modes having large attenuation coefficients, in a multimode chalcogenide straight fiber has been proposed for achieving optimal dynamic range and high sensitivity of the sensors.

In this paper, we apply this approach for analysis of evanescent modes of a fiber bend immersed into a liquid consisting of a diesel fuel and an antigel additive, which is used to reduce the oil freezing temperature. Absorption coefficients of the 0 - 1 vol.% solution were evaluated and then used in numerical modeling of evanescent modes of a loop probe. The loop probe were fabricated from a specially designed fiber having low optical losses at wavelengths $\lambda$ in the range of $3 - 10$ $\mu m$. Transmittance of the probe immersed into the liquid analyte was measured and compared with the results of numerical modeling.

2. Spectroscopic measurements

For evaluation of absorption coefficients $\alpha$ of the constituents Fig. 1(a), transmittance spectra $T(\lambda )$ of bulk samples poured into a thin cell were measured by means of a Fourier-Transform Infrared Spectrometer (FTIR). The absorption band of the antigel additive with the maximum at $\lambda _a=7{.}83$ $\mu$m corresponding to esters has been chosen for chemical analysis. $T(\lambda _a)$ was measured at the antigel additive concentrations $C = 0 - 1$ vol.% and $\alpha$ was evaluated for each $C$ (Fig. 1(b), inset). As shown in Fig. 1(b), $\alpha (C)$ is well approximated by a linear function.

 

Fig. 1. Spectral dependencies of the absorption coefficients of the diesel fuel (black symbols) and the antigel additive (red symbols) measured in a cell (inset) with $d = 11$ $\mu$m (a); absorption coefficients of the solution at $\lambda _a= 7.83$ $\mu$m depending on the antigel additive concentration, $d = 80$ $\mu$m, inset: absorption coefficients at around $\lambda _a$ (b)

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To fabricate the loop probes, the special pure Ge$_{26}$As$_{17}$Se$_{25}$Te$_{32}$ chalcogenide glass was prepared by means of the chemical-distillation purification technique [9]. A single-index fiber of 180 $\mu$m diameter was drawn by using a crucible method. Total optical losses of the fiber were $< 1$ dB/m at $\lambda =5.5 - 8.5$ $\mu$m. This fiber was used to create a probe consisting of the fiber loop (three half-circles) with the loop radius $R_b=2$ mm and input and output straight tails each of the length $L = 10$ mm. The fiber loop was fabricated by bending the fiber around a heated ceramic rod [9]. Unpolarized light beam from a globar was focused by a 5 cm focus lens and launched coaxially into the input tail of the loop probe placed into a glass vessel filled with the liquid analyte Fig. 2(a). The probe transmittance was measured by using an MCT detector cooled by liquid nitrogen. For each $C$, the normalized transmittance $\tau =T_1/T_0$ was evaluated, where $T_1$ and $T_0$ were measured, respectively, with and without the liquid in the vessel. In Fig. 2(b), the normalized transmittance is shown for some concentrations of the antigel additive.

 

Fig. 2. Scheme of the experimental set-up for the FEWS (a); spectral dependences of the normalized transmittance of the loop probe (b). $R_b = 2$ mm, $L = 10$ mm

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3. Peculiarities of the evanescent modes of a fiber bend

For the theoretical treatment, we have used the COMSOL Multiphysics solver based on the finite elements analysis [10]. In a numerical model of a fiber bend, a COMSOL code provides Maxwell’s equations solution as a bend mode characterized by its complex-valued propagation constant and electromagnetic field distribution in a transverse cross-section of a bent fiber.

In accordance with the approach developed in COMSOL, electric and magnetic fields of a bend mode are expressed respectively, as $\vec {E}( r,\theta ,z,t )= \vec {E}( r,z ) \exp \left ( i\left ( \omega t-\gamma \theta \right ) \right )$ and $\vec {H}( r,\theta ,z,t )= \vec {H}( r,z ) \exp \left ( i\left ( \omega t-\gamma \theta \right ) \right )$ in a polar system $(r,\theta )$ shown in inset to Fig. 3(a). Here $\omega$ is the light frequency, $\gamma = \gamma _{\mathrm {re}} + i \gamma _{\mathrm {im}}$ is angular parameter that is complex-valued due to radiative losses even if there is no absorption in or outside the fiber [11]. If a fiber bend is immersed into an absorbing medium, $\gamma _{\mathrm {im}}$ increases due to the external absorption. Power of each evanescent mode decreases as $\exp ( -2 \gamma _{\mathrm {im}} \theta )$ [8].

 

Fig. 3. Calculated $\gamma _{\mathrm {im}}$ of the $\mathrm {HE}_{\mathrm {1m}}$ $||$ modes of the fiber bend depending on $R_b$, inset: scheme of the bend in a polar system (a); $P_{ex}/P$ calculated for the first 300 bend modes by using the COMSOL solver, $R_b = 2$ mm (b). $C = 1$ vol.%

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Refractive index of the solution, in accordance with the Bouguer-Lambert-Beer law, was taken equal to $1.4+i \alpha /2 k$, where $k=2\pi /\lambda _a$, and $\alpha$ was defined by using the calibration curve in Fig. 1(b). The calculations were carried out for the fiber with the core radius $a = 90$ $\mu$m and $n_{\mathrm {co}} = 2.8$. For the bend modes, we used the same notations as accepted in the theory of straight optical fibers [8]. As the heat radiation was launched coaxially with the input straight tail of the probe, we assume that $\mathrm {HE}_{\mathrm {1m}}$ modes were mainly excited in the tail [8]. Each doubly degenerate mode of a straight fiber was split at a bend to a pair of modes having orthogonal polarizations [11]. Electrical field of one mode is oscillating in the bend plane ($||$ mode) and of another mode — perpendicular to the plane ($+$ mode). At first, absolute value $|\vec {E}|$ of electric field of a bend mode was plotted over the fiber cross-section at large $R_b$ when the bend mode looks similar to the straight fiber mode. Then $R_b$ was gradually decreased, $|\vec {E}|$ profile was plotted and analyzed visually for the mode identification.

It has been revealed that for a given $R_b$, $\gamma _{\mathrm {re}}$ of $\mathrm {HE}_{\mathrm {1m}}$ mode decreases linearly with $m$, and $\gamma _{\mathrm {im}}$ grows with $m$. As shown in Fig. 3(a) for $||$ modes, $\gamma _{\mathrm {im}}$ of a mode generally grows with $R_b$ (parameters of the modes with $m>5$ are not shown for $R_b < 10$ mm because here visual identification of the modes is difficult). At $R_b >10$ mm, $\gamma _{\mathrm {im}}$ of the modes having diverse $m$ differ several times. In fact, $\gamma _{\mathrm {im}}$ is determined by the amount of the power $P_{ex}$ of radiation propagating outside the fiber core. In Fig. 3(b), $P_{ex}/P$, where $P$ is the total power of the mode is shown for the sequence of 300 modes. Magnitudes of $\gamma _{\mathrm {im}}$ of the $+$ modes (not shown in Fig. 3(a)) are less than those of the $||$ modes and are more scattered. In the variety of the bend modes, $\mathrm {HE}_{\mathrm {1m}}$ modes are shown by colored circles. The mode sequence number $j$ in Fig. 3(b) is defined as it appears at the output of the COMSOL solver.

Longitudinal propagation constant of a straight fiber mode ${{\beta }^{s}}=\beta _{\mathrm {re}}^{s}+i \beta _{\mathrm {im}}^{s}$. At a fiber bend, the propagation constant varies over the fiber cross-section [11]. In the COMSOL model of the fiber bend, $\beta ^b=\gamma /r_0=\beta _{\mathrm {re}}^{b}+i \beta _{\mathrm {im}}^{b}$ is evaluated, where $r_0= \int r P ds / \int P ds$ is obtained by integration over the fiber cross-section of the Pointing vector component $P$ perpendicular to the plane of the cross-section. In our calculations, $r_0 \approx R_b$ with 1-2% deviations.

In inset to Fig. 4(a), $\beta ^b$ and $\beta ^s$ of the 300 modes of the bent fiber and of the same straight fiber are shown on a complex plane. Magnitude of $\beta ^b_{\mathrm {re}}$ of each mode is less than $\beta ^s_{\mathrm {re}}$ of the same mode of the straight fiber [11]. The higher-order modes have smaller $\beta ^b_{\mathrm {re}}$ and greater $\beta ^b_{\mathrm {im}}$, similar to evanescent modes of the straight fiber. In Fig. 4(a), $\beta ^b$ are plotted for modes with $j > 300$, including $\mathrm {HE}_{\mathrm {1m}}$ modes with $m = 10$, 20, 30 (red circles) and $\beta ^s$ of the same modes (red squares). The difference between attenuation coefficients $2\beta ^b_{\mathrm {im}}$ and $2\beta ^b_{\mathrm {im}}$ of the bent and straight fiber modes increases with $m$.

 

Fig. 4. Complex plane of the propagation constants of the bend modes ($\beta ^b$,circles) and of the same straight fiber ($\beta ^s$,squares), $R_b = 2$ mm, inset: magnified area framed below, $\mathrm {HE}_{\mathrm {1m}}$ modes are marked by color (a); imaginary parts of $\beta ^b$ of the $\mathrm {HE}_{\mathrm {1m}}$ modes ($m =1,5$) depending on $R_b$, inset: profiles of $|\vec {E}|$ of the $\mathrm {HE}_{\mathrm {15}}$ mode in the fiber cross-section at various $R_b$, arrows point out the electric field oscillations in the fiber cross-section (b). $C = 1$ vol.%

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In Fig. 4(b), $\beta ^b_{\mathrm {im}}$ of $\mathrm {HE}_{\mathrm {1m}}$ modes of the bend decrease with $R_b$ and approach $\beta ^s_{\mathrm {im}}$ at $R_b>50$ mm. Analysis of $|\vec {E}|$ profiles shown in inset to Fig. 4(b) reveals that radial order of a bend mode keeps the same as of the straight fiber mode, but azimuthal order is significantly disturbed.

Without an external absorption, $\beta _{\mathrm {im}}^b=0$ are of the same order of magnitude as $\beta _{\mathrm {im}}^s \sim 10^{-16}-10^{-12}$ mm$^{-1}$ calculated by the COMSOL solver. Then we can conclude that in the single-index fiber with large $n_{\mathrm {co}}$, the bending losses are quite small and $\beta _{\mathrm {im}}^b$ of the modes of the fiber bend immersed into the solution are determined mainly by the external absorption.

4. Discussion

Each mode excited in the input straight tail of the probe shown in Fig. 2(a) experiences spatial transformation to the bend modes at the fiber cross-section marked as b1. The coupled bend modes reach the cross-section b2 and transform back to the straight fiber modes. In the normalized transmittance $\tau$ (as defined in Section 2), power losses at b1 and b2 cross-sections are reduced, but the bend modes coupling is to be taken into account in computer modeling of the total electromagnetic field propagation. For a multimode fiber, this complex computational problem requires a special study. At the first stage of this research, it is important to reveal the changes in an evanescent mode field due to the fiber bending.

In assumption that the mode coupling is weak, we can analyse absorbance of the bend modes separately. For each $\mathrm {HE}_{\mathrm {1m}}$ mode, the absorbance $A^b=-\log (T^b/T_0^b)$ of a fiber loop, where transmittance $T^b=\exp (-2\gamma _{im} \theta _L)$ with $\theta _L=N \pi$ ($N$ is the number of half-turns of the loop) was evaluated, $T_0^b$ being evaluated without the absorbing liquid. We have found that due to the large refractive index of chalcogenide glass, the bending losses in air are negligibly small and $T_0^b\approx 1$ even at small $R_b$. As is shown in Fig. 5(a), $A^b$ of a mode with $m < 6$ does not depend significantly on $R^b$. For a mode with larger $m$, $A^b$ grows with $R_b$ due to $\gamma _{im}$ increase Fig. 3(a). For comparison, absorbance $A^s = -\log (T^s)$ of $\mathrm {HE}_{\mathrm {1m}}$ mode of a straight fiber of the length $L = N \pi R$ was evaluated, where $T^s=\exp (-2\beta _{im}^s L)$. As is shown in Fig. 5(b), at given $C$ and $N$, for each mode, $A^b > A^s$. With $C = 1\%$, for $m = 5$, 20 and 30, $A^b/A^s = 1.2$, 1.7 and 2.2, respectively. This proves that attenuation coefficients of the higher-order modes increase greater due to the bending, while the azimuthal dependences of the modes with larger $m$ (shown in inset to Fig. 5(b)) are not so disturbed as those of the lower-order modes (shown in inset to Fig. 4(b)) at the bend with $R_b = 2$ mm. As $A^b = 2 \gamma _{im} N \pi /\ln (10)$, it grows linearly with $N$ as is proved by analysis of the plots obtained for the mode with $m =30$ in Fig. 5(b).

 

Fig. 5. Absorbance of the loop bend calculated for $\mathrm {HE}_{\mathrm {1m}}$ $||$ modes: a) $A^b$ depending on $R_b$, $N = 3$, $C = 1$ vol.$\%$; b) $A^b$ (circles), $A^s$ (columns) depending on $C$, $R_b = 2$ mm, inset: $|\vec {E}|$ profiles of the mode with $m = 30$ in a cross-section of the straight (S) and of the bent fiber, arrows point out the electric field oscillations; c) $A^p$ of the probe shown in Fig. 2(a) (circles) depending on $C$, the measured absorbance (squares), $R_b = 2$ mm

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Transmittance of the probe shown in Fig. 2(a), $\tau ^p= T/T_0 = T^b T_1^s T_2^s$, where $T_1^s$ and $T_2^s$ is, respectively, transmittance of the input and output straight tails. The absorbance $A^p= -\log (T^p)$ evaluated for $\mathrm {HE}_{\mathrm {1m}}$ modes is shown in Fig. 5(c) in comparison with the measured absorbance. The measured dependence is steeper than $A^p(C)$ calculated for the individual modes. As is mentioned above, total transmittance of a multimode fiber bend is not equal to a sum of individual modes transmittances averaged over the number of modes because of the modes coupling. At this stage of the research, we can notice via analysis of Fig. 5(c) that the character of the evanescent modes coupling depends on $C$, i.e. on an absorption coefficient of the liquid analyte. Impact of the higher-order modes into the total absorbance increases with $C$.

5. Conclusion

In this work, electromagnetic theory of optical fibers has been firstly applied in theoretical analysis of the FEWS sensor with a loop probe. Modeling of light propagation in a multimode fiber bend is a complex computational problem mostly because of the fiber modes coupling at the bend. At the first stage of the modeling as applied to the FEWS sensor design, properties of individual evanescent modes of the bend are to be revealed. By using a numerical method for evaluating electromagnetic fields of the fiber bend modes, we have found that attenuation coefficients grow with radial and azimuthal orders of the modes. These coefficients are larger than those of the same straight fiber modes and the difference between them is greater for the higher-order modes. As we have shown earlier by the example of a straight fiber probe [12], evanescent modes with larger attenuation coefficients are more suitable for optimization of the probe sensitivity, dynamic range and detection limit. As $||$ mode of the loop probe has greater attenuation coefficient than the same $+$ mode, then at the probe input, it is reasonable to use a light beam polarized in the bend plane. Results of this work confirm that selective excitation of higher-order modes of a loop probe can be used to optimize its functionality as well as of a straight probe. The fiber bending definitely provides better functionality of a FEWS sensor with a heat source of incoherent radiation. With a coherent laser radiation, it is possible to create conditions for excitation of higher-order modes having large attenuation coefficients.

The absorbance, calculated for individual modes of the fiber loop embedded into an absorbing liquid, linearly grows with the number of the fiber half-turns. For low-order $\mathrm {HE}_{\mathrm {1m}}$ modes, the absorbance is weakly depending on the bend radius, but with increase of a mode order, the absorbance grows with the radius due to the increase of the mode attenuation coefficient and of the mode path length in the bent fiber. However, a compact loop probe with the bend radius less than 5 mm is more practical. Determination of the bend radius for a loop probe depends also on technological limitations of the probe fabrication. The probes fabricated with the bend radius less than 2 mm by means of the technology described in this paper, have low quality.

The results obtained in this paper provide a base for further development of the computer model of electromagnetic waves in the FEWS sensor.