1. Introduction

The analysis of physiological processes and early detection of certain diseases [1] requires sensors capable of probing subtle changes in temperature [2], pH level [3,4], and concentration of biological fluids and gases [3] in hard-to-reach places inside living organisms such as the brain [5], spinal cord [6], ovarian tract [7] and blood vessels [8]. Such changes can often be sensed as small shifts in the optical refractive index of bodily fluids and tissues [8].

Microstructured exposed-core optical fibres (ECFs) provide a broad range of optical properties demanded by biomedical refractive index sensors intended to operate inside living organisms [9–12]. ECFs confine and guide light in a small volume of a dielectric material surrounded by longitudinal air holes (Fig. 1). One of the holes is open along the entire length of the fibre, which allows using it as a sample chamber where a portion of the guided light evanescently extends above the fibre and provides light-matter overlap and enhanced interaction required for sensing.

 

Fig. 1. (a) Schematic of an ECF coated with a dielectric layer of thickness $h$, length $\ell$ [see (c)], and refractive index $n_{dl}$. (b) Optical intensity profile in the fibre with the $100$-nm-thick dielectric layer and $n_{dl}=2$. The fundamental guided mode confined in the Y-shaped core and a higher-order mode localised in the dielectric layer can be seen. (c) Longitudinal cross-section of the ECF and schematic of the mode behaviour. The mode of the bare ECF splits into the two guided modes due to the dielectric layer. The two modes co-propagate, re-couple and interfere near the output edge of the fibre.

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Here, by considering a thin dielectric coating of an ECF (Fig. 1), we show that the guided mode can be split into a surface sensing mode and a mode that is isolated from the surface. The isolated mode is immune to environmental changes, but the surface mode is sensitive to refractive index shifts in the outer medium across the entire fibre length. We show that interference between the two modes can be used to create a fibre-optic sensor capable of detecting shifts in the optical refractive index with $60,000$ rad/RIU-cm sensitivity. Such sensitivity is comparable with that of two-arm interferometers [13,14] and two-mode one-fibre interferometers based on elliptical core [15], mismatched core [16], liquid-crystal-clad [17], and photonic crystal [18] fibres.

2. Two-mode microstructured exposed core fibre

We consider a standard ECF structure [19] (Fig. 1) that consists of a Y-shaped silica core (the refractive index $n_{co}=1.4607$, $2.2$ $\mu$m size) formed by three elliptical air holes ($n_{air}=1$) [9–11,19,20]. Two of these holes are fully enclosed by silica, but the third hole is open such that the top surface of the Y-shaped core can be accessed from the outer space across the fibre length. The exposed surface of the Y-shaped hole is covered by a dielectric layer of uniform thickness $h$ and optical refractive index $n_{dl}$. We consider $n_{dl}$ from $1$ to $2.5$ because the dielectric layer deposited on top of a realistic ECF can be made of Teflon ($n_{dl}=1.36$), silica glass ($n_{dl}=1.46$), a polymer ($n_{dl}=1.49$), tellurite glass ($n_{dl}=2$) or other high refractive index materials [8,20–22]. The optical properties of such fibre originate from the small Y-shaped core with the dielectric layer (inset Fig. 1) and therefore only this region is considered in our analysis.

Figure 2 shows the dispersion characteristic of the ECF and the representative modal optical intensity profiles in the $xy$-plane calculated with an in-house finite-difference time-domain (FDTD) method [23]. The same results would be obtained using other simulation techniques. The wavelength is $532$ nm and the thickness of the dielectric layer is $h=100$ nm. Qualitatively similar results were obtained for all wavelengths across the visible spectral range as well as for the fibres with the other experimentally accessible [20] dielectric layer thicknesses $50-200$ nm.

 

Fig. 2. Dispersion characteristic of the ECF with the $100$-nm-thick dielectric layer. Themap is composed of individual power spectra calculated for $n_{dl}{=}1\cdots 2.5$. The peaks in these spectra trace dispersion curves, but their magnitude corresponds to the relative fraction of the power in each guided mode (encoded as the intensity in red). The insets show the modal intensity profile in the $xy$-plane of the fibre.

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The bare fibre ($n_{dl}=1$) supports two guided modes with the effective refractive indices $n_{eff}=1.183$ and $n_{eff}=1.377$. The mode with $n_{eff}=1.377$ is the fundamental mode and the other mode is a higher-order guided mode. At $n_{dl}=1$, the two modes behave independently – their resonance peaks are separated and have different magnitudes and linewidths. However, because of a strong dependence of $n_{eff}$ of the higher-order mode on $n_{dl}$, the two modes hybridise at $n_{dl} \approx 1.7$, which is evidenced by hybridisation of their modal intensity profiles (Fig. 2), nearly equal magnitude of power density peaks, and avoided dispersion curve crossing [24].

At $n_{dl}>1.8$, the hybridisation disappears and the two modes again become independent. The fundamental mode regains its original profile, but that of the higher-order mode becomes completely different – the light is localised in the dielectric layer where the refractive index is higher than that of the core. At $n_{dl} > 1.7$ there also appears the second higher-order mode with the light localised at the interface between the Y-shaped core and air. Initially, the fraction power carried by this mode is low, but it gradually increases as the value of $n_{dl}$ is increased.

In the following, we will consider the regime of $n_{dl} > 1.8$ where the guided modes strongly confine light in the core and the dielectric layer. We will not consider the regime of mode anti-crossing ($n_{dl} \approx 1.7$), although that regime would be suitable for other applications [24].

3. Analytical model of sensitivity

As the light propagates along the fibre section with the dielectric layer of length $\ell$ [Fig. 1(c)], a phase difference arises from the difference in propagation constants of the fundamental mode in the Y-shaped core, $\beta _{co}$, and the higher-order mode in the dielectric layer, $\beta _{dl}$. A fraction of the power in the higher-order mode can be coupled back into the fundamental mode. The out-of-phase component results in an interference effect that can be observed as an oscillating attenuation measured as a function of wavelength at the output edge of the fibre.

We use the perturbation theory [25] where we assume that the refractive index of the dielectric layer is perturbed as $n_{dl} + \Delta n_{dl}$, but that of the fibre core remains unchanged. This idealised model allows us to validate the analytical formalism presented below, and it also corresponds to a practical scenario of small temperature changes induced mostly in the dielectric layer by living biological cells or micro-droplets of liquids placed on top of the fibre.

The phase of fringes arising due to the interference effect is $\phi = \frac {2\pi \ell \Delta n_{eff}}{\lambda } = \ell \Delta \beta$, where $\lambda$ is the wavelength of light in free space, $\ell$ is the length of the fibre section with the dielectric layer [Fig. 1(c)], and $\Delta \beta {=} \beta _{co} {-} \beta _{dl}$ is the difference in the propagation constants of the interfering modes. Under the condition of sufficiently small $\Delta n_{dl}$ we obtain $\beta _{dl} = \bar {\beta }_{dl} + k \bar {\eta }_{dl} \Delta n_{dl}$, where $\bar {\beta }_{dl}$ and $\bar {\eta }_{dl}$ are the unperturbed propagation constant and the fraction (with respect to the total amount carried by all modes) power of the mode guided in the dielectric layer, respectively, and $k$ is the wavevector. Thus, by following [16], the refractive index change $\Delta n_{dl}$ can be related to the measurable change in phase $\Delta \phi$ as $\Delta n_{dl} \approx \frac {\lambda \Delta \phi }{2\pi \bar {\eta }_{dl} \ell }$.

In Fig. 3(a) we plot $\Delta n_{dl}$ as a function of $\Delta \phi$ for several representative fibre section lengths $\ell$. We consider a dielectric layer material with $n_{dl}=2$, which in a practice would correspond to tellurite glass [21]. We observe that by using tens of centimeter long sections of the fibre, one could sense refractive index changes of ${\sim } 10^{-6}$ by using a single interferometer arm for the interfering modes and without disturbing the mode propagating in the core of the fibre. Shorter fibre sections can also be used when a smaller sensitivity is acceptable.

 

Fig. 3. (a) Change in the optical refractive index of the dielectric layer of the ECF, $\Delta n_{dl}$, as a function of the phase change $\Delta \phi$ for several representative lengths $\ell$ defined in Fig. 1(c). The thickness of the dielectric layer is $100$ nm and $n_{dl} = 2$. For example, at $\ell = 150$ cm one could sense refractive index changes of ${\sim } 10^{-6}$ leading to the change of phase $\Delta \phi = \pi$. (b) Phase sensitivity as a function of $n_{dl}$.

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We also calculate the phase sensitivity [14,16] as $S_{\phi } = \frac {\Delta \phi }{\Delta n_{dl} \ell }$. For the practical scenario [16,21] of $\Delta \phi = \pi$ for $n_{dl} = 2$ we obtain $S_{\phi } \approx 17,000$ rad/RIU-cm. This value is of the same order of magnitude as $S_{\phi }$ predicted for a slow-light Mach-Zehnder interferometer [14]. In Fig. 3(b), we plot $S_{\phi }$ as a function of $n_{dl}=1.8\cdots 2.4$. The model predicts a gradual decrease in $S_{\phi }$ from $60,000$ rad/RIU-cm to $3,000$ rad/RIU-cm, which is consistent with the diverging behaviour of the dispersion curves of the fundamental and the first higher-order modes (Fig. 2).

4. Rigorous numerical simulations

We numerically verify the sensitivity in Fig. 3. FDTD simulations of macroscopic fibre sections are impractical due to prohibitive computational time requirements. However, the linearity of our model allows maintaining the same value of $S_{\phi }$ by increasing $\Delta n_{dl}$ and decreasing $\ell$. We verified that $\Delta n_{dl} \le 0.1$ does not to disrupt the validity of the approximations made in the analytical model. Thus, we choose $\ell =50$ $\mu$m and $\Delta n_{dl} \approx 3.66 \times 10^{-2}$.

Figure 4(a) shows the fringe pattern calculated by collecting and Fourier-transforming the light emitted from the output edge of the fibre [Fig. 1(c)]. The observed fringe shift $\Delta \phi \approx \pi$ at $\sim 534$ nm is in good agreement with the predictions of the analytical theory. The small shift of the fringe from the nominal wavelength $532$ nm is an artefact caused by the finite resolution of the Fourier transformation of the time-domain signal produced by the FDTD simulation.

 

Fig. 4. (a) Calculated fringe pattern produced by the $\ell =50$ $\mu$m fibre section with the dielectric layer [$\ell$ is defined in Fig. 1(c)] with the unperturbed refractive index of the dielectric layer ($n_{dl}=2$, solid curve) and the perturbed refractive index ($n_{dl}+\Delta n_{dl}$, dashed curve). In agreement with the analytical theory, the phase shift $\Delta \phi \approx \pi$ is observed at $\sim 532$ nm. (b) Optical intensity distribution in a short region of the fibre.

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The optical intensity profile along the length of the fibre [Fig. 4(b)] shows a periodic energy exchange between the core and dielectric layer. The period of the interference effect equals ${\sim }2.0$ $\mu$m, which is in agreement with the theoretical value calculated for $\lambda =532$ nm and $n_{dl}=2$ as $\lambda /\Delta n_{eff}$ using $\Delta n_{eff}$ obtained from the dispersion characteristic in Fig. 2.

We also show that the investigated ECF is highly sensitive to changes in the refractive index, $n_{out}$, of the medium located above the dielectric layer. In this case, the refractive indices of both fibre core and dielectric layer remain unchanged, but that of the outer medium is perturbed as $n_{out} + \Delta n_{out}$. This scenario is difficult to analyse analytically because of the need to calculate leaky-mode losses [26]. Hence, we employ the FDTD method. Figure 5 shows the fringe pattern calculated for a $50$-$\mu$m-long fibre section with the dielectric layer. We use $\Delta n_{out} = 9.2 \times 10^{-2}$ and $n_{dl}=2$. We observe a phase shift of $\sim \pi$ and we obtain $S_{\phi } = 6,830$ rad/RIU-cm.

 

Fig. 5. Fringe pattern produced by the $50$-$\mu$m-long fibre section with the dielectric layer with the unperturbed (solid curve) and uniformly perturbed (dotted curve, $\Delta n_{out} \approx 9.2 \times 10^{-2}$) refractive index of the medium located above the dielectric layer. The refractive indices of the core and dielectric layer, $n_{dl}=2$, are constant.

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The results of our analysis should not be affected by optical losses in the fibre core and the dielectric layer. For example, optical losses in a $150$-cm-long fibre section without dielectric layer are about $3\%$ [19]. Optical losses in the tellurite layer manufactured using high purity raw materials and an improved film fabrication technique can be as low as $3\%$ [27]. Furthermore, the signal attenuation should be decreased by the fact that in the tellurite layer the light forms hot spots [Fig. 4(b)]. Overall, the attenuation can decrease the intensity of the signal but it would not affect the interference behaviour predicted by our theory and simulations.

Thus far, in our analysis we considered a relatively simple system with $n_{out}=1$. However, qualitatively similar results will be obtained for $n_{out}=1.33\cdots 1.6$, which corresponds to the refractive index of most common biological fluids and cells [28]. Due to high computational complexity, the analysis of interference processes in a liquid-immersed fibre is possible only with numerical techniques. However, qualitative behaviour of light in the dielectric layer can be predicted analytically. The refractive index of liquids and cells is smaller than $n_{dl}=2$, which ensures the existence of guided modes in the dielectric layer. The sensitivity is proportional to the fraction of the power evanescently coupled to the medium above the dielectric layer [25]. This fractional power increases as the refractive index of the outer medium is increased [29]. However, this also results in shorter propagation lengths at which strong interference fringes would be observed, because of radiation losses and absorption by biological fluids [28].

5. Conclusions

Our simulations have demonstrated that we can make a single-arm interferometric sensor by coating the core of a microstructured exposed core fibre with a high refractive index material. We show that the proposed fibre structure can guide light in both the core and coating, thereby satisfying conditions for strong mode interference and providing the sensitivity of up to $60,000$ rad/RIU-cm. Single-arm fibre-optical interferometers are simpler and more robust than those based on two arms, and therefore the investigated fibre structure should be especially suitable for sensing of small refractive index shifts in living organisms.