1. Introduction

The resonant coupling between a core-guided mode and a plasmonic mode in photonic crystal fiber (PCF) or microstructured optical fiber (MOF)-based plasmonic sensor is based on the phase or loss matching condition [1,2]. The MOF-based plasmonic sensors have made significant progress in last few years and different sensor designs such as hybrid mechanism [3], D-shaped fiber sensor [4], and multicoating PCF [5] have been reported. In a recently-reported work [6], a surface plasmon resonance (SPR)-based biosensor utilizing hexagonal lattice PCF exhibiting birefringent behavior has been investigated by using a finite element method (FEM) with perfectly matched layers (PML). The above sensor was reported to have spectral sensitivity (S_λ) of nearly 2000 nm/RIU when the analyte refractive index (n_a) is varied from 1.36 to 1.37, and the maximum amplitude sensitivity (S_A) of 320 RIU⁻¹ when n_a = 1.36. However, the maximum loss depth was reported to be only 42 dB/cm for n_a = 1.36.

Another similar sensor structure [7] reported to achieve S_λ = 2200 nm/RIU for n_a varying from 1.35 to 1.36, and S_A = 241 RIU⁻¹ for n_a = 1.36. For the above configuration, the maximum loss depth is 157 dB/cm for n_a = 1.36. In another similar sensor structure [8] with core modes I and II (x- and y- polarization), the following local spectral sensitivities were reported for a corresponding variation of n_a from 1.36 to 1.361: S_λ (I, x) = 3100 nm/RIU, S_λ (II, x) = 3300 nm/RIU, S_λ (I, y) = 3000 nm/RIU, and S_λ (II, y) = 3100 nm/RIU, which are very close to one another. The above sensor structure reported the losses as: α (I, x) = 68.4 dB/cm, α (II, x) = 195.7 dB/cm, α (I, y) = 12.4 dB/cm, and α (II, y) = 33.1 dB/cm at the resonant wavelengths. It is distinctively clear that the loss values were significantly different from one another (particularly, type II, x-component), which indicated that in this sensor structure, there is a stronger interaction between the core and plasmonic modes of type II near the phase matching point for the x-component of the electric field as compared to the y-component. These results are in contradiction with the results reported in other works [6,7].

In a recent work [9], a microstructured birefringent fiber-based plasmonic sensor containing 14 holes consisting of two hexagonal lattice rings with five of the central horizontal air holes suppressed, and with gold and analyte layers surrounding the fiber was proposed. Due to birefringent nature, this configuration supported two types (I and II) of resonant modes. The type I core guided mode resonance of such structure was highly advantageous as it was able to achieve considerably higher S_λ (7000.0 nm/RIU), very large figure of merit, i.e., FOM (573.8 RIU⁻¹), and higher S_A (886.9 RIU⁻¹).

In another recent work by Rifat et al. [10], a solid-core (radius 1.4 μm) PCF-based SPR biosensor was reported with 14 air holes arranged in a two-ring hexagonal lattice (6 holes with radius 0.6 μm in the first ring and 8 holes with radius 0.8 μm in the second ring), and the center-to-center distance (d) of 2 μm between air holes. Further, these holes were inserted in a SiO₂ substrate (radius r_g = 5 μm) surrounded by a gold layer (thickness t_g = 30 nm), which was finally covered by an analyte layer (thickness t_a = 1.2 μm). The spectral and amplitude sensitivities were determined using FEM with PML. The above sensor achieved the S_λ values to be 2000 nm/RIU (n_a: 1.33 to 1.34), 3000 nm/RIU (n_a: 1.34 to 1.35), 3000 nm/RIU (n_a: 1.35 to 1.36), and 4000 nm/RIU (n_a: 1.36 to 1.37). The maximum S_A values was reported to be 190 RIU⁻¹ (when n_a = 1.33), 397 RIU⁻¹ (when n_a = 1.35), and 478 RIU⁻¹ (when n_a = 1.36). However, in the above work [10], the higher loss depth is merely 60 dB/cm for n_a = 1.37. It is important to mention that in the above-discussed works [6,7,10], the orientation of the electric field in plasmonic modes is from up to down in the left part and from down to up in the right part of the fiber for x-polarization, and from down to up in the left and right parts of the fiber for y- polarization. However, the more refined set of x- and y-polarizations can result in higher sensitivities and loss depth, which can be explained by using FEM with PML, if applied correctly.

The present work reports on the sensing characteristics of two core (central and lateral) modes (x- and y-polarizations) of a new birefringent plasmonic biosensor based on solid-core MOF analyzed by using FEM with the matched boundary conditions [11]. A detailed analysis of the sensor design is carried out in terms of birefringence, the variation of Poynting vector, and modal characteristics near phase or loss matching points. The above analysis culminates at an in-depth evaluation of the sensor’s performance parameters.

2. Microstructured plasmonic optical fiber structure

The proposed SPR biosensor is based on an MOF (Fig. 1

 

Fig. 1 (a) Cross section of a fiber consisting of 14 air holes (d = 2 μm) with radius r = 0.5 μm, which are inserted in a SiO₂ core (radius r_g = 5 μm) surrounded by a gold layer (thickness t_g = 40 nm) and an analyte layer. (b) The orientation of the dominant electric field for different core and plasmonic modes, which are used for FEM simulations.

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Further, the electric field orientation in plasmonic modes [(II, b) and (II, y)] is comparable (Fig. 1) with that for the corresponding plasmonic modes [x- and y- polarization], respectively, defined in the references [6,7,10]. For simulation of the proposed sensor, the wavelength dependence of the refractive index (RI) of silica substrate and gold layer are adapted from the references [12–14] and [15], respectively.

3. Results and discussion

3.1 Analysis of sensor design in view of MOF birefringence

Figure 2(a)

 

Fig. 2 Contour plot of the z-component S_z (x, y) of Poynting vector at the phase matching points: (a) λ (I, x) = 0.629 μm, (b) λ (II, x) = 0.629 μm, (c) λ (I, y) = 0.624 μm, and (d) λ (II, y) = 0.624 μm. All the figures belong to n_a = 1.33.

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By analyzing both the contour plots together, it can be readily observed that the real and imaginary parts of n_eff for the modes (I, x) and (II, x) are very close to each other at λ = 0.629 μm. Thus, there is a resonant coupling between (I, x) and (II, x) core modes at the phase matching point.

Figure 2(c) shows the contour plot of S_z(x,y) at the phase matching point (λ = 0.624 μm) for the mode (I, y, β/k = 1.451197 + 0.00006310903i) when n_a = 1.33. Figure 2(d) depicts the S_z(x,y) plot for the mode (II, y, β/k = 1.451057 + 0.0003021366i) at the same values of λ and n_a. One can observe from Fig. 2 that at small value of n_a and at resonant wavelength, the imaginary part of n_eff of (I, x) mode at λ = 0.629 μm is larger than that of the (I, y) core mode at λ = 0.624 μm. It further consolidates the effect of fiber birefringence at the phase matching point.

Figures 3(a) and 3(b)

 

Fig. 3 Contour plot of the z-component S_z (x, y) of Poynting vector at the phase matching points for: (a) core mode (II, x) at λ = 0.596 μm, (b) plasmon mode (II, x) at λ = 0.596 μm, (c) core mode (II, y) at λ = 0.595 μm, and (d) plasmon mode (II, y) at λ = 0.595 μm. All the figures belong to n_a = 1.33.

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Figures 4(a) and 4(b)

 

Fig. 4 Contour plot of the z-component S_z (x, y) of Poynting vector of the core modes at the loss matching points: (a) λ (III, x) = 0.591 μm, (b) λ (I, x) = 0.591 μm, (c) λ (III, y) = 0.590 μm, and (d) λ (I, y) = 0.590 μm. All the figures belong to n_a = 1.33.

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Further, Figs. 4(c) and 4(d) represent the S_z(x,y) plots at the loss matching point (λ = 0.590 μm) for the core guided mode (III, y, β/k = 1.452564 + 0.00002272466i) and core guided mode (I, y, β/k = 1.452892 + 0.00002381091i) when n_a = 1.33. It can be noted from the above plots that at the loss matching point, the imaginary parts of n_eff for the modes (III, x) and (I, x) are very close to each other. The same pattern is evident for the modes (III, y) and (I, y) also.

3.2 Modal analysis near phase matching points

Figure 5(a)

 

Fig. 5 Real part of the effective index versus the wavelength for the core modes (I, x, green), (I, y, red), (II, x, blue), (II, y, cyan), (III, x, magenta), and (III, y, black) near the phase (λ (I, y) = 0.735 μm) or loss (λ (I, x) = 0.747 μm, λ (II, x) = 0.692 μm, and λ (II, y) = 0.693 μm) matching points when n_a = 1.37. The dashed lines (blue and cyan) are for the plasmonic modes. The brown points are for a fictive intersection (due to the avoided-crossing effect [16,17]) of the dispersive curves near the loss matching wavelengths.

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More precisely, the phase matching points are: λ (I, x) = 0.747 μm, λ (I, y) = 0.735 μm, λ (II, x) = 0.692 μm, and λ (II, y) = 0.693 μm. Figure 5(b), in addition to above-mentioned core modes, contains the spectral variation of plasmon modes (II, x) and (II, y) as well. In continuation, Fig. 6(a)

 

Fig. 6 (a) Imaginary part of the effective index versus the wavelength near the phase matching points: λ = 0.629 μm for the core mode (I, x, green), λ = 0.624 μm for the core mode (I, y, red), λ = 0.591 μm for the core mode (III, x, magenta), and λ = 0.59 μm for the core mode (III, y, black) when n_a = 1.33. The (imaginary) effective index for the mode (II, x, blue) is close to that of the mode (I, x, green) at λ = 0.629 μm. (b) Spectral variation of loss near the phase matching points: λ = 0.747 μm for the core mode (I, x, green), λ = 0.735 μm for the core mode (I, y, red), λ = 0.675 μm for the core mode (III, x, magenta), and λ = 0.669 μm for the core mode (III, y, black) when n_a = 1.37. The loss for the core mode (II, x, blue) is close to that of the core mode (I, x, green) at λ = 0.747 μm.

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Fig. 7 (a) Variation of imaginary part of the effective index with wavelength near the phase matching points: λ = 0.596 μm for the mode (II, x, blue) and λ = 0.595 μm for the mode (II, y, cyan) when n_a = 1.33. The dashed lines (blue and cyan) are for the plasmonic modes. These lines show an inflection point at the resonant wavelength for the core modes (II, x) and (II,y). (b) Variation of imaginary part of the effective index with wavelength for the core modes (I, x, green) and (I, y, red) near the phase matching points for two analyte RI values (n_a = 1.33 and n_a = 1.331).

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Figure 8(a)

 

Fig. 8 (a) Variation of imaginary part of the effective index with wavelength near the phase matching points: λ = 0.596 μm for the mode (II, x, blue) and λ = 0.595 μm for the mode (II, y, cyan) when n_a = 1.33. Imaginary part of the effective index for the mode (II, x, blue) at λ = 0.6305 μm is close to that of the mode (I, x, green) at λ = 0.629 μm. (b) Variation of imaginary part of the effective index with wavelength near the phase matching points: λ = 0.692 μm for the mode (II, x, blue) and λ = 0.693 μm for the mode (II, y, cyan) when n_a = 1.37. Apparently, imaginary part of the effective index for the mode (II, x, blue) is close to that of the mode (I, x, green) at λ = 0.747 μm. Also, the corresponding values of (imaginary) effective index for (II, y, cyan) and (I, y, red) modes are close to each other at λ = 0.735 μm. The dashed lines (blue and cyan) are for the plasmonic modes.

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Further simulation results indicate that for n_a ≥ 1.39, the shape of the resonance curves for the modes (II, x) and (II, y) get enlarged due to simultaneous excitation and a strong interaction with other core and plasmon modes such as (II, a) and (II, b).

3.3 Sensor performance analysis

Figures 9(a) and 9(b)

 

Fig. 9 Spectral variation of amplitude sensitivity for (a) guided mode I, and (b) guided mode II when n_a = 1.37. From these plots, it can be observed that the maximum values of the amplitude sensitivity are S_A (I, x) = 215.0 RIU⁻¹ for λ = 0.752 μm, S_A (I, y) = 328.9 RIU⁻¹ for λ = 0.741 μm, S_A (II, x) = 227.7 RIU⁻¹ for λ = 0.744 μm, which is close to the resonant wavelength λ (I, x) = 0.747 μm, and S_A (II, y) = 271.0 RIU⁻¹ for λ = 0.734 μm, which is close to the resonant wavelength λ (I, y) = 0.735 μm.

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The maximum values of S_A for (I, x) and (I, y) modes are 215.0 RIU⁻¹ (at λ = 0.752 μm) and 328.9 RIU⁻¹ (at λ = 0.741 μm), respectively. For (II, x) mode, the maximum S_A is 227.7 RIU⁻¹ at λ = 0.744 μm, which is close to the resonant wavelength λ (I, x) = 0.747 μm. Likewise, for (II, y) mode, the maximum S_A is 271.0 RIU⁻¹ at λ = 0.734 μm, which is close to the resonant wavelength λ (I, y) = 0.735 μm. The S_A for the mode (II, x) increases by a margin of 100.0 RIU⁻¹ when the wavelength increases from 0.700 μm to 0.744 μm. A nearly equal increase (100.0 RIU⁻¹) in S_A is observed for (II, y) mode when the wavelength increases from 0.700 μm to 0.734 μm. Both the above points indicate towards an important role of longer wavelength in sensing performance. For a higher value of n_a (i.e., 1.39), the maximum values of the amplitude sensitivity become S_A (I, x) = 363.6 RIU⁻¹ at λ_A = 0.891 μm and S_A (I, y) = 560.6 RIU⁻¹ at λ_A = 0.875 μm. For (II, x) mode, the maximum S_A is 349.1 RIU⁻¹ at λ_A = 0.880 μm, which is the resonant wavelength for (I, x) mode. Similarly, the maximum S_A for (II, y) mode is 326.0 RIU⁻¹ at λ_A = 0.863 μm, which is the resonant wavelength for (I, y) mode.

The local spectral sensitivities, calculated for an increase of n_a with 0.001, are increased with n_a as following:

Table 1

Table 1. Values of δλ_res [nm], δλ_0.5 [nm], S_A [RIU⁻¹], α [dB/cm], P₁, P₂, Δλ [μm] and λ [μm] corresponding to the core modes (I, x) and (I, y) for three values of the analyte RI (n_a = 1.33, n_a = 1.37 and n_a = 1.39).

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Table 2. Values of δλ_res [nm], δλ_0.5 [nm], S_A [RIU⁻¹], α [dB/cm], P₁, P₂, Δλ [μm] and λ [μm] corresponding to the core modes (II, x) and (II, y) for three values of the analyte RI (n_a = 1.33, n_a = 1.37 and n_a = 1.39).

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Finally, Fig. 10

 

Fig. 10 Spectral variation of imaginary part of the effective index near the loss matching points: (λ = 0.591 μm when n_a = 1.33 and λ = 0.675 μm when n_a = 1.37) for the core mode (III, x, magenta) and (λ = 0.590 μm when n_a = 1.33 and λ = 0.669 μm when n_a = 1.37) for the core mode (III, y, black).

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It is evident from the simulation results that resonant interaction between the modes (I, x) and (III, x) is stronger for a large value of n_a. The same behavior is prevalent for the modes (I, y) and (III, y).

When the light is incident from the center core, multiple supermodes will be simultaneously excited and the supermodes are superimposed on each other, resulting in coupling of light among different cores. The effect is important for the modes (II, x), (II, y), (II, a) and (II, b) when n_a is large where the imaginary parts of these modes are close. Thus for λ (II, x) = 0.692 μm when n_a = 1.37 we obtain (II, x, β/k = 1.444337 + 0.001316i), (II, y, β/k = 1.444012 + 0.001264i), (II, a, β/k = 1.444371 + 0.001297i), and (II, b, β/k = 1.444024 + 0.001320i). The modes (I, x) and (I, y) can be resolved because the modes (I, a) and (I, b) are absent. Thus, the effective indices for the set of the core modes from Fig. 1(b) and λ (I, x) = 0.747 μm when n_a = 1.37 are: (I, x, β/k = 1.445899 + 0.0001825724i), (I, y, β/k = 1.445819 + 0.00007787901i), (II, x, β/k = 1.445320 + 0.0001848001i), (II, y, β/k = 1.445414 + 0.0002952073i), (II, a, β/k = 1.445698 + 0.0003379079i), (II, b, β/k = 1.445551 + 0.0003085699i), (III, x, β/k = 1.445078 + 0.00001439594i), (III, y, β/k = 1.445127 + 0.000008063464i), (III, a, β/k = 1.445123 + 0.0000371104i), (III, b, β/k = 1.445334 + 0.00006953048i). The effective indices for the set of the core modes from Fig. 1(b) and λ (I, y) = 0.735 μm when n_a = 1.37 are: (I, x, β/k = 1.445938 + 0.0001437943i), (I, y, β/k = 1.446262 + 0.0001490737i), (II, x, β/k = 1.446487 + 0.0003009482i), (II, y, β/k = 1.446180 + 0.0003000691i), (II, a, β/k = 1.446387 + 0.000419959i), (II, b, β/k = 1.446252 + 0.0004117052i), (III, x, β/k = 1.445610 + 0.00001603232i), (III, y, β/k = 1.445653 + 0.000008955382i), (III, a, β/k = 1.445663 + 0.00003296468i), (III, b, β/k = 1.445861 + 0.00004292702i.

4. Summary

FEM with the matched boundary conditions is applied to analyze the performance of a plasmonic biosensor based on a birefringent solid-core microstructured optical fiber with three core modes and a plasmonic mode (x- and y- polarizations). The interaction between the central core mode (x- and y- polarizations), which shows up as a supermode, and a plasmonic mode is realized by an intermediate lateral core mode. Thus, the phase and loss matching conditions between a core mode and a plasmonic mode are applicable to the resonant interaction between two core modes. The transmission losses of the lateral core modes are larger than those of the corresponding central core modes for the x- and y- polarization components. A very interesting result is that the maximum value S_A (II, x) = 227.7 RIU⁻¹ of the amplitude sensitivity for the second (lateral) core mode (x- polarization) is at a wavelength λ = 0.744 μm, which is close to the resonant wavelength λ (I, x) = 0.747 μm of the first (central) core mode when the refractive index of the analyte is n_a = 1.37. Also, the maximum value S_A (II, y) = 271.0 RIU⁻¹ of the amplitude sensitivity for the second core mode (y- polarization) is at a wavelength λ = 0.734 μm, which is close to the resonant wavelength λ (I, y) = 0.735 μm of the first core mode. Further, for a higher value of n_a (i.e., 1.39), the maximum values of the amplitude sensitivity are S_A (I, x) = 363.6 RIU⁻¹ for λ_A = 0.891 μm, S_A (I, y) = 560.6 RIU⁻¹ for λ_A = 0.875 μm, S_A (II, x) = 349.1 RIU⁻¹ for λ_A = 0.880 μm, which is the resonant wavelength λ (I, x), and S_A (II, y) = 326.0 RIU⁻¹ for λ_A = 0.863 μm, which is the resonant wavelength λ (I, y). All the core modes, i.e., (I, x), (I, y), (II, x), and (II, y), and plasmonic modes, i.e., (II, x) and (II, y), can be used to describe the sensing characteristics of the optical fiber sensor. However, these modes can be best exploited (Tables 1 and 2) depending on the value of n_a.