Introduction

Surface plasmon resonance (SPR) is an optical sensing technique offering many advantages such as high sensitivity and real-time detection [1]. Combined with biometric elements, SPR sensors can be used in the analysis of biomolecular interactions [2, 3]. However, conventional SPR sensors consist of thin noble metal films deposited on the substrate [4], and the relatively large confinement loss and low penetration depth hamper the detection of biological macromolecules or thick ligand layers. In comparison with the single-mode or multi-mode fibers [5, 6], photonic crystal fibers (PCFs) are regarded to be excellent SPR sensors due to their outstanding characteristics and design flexibility which enables optimization to enhance the interactions between the sample and light [7]. Hence, owing to the small size and high sensitivity, PCF-SPR sensors have attracted much attention in medical monitoring, biochemical analysis, and food testing [8,9,10].

In order to improve the performance of PCF-SPR sensors, several methods have been proposed. The main strategies are modification of the PCF structure and selection of different plasmonic materials. PCF structures with different layouts of air holes and polished optical fibers have been suggested, for example, quasicrystal arrangements [11], elliptical air holes [12], rectangular air holes, and U-shape [13], V-shape [14], D-shape [15], and multi-shape [16] PCFs. Spurred by recent development of coating technology and new materials, PCF-SPR structures are no longer limited to single-layer metals such as gold or silver. Metal oxides like titanium dioxide (TiO2), zinc oxide (ZnO), and magnesium fluoride (MgF2) can be combined with metal layers to form multi-layer coatings to improve the sensing properties.

Singh and Prajapati have proposed a gold/MoS2/graphene D-shape PCF refractive index (RI) sensor [17]. In this device, graphene is deposited to mitigate oxidation and offer better analyte adsorption. Owing to the high surface adsorption efficiency and excellent bandgap tunability of MoS2, the wavelength sensitivity and FOM are 14,933.34 nm/RIU and 401.05 RIU−1, respectively. Liang et al. have analyzed a D-type PCF-SPR refractive index sensor consisting of a silver-graphene/ZnO double-layer composite film [18] and demonstrated sensitivity and resolution of 6000 nm/RIU and 1.667 × 10−5, respectively. Wang et al. have reported a PCF-SPR sensor with Au-Ta2O5 and Ag-Ta2O5 coatings for dual analyte detection [19]. The Ta2O5 film shifts the working wavelength to the near-infrared region and strengthens the penetrating ability, consequently improving the sensing properties. However, the wavelength sensitivity and resolution are only 11,466 nm/RIU and 8.6 × 10−6. Islam et al. have designed Au-TiO2-based PCF-SPR sensors comprising ultra-small air holes for the four coupling channels [20] showing a figure of merit (FOM) of 509.27 RIU−1. In order to simultaneously detect two refractive indexes (RIs), an asymmetric dual RI D-type PCF-SPR sensor has been designed by Wang et al. [21]. In this asymmetric structure, Ag-TiO2 and Ag-ZrO2 are the dielectric materials and the wavelength sensitivity of the dual channel is 11,200 nm/RIU and 14,300 nm/RIU.

In this work, a novel symmetrical D-shape PCF-SPR sensor with Au-MgF2 double-layer film is proposed for refractive index (RI) sensing. The symmetrical D-shape PCF can enhance the coupling between the core mode and SPP mode; meanwhile, the novel Au-MgF2 double-layer film structure can effectively narrow the full width at half maximum (FWHM) of the loss spectrum and then improve the resolution and FOM of the sensor. The sensor is numerically simulated by the finite element method (FEM) using the COMSOL Multiphysics software. A circular perfect matching layer (PML) is employed as absorbing boundary condition, and the free triangle meshes are adopted in the simulation. A maximum wavelength sensitivity of 59,000 nm/RIU and amplitude sensitivity of 6076 RIU−1 are demonstrated.

Design and Theory

Sensor Structure

Figure 1a depicts the sectional view of the PCF-SPR sensor, and the PCF structure is depicted in Fig. 1b. The radius R of the PCF is 5.5 μm. The central air hole is removed to form the fiber core, while the four air holes in the first layer are replaced by two air holes to expand the coupling channel between the core and plasmonic film. Then, the PCF is designed as double-side D-shaped structure, which can be produced by polishing, and the height h is 4.2 μm. The MgF2 can better adhere to the surface of Au film, increases the penetration depth of electric field in the sensing medium, and enhances the interaction between plasmons and analyte, thus improving the performance of the proposed PCF-SPR sensor. The Au-MgF2 film with a total thickness of t = 44 nm and width of w = 7 μm can be deposited by magnetron sputtering or chemical vapor deposition (CVD) [22]. The distance Λ between two adjacent air holes is 2.2 μm, and the diameter of the uniform air hole d is 1.1 μm. The outer layer is the perfectly matched layer (PML) to absorb incident light.

Fig. 1
figure 1

a Cross-section of the PCF-SPR sensor and b structure of PCF

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The sensor is made of fused silica, and the dispersion relationship is given by the Sellmeier equation [23]:

$$n(\lambda ) = \sqrt {1 + \frac{{0.6961663\lambda^{2} }}{{\lambda^{2} - 0.0684043^{2} }} + \frac{{0.4079426\lambda^{2} }}{{\lambda^{2} - 0.1162414^{2} }} + \frac{{0.8974794\lambda^{2} }}{{\lambda^{2} - 9.896161^{2} }}}$$
(1)

where λ is the wavelength of the incident light in vacuum. Au is considered the ideal plasmonic material due to its high chemical stability. The complex dielectric permittivity can be calculated using the Drude-Lorentz model [24]:

$$\epsilon_{{{\text{Au}}}} = \epsilon_{\infty } - \frac{{\omega_{{\text{p}}}^{2} }}{{\omega^{2} + i\omega \omega_{\tau } }}$$
(2)

where ωp = 1.36 × 1016 (rad/s) and ωτ = 1.45 × 1014 (rad/s) are the plasmonic frequency and scattering frequency of electrons, respectively. The dielectric constant ε∞ at high frequencies is 9.75, ω = 2πc/λ stands for the angular frequency of incident light, and εAu is the permittivity of the gold film. Therefore, the complex dielectric permittivity εAu changes with the incident wavelength. The refractive index of MgF2 is described by [25]

$$n_{{{\text{MgF}}_{{2}} }} (\lambda ) = \sqrt {1 + \sum\limits_{i = 1}^{3} {\frac{{C_{i} \lambda^{2} }}{{\lambda^{2} - \lambda_{i}^{2} }}} }$$
(3)

where C1 = 0.48755108, C2 = 0.39875031, C3 = 2.3120353, λ1 = 0.04338408, λ2 = 0.09461442, and λ3 = 23.793604.

Analysis of the Sensor

The confinement loss affects the PCF-SPR sensor and is defined by the following equation [10]:

$$\alpha_{{{\text{CL}}}} (dB/cm) = \frac{20}{{\ln (10)}} \times \frac{2\pi }{\lambda } \times {\text{Im}} (n_{{{\text{eff}}}} ) \times 10^{4}$$
(4)

where Im(neff) refers to the imaginary part of the effective RI. Figure 2a shows the dispersion relationship and confinement loss spectra of the sensor when the analyte RI is na = 1.40. The red solid curve and dashed curve represent the real part of the effective RI of the y-polarized core mode and SPP mode, respectively. At the intersection point, the phase-matching conditions for surface plasmon resonance are satisfied, and the energy of the core mode transfers gradually to the SPP mode as shown in Fig. 2b. Therefore, the confinement loss spectrum shows a loss peak. The black solid line describes the confinement loss of the y-polarized fundamental mode, and similarly, the confinement loss of the x-polarized fundamental mode is shown by the blue solid line. The SPR effect of y-polarization is stronger than that of x-polarization, and more energy couples to SPP mode. Therefore, the electric field intensity of y-polarization is smaller than that of x-polarization, as shown in Fig. 2b and c, and y-polarization is chosen in our subsequent analysis.

Fig. 2
figure 2

Dispersion relationship of the core mode and SPP mode for the analyte RI na = 1.40: a confinement loss spectra, b electric field distribution of y-polarization, and c electric field distribution of x-polarization at the phase-matching point

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Minor variations in the analyte refractive index change the effective refractive index of the SPP mode resulting in a shift in the phase-matching point. It means that the loss spectrum will change with the analyte refractive index. The resonant peak wavelength of the loss spectrum can be used to measure the refractive index of the analyte. As shown in Fig. 3a, the loss spectra exhibit redshift as the refractive indexes of the analyte increase. Since the core mode is confined in the fiber core, its effective refractive index is not influenced by the analyte. But the real part of the effective indexes of the SPP mode increases with increasing analyte RIs. Therefore, the phase-matching intersection shifts to a longer wavelength. The fitted polynomial curve of the resonant wavelength versus analyte RI is depicted in Fig. 3c. It can be seen that the resonance wavelengths increase slowly in the RI range of 1.25–1.42 and then increase rapidly. This is because the refractive index difference between the core mode and SPP mode decreases significantly with the increase of analyte RI, resulting in the dramatic increase of the resonance wavelength [26]. In addition, the limitation of the PCF structure on the core mode decreases with increasing wavelengths, leading to enhanced surface plasmon resonance effects. Therefore, the peak loss increases gradually. The wavelength offset of the loss peak is utilized to estimate the wavelength sensitivity of the sensor by the following relationship [27]:

$$S_{{\text{w}}} (nm/RIU) = \frac{{\Delta \lambda_{{\text{p}}} }}{{\Delta n_{{\text{a}}} }}$$
(5)

where Δna is the change of the analyte RI and Δλp is the wavelength shift of the loss peak. It is noted that the bimodal spectrum appears as RI = 1.43. The two peaks are caused by complete coupling and incomplete coupling between the core mode and SPP mode [28]. The resolution of the PCF-SPR sensor is defined as follows [29]:

$$R = \frac{{\Delta n_{{\text{a}}} \times \Delta \lambda_{{{\text{min}}}} }}{{\Delta \lambda_{{\text{p}}} }}$$
(6)

where Δλmin is the minimum wavelength resolution which is set to be 0.1 nm.

Fig. 3
figure 3

a Confinement loss spectra and b amplitude sensitivity for different analyte RIs. c Fitted polynomial curve of the resonance wavelengths and FOM for different analyte RIs

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The amplitude interrogation method is adopted to determine the properties of the sensor, and the expression is shown in the following [30]:

$$S_{{\text{a}}} (RIU^{ - 1} ) = - \frac{1}{{\alpha_{{{\text{CL}}}} }} \times \frac{{\Delta \alpha_{{{\text{CL}}}} }}{{\Delta n_{{\text{a}}} }}$$
(7)

where αCL is the loss at the wavelength and ΔαCL is the loss difference between two adjacent RIs. The amplitude sensitivity for RIs between 1.25 and 1.41 is shown in Fig. 3b, and it can be inferred that the maximum amplitude sensitivity is 6076 RIU−1 in the RI range of 1.41–1.42.

The figure of merit (FOM) is commonly used to assess the detection accuracy of sensors and is expressed by the following relationship [31]:

$$FOM(RIU^{ - 1} ) = \frac{{S_{{\text{w}}} }}{FWHM}$$
(8)

where FWHM is the full width at half maximum of the loss spectrum. It means that higher wavelength sensitivity and narrower linewidth represent better sensing properties. The FOM curve for different RIs is displayed in Fig. 3c. The PCF-SPR sensor exhibits higher FOM that reaches 2033 RIU−1 as the RI is changed from 1.42 to 1.43.

Optimization and Discussion

The structural parameters of the coating materials Au-MgF2 influence the properties of the sensor. The thicknesses of the Au and MgF2 films are first optimized. For \(t_{{MgF_{2} }}\) = 14 nm and for Au film thicknesses tAu between 20 and 40 nm, the wavelength sensitivity and FOM are calculated for RI = 1.42 ~ 1.43. As shown in Fig. 4a, the wavelength sensitivity and FOM reach the maximum values when tAu = 30 nm. Similarly, Fig. 4b exhibits the wavelength sensitivity and FOM for different \(t_{{MgF_{2} }}\) of the MgF2 films with thicknesses between 10 and 18 nm. The properties of the sensor are excellent when \(t_{{MgF_{2} }}\) = 14 nm. Because a thicker film weakens the coupling between the core mode and SPP mode [32], tAu = 30 nm and \(t_{{MgF_{2} }}\) = 14 nm are selected as the optimal thicknesses.

Fig. 4
figure 4

Wavelength sensitivity and FOM for different thicknesses: a Au layer and b MgF2 layer

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Figure 5a shows the loss spectra for RI = 1.42 (dashed lines) and RI = 1.43 (solid lines) as the width w of the coating layer is changed from 5.0 to 7.0 μm. When the width increases, the resonance peak of the loss spectra shifts toward a longer wavelength for na = 1.43, but the resonance wavelength remains basically unchanged for na = 1.42. Hence, the wavelength sensitivity is improved with increasing w. Figure 5b shows the wavelength shift Δλ and FOM, and w = 7 μm is the optimal width of the coating.

Fig. 5
figure 5

a Loss spectra for different coating widths and na = 1.42–1.43 and b resonance wavelengths of the loss spectra for na = 1.42–1.43 and FOM for na = 1.42

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For RIs of 1.42 and 1.43, the loss spectra for different air hole pitches Λ are shown in Fig. 6a and b. The wavelength sensitivity decreases with increasing Λ, and the FOM reaches the maximum of 1668 RIU−1 at Λ = 2.2 μm. Therefore, the optimal air hole pitch is determined to be 2.2 μm. The impact of the air hole diameter d on the properties of the sensor (RI = 1.42 ~ 1.43) is illustrated in Fig. 7. With increasing diameter d, the FWHM of the loss spectra widens and the intensity of the resonance coupling decreases when na = 1.42. It is because a larger diameter d gives rise to better confinement to the core mode. Figure 7b compares the wavelength sensitivity and FOM, and the sensor has the best properties when d = 1.1 μm.

Fig. 6
figure 6

a Loss spectra for different air hole pitches and na = 1.42–1.43 and b resonance wavelengths of the loss spectra for na = 1.42–1.43 and FOM for na = 1.42

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Fig. 7
figure 7

a Loss spectra for different air hole diameters and na = 1.42–1.43 and b resonance wavelengths of the loss spectra for na = 1.42–1.43 and FOM for na = 1.42

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On the basis of the aforementioned analysis, the optimal parameters of the PCF-SPR sensor are determined as follows: w = 7 μm, h = 4.2 μm, Λ = 2.2 μm, d = 1.1 μm, tAu = 30 nm, and \(t_{{MgF_{2} }}\) = 14 nm. The properties of our sensor are compared with those of similar sensors, as shown in Table 1. It can be seen that the sensing structure and coating materials directly influence the performance. Compared with other sensors, our sensor shows a maximum wavelength sensitivity of 59,000 nm/RIU and FOM of 2033 RIU−1, which are better than those reported recently.

Table 1 Comparison of the properties of our sensor and those containing multi-layer coatings reported recently
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Conclusion

A PCF-SPR sensor with a dual Au-MgF2 layer is designed and analyzed by the finite element method. The maximum wavelength sensitivity and amplitude sensitivity are 59,000 nm/RIU and 6076 RIU−1, respectively, in the refractive index range of 1.25–1.43. Moreover, the figure of merit (FOM) is improved to 2033 RIU−1. The sensor can thus detect smaller changes in the refractive index of the analyte with a resolution of 1.69 × 10−6 RIU. This PCF-SPR sensor has large application potential in RI sensing, and the strategy also provides insights into the future development of RI sensors with high resolution.