Introduction

In recent years, the immense development of a new type of photonic crystal fiber (PCF) with an array of air holes along the propagation direction has more attractive research work going worldwide. PCFs can be divided into a two broad classes according to whether they use index guiding (guides light by total internal reflection (TIR) between a high-refractive-index core and a low-refractive-index cladding region) or band gaps (guides light by the photonic band gap in a low-refractive-index core region) for optical confinement. PCF sensors have more attractive applications like optical sensors [1, 2], surface plasmon resonance (SPR) sensors [3,4,5,6,7,8], biosensor [9], humidity sensor [10], optical amplifier [11], temperature sensor [12] and pressure sensor [13]. PCF sensors are now a newfangled subject of field study due to the various advantages offered by PCFs such as high confinement compare to conventional optical fiber, flexible design and limit the distortion [14]. The theoretical concept of surface plasmons was introduced by Ritchie [15]. Using a related concept, Otto [16] and Kretschmann [17] have established sensing-based techniques. Surface plasmon is a near-field electromagnetic interaction formed on the surface of the metal and dielectric. When light waves are incident to the metal and the dielectric interface, a collective oscillation of free electrons on the surface of the metal occurs, and the electromagnetic waves couple to the free electron; then, surface plasmon forms. For example, if the frequency of electron oscillation and the frequency of the incident wave are consistent, it will produce resonance. Sensor with gold-coated core has been developed in optical fiber, which excites surface plasmon [18]. Resonance peak will be shifted at phase matching condition. Particular wavelength is called phase matching wavelength when phase matching occurs it. Loss spectrum of SPP and fundamental mode will match at phase matching wavelength. This is the basic condition for sensing. In recent times, many researchers have to design photonic crystal fiber SPR sensors, because these sensors offers high sensitivity, structural reduction, robust, fabrication flexibility and lower cost [19]. Two-dimensional photonic lattices of gold and silver nanowires were developed and fabricated [20].

In this paper, surface plasmon resonance photonic crystal fiber biosensor is proposed. Gold-coated air hole is placed in the center of PCF. Gold is acted as a sensing layer. Next to the center air hole is filled with unknown analyte. This analyte is reacting with the surface plasmon. This leads to phase matching condition. Gold is the suitable plasmonic material compared to silver, because it has high resonance peak shift [21]. PCF with boundary condition is simulated using finite element method.

Structure and theory

The simplified graphic representation of the designed photonic crystal fiber is shown in Fig. 1. The diameter of small air holes (d1) and large air holes (d2) are 1 μm and 1.6 μm, respectively. Pitch can be fixed with the value of 2 μm.

Fig. 1
figure 1

Simplified picture representation of the designed PCF structure

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Blue shaded air hole is filled with gold nanowire. Cladding is filled with the material silica. Red shaded air hole is filled with analyte. Other holes are filled with air. Electric field is applied to the PCF, i.e., light is pass through the fiber; metal–dielectric interface leads to surface plasmon generation. Phase matching occurs means resonance peak will be shifted. Fundamental mode and SPP mode has high-loss spectrum. Due to this phase matching condition, maximum loss occurs at particular wavelength [22].

Coupling between SPP and core modes is represented by coupling theory [23]:

$$ \frac{{{\text{d}}E_{1} }}{{{\text{d}}z}} = i\beta_{1} E_{1} + i\kappa E_{2} $$
(1)
$$ \frac{{{\text{d}}E_{2} }}{{{\text{d}}z}} = i\kappa E_{1} + i\beta_{2} E_{2} $$
(2)

Here, E1 and E2 are mode fields of the core guided mode and plasmon mode; β1 and β2 are propagation constants of guided mode and plasmon mode, respectively. Direction of the propagation is along z direction. Strength of coupling is denoted by k. Difference in propagation constant of core and plasmonic mode is zero when phase matching occurs. To reduce the energy loss, PML (perfectly matched layer) is applied to the structure.

Pure silica is used as background material, and wavelength-dependent Sellmeier equation of the material is [24]

$$ n\left( \lambda \right)^{2} = 1 + \frac{{A_{1} \lambda^{2} }}{{\lambda^{2} - \lambda_{1}^{2} }} + \frac{{A_{2} \lambda^{2} }}{{\lambda^{2} - \lambda_{2}^{2} }} + \frac{{A_{3} \lambda^{2} }}{{\lambda^{2} - \lambda_{3}^{2} }} $$
(3)

Here, A1 = 0.696 1663, A2 = 0.407 9426, A3 = 0.897 479, λ1 = 0.068 404, λ2 = 0.116 2414 and λ3 = 9.896 161 are the coefficients for the material, and operating wavelength is \(\lambda\). Similarly, the material dispersion of gold is also categorized by the Drude–Lorentz model [22] and could be stated as:

$$ \varepsilon \left( \omega \right) = \varepsilon_{\infty } - \frac{{w_{p}^{2} }}{{w^{2} + i\Gamma_{p} w}} - \frac{{f_{1} w_{1}^{2} }}{{w^{2} - w_{1}^{2} - i\Gamma_{1} w}}. $$
(4)

Here, w is angular frequency. At high frequency, the permittivity of gold is ε∞. Plasma frequency and damping frequency by Drude model are wp and Γp. f1 is the weighting factor. Lorentz oscillator has the frequency is w1, and spectral width is Γ1. Table 1 lists the parameters and values of gold dispersion equation.

Table 1 Parameters and their values of gold dispersion equation
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The wavelength sensitivity is calculated by using wavelength interrogation method [25].

$$ S(\lambda ) = \frac{{\partial \lambda_{{{\text{peak}}}} }}{{\partial n_{{\text{a}}} }}\;({\text{nm/RIU}}) $$
(5)

Here, \( \partial \lambda_{{{\text{peak}}}}\) depends on the analyte refractive index. At resonance condition, loss peaks are high. Different loss peaks are occurring for different filled analyte refractive indices. \(\partial n_{{\text{a}}}\) is the analyte refractive index difference. Amplitude sensitivity of the sensor is calculated by using the following equation. This technique of calculation is called amplitude interrogation technique [26].

$$ S_{A} (\lambda ) = - \frac{1}{{\alpha \left( {\lambda , n_{{\text{a }}} } \right)}}\left( {\frac{{\partial \alpha \left( {\lambda , n_{{\text{a }}} } \right)}}{{\partial n_{{\text{a}}} }}} \right)\,({\text{RIU}}^{ - 1} ) $$
(6)

Here, \(\alpha \left( {\lambda , n_{{\text{a }}} } \right)\) is the confinement loss of the fundamental core mode and depends on wavelength. \(\partial \alpha \left( {\lambda , n_{{\text{a }}} } \right)\) is the difference of confinement loss.

The following equation represents the resolution (R) of sensor [27, 28]

$$ R = \frac{{\partial n_{{\text{a}}} \cdot \partial \lambda_{\min } }}{{\partial \lambda_{{{\text{peak}}}} }} $$
(7)

Dual-core PCF-based SPR sensor is proposed in this paper. Finite element method (FEM) is used to study the proposed sensor. Perfectly matched layer (PML) with PCF structure is used to avoid the energy loss. Analyte refractive index is taken from 1.33 to 1.35. In addition, thickness of gold layer is varied for obtaining the better performance of the sensor.

Numerical Results and Discussion

Figure 2 shows the optical field distribution of the proposed sensor. It is clear that the loss of y-polarized mode is higher so that maximum energy loss arises at y-polarization compared to x-polarization and it is the evident for maximum energy transferred to cladding region. One more is that x-polarization takes place when more energy is accumulated in core region. Figure 2e demonstrates the real part of effective index of core mode, and their losses are depend on wavelength. Confinement loss is calculated by using the formulae as stated in [27].

$$ C_{{\text{L}}} = 8.686 \times \left( {\frac{2\pi }{\lambda }} \right) \cdot {\text{Im}} \left( {n_{{{\text{eff}}}} } \right) \times 10^{4} \,\left( {\text{dB/cm}} \right) $$
(8)
Fig. 2
figure 2

Optical field distribution of the proposed dual-core PCF ax-polarized and cy-polarized fundamental mode, bx-polarized, dy-polarized SPP mode and e confinement loss for y-polarized mode at analyte RI of 1.33 with tg = 0.2 μm

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Here, imaginary part of effective refractive index is Im (neff) and \(\left( {\frac{2\pi }{\lambda }} \right) = k_{0}\) is the free space wave number. Two regions will act as a core due to the RI variation between the center and surrounded cladding (Table 2).

Table 2 Performance analysis comparison of the proposed sensor with existing sensors
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For x-polarization, the maximum loss occurs at wavelength of 1.45 μm, 1.55 μm and 1.6 μm for the analyte refractive index of 1.33, 1.34 and 1.35, respectively. Confinement loss is high at 1.5 μm, 1.55 μm and 1.6 μm for the RI of analyte of 1.33, 1.34 and 1.35, respectively, for y-polarization. Wavelength sensitivity is 10,000 nm/RIU and 5000 nm/RIU for the RI of analyte is 1.33 and 1.34, respectively, for x-polarization. Wavelength sensitivity is 5000 nm/RIU and 5000 nm/RIU for the corresponding analyte RI of 1.33 and 1.34, respectively, for y-polarization (Fig. 3).

Fig. 3
figure 3

Loss spectrum of ax-polarized, by-polarized modes with tg = 0.2 μm

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In Fig. 4, the amplitude sensitivity is 235 RIU−1, 271 RIU−1 and 334 RIU−1 for analyte RI of 1.33, 1.34 and 1.35, respectively, for x-polarization. For analyte RI of 1.33, 1.34 and 1.35, amplitude sensitivity is 305 RIU−1, 340 RIU−1 and 394 RIU−1, respectively, for y-polarization. The technique to calculate amplitude sensitivity is called amplitude interrogation technique (Fig. 5).

Fig. 4
figure 4

Amplitude sensitivity of ax-polarized, by-polarized modes with tg = 0.2 μm

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Fig. 5
figure 5

Loss curve of ax-polarized, by-polarized modes with tg = 0.3 μm

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The proposed biosensor has reasonable resolution with good sensitivity.

Conclusion

SPR biosensor-based dual-core photonic crystal fiber (DC-PCF) has been investigated in this paper. Investigation of the sensor is done by using finite element method. From the simulation result, wavelength sensitivity is obtained around 10,000 nm/RIU and 5000 nm/RIU with analyte refractive index of 1.34 for x-polarization and y-polarization. 334 RIU−1 and 394 RIU−1 are the amplitude sensitivity for x-polarization and y-polarization, respectively. Resolution for both polarizations is 2 × 10–5 RIU. So, the proposed biosensor can detect the analyte of low refractive index at different wavelengths.