A theoretical study of a plasmonic sensor comprising a gold nano-disk array on gold film with a SiO₂ spacer^*

Figure 2(a) shows the reflection spectrum of the structure simulated by the FDTD method in a wide wavelength range, from 500 nm to 2000 nm. In this case, the diameter of the nano-disks is 280 nm, the period is 720 nm, and the background index (analyte refractive index) is 1.30. As shown in the figure, three main resonant reflection modes can be found in the curve, located at 740 nm (mode 1), 964 nm (mode 2), and 1453 nm (mode 3). It is apparent that the surface plasmon polaritons can be effectively excited in this composite structure. The incident energy is coupled to LSP via single nano-disks, which interact with adjacent disks both in the upper space and through PSP. In addition, the incident energy is transferred to PSP via the two-dimensional (2D) grating. Furthermore, coupling between LSP and PSP enhances the energy of incident light transfer to SPPs.^[53] The strong coupling results in very low reflectivity of the reflection spectrum at resonant wavelength. The nano-disk array acts as a 2D grating, which provides additional momentum and couples the incident plane wave into PSP. The additional momentum is given by the following equation^{[54, 55]}

$$\begin{eqnarray}&&G=\displaystyle \frac{2\pi }{d}\sqrt{{n}^{2}+{m}^{2}},\end{eqnarray} \tag{ 2 }$$

where d is the period constant and n and m are integers. As shown in Fig. 2(a), mode 1 and mode 2 correspond to (1, 1) and (1, 0), respectively, and localized surface plasmons are predominant in the last resonant mode.^[54]

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To further examine the structural features, we calculated the electric-field distribution of mode 2 and mode 3 on the xz plane, along the diameter of the nano-disk. As shown in Figs. 2(b) and 2(c), the existence of SPPs is clearly indicated, and the electric-field enhancement in modes 2 and 3 is caused by strong coupling between LSP and PSP. It is noteworthy that both mode 2 and mode 3 are the result of LSP and PSP coupling. The main difference between them is that mode 2 is dominated by PSP, whereas mode 3 is dominated by LSP.

This is how our structure works as a reflective index sensor. The change in the analyte RI on top of the arrays causes a resonance peak shift, and the analyte RI can be sensed by detecting the position of the resonance peak. The RI sensitivity and the FOM are often used for quantifying the performance of a reflective index sensor. A good RI sensor should have both high sensitivity and a high FOM. The RI sensitivity is defined as $S={\rm{\Delta }}\lambda /{\rm{\Delta }}n$ (${\rm{\Delta }}\lambda$ is the shift of the resonance peak wavelength and ${\rm{\Delta }}n$ is the refractive index change of the analyte) and the FOM is defined as FOM = S/FWHM. As shown in Fig. 2(a), mode 1 is insufficient for RI sensing due to its weak resonance intensity (the reflectivity is around 0.5), compared with other resonance modes. Modes 2 and 3 have almost the same strong resonance intensity (the reflectivity is around 0). However, the FWHM of modes 2 and 3 is approximately 9 nm and 122 nm, respectively. Considering the relationship between the FOM and the FWHM (a small FWHM results in a high FOM), it is reasonable to choose mode 2 as the sensing peak, as it has the narrowest line width and therefore the better resonance intensity.

Thus, we selected mode 2 to be our research object and we examine the near infrared wavelength range from 900 nm to 1100 nm (where mode 2 belongs), to study the influence of the analyte RI on the resonance mode. Six reflection spectra with different analyte RI and the same structure parameters are presented in Fig. 3(a). From these curves, we find that the resonance peaks show red shifts as the RI increases, and all the spectra have almost the same FWHM and dip depths. This is because a slight change in the reflective index does not affect the resonance mechanism of surface plasmon polaritons in the structure. A good linear relationship (between the position of mode 2 and the RI of the analyte) is observed in Fig. 2(b). In order to calculate the sensitivity and the FOM, we plot the position of mode 2 as a function of the analyte RI in Fig. 3(b). The sensitivity and the FOM are calculated to be 690 nm/RIU and 76.6 RIU⁻¹, respectively, for a period of 720 nm and a diameter of 280 nm. The FOM of this plasmonic sensor is quite high compared to those of other plasmonic RI sensors^[24–29] because of the narrow line width. However, this is not the best possible performance of the sensor. The sensor can be optimized by regulating some of the structure parameters to achieve a higher sensitivity, as discussed below. Thus, our sensor does not have strict structure parameter requirements and is able to work in a relatively rough process.

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When the period is fixed, the resonance peaks change approximately linearly with the RI of the analyte and the sensitivity is fixed, according to the definition of sensitivity. Therefore, we investigated the sensitivity of plasmonic sensors of different periods, to determine the relationship between them. An interesting phenomenon was observed. When the period was increased, the position of the resonance wavelength (mode 2) changed evenly with the red shift. This provides a way of controlling the RI sensor. By changing the period to an appropriate size, we can choose the infrared band needed to detect the RI of the analyte, based on the parameters of the light source. The sensitivity of the plasmon sensor as a function of the period is shown in Fig. 4(a), with the diameter of the nano-disks kept at 280 nm. It can be observed that an increase in the periodicity leads to an increase in sensitivity. The RI sensitivity can reach 853 nm/RIU at a period of 900 nm. The FOM of the plasmon sensor as a function of the period is shown in Fig. 4(b). As the period increases, the FOM increases in nonlinearity and a high value of FOM = 126 RIU⁻¹ is achieved when the period is 900 nm. From the formula for the FOM, we can extract two explanations for this increase in the FOM: an increasing RI sensitivity and a decreasing FWHM. These two reasons are both related in this plasmon sensor.

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It seems that we can get a higher sensitivity by applying a larger period, but in fact, as we continue to increase the period, mode 2 moves to a longer wavelength and ultimately interacts with mode 3. Reflection spectra for the same nano-disk diameter of 280 nm and background RI (analyte) of 1.30, with varied periods, are shown in Fig. 4(c). Figure 4(b) clearly indicates that mode 2 is sensitive to the period, whereas mode 3 barely changes. It also clearly demonstrates how mode 2 interacts with mode 3, when mode 2 and mode 3 have almost the same resonance wavelength. If we continue to increase the period, an anti-crossing behavior can be observed from the spectra.^[54] As shown by the blue dashed line in Fig. 4(c), when the period reaches 1100 nm, the FWHM of mode 2 is obviously increasing, and the reflectivity is not close to zero, unlike for other periods under 1100 nm. The same problem becomes more palpable when we increase the period to 1150 nm, as shown by the solid black line in Fig. 4(c). In others words, when modes 2 and 3 interact with each other, the advantages of mode 2 for RI sensing (low reflectivity and a narrow FWHM) are lost. Accordingly, although increasing the period can bring higher RI sensitivity, this structure does not provide the same advantages as a plasmonic RI sensor when the period is too large.

Another important geometric parameter of this structure is the diameter of the nano-disks. With the period fixed at 720 nm, the RI sensitivity as a function of the diameter of the nano-disks is shown in Fig. 5(a), with the diameter of the nano-disks increasing from 200 nm to 360 nm, in step sizes of 20 nm. A slight fluctuation in the RI sensitivity can be observed, but ultimately the diameter of the nano-disks has little effect on the RI sensitivity. This further demonstrates that our structure can work as a plasmonic RI sensor even in a relatively rough process. Figure 5(b) shows the FOM as a function of the diameter of the nano-disks. The FOM reaches the highest value of 85 RIU⁻¹ when the diameter of the nano-disks is 240 nm. This may be attributed to the enhancement factor reaching a maximum when the diameter is 240 nm.^[55] Strong resonance leads to a narrow FWHM and a high FOM.

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