1. Introduction

Surface plasmon polaritons (SPP) are transverse magnetic (TM) polarized optical surface waves that propagate along a metal–dielectric interface with fields that peak at the interface and decay away exponentially into both medium [1]. Recently, optical devices based on SPP have attracted much attention because of their ability of confining and guiding electromagnetic waves in subwavelength scales, which allow for the manipulation and guidance of optical signals in very compact, fast and power-efficient devices. Among these devices, SPP sensors have been receiving continuously growing attention due to their high sensitivity to the variations in the optical properties of the dielectric adjacent to the metal layer. So far, a number of SPP sensors based on various detection schemes, e.g., angular interrogation [2,3], wavelength interrogation [4–6], or intensity measurement [7–9], have been reported.

To date, SPP modes supported by metal slabs, such as long-range SPP (LRSPP) mode and short-range SPP (SRSPP) mode in insulator-metal-insulator (IMI) slab, have been studied thoroughly [10–12]. The losses of the LRSPP mode in a symmetric IMI slab can be infinitely small, but it also requires the metal layer to be infinitely thin, which is not practical in application. The LRSPP mode in an asymmetric IMI slab can exhibit infinitely small losses with finite metal layer thickness, since the fields approach those of a plane wave in the higher index medium that compensates for dissipation in the metal [11]. But the losses are not sensitive to the insulator refractive index change, which limits the application of the structure as SPP sensory devices.

In this work, the dispersion relation of an air-silver-silicon-silver-fluid (air-Ag-Si-Ag-fluid) five-layer slab is analyzed theoretically. Some special SPP modes whose energy penetrates deeply (typically, tens of microns or more) into the fluid, called as the super-long range SPP modes, are found with their losses being extremely small and sensitive to the fluid refractive index change as the modes operate near their interspace cut-off regions, where the dispersion curves are non-continuous. This phenomenon can be applied to sensory devices with the refractive index detection. Therefore, a SPP sensor based on intensity measurement for fluid refractive index detection is proposed in this study. It exhibits large scale of linear detection (e.g., 0.08, for 1550 nm ~1.33 to 1.41), high resolution (e.g., 7.9 × 10⁻⁶ Refractive Index Units (RIU)) and compact in size of 200 μm in length.

2. Super-long range surface plasmon polaritons modes and Sensor design

2.1 Super-long range surface plasmon polaritons modes

Our SPP sensor consists of an Ag-Si-Ag slab and a flow cell filled with the detecting fluid on top of the waveguide, as shown in Fig. 1 . The two Ag layers have the same thickness of 30 nm, with the waveguide Si layer sandwiched between them. The variable d_Si and n_D denote the Si layer thickness and the refractive index of the detecting fluid, respectively. The working wavelength is fixed at 1550 nm with the Si refractive index of 3.518 and the Ag permittivity of −86.6424 + 8.7422i [13]. The SPP modes are propagating along the z axis with the device length of L. The sensor can be thought as an air-Ag-Si-Ag-fluid five-layer slab in x-z plane since it is thick enough in y axis.

 

Fig. 1 The geometry of the SPP sensor consisting of an Ag-Si-Ag slab and a flow cell filled with the detecting fluid.

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Optical waveguide theory [14] and transfer matrix method are applied to calculate the complex effective index of our five-layer slab, with its real part and imaginary part representing the mode propagation constant and losses, respectively. As SPP are TM waves which contain three electromagnetic component-H_y, E_x and E_z, we solve the scalar Helmholtz equation for TM mode in each layer as the following:

In formula (2), ω is the angular frequency. E_y, H_x and H_z components vanish for TM mode. By applying the continuity of the tangential components H_y and E_z in the boundary [14], we obtain the relations between each adjacent layer. Then the total transfer matrix M can be obtained via applying the transfer matrix method. By solving the equation |M| = 0, the complex effective index is obtained.

We choose water first as our detecting fluid with refractive index of 1.33. Figure 2 and Fig. 3 illustrate the dispersion relations of our five-layer slab with different Si layer thicknesses ranging from 0 nm to 800 nm. The real part of the mode complex effective index, which we call the mode index below, is illustrated in Fig. 2. This figure illustrates that the five-layer slab supports more TM modes as the Si layer thickness increases. As the Si layer thickness approaches 0 nm, TM₁ and TM₂ modes exhibit the mode index approaching water refractive index and air refractive index, respectively. These results suggest that an Ag/water interface SPP and an Ag/air interface SPP are formed, respectively. When the Si layer is thicker, higher order modes will form the Ag/water and the Ag/air interface SPP instead of TM₁ and TM₂ modes, with their mode indices approaching water refractive index and air refractive index, respectively. However, these SPP modes exhibit some differences between their fields in the Si layer. Figures 4(a) -4(d) show the z direction normalized energy flows-S_z component field of four points marked in Fig. 2 with a (TM₂ mode with d_Si = 300 nm), b (TM₄ mode with d_Si = 700 nm), c (TM₃ mode with d_Si = 300 nm) and d (TM₅ mode with d_Si = 700 nm), respectively. It is found from Figs. 4(a) and 4(b) that the mode energy of points a and b are focused at the Ag/water interface, while some differences appear that the mode of point b exhibits more standing waves in the Si layer because of a thicker Si layer (point b with d_Si = 700 nm to point a with d_Si = 300 nm). The same phenomenon can also be found when comparing the field of point c to point d. These results show that the Ag/water interface SPP and Ag/air interface SPP formed by higher order modes exhibit more standing waves in the Si layer.

 

Fig. 2 The real part of the complex effective index with different Si layer thicknesses of the air-Ag-Si-Ag-water five-layer slab, both the Ag layer thicknesses are fixed at 30 nm. Three so-called interspace cut-off regions, which represent the discontinuous regions of the dispersion curves, are marked with A, B and C, respectively.

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Fig. 3 The imaginary part of the complex effective index with different Si layer thicknesses of the air-Ag-Si-Ag-water five-layer slab, both the Ag layer thicknesses are fixed at 30 nm. Three so-called interspace cut-off regions, which represent the discontinuous regions of the dispersion curves, are marked with A, B and C, respectively.

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Fig. 4 Normalized S_z field of (a) TM₂ mode with d_Si = 300 nm (b) TM₄ mode with d_Si = 700 nm (c) TM₃ mode with d_Si = 300 nm (d) TM₅ mode with d_Si = 700 nm in the air-Ag-Si-Ag-water five-layer slab. The four cases are also marked in Fig. 2 as points a, b, c and d, respectively.

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It is important to note that, as the Si layer thickness increases, an unique phenomenon appears in which the dispersion curves of higher order modes are non-continuous: TM₂ mode appears to be cut-off around the mode index close to the water refractive index where we name as the interspace cut-off region (see region A in Fig. 2 and Fig. 3), while other higher order modes (e.g., TM₃ mode) have two interspace cut-off regions around the mode index close to the air refractive index and the water refractive index (see region B and C in Fig. 2 and Fig. 3), respectively. Here in these calculations, ‘interspace’ means the space where the dispersion curves are non-continuous, and we regard ‘cut-off’ as the region where the mode do not have solutions with positive imaginary effective index. But the mode may have solutions with negative imaginary part of effective index which require growing waves and we will discuss this more thoroughly later. To the best of our knowledge, the so-called interspace cut-off region has not been reported before. The possible origins of the interspace cut-off region lie in the thinness of the Ag layer, which we will also discuss later and we first put emphasis on the properties of the mode near the interspace cut-off region, which can be found in Fig. 3. This figure shows the imaginary part of the mode effective index of our five-layer slab. Since TM₄ mode and other higher order modes have the same properties as TM₃ mode, they are not given in Fig. 3, in which we find that the mode losses become infinitely small when the mode operates near the interspace cut-off region.

From the field’s point of view, all the modes near their interspace cut-off regions exhibit radiative fields into water since all their mode indices are lower than the water refractive index. Figure 5(a) shows the H_y component field of TM₃ mode with Si layer thickness of 401 nm, where the mode is near the interspace cut-off region C. Inset of Fig. 5(a) shows the local field divided by three regions (air, Ag-Si-Ag and water) with two red dashed lines. The effective index of the mode is calculated to be 1.2485 + 0.0052i with a damping attenuation field of about 17.9 μm into water due to the losses of the mode. As TM₃ mode approaches the interspace cut-off region C, the mode losses approaches zero since more and more energy propagates in the lossless medium water, which is also the reason why we call this the super-long range SPP mode. The so-called super-long range SPP mode can exhibit infinitely small losses with finite Ag layer thickness, which is the same as the case of the LRSPP mode in an asymmetric IMI slab, and both the cases of lossless limit require an infinite plane wave in the higher index medium that compensates for dissipation in the metal [11].

 

Fig. 5 (a) Normalized H_y field of TM₃ mode with Si layer thickness of 401 nm and Ag layer thickness of 30 nm in the air-Ag-Si-Ag-water five-layer slab. Inset shows the local field divided by three regions (air, Ag-Si-Ag and water) with two red dashed lines. (b) The real part of the complex effective index with different Si layer thicknesses, where the Ag layer thickness is changed to 70 nm with other parameters remained the same. (c) (d) Normalized S_z field of TM₃ mode with Si layer thickness of (c) 401.998 nm (d) 412.745 nm and Ag layer thickness of 30 nm. The two cases are marked in Fig. 2 as point e and f, respectively.

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Here we make another simulation to compare the super-long range SPP mode in our structure with the LRSPP mode in the asymmetric IMI slab calculated by Burke et al. [11]. In their work, they called the LRSPP mode as the Symmetric bound (S_b) mode. They found that there is a critical metal layer thickness h_C, above which the S_b mode corresponds to a bound wave. But when the metal layer thickness is below h_C, the bound wave no longer exists and instead of which, a growing wave solution appears which require both the amplitude grow with propagation distance and the fields in the higher index medium grow exponentially away from the metal layer. The growing wave solution exhibits a negative imaginary effective index which is only logical in mathematics. So we regard the region with growing wave solution as a cut-off case because it is not a real solution in physics. To compare with Burke et al.’s work, we calculate the growing wave solutions around the interspace cut-off region C (see Fig. 2) and the result is shown in Fig. 6(a) and 6(b). This Figure shows the real (Fig. 6(a)) and imaginary (Fig. 6(b)) part of effective index with Si layer thicknesses around interspace cut-off region C. Other than Fig. 2 and Fig. 3, the growing wave solutions which exhibit negative imaginary effective index are included. The red lines correspond to TM₃ mode, while the blue lines correspond to the growing wave solutions that take up the space of interspace cut-off region C to the moment. Two green points named as h_C1 and h_C2 represent the critical Si layer thicknesses in the boundary between TM₃ mode and the mode with growing wave solutions. Both our points h_C1, h_C2 and Burke et al.’ point h_C represent the lossless limit that requires a plane wave in the higher index medium. So when comparing with Burke et al.’ result, our design effectively squeezes the region with growing wave solutions to very small ranges of Si layer thicknesses. Again, since the growing wave solutions are not real solutions in physics, they are not shown in Fig. 2 and Fig. 3.

 

Fig. 6 (a) The real and (b) imaginary part of complex effective index with different Si layer thicknesses around interspace cut-off region C in Fig. 2, both the Ag layer thicknesses are fixed at 30 nm. The growing wave solutions are also included. The red lines correspond to TM₃ mode, while the blue lines correspond to the growing wave solutions that take up the space of interspace cut-off region C to the moment. Two green points named as h_C1 and h_C2 represent the critical Si layer thicknesses in the boundary between TM₃ mode and the mode with growing wave solutions.

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Then we discuss the possible cause of the interspace cut-off region. In Fig. 2 and Fig. 3, as the Si layer thickness increases near any interspace cut-off region, the focused energy will be transferred from the inner Si layer to the outer water. To better demonstrate this, we also choose TM₃ mode and interspace cut-off region C for instance. We first calculate the two critical interspace cut-off Si layer thicknesses both from the left side and the right side of interspace cut-off region C, and they are found to be 401.998 nm and 412.745 nm (see points e and f in Fig. 2, also points h_C1 and h_C2 in Fig. 6, precision of 0.001 nm), respectively. It means that TM₃ mode is interspace cut-off with the Si layer thickness ranging from 401.999 nm to 412.744 nm. The S_z field of points e and f are given in Fig. 5(c) and 5(d), respectively. By comparing Fig. 5(c) with 5(d), we find that more energy are focused in the inner Si layer in the field of point e than that of point f, and from point e to point f the energy are transferred from the inner Si layer to the outer water. From the field’s point of view, there should be a transition from point e to point f with the Si layer thickness increasing from 401.998 nm to 412.745 nm, but the mode is interspace cut-off.

So we make some other simulations to analyze the forming of the interspace cut-off region and we find that the interspace cut-off region will disappear if we increase the Ag layer thickness to some extent (e.g., 70 nm) with other parameters remained the same. The result is shown in Fig. 5(b), which illustrates the real part of effective index with different Si layer thicknesses where the Ag layer thickness is 70 nm. It can be found from Fig. 5(b) that the dispersion curves are continuous, which indicates that the possible cause of the existence of the interspace cut-off region in Fig. 2 may lie in the thinness of the Ag layer. In Fig. 2, the Ag layer is not thick enough to supply enough losses thus most of the mode energy flows propagate in water near interspace cut-off region (see S_z field of both point e and point f in Fig. 5(c) and 5(d)), and the fields approach those of a plane wave in water together with the interspace cut-off of the mode. However, with a thicker Ag layer, less energy is supplied by the plane wave in water than is dissipated in the Ag layer, and the energy can be transferred from the inner Si layer to the outer water continuously with the Si layer thickness increasing. The disappearance of the interspace cut-off region with thicker Ag layer (e.g., 70 nm) strongly supports our explanations.

2.2 Sensor design

Here we are very interested in the mode properties around the interspace cut-off region, where the mode losses are very sensitive to the Si layer thickness variation. By taking TM₃ mode for instance, as the Si layer thickness increases from 401 nm to 401.9 nm, the imaginary part of effective index falls from 0.0052 to 0.0004, which means that the mode can propagate 13 times longer in z direction. Since most of the mode energy propagates in water, the mode losses will also become sensitive to fluid refractive index change, which is the basic theory foundation of our SPP sensor.

Figure 7 shows the attenuation of TM₃ mode with four different fluid refractive index n_D around water. The attenuation is calculated by: $\text{α}\left(\text{dB}/\text{μm}\right)\text{ = -10lg}\left\{\text{exp}\left[\text{-Im}\left({\text{n}}_{\text{eff}}\right)\text{4π}/\text{1.55}\right]\right\}$. We choose the Si layer thickness around the second interspace cut-off region of TM₃ mode (see region C in Fig. 2 and Fig. 3) for calculation. The four different detecting fluid refractive indices are 1.33, 1.3305, 1.333 and 1.335 with red, green, blue and purple lines illustrated in Fig. 7, respectively. It can be seen clearly that with a properly chosen thickness of Si layer and device length, we can distinguish two fluids whose refractive index are very close to each other. This is caused by the mode characteristics that the attenuation curve is extremely sharp near the interspace cut-off region. For instance, with a Si layer thickness of 401.99 nm, the attenuation of n_D = 1.33 and n_D = 1.3305 are α₁ = 0.0005 dB/μm and α₂ = 0.0035 dB/μm respectively (see the black double arrow in Fig. 7). As a result, with a device length of only 3.3 μm (0.01 dB/(α₂-α₁)), this sensor can distinguish the two fluids. Where we assume that the resolution of the power measurement is 0.01 dB.

 

Fig. 7 Relation between the attenuation of TM₃ mode and Si layer thickness with four different fluid refractive index: n_D = 1.33 (red line), n_D = 1.3305 (green line), n_D = 1.333 (blue line) and n_D = 1.335 (purple line).

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Moreover, our target lies in high resolution and large scale of refractive index detection, but there are always conflictions between resolution and detection scale. With a longer device, we can distinguish smaller difference of fluid index, but the larger attenuation induced by the longer device will decrease the scale of refractive detection, since the output power is too tiny to detect. So in consideration of resolution, compact in size and scale of detection, the device length is fixed at 200 μm. We choose TM₃ mode for sensing, as it exhibits a symmetric field distribution, which is easy for coupling in (see Fig. 5(a), TM₃ mode exhibits a symmetric field distribution in the Si layer). For simplicity, we ignore the input mode coupling losses, which means the output power only depends on the losses of TM₃ mode in our SPP sensor. The output power is calculated as: P_out = P_inexp(−2αL), where P_in = 1 mW, L = 200 μm, α = Imag(n_eff) × 2π/λ.

Figure 8 shows the relation between output optical power of TM₃ mode and fluid refractive index n_D with three different Si layer thicknesses: d_Si = 402 nm (blue line), d_Si = 405 nm (red line) and d_Si = 410 nm (green line). Three linear dashed curves are given to compare with the curves of the detection so as to show the linear properties of the detection. According to the blue line and blue dashed line, the average sensitivity in the range of n_D = 1.33 to n_D = 1.41 can be as high as 1262 dB/RIU, which is equal to the slope of the blue dashed line. The sensitivity is calculated by: SA(RIU⁻¹) = Δα/Δn_DL [15]. Since our SPP sensor is based on the intensity measurement, if the resolution of the power measurement is 0.01 dB, the minimum detectable refractive index change can be as small as 7.9 × 10⁻⁶ RIU. In the case of n_D < 1.33, TM₃ mode is interspace cut-off, while in the case of n_D > 1.41, the linear properties of the detection gradually vanish. In spite of this, the result shows that in comparison with other sensors based on intensity measurement for refractive index detection [7,9,15], our designs exhibit larger scale of detection (e.g., 0.08, for 1550 nm ~1.33 to 1.41) and higher resolution (e.g., 7.9 × 10⁻⁶ RIU) with a compact device size (length of 200 μm).

 

Fig. 8 Relation between output optical power of TM₃ mode and fluid refractive index n_D with three different Si layer thicknesses: d_Si = 402 nm (blue line), d_Si = 405 nm (red line) and d_Si = 410 nm (green line). The device length is fixed at 200 μm. Three linear dashed curves are given to compare with the curves of the detection so as to show the linear properties of the detection which is, more specifically, scale of 1.33-1.41 for d_Si = 402 nm, scale of 1.415~1.485 for d_Si = 405 nm and scale of 1.552-1.592 for d_Si = 410 nm.

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Furthermore, the sensing range can be adjusted by changing the Si layer thickness, which can also be concluded from Fig. 8. The red line corresponds to the Si layer thickness of 405 nm and as seen from it, good linear properties are obtained with a detecting range of n_D = 1.415 to n_D = 1.485 but the sensitivity falls to 1089 dB/RIU. So we find that although the mode losses are very sensitive to the Si layer thickness change (see Fig. 7) which make the device difficult to be fabricated, small tolerance of the Si layer thickness will not affect the sensing range and the sensitivity too much (see Fig. 8). The sensitivity can be compensated by increasing the device length, and the sensing range may be adjusted by changing the Ag layer thickness. This will be the topic of our next calculation.

3. Conclusion

In conclusion, we have analyzed the dispersion relation of an air-Ag-Si-Ag-fluid five-layer slab theoretically, where some super-long range SPP modes are found with their losses being extremely small and sensitive to fluid refractive index change when operating near their interspace cut-off regions. By applying this phenomenon in detecting the fluid refractive index change, a SPP sensor based on intensity measurement is proposed. For refractive index detection around water, scale of linear detection as large as 0.08 and resolution as high as 7.9 × 10⁻⁶ RIU can be obtained with a compact device size of 200 μm in length. Sensing of other range of refractive index higher than water can be achieved by increasing the Si layer thickness.