1. INTRODUCTION

Surface plasmon (SP) resonance is a powerful and well-established technique for measuring variations in refractive index. It does suffer some drawbacks, however, in that the angle to excite SPs is rather high, which makes imaging challenging for multipoint imaging in prism-based systems and for objective lens excitation requires very high numerical aperture (NA). The present paper is a numerical and experimental investigation of the performance of sandwich-like structures for refractive index measurement in various structures, such as Otto, Fabry–Perot (FP), and a hybridized SP structure, which we call Otto–Kretschmann for reasons that will become apparent.

Figure 1 summarizes three structures: (a) the Otto structure [1], (b) the Kretschmann structure [2], and (c) the hybrid structure we call Kretschmann–Otto (KO). We can see that for the Otto structure the plasmon is excited on the upper gold surface by the evanescent wave emerging from the glass surface. In all cases, the upper surface may be considered to be thick, which, in practice, means thicker than about 100 nm. In the Kretschmann arrangement the SP is excited through the gold layer, typically about 50 nm, and in the KO the evanescent field from the first gold layer excites an SP that hybridizes with the SP on the upper surface. It should be pointed out that the KO configuration also produces a wave below the critical angle for the glass/sample region that may be thought of as a zeroth-order waveguide or FP mode. These terms are used interchangeably although it is more natural to think in terms of one or the other depending on propagation direction of the waves in the sample region. For simplicity, we will refer to the configuration in Fig. 1(c) as KO both above and below the critical angle. In this paper we consider separations between the gold layers of between approximately 200 nm and 1000 nm.

 

Fig. 1. Schematic diagrams of the (a) Otto configuration, (b) Kretschmann configuration, and (c) double metallic layered waveguide structure (Kretschmann Otto).

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Experimental study of Otto and KO structures is demanding since they require the two layers to maintain good parallelism over the extent of the measurement. There have been several studies where Otto type excitation is performed but these studies usually use fixed structures [3] between the layers, which does not allow a detail study of the effects of variation of gap separation in a single experiment. The standard prism-based excitation is somewhat inconvenient as it does not allow simple changes of incident angle and, moreover, the optical beam width is rather wide, which places even more severe demands on the required degree of parallelism of the two planes. Moreover, the geometry of the prism and the extreme angle of incidence degrade imaging performance [4]. One study of these double-layer structures used prism excitation and obtained useful results, but their system made it difficult to measure gap widths below around 500 nm [5]. In the present studies, we show significant results in terms of enhanced responsivity at much closer separations of the order of 200 nm by extending the idea of using back focal plane (BFP) observation through a microscope objective [6,7]. This allows us to observe modes and sensing regimes that were not possible at these larger separations.

For the reasons explained above, we wish to vary the separation of the two surfaces to explore the changes in behavior. For this reason two free-standing surfaces with well-controlled distance adjustment are needed. Objective-based BFP measurements are very attractive as they allow measurement of a wide range of angles (as well as polarization) in a single measurement. Moreover, and perhaps crucially, the area of examination is localized to the field of view of a high numerical aperture objective, so the demands on parallelism are much less severe compared to the case of prism excitation. In the experimental section we will discuss the experimental procedure required to perform these measurements including the data processing procedure as well as the simple but effective method of mounting the samples.

The purpose of this paper is first, to elaborate on a simple experimental procedure that is useful to explore similar structures not explicitly considered here and, second, to explore the behavior of guiding structures when the layer separation is in the deep sub-micrometer regime.

2. STRUCTURE SIMULATIONS AND THEORY

We now look at the main focus of our experimental observations, the KO configuration, in more detail. In the KO structure, the light energy is confined in the dielectric core, sandwiched between metal layers. The thickness of the cavity can be adjusted to various thicknesses to produce different BFP distributions in Fig. 2 according to different demands. Figure 2(a) is the relation of reflective intensity of $p$ polarization between incident angle and gap distance, whereas Fig. 2(b) shows the corresponding reflectivity for $s$ polarization. The thickness of the lower gold layer was set to 50 nm and the upper gold was considered to be semi-infinite; the dielectric separating the gold regions was air. It can be seen immediately that for the $p$ polarization when the separation is large the dip corresponding to the SP is clearly visible and the response is indistinguishable from the Kretschmann configuration; when the separation goes below about 1000 nm, the upper and lower surfaces start to interact and the modes become hybridized, with one excited at lower incident angles and the other at higher. The lower SP mode changes continuously into a FP mode below the critical angle as can be easily proved from Eq. (2). Even though the change in excitation is continuous the wave within the gap layer becomes propagating. So, below the critical angle the wave is an FP mode not directly related to the SP. For $s$ polarization almost no light penetrates the 50 nm gold layer above the critical angle, so the reflectivity is uniform. There are, however, a rich range of FP modes (including the zeroth-order mode) analogous to those for $p$ polarization below the critical angle. Figures 2(c)–2(f) show calculated BFP distributions under a 1.3 NA objective lens for horizontal linear input polarization at different gap distances, where the horizontal direction gives the $p$-modes and the vertical direction the $s$-modes and the combinations of the two at intermediate angles. It should be noted that for horizontal linear input polarization, the radial position in the BFP corresponds to the sine of the incident angle and the horizontal direction corresponds to pure $p$ polarization and the vertical direction to pure $s$ polarization. At intermediate angles the response is a sum of the two polarization states, as discussed in more detail in Section 5.

 

Fig. 2. (a) Variation of reflectivity with the product of sine of the incident angle and the index of the glass and gap distance for the double-layered structure for $p$ polarization. (b) Variation of reflectivity with the product of sine of the incident angle and the index of the glass and gap distance for the double-layered structure for $s$ polarization. (c) Calculated BFP distribution at gap $\text{distance}=300\mathrm{nm}$. (d) Calculated BFP distribution at gap $\text{distance}=400\mathrm{nm}$. (e) Calculated BFP distribution at gap $\text{distance}=600\mathrm{nm}$. (f) Calculated BFP distribution at gap $\text{distance}=800\mathrm{nm}$. The vertical dashed line in (a) corresponds to the angle for excitation of the conventional surface plasmon.

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Both the waveguide modes below the critical angle and the SP modes above can be divided into symmetric and anti-symmetric distributions. When the waveguide condition is obtained in Eq. (1) a mode can be excited [8]. In Fig. 1(c), this phase shift includes the phase shifts due to propagation and reflection from the lower, $\mathrm{\Delta }{\varphi }_1(\theta )$, and upper interfaces, $\mathrm{\Delta }{\varphi }_2(\theta )$, in Eq. (1). $2k_{2}d·\cos (\theta )$ is the phase change caused by light propagation in sample region, $d$ is the gap distance, $\theta$ is the angle between the plane wave and optical axis in the cavity, and $k_2$ is the wavenumber in region 2. Even when the sample medium is sandwiched between gold on both sides, this structure cannot be regarded as perfectly symmetric because the gold film is very thin to allow coupling, thus the coverslip will also contribute to the phase shift at the lower interface. Thus $\mathrm{\Delta }{\varphi }_1(\theta )$ and $\mathrm{\Delta }{\varphi }_2(\theta )$ in Eq. (1) are not equal:

It is useful to look at the case where both layers are semi-infinite which represents the limit of Eq. (1) as the layer thickness increases. The dispersion relation of these two modes can be derived from Maxwell’s equation to obtain Eq. (2). The derivation is given in Appendix A, but it should be noted that this equation is equivalent to that given in [9], albeit in a different form:

$A=e^{i{k}_{z2}·d}$ for the symmetric mode,

$A=-e^{i{k}_{z2}·d}$ for the anti-symmetric mode.

$k_{z2}$ is the wavenumber in region 2 along the $z$ direction. We can see that just above the critical angle $k_{z2}$ is small and imaginary, so $A$ tends to 1. Just below the critical angle $k_{z2}$ is small and real, so $A$ again tends to 1. In either case we can see for the anti-symmetric mode that $1-A$ mode tends to zero, so the response is continuous across the critical angle.

We now look at the field distributions for these two modes for a 610 nm gap distance with sample region filled with air ($n=1$) as example (Fig. 3), which are plotted using Fresnel equations. For a 610 nm thick cavity, 40.6° and 44.68° related to the dip positions for the anti-symmetric mode and symmetric mode, respectively. Since we are only looking at the $p$ polarization (TM mode) we look at the $H_y$ component of the field only. We see from Figs. 3(a) and 3(c) that the field peaks at the gold layers, and the different phase relations between the upper and lower waves is clearly seen in Figs. 3(b) and 3(d). These effects are borne out in the line graphs of Figs. 3(e)–3(h).

 

Fig. 3. Field distributions and phase relations of KO when the gap distance is 610 nm; 40.6° relates to the anti-symmetric mode and 44.68° relates to the symmetric mode. (a) Field distribution of $|H_y|$ component for the anti-symmetric mode; (b) phase of $H_y$ component for the anti-symmetric mode; (c) field distribution of $|H_y|$ component for the symmetric mode; (d) phase of $H_y$ component for the symmetric mode; (e) profile of (a) along the $z$ direction; (f) profile of (b) along the $z$ direction at $x=0.0725\mathrm{\mu m}$; (g) profile of (c) along the $z$ direction; (h) profile of (d) along the $z$ direction at $x=0.0725\mathrm{\mu m}$. In (e)–(h) the negative region represents the coverslip, the region between the first and second dashed line represents the gold layer, between the second and third dashed line we have the gap, and beyond the third dashed line is the thick gold.

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3. RESPONSIVITY SIMULATIONS

In this section, we compare the responsivity of the KO structure with conventional Kretschmann configuration for different modes and different incident angles to demonstrate the enhancement of responsivity. In order to understand responsivity, we need to apply different criteria dependent on whether we wish to consider the effect of a change in bulk reflectivity or the effect of a thin layer of analyte deposited on the (lower) gold surface.

A. Bulk Responsivity

In both bulk and layer responsivity we consider the $k$-vector of the wave mode to be the measurand. So, we can define the responsivity as

B. Layer Responsivity

Here we consider the responsivity of a very thin layer of sample on the gold film in both the Kretschmann and KO configurations. The additional relative path length in wavelengths introduced by the layer can be represented as $\varphi$, where $\varphi =(n_{\text{layer}}-n_{\text{ambient}})h/\lambda$, where $n_{\text{layer}}$ is the refractive index of the layer, $n_{\text{ambient}}$ is the refractive index of the sample region, $h$ is the thickness of the layer, and $\lambda$ is wavelength. The responsivity is then defined as

Note that the assumption that the change in the $k$-vector in the propagation direction depends only on the product of index contrast and thickness is only valid for $\varphi \ll 1$; we have verified this for various combinations of $\mathrm{\Delta }n=(n_{\text{layer}}-n_{\text{ambient}})$ and $h$. We examine the range of validity of this assumption in Fig. 4(a). In this figure, we calculate the responsivity for specific values of $\mathrm{\Delta }n$ and $h$ and the corresponding values of $\varphi$. The contours represent the locus of $\mathrm{\Delta }n$ and $h$ where the use of a single parameter leads to 5%, 10%, and 20% errors, respectively. For instance, all positions below the green line lead to less than 5% error.

 

Fig. 4. (a) Comparison of various responsivity errors for the zeroth mode; (b) comparison of bulk responsivity between different modes; (c) comparison of binding responsivity between different modes. In (b) and (c) the responsivity of the pure Kretschmann mode is shown on the $x$-axis; in fact, this mode is excited beyond the critical angle, so cannot be shown directly on the graphs.

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In this paper, gold has been used as the plasmonic material having the complex permittivity, obtained using the Drude’s model [10], of ${ϵ}_{\text{gold}}=-11.8195+1.2366i$ at $\lambda =633\mathrm{nm}$.

The different modes show different responsivity values at different gap distances. It is important to point out that most papers record the responsivity in degrees, the change in the sine is the more physically significant measure since it is proportional to the change in the $k$-vector. In our case where we need to compare responsivities at widely separated incident angles the difference in this measure is very important.

Figures 4(b) and 4(c) respectively show the responsivity of the KO structure as (i) bulk responsivity $R_{\text{bulk}}=\frac{d{k}_{\text{mode}}}{d{n}_{\text{sample}}}$ and (ii) layer responsivity $R_{\text{layer}}=\frac{d{k}_{\text{mode}}}{d\varphi }$. In both cases the $x$-axis represents the incident angle coming from the lower medium, for the condition where the sample region is pure vacuum. Since the angle where the modes are excited depends on the separation, each value of $n\sin \theta$ corresponds to a particular value of separation, $d$; see Fig. 2. The corresponding value of $d$ differs for each mode. The plots show the responsivity of the excited mode for the original separation. Responsivity changes with mode, angle, and gap distance are shown is Figs. 4(b) and 4(c) for bulk and layer responsivity, respectively. The dashed-dotted blue line is for the zeroth mode, the solid green line is for the first mode, and dashed red line is for the second mode. Both bulk and layer responsivity in Figs. 4(b) and 4(c) show that responsivity increases at smaller incident angles and the highest responsivity for the zeroth mode is at close to normal incidence, which can be as much as 16 times and 5 times greater than the single-layer Kretschmann arrangement for bulk and layer responsivity, respectively. The values of bulk and layer responsivity for the Kretschmann configuration are calculated as 1.1 and 3.54, respectively, and they are plotted as purple crosses in Figs. 4(b) and 4(c) to compare with the values for various modes in the KO structure. It should be noted that the purple crosses in these two plots are labeled as their corresponding responsivities for comparison purpose; they are excited at their excitation angles, which are above the critical angle. We can conclude therefore that the FP is a more responsive method to measure both the deposition of a layer and the bulk index changes compared to SP with the Kretschmann configuration provided that the analyte can be introduced to the cavity in a convenient manner.

4. EXPERIMENT SETUP

To examine the behavior of the wave modes over a wide range of incident $k$-vectors both above and below the critical angle, we used BFP observation to access the response over a wide range of incident angles and polarizations.

Figure 5 shows a simplified schematic of the optical system. A 633 nm He–Ne laser was used as the illumination source and a 1.3 NA oil immersion objective (CFI Plan Fluor $100\times$ Oil, Nikon) was used to excite the different wave modes. The galvanometer mirror (GVS011, Thorlabs) was placed conjugate with the BFP of oil immersion objective to ensure light remains in focus at the sample plane when illuminating different regions of the sample. A pellicle beam splitter was used to separate the illumination and imaging paths. The light from the sample was detected with a CCD camera, which is located at the conjugate plane of the BFP of the immersion objective. When plane gold samples were used the rotating diffuser was used to reduce speckle noise; however, for spatially resolved measurements it was removed so that the beam came to a focus at the sample plane.

 

Fig. 5. Simplified schematic showing the operation of the system.

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Figure 6 shows the operation of the system schematically. To explore the behavior of the separation between two metallic layers and to compare with simulation results, our sensor consisted of two free-standing gold-plated glass parts. One is gold film sputtered on a cover glass fixed on the focal plane of the immersion objective lens, and the thickness of the gold was approximately 50 nm in order to obtain lowest reflectivity. The upper gold sample was evaporated to 200 nm thickness so that it behaves as bulk gold. The system is connected to a coarse actuator, D, a fine stage C and kinematic mounts to give five-axis rotation. There are several main technical constraints that need addressing during the experiment.

 

Fig. 6. Schematic of the sensing part, showing key mechanical components.

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A. Minimizing the Initial Gap Distance of the Sample Region

A key part feature of the experimental results is the ability to examine small gap separations down to approximately 200 nm. With a large upper gold layer this is very challenging as it places severe demands on the parallelism of the layers, and also, small amounts of residue on the surface will prevent close approach of the two layers. The solution is to use very small pieces of gold on the upper layer, obtained by fracturing a coated coverslip. The fragments used were no more than 500 μm across. The BFP measurement allows this since the field of view is around 250 μm, such small area observation is not really practical with a prism-based measurement.

B. Parallelism of the Two Gold Layers

From Fig. 6 we see that the bulk gold-plated part is attached on one side of a needle, which is connected with a $x–y$ translation stage mounted on a five-axis kinematic mount KM100B/M (Thorlabs). The angle of the bulk gold surface can be adjusted to be parallel to the 50 nm gold film structure or the focal plane of the objective lens. Meanwhile, the combination of these two mounts can ensure the alignment of the bulk gold is over the illumination beam. By carefully adjusting the angle and its $x–y$ position, a circular and symmetrical BFP with waveguide modes within the critical angle range could be observed. The $x–y$ actuator has two purposes: first, it ensures that two gold layers are properly aligned over the illumination beam and, second, it can ensure that the two samples are parallel. Moving the objective a small distance in $x–y$ allows one to observe the change in the mode pattern, which allows one to assess if the layers are moving closer or further away. This indicates the direction of tilt that can be corrected in both $x$ and $y$ directions until satisfactory parallelism is achieved. This approach ensured that the separation changed by less than about 10 nm over the field of view. When the diffuser was removed, however, this approach could not be used because of the requirement to maintain the scan mirror in a plane conjugate to the BFP, with the consequence that the tilt for the spatially resolved measurements was not as well controlled as for the uniform samples.

C. Precise Nanoscale Control of the Gap Distance

Once the parallelism of two gold structures was achieved, we were able to vary the gap distance to study the behavior in the BFP. It can be seen from Fig. 6 that the bulk gold-plated glass slide was stuck on one side of the needle and taped on a $x–y$ translation stage connected with a five-axis kinematic mount (KM100 /M, Thorlabs) to allow tip-tilt control of the small area gold film. This was fixed on a piezo linear stage P-621.1 CD driven by -E665 (Physik Instrumente) to realize the fine tuning with 1 nm resolution and connected to a translation stage M-126.DG driven by C-863 (Physik Instrumente) to support the coarse adjustment with a resolution of 8 nm. By combining these two stages (C and D in Fig. 6), changes in gap separation may be controlled to 1 nm precision over several micrometers. The absolute separation is not known directly, however, and the procedure to address this is described in Section 5.

Essentially, the system is a mechanical scanned microscope with a galvanometer mirror inserted at the conjugate plane of the BFP of the immersion objective to control the illuminated region on the sample by rotating to different angles. In our system, the scanning distance is 25 μm, which matches the period of bovine serum albumin (BSA) grating pattern. The galvanometer mirror, motors, and CCD cameras were controlled by MATLAB and all data were processed with MATLAB. Detailed simulation results and analysis are presented in Section 6.

D. Sample Preparation

Samples were prepared as three parts: 50 nm gold film (with 1 nm Cr adhesion layer) on top of cover glass, 200 nm gold on top of cover glass to act as the bulk gold layer. For the spatial resolved measurement, a stamped BSA grating was also deposited onto the 50 nm gold sample. The BSA gratings were stamped with a Sylgard stamp using a standard stamping process described elsewhere [4]. Figure 7 describes the sample structure with the BSA grating on top of the gold surface.

 

Fig. 7. Sample structure.

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5. DATA PROCESSING

BFP distributions of the Otto configuration and double metallic layer waveguide structure were obtained using our optical system described in Fig. 5 with the diffuser inserted for all the experiments except those on the grating sample described in Section 4. A single line trace is somewhat noisy, so we need to apply some data processing to utilize the available information from the BFP.

A. Determine the Center Position of the BFP

There are many different methods and algorithms that can be employed to get the center position of a circular BFP, such as Hough transformation [11] and least-squares [12]. Here we choose a simple but effective way to locate the position; its accuracy can be examined in the next step. In Fig. 8(a), we summed all rows together to get groups of symmetrically located dips due to various modes for $p$ polarization in Fig. 8(b). The middle position of each group should be the same and it is the center along the $x$ direction ($x_0$); the $y_0$ can also be located using the same method by employed the $s$ polarization modes [Fig. 8(c)]. The minimum positions of dips are determined by a third-order polynomial curve fitted to several data points so that the position of the center could be located to a fraction of a pixel. In Figs. 8(b) and 8(c), small insets within them illustrate the method to find the actual dip position of using third-order polynomial curve fitting, and crosses represent the raw data while the solid line is for fitted curve.

 

Fig. 8. (a) Schematic of BFP, (b) summation of data in the $x$ direction, and (c) summation of data in the $y$ direction.

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B. Recover the Depth of Dip and Increase Signal-to-Noise Ratio

The signal along the horizontal axis represents pure $p$ polarization, whereas the vertical direction represents $s$ polarization. Any direction at an angle, $\varphi$, to the horizontal gives a combination of the two polarizations so that the output may be written to within a proportionality constant, thus

To recover the line response a line along $\varphi =0$ may be used, but this does not utilize the signal along other directions and therefore does not give the best available SNR. In order to recover a better estimate of the value of mode ${|r_p|}^2$, we apply an approach similar to that presented by Greivenkamp [13] where we estimate the least-square errors with respect to the parameters ${|r_p|}^2$, ${|r_s|}^2$, $\frac{1}{2}|r_p||r_s|\cos \theta$; from this a matrix equation is formed which gives a less noisy representation of the desired signal.

C. Convert the Relative Gap Distance Value of the Structure in Experiment to Absolute Separation

Since the absolute separation between the two gold surfaces is not known, we chose the best match between theory and experiment for one value only and assigned the value of the separation to be equal to that theoretical value. All other values of the separation are determined from the experimentally determined change in separation obtained from the position sensors of the precision actuator.

6. EXPERIMENTAL RESULTS AND DISCUSSION

A. Otto Configuration

Before exploring the double metallic layered waveguide structure, BFP distributions from the Otto configuration were obtained and the behavior of the Otto configuration with air gap width was studied. Although the Otto configuration is not the main thrust of the paper, there are few (if any) experimental studies with varying separation at visible wavelengths reported, so it is useful to validate the results on a more familiar system. It has been known that the gap distance strongly affects the coupling to SPs traveling at the metal/air interface [14]. In practical application, a more pronounced minimum results in a better SNR. The optimum air gap, corresponding to the deepest dip, depends strongly on the wavelength and the medium adjacent to the gap. Using the Fresnel equations and parameters appropriate to a 633 nm wavelength, a gap of 592 nm gives the deepest dip in air. At this thickness, maximum coupling to SPs occurs at an incident angle of 43.31 deg from a medium with 1.52 refractive index and the $p$-polarized reflectivity drops close to zero. Smaller gaps result in a broader and less pronounced minimum. Air gaps larger than the optimum value give less distinct minima as well, but the weak coupling ensures a sharp resonance with a narrow dip. A series of BFP distributions for the Otto structure were obtained using our system by controlling the gap distance. Four images with gap distance from large to small were listed in Fig. 9 from (a) to (d) with $d_a=690\mathrm{nm}$, $d_b=592\mathrm{nm}$, $d_c=470\mathrm{nm}$, and $d_d=390\mathrm{nm}$.

 

Fig. 9. (a)–(d) Experimental BFP distributions of the Otto configuration with air gap equal to 690 nm, 592 nm, 470 nm, and 390 nm from (a) to (d). (e) $p$-polarized reflectivity for (a)–(d).

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Figure 9(b) is evaluated as the BFP with optimum air gap (592 nm) which has the smallest minimum reflectivity among all data. It can be seen from Fig. 9(e) that an air gap smaller than the optimum value [Figs. 9(c) and 9(d)] shows wider and shallower dip while that with a larger air gap [Fig. 9(a)] presents less distinct minimum values, which are all matched with theory and simulation result. Moreover, the dip position moves to slightly smaller incident angles with decreasing separation as predicted by theory. All line traces were obtained using the least-squares processing algorithm described in the previous section.

B. Double Metallic Layered Waveguide Structure

Figure 10 shows experimental and simulated BFP distributions obtained from a double metallic layered waveguide structure with 50 nm gold film adhered to a 1 nm chromium layer coated on 1.52 refractive index coverslip in proximity to a thick (200 nm) gold layer attached to the actuator system described in Section 4. The actuator allows the air gap spacing to be controlled. Figures 10(a)–10(e) shows the experimental results and Figs. 10(f) to 10(j) are the corresponding simulations. By comparing the simulated BFP distributions with their corresponding experimental results in Fig. 10, we can see that the change in separations measured are in excellent agreement. It should be noted that with a 227 nm gap distance [Figs. 10(a) and 10(f)], the anti-symmetric mode does not exist. For this reason only the symmetric mode with an excitation angle larger than the angle for an SP in the normal Kretschmann is observed in the BFP. As the separation increases, the anti-symmetric mode appears [Figs. 10(b)–10(e)] from the center (0° incident angle) and will finally merge with the symmetric mode as the normal SP mode when there is no interaction between these two gold layers.

 

Fig. 10. BFP distributions obtained from the double metallic layered waveguide structure at various gap distances. (a)–(e) Experimental BFP distributions. (f)–(j) Corresponding simulated BFP distributions. Horizontal direction is $p$ polarization and vertical direction is $s$ polarization. The abbreviations labeled in (a) to (e) are SY for symmetric mode, 0P for $p$-polarized zeroth mode, and 0 S for $s$-polarized zeroth mode. It should be noted that the $p$-polarized zeroth mode (below critical angle) and anti-symmetric mode (above critical angle) are continuous across the critical angle.

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C. Binding Response for Double Metallic Layered Waveguide Structure

In this section, we explore the binding responsivity of this structure under various gap distances experimentally. Before employing a grating sample and a system with scanning function, we use a uniform sample layer deposited on gold film and a bare gold sensor to get two series of BFP distributions by controlling the gap distances changing in 10 nm steps. The sample layer we deposit on the gold surface is BSA with effective a refractive index of 1.4 [15] and its thickness is 15 nm, which is evaluated using BFP distribution by comparing it with the bare gold sensor. Since the values are outside the range where the single optical path length parameter, $\varphi$, could be used (see Section 3) with sufficient accuracy, both the height and refractive index were used in the model to compare with the experiment. In Fig. 11, gap distances equal to 900 nm [Figs. 11(a)–11(d)] and 620 nm [Figs. 11(e)–11(h)] are selected as examples to illustrate the experimental results for binding responsivity. The detailed comparisons of the dip positions for various experimental conditions are shown in Fig. 12.

 

Fig. 11. Experimental and simulated BFP distributions for the reference region and BSA region under (a)–(d) 900 nm gap distance and (e)–(h) 620 nm gap distance, respectively.

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Fig. 12. Comparison of dip positions between simulated data and experimental data for different wave modes. Solid blue line and dashed red line show the simulated dip position for the reference region and BSA region, respectively, for zeroth order, first order, and symmetric order from (a) to (c); blue crosses and red open circles represent the experimental reference region and BSA region results for corresponding orders from (a) to (c).

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Figure 12 shows a comparison of the dip positions between simulated results and experimental data. The blue color represents the bare gold surface and the red color shows the results with the additional BSA layer surface. The solid line and dashed line are the simulation curves of the bare gold surface and with the additional BSA layer surface, respectively. Crosses and open circles give the experimental data of the bare gold surface and with the additional BSA layer surface, respectively. It can be seen from Fig. 12 that all experimental data agree well with simulated results. For the zeroth order and first order, they possess the highest responsivity at a small angle region and will decrease with increasing angle.

The separate series of BFP distributions under various gap distances for the bare gold sensor and BSA layer sensor match with the theoretical results and present high responsivity at small incident angles range. This enhanced sensitivity is more pronounced for the lower-order modes.

It is useful to demonstrate this sensitivity in a single spatially varying sample. In this series of experiments, we used the grating sample to compare the response of a coated and uncoated region. The scanning galvanometer was used to scan the image points at the sample surface, so that the diffuser was removed. The sample was stamped with a BSA protein grating on top of a 50 nm gold film and its schematic is shown in Fig. 7. The thickness of the BSA region is evaluated as 26.8 nm by comparing the dip positions of bare gold and BSA region and based on 1.4 refractive index for BSA. The region of reference and the region with 26.8 nm thick BSA are illuminated successively by rotating the scanning mirror to corresponding angles. With the spatially resolved sample the agreement between modeling and experiment, although satisfactory, was not as good as the case for the uniform sample. This arose because of the slightly poorer parallelism that could be achieved and the more noisy BFP due to the presence of coherence noise. The BFP distributions for the grating sample in our experiments showed an offset gap distance between the reference and BSA region, which means the bulk gold part is not perfectly parallel to the 50 nm gold film sensor. By calibrating at one defocus position we estimate that the difference in height between the coated and uncoated regions was 20 nm. This value was then applied to all subsequent data. The slant angle between the two layers can be calculated as $\mathrm{atan}(20\mathrm{nm}/25\mathrm{\mu m})\approx 0.046°$; the comparison between simulation results and experimental data are shown in Fig. 13. It can be observed that experimental data agreed better with the simulated data at larger angles [Figs. 13(a)–13(c)] than smaller angles [Fig. 13(d)]. One reason for this is simply that the small angle range is more responsive, thus amplifying small discrepancies.

 

Fig. 13. Comparison of dip positions between simulated data and experimental data for different orders. The solid blue and dashed red lines show the simulated dip position for the reference region and BSA region, respectively, for zeroth order, symmetric order, first order, and second order from (a) to (d); blue crosses and red open circles represent the experimental reference and BSA coated regions, respectively, for different modes as indicated in the figures.

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7. DISCUSSION AND CONCLUSIONS

In the sections above, we have shown by both theory and experiment that the change of the $k$-vector of the detected wave can be made more responsive to a change in refractive index or the presence of a layer compared to the surface plasmon in the Kretschmann configuration. The comparisons of both bulk and layer responsivity for the KO structure and Kretschmann configuration are listed in Table 1.

Table 1. Comparison of Responsivity

View Table

The achievable sensitivity of the system while related does not correspond directly to the responsivity, which depends on the precise mechanism of detection. Let us take one specific example (zeroth mode at normal incident) where the responsivity as defined for bulk index and layered sample is given by Eqs. (3) and (4). In this case the responsivity of the signal compared to the SP Kretschmann configuration is a factor of 16 and 5 times greater, respectively. On the other hand, the dip width in terms $k$-vector is greater by a factor of 10. For SP sensors it is normally said that the narrower the dip the better the sensitivity, While this is generally true, the exact detection mechanism needs to be taken into account. We consider three detection mechanisms.

A. Simple Intensity Detection

If we use the naïve approach of simply fixing the angle and looking at the intensity change, then the sensitivity would depend on the product of the responsivity and the steepness of the dip:

In which case the relative sensitivity of the selected mode compared to the Kretschmann mode would be 1.6 for the bulk and 0.5 for the layer.

B. Fitting the Dip Position

In state-of-art surface plasmon resonance (SPR) sensor systems based on prism excitation the usual approach to measuring the position of the minimum is to acquire the response as a function of angle and fit a curve. This has the advantage of using a lot more data compared to the measurement of a single point. In this case the gradient of the dip is a factor, but the deleterious effect of the dip width is mitigated somewhat because a wider dip means that it is possible to detect more light.

Equation (5) in Lipson and Ran [16] confirms this intuition where the sensitivity is proportional to the gradient and of the reflectivity and the square root of the number of detected photons (if readout noise is neglected). Since the gradient of the reflectivity is proportional to the width of the dip, the number of photons detected can be considered to also be proportional to the width of the dip. We may expect that the relative sensitivity compared to the Kretschmann mode will be

C. $\mathsf{V}(\mathsf{z})$ Microscopic Detection

In recent years we and other groups [17,18] have used confocal detection where the phase change over a given path is measured. In this case the width of the dip has two slightly conflicting effects: a wide dip corresponds to stronger coupling, which gives a large signal; on the other hand, this signal will decay more rapidly. To a first approximation the sensitivity of these systems is proportional to the change of phase of the propagating surface wave, in case: $\text{Sensitivity}\propto \text{Responsivity}$. For this system the improvement in sensitivity will approach 16 and 5, respectively. It should be pointed out that no attempt has been made here to optimize the $Q$ of the resonator as our parameters were determined by the requirements to excite SPs efficiently, it is expected that dielectric structures could improve the performance still more.

Our optical system setup can realize the precise control of two parallel layers to within approximately 200 nm separation. This helps us to comprehend how the air gap width affects the SPR effect in the Otto configuration and also BFP distributions and responsivity behavior of the double metallic layered waveguide structure sensor. Both simulation and experimental results have been presented, which match well. It is shown that this double metallic layered sensor exhibits larger changes of $k$-vector to a given change of refractive index. Moreover, these changes are observed at small incident angles, which means that the system can be implemented with more inexpensive optics compared to conventional SPR. The experimental data shows very good match with simulated results for both BFP distributions at different gap distances and responsivity for various modes including waveguide modes and SPR modes. The system opens up the possibilities to study more complex systems, such as grating sandwich structures with variable separation.

APPENDIX A

Since we are interested in the modes that can be supported by the KO structure we use the simplified assumption that both metal media (medium1 and medium 3) are semi-infinite and the thickness of the dielectric region is $d$, as shown in Fig. 14.

 

Fig. 14. MIM structure.

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The field distribution of TM in three media are

(A1)$$H_y(z)=\{\begin{array}{ll}{H}_1{e}^{i{k}_{z1}·z}& z>\frac{d}{2}\\ H_2^{+}{e}^{-i{k}_{z2}·z}+H_2^{-}{e}^{i{k}_{z2}·z}& -\frac{d}{2}<z<\frac{d}{2}\\ H_3{e}^{-i{k}_{z3}·z}& z<-\frac{d}{2}\end{array}.$$

And the boundary conditions of surface 1 are

(A2)$$H_1{e}^{i{k}_{z1}·\frac{d}{2}}=H_2^{+}{e}^{-i{k}_{z2}·\frac{d}{2}}+H_2^{-}{e}^{i{k}_{z2}·\frac{d}{2}},$$

(A3)$$-\frac{k_{z1}}{{ϵ}_1}{H}_1{e}^{i{k}_{z1}·\frac{d}{2}}=\frac{k_{z2}}{{ϵ}_2}{H}_2^{+}{e}^{-i{k}_{z2}·\frac{d}{2}}-\frac{k_{z2}}{{ϵ}_2}{H}_2^{-}{e}^{i{k}_{z2}·\frac{d}{2}}.$$

The boundary conditions of surface 2 are

(A4)$$H_3{e}^{+i{k}_{z3}·\frac{d}{2}}=H_2^{+}{e}^{i{k}_{z2}·\frac{d}{2}}+H_2^{-}{e}^{-i{k}_{z2}·\frac{d}{2}},$$

(A5)$$\frac{k_{z3}}{{ϵ}_3}{H}_3{e}^{i{k}_{z3}·\frac{d}{2}}=\frac{k_{z2}}{{ϵ}_2}{H}_2^{+}{e}^{i{k}_{z2}·\frac{d}{2}}-\frac{k_{z2}}{{ϵ}_2}{H}_2^{-}{e}^{-i{k}_{z2}·\frac{d}{2}}.$$

When medium 1 and medium 3 are the same material and using $R_i$ to stand $k_{zi}/{ϵ}_i$, where $i$ is the medium number, solving Eqs. (A1)–(A5), we can have

(A6)$$e^{i{k}_{z2}·d}=\pm \frac{R_2+R_1}{R_2-R_1}$$

and

(A7)$$H_2^{+}=\pm H_2^{-}.$$

The positive-negative symbol indicates that there are two modes of surface plasmon in the metal-insulator-metal (MIM) structure, corresponding to $\pi$ phase difference.

According to Snell’s law, the $k_x$ of the three layers should be the same:

(A8)$${ϵ}_i{k}_0^2=k_{zi}^2+k_x^2,i=1,2,3.$$

Thus,

(A9a)$$k_x^2=\frac{{ϵ}_1{k}_0^2{ϵ}_2^2{(1+A)}^2-{ϵ}_2{k}_0^2{ϵ}_1^2{(1-A)}^2}{{ϵ}_2^2{(1+A)}^2-{ϵ}_1^2{(1-A)}^2},$$

in which

(A9b)$$A=e^{i{k}_{z2}·d}\text{for}{H}_2^{+}=H_2^{-}(\text{symmetric mode}),$$

(A9c)$$A=-e^{i{k}_{z2}·d}\text{for}{H}_2^{+}=-H_2^{-}(\text{anti-symmetric mode}).$$

Therefore, in the KO structure, when two gold surfaces are close enough, SPR modes excited at the bulk gold surface and at the gold film surface facing each other are hybridized and form two surface plasmon modes: anti-symmetric modes and symmetric modes.

Funding

ITF grant (GHP/010/14SZ); Engineering and Physical Sciences Research Council (EPSRC) (EP/G037345/1, EP/L016362/1); PolyU Strategic Importance (1-ZE23); Ningbo Science and Technology Bureau; International Doctoral Innovation Centre; University of Nottingham.

Acknowledgment

We gratefully acknowledge the partial support of ITF and Mengqi Shen acknowledges the financial support from the International Doctoral Innovation Centre, Ningbo Science and Technology Bureau, and the University of Nottingham. This work was also partially supported by the UK EPSRC. We also acknowledge the support of the PolyU Postdoctoral Fellowship award for SP and PolyU Strategic Importance for SL. Mengqi Shen would like to thank Dr. C. W. See from the University of Nottingham, UK, for continuous support.

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