1. Introduction

Surface plasmon (SP) refers to a longitudinal electron concentration wave formed at metal dielectric interface. SP resonance (SPR) is sensitive to surface states and therefore has been widely used as a biosensor. SPR biosensing suffers from moderate sensitivity due to the label-free nature. A large part of research on SPR biosensing, therefore, has been focused on the improvement of detection sensitivity using various techniques, for example, phase-sensitive detection [1–3], mediation by metallic nanoparticles [4–6], localization of near-field [7–12], and colocalization with biomolecular interaction [13–15] as well as multimodal approaches [16,17]. SPR biosensing has also been used, for instance, to confirm cellular adhesion in cell microscopy [18–21].

Most of these efforts have been made on flat substrates and studies of excitation and propagation of SP on a curved surface have been conducted largely in the context of SP waveguides [22,23]. Oftentimes, this has set a limit in the way that SPR is applied, while demand for highly sensitive real-time optical sensors in portable and/or wearable platform that may involve curved surface has been increasingly strong, for example, in regard to SPR on patch-type flexible substrates [24,25]. In the past, SPR structure on a spherical surface was studied for scan-less angular interrogation [26], and curvature effect in conjunction with the performance of wire-grid polarizers [27]. Also, the effect of curved and dispersion relation of SP waves propagating along a curved interface between metallic cylinder and dielectric ambiance was described [28]. In addition, curvature effects were examined when SP is coupled by metal-coated dielectric probe in curved thin-film tunneling geometry [29]. Propagation of spoof SP was proposed and analyzed on corrugated and flexible bent surface in the microwave frequency range [25,30], while waveguide propagation of long-range SP polariton (SPP) was explored on curved surface [31,32]. In comparison, propagation loss of SPP on curved graphene substrates was investigated [33].

On the other hand, feasibility of plasmonic sensing capability in in vivo applications based on fiber-optic endoscopy has drawn significant interests, to which investigation of plasmon characteristics on a curved surface can be particularly crucial. Partly, a limit arises from the nature of SPR sensing where detection is performed on flat surface deposited with thin metal film. For this reason, SPR biosensing intended for endoscopy has been restricted to deposition of metallic films at the tip of optical fiber [34–38], hollow optical fiber [39], fiber-optic heterostructures [40–42], a section of fiber cladding polished for immobilization with probe biomolecules [43–47], or fiber with localized SPR [48]. Effect of various parameters such as metal, its thickness, and taper profiles on the detection properties of fiber optic SPR sensors has also been investigated [49–53]. In addition to fiber-based SPR sensors, plasmonic structures have been implemented on flexible substrates that form a curved surface, for example, by impregnating metal nanoparticles in polymer substrates [54,55].

In this study, we intend to study SPR characteristics on curved surface and to explore direct influence of curved shape of surface and its curvature on SPR detection characteristics. Assuming light incidence in plane waves, effects of curved surface may be approximated as dispersion of light momentum, to the first degree, with the extent of dispersion correlated to the surface curvature. Detailed effects may depend on many parameters such as conditions of light incidence. Understanding these effects can help gain an insight into maintaining desirable sensing qualities or even improving them in diverse environment. The results of this study, therefore, can be extremely important not only for extending the applicability of traditional SPR biosensing, but also for putting forward a novel structure with advanced sensor characteristics.

2. Methods and models

For the convenience of analysis, curved surface was assumed to be cylindrical in two configurations of light incidence: in the parallel incidence, incident light wave vector is contained in the plane spanned by the normal of the curved surface as shown in Fig. 1(a). On the other hand, the incident wave vector cannot be contained in the perpendicular light incidence in Fig. 1(b). Curvature radius r ranges between the minimum obtained from that of multi-mode optical fiber with diameter ϕ = 450 μm and maximum corresponding to flat surface, i.e., 225 μm < |r| < ∞. Initially, SPR detection on a surface with spatially uniform single curvature that is either positive or negative was considered: curved surface with positive curvature is based on convex mirror reflection as shown in Fig. 1(a) and 1(b). In contrast, one with negative curvature uses concave mirror surface shown in Fig. 1(c) and 1(d). In a later part, the analysis is extended to SPR detection on spatially complex multi-curvature surface. It is assumed that the curved surface is supported by single SF10 glass substrate without core/cladding structure. The curvature effects for other glass substrates such as BK7 and SF11 are to be largely in line with the case of SF10 if the angle of incidence is appropriately adjusted in the model for momentum matching. The curved glass substrate is uniformly deposited with 50-nm thick gold.

 

Fig. 1 Schematic illustration of the curved surface used for numerical calculation: (a) parallel and (b) perpendicular light incidence with positive curvature (r > 0). In contrast, (c) parallel and (d) perpendicular incidence with negative curvature (r < 0). Thick solid arrows represent the direction of light incidence. In the parallel light incidence, wave vector is contained in the yz plane with p-polarization in the yz plane and s-polarization parallel to x-axis. In the perpendicular incidence, wave vector is contained in the xz plane with p-polarization in the xz plane and s-polarization parallel to y-axis. Curved surface is approximated in segments (seven shown in the schematic and 13 used in the calculation).

Download Full Size | PDF

The calculation is based on wavelength scanning SPR detection. The angle of light incidence θ_sp was fixed at 55° as shown in Fig. 1 with respect to the plane of center normal (xz plane in the parallel incidence and xy plane in the perpendicular incidence). Both p and s polarization have been considered as shown in Fig. 1. Note that multiple SPR peaks were exhibited for hybrid plasmon modes in optical fiber [51]. Material constants for the dispersion of SF10 glass and gold were taken from reference [56].

Numerical calculation of the parallel and, particularly, the perpendicular incidence in 3D imposes significant computational overload. Therefore, the calculation has been simplified by segmentation based on assumptions that a curved surface varies slowly and can be modeled as piecewise linear segments with an identical length [27]. If a segment is infinitesimally small in an infinite number, piecewise linear segments become identical to an ideal curved surface. In a very small segment that corresponds to high curvature involving sharp bends, however, excited SP would interact with SP excited in neighboring segments and SPP waves may radiate energy into radiation loss [57], which cannot be modeled by segmentation-based analysis. For this reason, each segment is assumed to be sufficiently larger than SP propagation length, so that light transmitted through neighboring segments can be approximated as incoherent superposition [58]. Compared to conventional numerical computation methods such as rigorous coupled-wave analysis (RCWA) and finite difference time domain (FDTD) method, the segmented wave analysis is extremely simple and thus drastically fast. Also, the approach can be easily extended into multi-dimensional calculation. In addition, we emphasize that the segmentation offers more insights and is more intuitive than direct numerical analysis as we shall see later. Because collimated plane-wave light incidence becomes un-collimated upon reflection by the curved surface in a manner similar to the reflection by a concave or a convex mirror, our calculation effectively assumes a detector to be capable of collecting all the light rays formed by the reflection and detecting a range of wave vectors upon reflection. In other words,

Resonance shift was calculated with DNA hybridization as a model biomolecular interaction which is assumed to form a homogeneous dielectric layer with thickness equal to the length of DNA oligomers. Optical parameters of 24-mer single-stranded DNA (ssDNA) and double-stranded DNA (dsDNA) were taken as constant respectively to be n_ssDNA = 1.449 and n_dsDNA = 1.517 with 9.32 nm thickness [59]. DNA hybridization was assumed to take place under buffer ambience (n_amb = 1.33).

3. Results and discussion

3.1 Momentum-matching on single curvature surface

SPR is created by momentum matching between an incident photon and SP. When the angle of incidence is equal to the resonance angle in the thin film, i.e., θ_in = θ_sp, SP momentum, K_SP, is given by

Theoretical insight can be gained from segmentation: under segmentation, curved surface with a fixed incident wave vector may be treated as a flat surface with dispersive light incidence in a range of wave vectors, where the range is determined by the surface curvature and configuration of light incidence. In other words, each segment is characterized by a slightly different incidence angle at which the momentum-matching takes place. If the center segment is indexed with m = (N + 1)/2 (N: an odd integer) and light is incident onto the center segment at resonance, the angle between the normal of the (m + i)th segment and the incident light is equal to

3.2 Resonance characteristics on single curvature surface

Figure 2 presents the resonance curves for the parallel and the perpendicular incidence, as the curvature varies in the positive range. Figure 2(a) and 2(b) show that changes in the parallel incidence are much more drastic to the curvature in the sense that SPR quickly disappears with curvature while it remains robust in the perpendicular incidence [Fig. 2(b)]. This is related to what was suggested in the above analysis based on r_λ, i.e., the more direct effect of curvature in the parallel incidence, in which contribution of plasmon resonance in different parts quickly dissipates as the curvature increases and less number of segments participate in the resonance. This is much less dramatic in the perpendicular incidence. Even at a large curvature, most of curved surface contributes to the resonance. As expected, SPR is not observed for s-polarized light incidence in the parallel incidence [Fig. 2(c)]. For the perpendicular incidence, however, light polarization is not maintained as fixed for all segments: s-polarized light at the center segment becomes slightly p-polarized, especially for those segments far from the center. This effect is more prominent if the curvature radius is smaller, i.e., for more curved surface, and appears as reduced reflectivity in Fig. 2(d). For reference, we have also confirmed the resonance characteristics calculated without segmentation by numerical methods such as RCWA to be in good agreement with the segmentation. In terms of run-time, segmentation-based calculation took typically less than 20 seconds for each segment, thus between four and five minutes for the total 13 segments. In contrast, it took more than 160,000 seconds using 2D RCWA for a 100 μm arc length. In other words, although the absolute amount of time depends on specific computational resources put to the calculation, segmentation was much faster by more than 1600 times and the reduction of computation time is even more drastic if compared to 3D methods: considering that more than 100 sets of parameters were calculated, segmentation allowed significant reduction of run-time, enabling calculation for various surface structures in a reasonable time.

 

Fig. 2 Resonance curves with respect to the curvature radius in the range of r > 225 μm: (a) parallel and (b) perpendicular incidence with p-polarized light incidence. Filled square symbols follow the resonance at each curvature. Respectively same incidence in (c) and (d) with s-polarization. Arrows represent a decrease of curvature radius (increased curvature) from flat surface (r = ∞) to r = 225 μm.

Download Full Size | PDF

For surface with negative curvature, the concept of segmentation suggests that Eqs. (1)-(6) remain essentially valid except for the segment index i running from + (N – 1)/2 to –(N – 1)/2. In other words, curvature effects should be identical to what may be observed on the surface with positive curvature as long as all the reflected light rays are captured at detector. What ultimately distinguishes the surface with negative curvature from that of positive curvature may be the power to collect all the light rays. In the case of negative curvature, the collection of all light rays can be difficult because a detector needs to capture diverging rays with a finite pupil size. Interestingly, Fig. 2 indicates that resonance characteristics can improve by missing light rays from far segments assuming an on-axis detector since these segments may cause resonance broadening. This possibility will be investigated in more detail in Discussion.

Resonance characteristics, such as resonance wavelength λ_sp and width of resonance characteristics δλ_sp, with respect to curvature were obtained for the two incidences as presented in Fig. 3. Here, δλ_sp is defined as the difference of wavelengths at which the reflected intensity is equal to the average of the intensity at critical wavelength and the minimum intensity at resonance. In the parallel incidence, resonance wavelength does not change significantly at a long curvature radius, which blue-shifts monotonically when the curvature radius decreases below r = 7500 μm as shown with a black arrow in Fig. 3(a). The blue-shift of resonance wavelengths with curvature was reported previously for localized SPR [60]. In contrast, resonance wavelength remains almost unchanged for much wider range of curvature radius in the perpendicular incidence, which blue-shifts appreciably below r = 1500 μm (red arrow in Fig. 3(a)). This result confirms that the parallel incidence is more sensitive to curvature, since plasmon resonance in the parallel incidence is dominantly established in the center segment. This is not the case in the perpendicular incidence, where momentum-matching to produce plasmon takes place in many segments, therefore the change of resonance wavelength with respect to curvature can be more gradual.

 

Fig. 3 (a) Resonance wavelength λ_sp, (b) width of resonance characteristics δλ_sp (filled symbols, left), and reflected intensity at resonance R_sp (open symbols, right) in the parallel and the perpendicular incidence as the curvature is varied on a logarithmic scale. Inset in (a) shows λ_sp with respect to curvature which is an inverse of curvature radius. Arrows in (a) represent inflection points for the parallel (black) and the perpendicular incidence (red).

Download Full Size | PDF

Note that correlation between λ_sp and sin(θⁿ_in)_max, which represents plasmon momentum associated with the maximum segmental angle of incidence, is very strong at −0.978 and −0.933 for parallel and perpendicular incidence, respectively. This is reflected in the good linearity between curvature and λ_sp shown in the inset of Fig. 3(a) and suggests that curvature effect on resonance shift is dominated by light momentum dispersion and that the dominance is slightly stronger in the parallel incidence.

Figure 3(b) shows resonance width δλ_sp and reflection intensity at resonance R_sp with curvature. For both parallel and perpendicular incidence, R_sp increases as the curvature rises. R_sp is higher in the parallel incidence for all curvatures, which indicates larger damping of SP due to relatively poor momentum-matching [61]. On the other hand, δλ_sp remains almost constant in the perpendicular incidence, although it increases slightly at a very high curvature. This is in direct contrast with the behavior shown in the parallel incidence, where the resonance broadens significantly with higher curvature, which is then followed by a sudden drop in the width. The narrowed resonance at very high curvature, e.g., r = 1500 μm, may not be translated into desirable sensor characteristics because of high R_sp. In this regard, perpendicular incidence exhibits more robust sensor performance in the presence of curvature changes.

Overall, resonance wavelength (λ_sp) and width of resonance characterstics (δλ_sp) may change significantly in response to curvature variation. Figure 3 shows that the change is much more significant for the parallel incidence. In the range of curvature radius that we considered, the shift of λ_sp and δλ_sp between maximum and minimum amounts to be 56 nm and 101.9 nm for the parallel incidence vs. 18.5 nm and 12.5 nm for the perpendicular incidence.

Understanding of the effect of curvature on the specific molecular detection characterstics was attempted with DNA hybridization as a model interaction, as shown in Fig. 4. It is clear that the resonance shift for DNA hybridization is in general similar to that of bare substrate as shown in Fig. 3(a) and decreases at higher curvature. The results also show a threshold curvature below which resonance shift decreases significantly (marked by arrows in Fig. 4). Once the curvature radius falls below this threshold, resonance shift becomes much smaller in the parallel incidence than in the perpendicular incidence. Note that resonance shift slightly increases at very high curvature when ssDNA immobilizes in the perpendicular incidence (see the red circle in Fig. 4). This arises from the light incidence that becomes s-polarization like for segments away from the center. In other words, if incident light can be maintained to be p-polarized for all segments, resonance wavelength shift would decrease monotonously with higher curvature. For DNA hyridization, as the curvature radius decreases from r = ∞ to 1500 μm, the resonance wavelength shift (Δλ_sp) is reduced by 58% in the parallel incidence, compared to no visible decrease in the perpendicular incidence If the curvature increases further, SPR disappears in the parallel incidence, while the shift in the perpendicular incidence decreases by 17% from SPR on flat surface. Interestingly, the detection sensitivity D, which may be defined as the resonance shift per unit index change, i.e., D = Δλ_sp/Δn, was shown to become a little higher with curvature than on flat surface with a mximum at 183 nm/RIU for DNA hybridization (see Fig. 4).

 

Fig. 4 Resonance wavelength shift Δλ_sp produced by DNA immobilization and hybridization in the parallel and the perpendicular incidence. Arrows represent threshold curvature below which resonance shift decreases significantly. The red circle is the area in which the shift increases despite increased curvature.

Download Full Size | PDF

Figure 5 shows plasmon momentum at i-th segment Kⁱ_SP = (2π/λ_sp)sinθⁱ_in in the parallel and the perpendicular incidence. First of all, when momentum-matching takes place, the difference of momentum from that of the center becomes larger at a segment farther from the center and the range of plasmon momentum that is matched in contributing segments increases with a smaller curvature radius for both incidences. In the parallel incidence, the range is much wider than in the perpendicular incidence, which is because the angular range of incidence onto segments is much larger in the parallel incidence, when we compare Eqs. (3) and (4). For example, the momentum variance, which may be defined as the difference between maximum and minimum momentum matched among the segments, is ΔK^prl_SP = 2.15 μm⁻¹ and is ΔK^ppn_SP = 0.0119 μm⁻¹ at r = 3750 μm (see Fig. 5), thus the relative ratio ΔK^prl_SP/ΔK^ppn_SP = 181. In other words, momentum matching occurs in a much wider range in the parallel incidence by two orders of magnitude, which causes far broader and longer resonance shift. Note also that in the perpendicular incidence, the distribution of matched momentum is symmetric with respect to the center segment, i.e., λⁱ_sp = λ^-i_sp (i = –(N – 1)/2 ~(N – 1)/2), which is not the case in the parallel incidence. The trend as a result of DNA hybridization at each segment is almost identical to Kⁱ_SP in ambiance shown in Fig. 5, except momentum matching at a lower momentum, i.e., at a longer resonance wavelength, with immobilization and hybridization.

 

Fig. 5 Segmental plasmon momentum at SPR with curvature radius: (a) parallel and (b) perpendicular incidence. Inset shows segment index running between 1 and 13 with 7 as the center (13 segments in total). Calculation of SPR was performed in water ambiance without DNA. Momentum variance ΔK_sp is also shown at r = 3750 μm.

Download Full Size | PDF

3.3 Extension to multi-curvature surface

Above analysis can be easily extended to SPR detection on a more complex surface with multiple curvatures. As an example of such a multi-curvature surface, consider a structure which consists of sub-surfaces with positive and negative curvature, i.e., curvature radius is fixed at r = 225 μm on the left half while it is varied on the right half for |r| ≥ 225 μm, i.e., −4.44 mm⁻¹ ( = −1/225 μm) < curvature < 4.44 mm⁻¹ ( = 1/225 μm). Segmentation analysis allows resonance characteristics of the structure to be easily calculated from Eq. (1).

First of all, resonance curves for the parallel incidence are presented in Fig. 6(a). The direction of the arrow represents an increasing curvature from −4.44 to + 4.44 mm⁻¹. The surface of interest is shown as the inset next to the arrow. The resonance dip red-shifts to a longer wavelength with curvature except at high curvature in the positive range. Note that resonance at r = ∞ is broader and shallower than in Fig. 2(a) because of partially curved surface on the left half. For the perpendicular incidence shown in Fig. 6(b), drastically different trends are observed. With a decrease in |r|, i.e., an increase in the curvature magnitude (increase of curvature shown as an arrow in Fig. 6(b)), resonance blue-shifts. In other words, an increase of curvature (less negative curvature), while it is negative, red-shifts resonance wavelengths for both parallel and perpendicular incidence. If the curvature is positive, however, a curvature increase prompts a red-shift in the parallel incidence and a blue-shift in the case of perpendicular incidence, as presented in Fig. 6(c) and also in terms of curvature radius in Fig. 6(d). The apparently opposite behavior between parallel and perpendicular incidence for positive curvature is associated with momentum variance, i.e., momentum variance increases in the parallel incidence, which decreases in the perpendicular incidence. In the perpendicular incidence, it is noted that structures with an identical |r| present identical resonance characteristics, which therefore become symmetric with respect to r = 0.

 

Fig. 6 Resonance characteristics of multi-curvature surface: curvature radius on the lefthand side is fixed at r = 225 μm, while it varies in the range of |r| ≥ 225 μm. Resonance curves: (a) parallel and (b) perpendicular light incidence. Color scheme for (a) and (b) from r = −225 to + 225 μm in blue to red. Shift of resonance wavelengths (λ_sp) calculated on bare substrates under buffer ambiance: (c) with curvature and (d) in terms of curvature radius (black: parallel and red: perpendicular incidence). (e) Resonance shifts (Δλ_sp) due to immobilization of ssDNA and hybridization into dsDNA. Curvature radius at r = ∞ represents flat surface.

Download Full Size | PDF

Resonance shift on multi-curvature surface as a result of immobilization of ssDNA and hybridization into dsDNA were also calculated as shown in Fig. 6(e). As we observed in the case of single curvature surface, the results show that measured resonance shift changes largely depending on the curvature. If one measures biomolecular interactions using potentially (and undesirably) flexible substrates, Fig. 6(e) suggests that perpendicular light incidence provide a much more robust environment with lower sensitivity to curvature. For example, resonance shift due to immobilization on curved surface varies by 11% ( = 2/19) only for the perpendicular incidence, while it is as large as 157% ( = 16.5/10.5) for the parallel incidence. On the other hand, if one intends to obtain highest detection sensitivity, an optimum may exist with parallel incidence.

3.4 Discussion

For discussion, we extend the results into SPR detection on surface with arbitrary curvature, for example, on spherical concave or convex surface. Segmentation suggests that resonance should in general be even broader on spherical surface than what may be observed on cylindrical surface, because momentum matching occurs in a wider range assuming an identical single curvature and matched momentum becomes more dispersive on spherical surface. However, the relation between curvature and resonance characteristics may not be monotonous, as observed in Fig. 3.

Also worth to note is the effect of non-collimated wavefront. Collimated light incidence on curved surface is optically equivalent to non-collimated light incidence on a flat surface. In other words, non-collimated light incidence should result in resonance shift and broadening shown in Figs. 2 and 3. In fact, this suggests at a very interesting possibility that non-collimated light incidence on curved surface may alleviate the degradation that can be incurred by the curvature. For parallel light incidence, converging light on a convex surface may be able to minimize or completely remove the degradation if light wavefront is optimized to the surface curvature. In the case of perpendicular light incidence, non-collimated wavefront would also affect resonance characteristics. In general, the effect of curvature is significantly weaker than the case of parallel incidence, therefore effect of non-collimated wavefront is less prominent. These results and the overall trends observed in this paper would also hold in angle and phase scanning systems because the curvature effects are dominated by light momentum dispersion.

Despite limits of segmentation for high curvature with curvature radius |r| < 100 μm, the approach proposes an interesting possibility that curvature of individual segments may be customized for targeted resonance characteristics. Although higher curvature tends to decrease resonance shift with broader characteristics, the degradation may be minimized if detection is performed with light reflected at specific segments. For instance, in the parallel incidence shown in Fig. 5(a), one may capture light beams reflected by segments 1-7 while excluding those from segments 8-13. On the other hand, it is also suggested that use of diverging (converging) spherical light source should minimize resonance broadening for curved surface with positive (negative) curvature in both parallel and perpendicular incidence. The possibility of modulating resonance characteristics opens up novel plasmonic sensing capabilities, especially by combining with emerging concepts such as metamaterials.

4. Concluding remarks

We have explored surface curvature effects on SPR biosensor characteristics based on curvature segmentation. The segmentation allows partial contribution of segments that approximate curved surface to be assessed in an intuitive manner. It was shown that higher curvature in general leads to broader resonance characteristics and reduces resonance shift almost linearly. The extent of the curvature effects was more drastic in the parallel incidence. The approach was extended to investigate complex multi-curvature surface structures. The analysis can be of fundamental importance in fiber-based in vivo applications and for plasmonic devices using flexible substrates.