Open Access
Add to BibSonomy
Abstract
In this report, we explore a segmentation-based approach for the calculation of surface plasmon resonance (SPR) on the curved surface with high curvature by modeling it as a contiguous array of finite segments. The approach would significantly facilitate the calculation with good accuracy because of the inherent nature that transfer matrix analysis can be used. Using the segmentation model, resonance characteristics at SPR were obtained as the curvature radius was varied. For validation of the segmentation, resonance wavelength (λSPR), reflectance at resonance (RSPR), and resonance width (δλSPR) were compared with the finite element method in the parallel and perpendicular light incidence. It was found that the results from the segmentation were in excellent agreement, λSPR in particular, while RSPR and δλSPR under parallel incidence showed disparity between the two models due to the short segmentation. Resonance of curved surface on the rigid and flexible substrate was compared and the overall trend was found to be almost identical. The segmentation is expected to provide a simple, fast, and efficient way for studying plasmonic devices with high curvature in flexible and wearable applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Flexible devices have been extremely important for emerging applications in both electronics and optics and have drawn significant attention in recent years [1–5]. Even if we narrow down the attention to optical elements, effects of curvature for flexible applications have been heavily investigated: since Rowland used a grating on a spherical concave surface [6], curvature effects have been explored for diffraction gratings [7], light-emitting diodes [8–10], optical fibers [11–13], plasmonic waveguides [14–20], wire-grid polarizers [21], microwave plasmonics [22,23], and liquid crystals [24].
In line with this trend, biosensors have been mounted increasingly on a flexible curved platform [25–30]. Of particular interest is biosensing based on surface plasmon resonance (SPR), which relies on the excitation of electron density waves referred to as surface plasmon (SP) formed between thin metal and dielectric layers. The resonance condition in SPR biosensing depends on the momentum matching of incident light to SP, which has been the basis to implement a SPR biosensor. However, label-free nature of SPR biosensing tends to compromise the performance, most notably, detection sensitivity. Most of the research regarding SPR biosensors have thus been focused on the improvement of sensitivity using amplification by metallic nanoparticles [31–33], phase-sensitive detection [34–36], combinatorial approaches [37,38], near-field localization [39–41], and field-matter colocalization [42–44]. In addition, SPR biosensing has been used to confirm cell membrane adhesion without labels [45–49]. Recently, wearable sensors for drug detection have been developed as a patch type using plasmonic metasurface and silver nanowire with surface-enhanced Raman scattering [50,51]. In the same manner, SPR biosensor can be applied to wearable sensors as a dermal patch. Hydrogel film, which can extract sweat and analytes from the body, and plasmonic metal film are integrated in the breathable polymer film and transferred to skin secured by breathable adhesive tape. When the light from a low power He-Ne laser is incident on a dermal patch, resonance spectra can be collected by spectrometers.
On a flexible platform, the performance of SPR biosensing is directly affected by surface curvature. Effects of curvature on SPR detection characteristics were investigated, wherein it was found that curved surface broadens momentum matching and in general degrades resonance shifts as a result of molecular interactions and dynamic range that may be obtained [52]. The calculation of optical characteristics on a flexible surface has been simplified by modeling curved surface with piecewise linear segments of finite length [21,49]. The approach transforms the task of computationally heavy 3D calculation into an array of extremely light 1D calculations. With an infinitesimally small segment in an infinite number, piecewise linear segmentation can mimic an ideal curved surface, i.e., calculation with a smaller segment would produce more precise results. However, a very small segment, if used to approximate high curvature with sharp bends, may not allow SP excited in neighboring segments to interact and account for radiation loss of SP polariton (SPP) waves [53]. The limit forced a segment to be significantly larger than SP propagation length and therefore segmentation-based analysis to be useful only for low curvature surface despite the strength of this model that permits fairly precise yet extremely simple approximation compared to more direct simulation methods [54].
In this study, we intend to minimize the requirement on the segment length, thereby extending the segmentation-based analysis to a much wider range of surface curvature. In doing so, we explore the conditions under which the disparity between the segmentation-based approach and finite element method (FEM), which provides more exact yet complicated numerical simulation, can be negligible. Understanding optical characteristics with high curvature surface can be greatly facilitated in a segmentation-based picture, not just for SPR sensors, but also for diverse optical devices on flexible surfaces.
2. Numerical model and method
2.1 Modeling curved surface
The optical configuration that we use for this study is in general in agreement with the prior model, i.e., surface of interest is assumed to have cylindrical curvature in two configurations with respect to light incidence as shown in Fig. 1(a): incident wave vector in the parallel incidence is contained in the plane spanned by the normal of the curved surface (‘normal plane’). In contrast, the incident wave vector is not contained in the normal plane for the perpendicular light incidence. The perpendicular incidence was found to present narrower resonance because segments form more uniform momentum-matching [47]. We have considered only positive surface curvature. An SF10 glass substrate is assumed to support the curved surface without any internal core/cladding structure and is uniformly coated with 50-nm thick gold for efficient excitation of SP [55–57].
Fig. 1. Schematic illustration of the curved surface used for numerical calculation: (a) segmentation-based model and (b) finite element method (FEM). Red cylinders represent beam profiles and thick black arrows indicate the direction of parallel and perpendicular light incidence. The parallel and perpendicular incident wave vectors are contained in the zx and yz plane. Curved surface is approximated in segments (7 shown in the schematic and 37 used in the calculation) in the segmentation-based model. Smooth curvature is modeled in FEM.
The calculation of SPR detection is based on wavelength scanning between λ = 500 and 800 nm with the angle of incidence θin fixed at 55°. θin is defined with respect to the plane of center normal (yz plane in the parallel incidence and zx plane in the perpendicular incidence). The polarization direction corresponding to s- and p-polarization is also shown in Fig. 1. In this calculation, polarization direction is fixed at p-polarized light for excitation of SP instead of s-polarized light. Refractive indices for SF10 glass and gold were obtained from a Ref. [58].
To investigate resonance shift for biosensors in the curved surface bio-molecule analyte is assumed to be DNA oligomers as a homogeneous dielectric layer with 9.32 nm thickness [59]. Refractive index of analyte is changed to 1.40, 1.45 and 1.50 which are values close to optical parameters of 24-mer single-stranded DNA and double-stranded DNA.
2.2 Segmentation-based analysis
In the segmentation-based analysis, the curved surface is divided into many short segments and each segment is modeled as an infinite plane thin film with a changed incident angle of light. We assume that piecewise short linear segmentation can mimic the high curvature surface. The boundary conditions between segments and related diffraction are ignored. Reflected light from different segments are averaged incoherently. Overall reflectance Rcurved is given by
In Eq. (1), N is the total number of segments for each curvature. ki is the wave vector incident onto the i-th segment. $\theta _{in}^i$ denotes the incident angle at the i-th segment and depends on the incidence configuration. Considering variation of incident light power to each segment, reflectance of the i-th segment (Ri) is reduced by an obliquity factor ($\textrm{cos}\theta _{in}^i$) of the segment. In other words, because of the segmentation, Rcurved is expressed as a discrete sum of segmental reflectance Ri which can be calculated from Fresnel coefficients at each wavelength. At each segment reflection coefficients according to wavelength are calculated using transfer matrix method. The electric field amplitudes between the layer 0 and M + 1 are expressed by
Each surface of curvature radius r is modeled of segments of length (Ls) defined as Larc/N ≈ 0.8 μm, which is much smaller than SP propagation length in plasmonic metals (typically 20 and 40 μm for silver and gold at λ = 633 nm wavelength) where Larc is the arc length of curved surface and N = 37. For efficient modeling of high curvatures surface the number of segments is optimized. The detailed optimization process will be described in the section of results and discussion. The chord length of curved surface (Lchord), which indicates the area where the light is incident, is fixed at 30 μm. The overall arc length of the curved surface (Larc) is varied, i.e., the segment length Ls used in each model correspondingly varies, depending on the radius. Finite arc length in the computation amounts effectively to the placement of an aperture that blocks off-axis reflection. For comparison of segmentation-based analysis with more rigorous yet computationally heavier approaches, resonance characteristics of curved surface in the parallel and perpendicular incidence were numerically calculated using FEM.
2 FEM model
In the FEM calculation, segments of surface were modeled to mimic a smooth curved surface, as shown in Fig. 1. FEM solves Maxwell’s equations numerically within spatially discretized small meshes using appropriate boundary conditions. In the case of parallel incidence, we simulated a simple 2D curved surface model using port boundary condition used for a Gaussian beam propagating from the leftmost boundary of SF10 substrate toward the curved surface with an oblique angle of incidence θin. Reflected light was obtained in the rightmost boundary of SF10 substrate. For the perpendicular incidence, a 3D curved surface model was used with an underneath port boundary condition for the propagation of a Gaussian beam. Floquet periodic boundary condition was applied to the front and back zx plane. Calculation domain depth to the y-axis was minimized to be as thin as possible for reducing use of computational resources. Scattering boundary conditions and perfectly matched layers were used for eliminating unexpected reflections. In the simulation, thicknesses of substrate are set up about 47 μm and 2 μm for parallel and perpendicular incidences. Reflectance was calculated using a ratio of incident light and reflected light power Pout/Pin integrating corresponding boundaries in the 2D model. In the 3D model, reflected light power was obtained by subtracting absorbed and leakage power from incident light power Pin. Absorbed power Pabs was integrated power dissipation over the whole volume of the curved metal film, while leakage power Pleak was calculated from Poynting vectors at either end of the curved metal film with respect to the yz plane. Reflectance for the perpendicular incidence can then be obtained as (Pin - Pabs - Pleak)/Pin.
2.4 Computational resources
Segmentation can reduce the run-time significantly compared to FEM. In the segmentation-based model, reflectance spectra are obtained by the transfer matrix method based on thin-film optics. Calculation of one segment took 0.12 seconds for sweeping the wavelength from λ = 500 nm to 800 nm with a 1-nm step, thus 4.5 seconds for the total 37 segments. For the calculation, a personal computer equipped with a 3.9-GHz dual-core CPU and 16 GB RAM was used. On the other hand, high computing power is necessary for FEM calculation. With a workstation equipped with a 3.0-GHz 48-core CPU and 1TB RAM, it took about 26 and 21 hours using FEM for the chord length Lchord = 30 μm in the wavelength range of λ = 500 ∼ 800 nm, respectively, for the parallel and perpendicular incidence with usage of 59 GB RAM. Based on the run-time, therefore, segmentation was found to be much faster by 20,800 and 16,800 times in the parallel and perpendicular incidence than FEM.
3. Results and discussion
3.1 Effect of curvature radius on the resonance characteristics
We have first varied the curvature radius (r) to understand the effect on SPR for the parallel and the perpendicular incidence using segmentation with a short segment length of about 0.8 μm, i.e., Ls ≈ 0.8 μm. Figures 2(a) and 2(b) present resonance characteristics for curvature radius varying from r = 100 μm to 3000 μm fixed at Lchord = 30 μm. The trend of resonance broadening with a blue-shift in the resonance wavelength (λSPR) is observed with a smaller curvature radius in the parallel incidence which involves momentum-matching with higher off-center wave vectors. In contrast, λSPR remains almost unchanged in the perpendicular incidence despite a change in the curvature radius. This is because a smaller number of segments contribute to SPR with an increase of curvature in the parallel incidence, whereas the whole curved surface participates in the resonance regardless of curvature radius in the case of perpendicular incidence.
Fig. 2. Resonance curves calculated by segmentation for (a) parallel and (b) perpendicular light incidence in p-polarization with respect to the curvature radius in the range of r = 100 ∼ 3000 μm when the chord length is fixed at Lchord = 30 μm. Resonance broadening with a blue-shift is clearly observed in (a). Resonance characteristics from flat surface (r = ∞) to curvature radius (r = 100 μm): (c) resonance wavelength λSPR (filled symbols, left), reflectance at resonance RSPR (open symbols, right), and (d) resonance width δλSPR. Inset in (c) shows λSPR with respect to the curvature (an inverse of curvature radius r) with linearly fitted lines (R2 = 0.947 and 0.953 for parallel and perpendicular incidence).
Figures 2(c) and 2(d) shows resonance wavelength (λSPR), reflectance at resonance (RSPR), and resonance width (δλSPR) as the curvature radius is varied. Here, resonance width is defined as the full-width-at-half-maximum of the resonance dip in reference to the critical wavelength. It is clear that λSPR is blue-shifted while RSPR increases monotonously with a shorter curvature radius, as shown in Fig. 2(c). Resonance wavelength shows good linearity with curvature ( = 1/r), as shown in the inset of Fig. 2(c), because plasmon momentum is strongly associated with the maximum segmental angle of incidence [53]. Variations of resonance wavelength and reflectance at resonance are more drastic in the parallel incidence than in the perpendicular incidence. An increase of reflectance at resonance is often associated with damping [60–62]: from the perspective of segmentation analysis, even if resonance is formed at a segment, other segments may not produce resonance and thus cause high damping, which dominates overall damping characteristics. Interestingly in the parallel incidence, Fig. 2(d) shows that the width (δλres) of a resonance dip initially increases with a shorter curvature radius, because each segment has a slightly different momentum-matching condition, then it decreases due to the high damping at the high off-axis segment. As the curvature radius decreases below r = 500 μm, the resonance dip of curved surface gradually disappears and the width of the resonance dip decreases from the maximum broad width in the parallel incidence. On the other hand, the width of resonance dip almost remains as a constant over the widest range of curvature radius in the perpendicular incidence. To sum up, the resonance of curved surface is more sensitive in the parallel incidence than the perpendicular incidence. In the range of curvature radius that we calculated, the differences of λSPR and δλSPR between maximum and minimum values to be 49 nm and 95 nm for the parallel incidence versus 5 nm and 1 nm for the perpendicular incidence. These results confirm the trends observed with large segments (Ls = 20 μm) [49], which shows that the resonance characteristics are less sensitive to segment length than presumed.
3.2 Validation and near-field analysis
What is as important as the resonance characteristics on curved surface is the validity of segmentation analysis using segments that are much smaller in size than SP propagation length. For the purpose of validation, we explored the effectiveness of segmentation-based analysis in comparison with FEM. We therefore used FEM to calculate reflectance in the case of curved SPR to investigate resonance characteristics varying curvature radius from r = 100 μm to 3000 μm in the parallel incidence with Lchord = 30 μm, as shown in Fig. 3(a). Resonance characteristics were found to blue-shift and be broadened as the curvature radius became shorter because of momentum matching with a higher center-off wave vectors, which is overall in excellent agreement with the segmentation-based analysis presented in Fig. 2(a). FEM results, which are assumed to provide exact solutions, have slight disparities to the segmentation-based results. Figure 3(b) shows the reflectance spectra calculated by FEM and segmentation-based model in the parallel incidence with curvature radius r = 3000, 500, 300, and 100 μm. While resonance wavelength calculated by FEM and segmentation-based model agree quite well in the whole range, reflectance at resonance (RSPR) shows a difference between the two models. The disparity increases as the curvature radius becomes shorter and the resonance characteristic disappears for r < 300 μm due to significant damping, which leads to reduced disparity in the reflectance spectrum between FEM and segmentation-based model.
Fig. 3. Resonance curves calculated by FEM for parallel light incidence in p-polarization with respect to the curvature radius in the range of r = 100 ∼ 3000 μm when the chord length is fixed at Lchord = 30 μm. (b) Resonance curves calculated by the segmentation for parallel light incidence in p-polarization compared with the exact results calculated by FEM. Curvature radius is varied at r = 3000, 500, 300, and 100 μm. Near-field intensity distribution (|E|2): (c) r = 3000, (d) 500, (e) 300, and (f) 100 μm for parallel light incidence in p-polarization. Scale bar: 5 μm.
The disparity can be explained in the near-field: Figs. 3(c)–3(f) presents the near-field intensity by FEM in the zx plane. For r = 3000 μm shown in Fig. 3(c), which represents almost planar surface, RSPR ≈ 0 from the segmentation-based model, as the incident light is efficiently coupled to excited SP wave under identical resonance conditions at each infinitesimal segment. Note that a small amount of incident light at the edge of the beam (marked by arrow) is reflected at the resonance wavelength (λ = 720 μm in this case), despite a long curvature radius r = 3000 μm. The reflection at the edge persists down to the curvature radius r = 300 μm, as shown in Figs. 3(d) and 3€. The increase of RSPR observed in FEM calculation compared to the case of segmentation is largely due to the reduction of SPP radiation loss at the finite beam edges. When the curvature radius is r = 100 μm, an extremely high curvature surface disrupts resonance characteristics of the whole curved surface as shown in Fig. 3(f). As a result, reflectance spectra show coincidence between FEM and segmentation-based model in Fig. 3(b).
Figure 4 shows resonance characteristics of curved surface in the perpendicular incidence using FEM to validate the results calculated by the segmentation-based model with short segments. Overall calculation of FEM is conducted with equivalent conditions used for the segmentation-based model. Figure 4(a) presents resonance characteristics calculated by FEM in the perpendicular incidence which are almost identical to the result of segmentation presented in Fig. 2(b). In the perpendicular incidence, curved surface contributes to SPR regardless of the curvature radius. Reflectance spectra of curved surface in both FEM and the segmentation-based model show little changes with r = 3000, 500, 300, and 100 μm, as shown in Fig. 4(b). In the case of r = 100 μm, FEM produced shorter resonance wavelength and slightly higher RSPR. This result can be explained in the near-field intensity distribution obtained by FEM. Figure 4(c)-(f) shows near-field intensity in the zx plane with curvature radius r = 3000, 500, 300, and 100 μm. Clearly, curved surface supports propagating SPPs in the gold surface in the perpendicular incidence, regardless of the curvature radius.
Fig. 4. Resonance curves calculated by FEM for perpendicular light incidence in p-polarization with respect to the curvature radius in the range of r = 100 ∼ 3000 μm when the chord length is fixed at Lchord = 30 μm. (b) Resonance curves calculated by the segmentation for perpendicular light incidence in p-polarization compared with the exact results calculated by FEM. Curvature radius is varied at r = 3000, 500, 300, and 100 μm. Near-field intensity distribution (|E|2): (c) r = 3000, (d) 500, (e) 300, and (f) 100 μm for perpendicular light incidence in p-polarization. Scale bar: 2 μm.
Direct comparison of resonance characteristics between FEM and the segmentation-based model is made in Fig. 5 for the parallel and the perpendicular incidence. First of all, resonance wavelength λSPR shows no significant difference in both cases, as shown in Fig. 5(a), although the difference is slightly larger for r ≥ 1500 μm in the parallel incidence. When the curvature radius falls below r = 200 μm in FEM and 100 μm in the segmentation-based model, the curved surface loses the resonance dip in the reflectance spectrum. In the perpendicular incidence, resonance wavelength remains almost unchanged with a slight blue-shift below r = 200 μm for both FEM and the segmentation-based model [marked with an orange arrow in Fig. 5(a)]. Both FEM and the segmentation-based model confirm that curvature affects resonance wavelength shift more in the parallel incidence than the perpendicular incidence.
Fig. 5. Resonance characteristics with respect to the curvature radius in the range of r = 100 ∼ 3000 μm calculated by segmentation vs. FEM for parallel and perpendicular incidence in p-polarization: (a) resonance wavelength λSPR, (b) reflectance at resonance Rspr, and (c) resonance width δλSPR (square: parallel and circle: perpendicular incidence, black: FEM and red: segmentation). Arrows in (a) and (c) represent inflection points of resonance characteristics. The difference is shown in yellow shade. (d) Variance of resonance curves between segmentation and FEM with respect to the curvature radius in the range of r = 100 ∼ 3000 μm for parallel and perpendicular incidence in p-polarization.
For the perpendicular light incidence, reflectance at resonance RSPR and resonance width δλSPR remain in good agreement between FEM and the segmentation, as shown in Figs. 5(b) and 5(c). However, a significant difference shows up in RSPR and δλSPR in the parallel incidence, whereas the overall trends agree well. RSPR decreases monotonically with an increase of curvature radius. Note also that RSPR is smaller with the segmentation than with FEM. This implies that damping is underestimated in the segmentation, which ignores the interaction of SPP radiation between neighboring segments and reduction of the radiation loss at high curvature as a short segment is used to model an infinite plane. On the other hand, δλSPR, when the curvature radius is reduced, increases before it drastically decreases at below 800 μm and 500 μm in the FEM and segmentation-based model [marked with gray and green arrow in Fig. 5(c)], i.e., resonance dip is broader for r < 700 μm and narrower for r > 700 μm out of segmentation-based model than FEM.
The disparity of RSPR and δλSPR observed in the parallel incidence demonstrates potential limitation of the segmentation-based model for investigating SPR on the curved surface. For quantitative evaluation of the disparity, we have calculated the variance of reflectance spectra between FEM and the segmentation-based model, i.e., the variance VAR is given by
Variance of reflectance spectra between FEM and the segmentation-based model is also calculated, varying segment length as shown in Fig. 6. For parallel incidence, at low curvature surface, where the curvature radius r ≥ 2000μm, variance has no big difference varying segment length, which means that a long segment length is enough to model curved surface. However, at high curvature surface, where the curvature radius r < 2000μm, variance gradually decreases as the segment length decreases and saturates when the segment length is below Ls = 1 μm as shown in Fig. 6(a). For perpendicular incidence, variance has a low value except r = 100 μm. When the curvature radius is 100 μm, as the segment length decreases variance decreases as shown in Fig. 6(b). Therefore, we need to choose a short segment length for a high curvature surface. We choose the optimized segment length about 0.8 μm with number of segments N = 37, because a short segment length increases the calculation time. In the condition of calculated curved surface, segmentation-based model using a short segment length Ls ≈ 0.8 μm is effective for the high curvature surface down to curvature radius r = 100 μm.
Fig. 6. Variance of resonance curves between segmentation and FEM with respect to the segment length Ls = 0.15 ∼ 10 μm for (a) parallel and (b) perpendicular incidence in p-polarization.
3.3 Detection characteristics for a biosensor
To exploit curved surface for a biosensor, reflectance spectra are calculated in terms of various refractive indices of analyte. Figure 7 shows resonance curves with respect to various refractive indices of nPBS = 1.33 (buffer ambience), nanalyte = 1,40, 1.45, and 1.50 (DNA hybridization) varying curvature radius at r = $\infty $, 2000, 1000, and 500 μm. The planar thin gold film with analyte shows resonance wavelength shift with continuous uniform shape of resonance dip. As the curvature radius decreases, resonance width and reflectance at resonance increases with high refractive index of analyte. Resonance shift of the biosensor for curved surface is almost maintained compared to planar structure when the curvature radius r = 1000 μm. However, resonance shift decreases when curvature radius r < 1000 μm. In the case of curvature radius r = 500 μm, resonance shift of analyte whose refractive index nanalyte = 1.50 in the curved surface shows ΔλSPR = 21 nm which is as low as 0.7-fold over planar gold thin film (ΔλSPR = 30 nm).
Fig. 7. Resonance curves with respect to various refractive indices of biomolecule analyte calculated by the segmentation for parallel light incidence in p-polarization. Curvature radius is varied at (a) r = $\infty $, (b) 2000, (c) 1000, and (d) 500 μm.
3.4 Extension to a flexible substrate
In this section, we extend the investigation of SPR characteristics of a curved surface to a flexible substrate using segmentation-based model and FEM. A flexible substrate is modeled in polyimide which is widely used to implement flexible optical devices such as OLEDs [63], solar cells [64], and biosensors [65]. Refractive index of polyimide was obtained from a Ref. [66]. The geometry of the curved surface is consistent with the case of SF10 substrate. Incident angle θin was changed from 55° to 54° due to the change of resonance angle conditions. SPR detection was investigated based on wavelength scanning between λ = 500 and 800 nm with p-polarized light for excitation of SP.
Figures 8(a) and 8(b) shows resonance characteristics of a flexible SPR device for curvature radius r = 100 ∼ 3000 μm, when the chord length is fixed at Lchord = 30 μm, using segmentation-based model and FEM for the parallel incidence. Compared to the case of SF10 substrate, resonance dip is more pronounced and sharper. Resonance wavelength (λSPR) is found to blue-shift with a shorter curvature radius in polyimide substrate while the dip is broadened because of momentum matching with off-axis wave vectors. We emphasize that the overall trend remains unchanged in the parallel incidence. Moderate disparity between segmentation-based model and FEM appears concerning reflectance at resonance (RSPR) because a finite flexible SPR structure causes incident light at the edge of the beam to be partly reflected at the resonance wavelength. SPP radiation loss is reduced at finite beam edges similar to the SF10 case. Figures 8(c) and 8(d) presents resonance characteristics of a flexible polyimide substrate calculated by segmentation-based model and FEM in the perpendicular incidence. The two methods show almost identical results, i.e., λSPR remains almost unchanged despite the change of curvature radius in the perpendicular incidence. Sharper resonance dip and blue-shifted resonance wavelength in the flexible polyimide substrate compared to SF10 cases are also observed in the whole range of curvature radius.
Fig. 8. Resonance curves with flexible polyimide substrate for parallel light incidence calculated by (a) segmentation and (b) FEM and for perpendicular light incidence calculated by (c) segmentation and (d) FEM in p-polarization with respect to the curvature radius in the range of r = 100 ∼ 3000 μm. The chord length is fixed at Lchord = 30 μm.
Comparison of resonance characteristics with flexible substrate between FEM and the segmentation-based model is also investigated to resonance wavelength (λSPR), reflectance at resonance (RSPR), and resonance width (δλSPR) as the curvature radius is varied. There is no significant difference in terms of resonance wavelength λSPR in both cases, as shown in Fig. 9(a), although the difference is slightly larger for r ≥ 1000 μm in the parallel incidence. Resonance wavelengths λSPR are around 698 nm and 695 nm in FEM and segmentation-based model above r = 1250 and 800 μm in the parallel incidence [marked with a gray and a green arrow in Fig. 9(a)]. When the curvature radius falls below r = 1250 and 800 μm in FEM and segmentation-based model, resonance wavelength is blue-shifted and the curved surface eventually loses the resonance characteristics below r = 200 μm in FEM and 100 μm in segmentation-based model. In the perpendicular incidence, resonance wavelength remains around λSPR = 694 nm and is slightly blue-shifted below r = 200 in both methods [marked with an orange arrow in Fig. 9(a))].
Fig. 9. Resonance characteristics with flexible substrate with respect to the curvature radius in the range of r = 100 ∼ 3000 μm calculated by segmentation versus FEM for parallel and perpendicular incidence in p-polarization: (a) resonance wavelength λSPR, (b) reflectance at resonance RSPR, and (c) resonance width δλSPR (square: parallel and circle: perpendicular incidence, black: FEM and red: segmentation). Arrows in (a) and (c) represent inflection points of resonance characteristics. The difference is shown in orange shade. (d) Variance of resonance curves with flexible substrate between segmentation and FEM with respect to the curvature radius in the range of r = 100 ∼ 3000 μm for parallel and perpendicular incidence.
Figures 9(b) amd 9(c) presents reflectance at resonance RSPR and resonance width δλSPR with a flexible substrate which shows approximately coincident tendency compared to the SF10 case, as shown in Figs. 5(b) and 5(c). For the parallel incidence, a significant difference of RSPR and δλSPR between FEM and segmentation-based model shows up, although there is great agreement for the perpendicular incidence. Due to the sharper resonance dip, δλSPR shows a narrower value on the whole calculation range of curvature radius [Fig. 9(c)], compared to the SF10 case [Fig. 5(c)]. However, the trend is almost equivalent between rigid (SF10) and flexible (polyimide) cases. δλSPR increases as the curvature radius decreases before it drastically decreases below r = 800 and 500 μm in the FEM and segmentation-based model [marked with a gray and a green arrow in Fig. 9(c)]. Variance of reflectance spectrum with flexible substrate between FEM and segmentation-based model is calculated based on Eq. (4) for quantitative evaluation of disparity, as shown in Fig. 9(d). In the perpendicular incidence, the variance is nearly zero. On the other hand, in the parallel incidence, the variance monotonously increases from r = 3000 to 500 μm and drastically decreases for r < 500 μm because of the disappearance of resonance characteristics. The variance is as high as 2.82-fold for the r = 500 μm over r = 3000 μm, i.e., a larger difference of variance arises according to the change of curvature radius in flexible polyimide substrate compared to SF10 case (1.86-fold).
4. Concluding remarks
We have explored the segmentation-based analysis to model SPR on the curved surface with high curvature by comparing the resonance characteristics such as λSPR, RSPR, and δλSPR with the exact solution calculated by FEM. Both rigid and flexible substrates were considered in the model. The results are overall in good agreement. RSPR and δλSPR showed relatively large disparity between the two methods in association with the finite segment length used in the segmentation, by which interaction of SPP radiation in neighboring segments is ignored. Overall trend was identical between rigid and flexible substrates. The approach provides an extremely fast and simple alternative to obtain curved SPR characteristics with reasonable precision and may be used in applications for flexible and wearable plasmonics.
Funding
National Research Foundation of Korea (2019R1A4A1025958, 2019R1A6A3A01096804); Korea Medical Device Development Fund (KMDF_PR_20200901_0088, KMDF_PR_20200901_0103).
Acknowledgments
K. Kang acknowledges the support by the BK21 program of School of Electrical and Electronic Engineering of Yonsei University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. I. Park, Y. Xiong, R. H. Kim, P. Elvikis, M. Meitl, D. H. Kim, W. Jian, J. Yoon, C. J. Yu, Z. Liu, Y. Huang, K. C. Hwang, P. Ferreira, X. Li, K. Choquette, and J. A. Rogers, “Printed assemblies of inorganic light-emitting diodes for deformable and semitransparent displays,” Science 325(5943), 977–981 (2009). [CrossRef]
2. M. H. Shih, K. S. Hsu, Y. C. Wang, Y. C. Yang, S. K. Tsai, Y. C. Liu, Z. C. Chang, and M. C. Wu, “Flexible compact microdisk lasers on a polydimethylsiloxane (PDMS) substrate,” Opt. Express 17(2), 991–996 (2009). [CrossRef]
3. R. H. Kim, D. H. Kim, J. Xiao, B. H. Kim, S. I. Park, B. Panilaitis, R. Ghaffari, L. Yao, M. Li, Z. Liu, V. Malyarchuk, D. G. Kim, A. P. Le, R. G. Nuzzo, D. L. Kaplan, F. G. Omenetto, Y. Haung, Z. Kang, and J. A. Rogers, “Waterproof AlInGaP optoelectronics on stretchable substrates with applications in biomedicine and robotics,” Nature Mater 9(11), 929–937 (2010). [CrossRef]
4. J. Pu, Y. Yomogida, K. K. Liu, L. J. Li, Y. Iwasa, and T. Takenobu, “Highly flexible MoS2 thin-film transistors with ion gel dielectrics,” Nano Lett. 12(8), 4013–4017 (2012). [CrossRef]
5. T. Georgiou, R. Jalil, B. D. Belle, L. Britnell, R. V. Gorbachev, S. V. Morozov, Y.-J. Kim, A. Gholinia, S. J. Haigh, O. Makarovsky, L. Eaves, L. A. Ponomarenko, A. K. Geim, K. S. Novoselov, and A. Mishchenko, “Vertical field-effect transistor based on graphene–WS2 heterostructures for flexible and transparent electronics,” Nature Nanotechnol. 8(2), 100–103 (2013). [CrossRef]
6. H. A. Rowland, “LXI. Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes,” Phil. Mag. Ser. 13(84), 469–474 (1882). [CrossRef]
7. M. C. Hutley, Diffraction Gratings (Academic, 1982), Chap. 7 Concave Gratings.
8. J.-S. Park, T.-W. Kim, D. Stryakhilev, J.-S. Lee, S.-G. An, Y.-S. Pyo, D.-B. Lee, Y. G. Mo, D.-U. Jin, and H. K. Chung, “Flexible full color organic light-emitting diode display on polyimide plastic substrate driven by amorphous indium gallium zinc oxide thin-film transistors,” Appl. Phys. Lett. 95(1), 013503 (2009). [CrossRef]
9. A. Epstein, N. Tessler, and P. D. Einziger, “Curvature effects on optical emission of flexible organic light-emitting diodes,” Opt. Express 20(7), 7929–7945 (2012). [CrossRef]
10. M. S. White, M. Kaltenbrunner, E. D. Głowacki, K. Gutnichenko, G. Kettlgruber, I. Graz, S. Aazou, C. Ulbricht, D. A. M. Egbe, M. C. Miron, Z. Major, M. C. Scharber, T. Sekitani, T. Someya, S. Bauer, and N. S. Sariciftci, “Ultrathin, highly flexible and stretchable PLEDs,” Nature Photon. 7(10), 811–816 (2013). [CrossRef]
11. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]
12. W. Belardi and J. C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013). [CrossRef]
13. L.-P. Sun, J. Li, Y. Tan, S. Gao, L. Jin, and B.-O. Guan, “Bending effect on modal interference in a fiber taper and sensitivity enhancement for refractive index measurement,” Opt. Express 21(22), 26714–26720 (2013). [CrossRef]
14. J.-W. Liaw and P.-T. Wu, “Dispersion relation of surface plasmon wave propagating along a curved metal-dielectric interface,” Opt. Express 16(7), 4945–4951 (2008). [CrossRef]
15. A. Passian, R. H. Ritchie, A. L. Lereu, T. Thundat, and T. L. Ferrell, “Curvature effects in surface plasmon dispersion and coupling,” Phys. Rev. B 71(11), 115425 (2005). [CrossRef]
16. W. Wang, Q. Yang, F. Fan, H. Xu, and Z. L. Wang, “Light propagation in curved silver nanowire plasmonic waveguides,” Nano Lett. 11(4), 1603–1608 (2011). [CrossRef]
17. Z. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76(1), 016402 (2013). [CrossRef]
18. P. Berini and J. Lu, “Curved long-range surface plasmon-polariton waveguides,” Opt. Express 14(6), 2365–2371 (2006). [CrossRef]
19. W.-K. Kim, W.-S. Yang, H.-M. Lee, H.-Y. Lee, M.-H. Lee, and W.-J. Jung, “Leaky modes of curved long-range surface plasmon-polariton waveguide,” Opt. Express 14(26), 13043–13049 (2006). [CrossRef]
20. T.-H. Xiao, L. Gan, and Z.-Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015). [CrossRef]
21. D. Kim and E. Sim, “Segmented coupled-wave analysis of a curved wire-grid polarizer,” J. Opt. Soc. Am. A 25(3), 558–565 (2008). [CrossRef]
22. X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, “Conformal surface plasmons propagating on ultrathin and flexible films,” Proc. Natl. Acad. Sci. 110(1), 40–45 (2013). [CrossRef]
23. X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013). [CrossRef]
24. N Karimi, A. Alberucci, O. Buchnev, M. Virkki, M. Kauranen, and G. Assanto, “Phase-front curvature effects on nematicon generation,” J. Opt. Soc. Am. B 33(5), 903–909 (2016). [CrossRef]
25. J. Rooney and E. A. H. Hall, “Designing a curved surface SPR device,” Sensors and Actuators B: Chemical 114(2), 804–811 (2006). [CrossRef]
26. C. Li, J. Han, and C. H. Ahn, “Flexible biosensors on spirally rolled micro tube for cardiovascular in vivo monitoring,” Biosens. Bioelectron. 22(9-10), 1988–1993 (2007). [CrossRef]
27. D. Pradhan, F. Niroui, and K. T. Leung, “High-performance, flexible enzymatic glucose biosensor based on ZnO nanowires supported on a gold-coated polyester substrate,” ACS Appl. Mater. Interfaces 2(8), 2409–2412 (2010). [CrossRef]
28. B. G. Choi, H. S. Park, T. J. Park, M. H. Yang, J. S. Kim, S.-Y. Jang, N. S. Heo, S. Y. Lee, J. Kong, and W. H. Hong, “Solution chemistry of self-assembled graphene nanohybrids for high-performance flexible biosensors,” ACS Nano 4(5), 2910–2918 (2010). [CrossRef]
29. S. Aksu, M. Huang, A. Artar, A. A. Yanik, S. Selvarasah, M. R. Dokmeci, and H. Altug, “Flexible plasmonics on unconventional and nonplanar substrates,” Adv. Mater. 23(38), 4422–4430 (2011). [CrossRef]
30. H. Shafiee, W. Asghar, F. Inci, M. Yuksekkaya, M. Jahangir, M. H. Zhang, N. G. Durmus, U. A. Gurkan, D. R. Kuritzkes, and U. Demirci, “Paper and flexible substrates as materials for biosensing platforms to detect multiple biotargets,” Sci. Rep. 5(1), 8719 (2015). [CrossRef]
31. L. He, M. D. Musick, S. R. Nicewarner, F. G. Salinas, S. J. Benkovic, M. J. Natan, and C. D. Keating, “Colloidal Au-enhanced surface plasmon resonance for ultrasensitive detection of DNA hybridization,” J. Am. Chem. Soc. 122(38), 9071–9077 (2000). [CrossRef]
32. S. Moon, D. J. Kim, K. Kim, D. Kim, H. Lee, K. Lee, and S. Haam, “Surface-enhanced plasmon resonance detection of nanoparticle-conjugated DNA hybridization,” Appl. Opt. 49(3), 484–491 (2010). [CrossRef]
33. S. Moon, Y. Kim, Y. Oh, H. Lee, H. C. Kim, K. Lee, and D. Kim, “Grating-based surface plasmon resonance detection of core-shell nanoparticle mediated DNA hybridization,” Biosens. Bioelectron. 32(1), 141–147 (2012). [CrossRef]
34. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach–Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004). [CrossRef]
35. P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007). [CrossRef]
36. A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011). [CrossRef]
37. B. Sepúlveda, A. Calle, L. M. Lechuga, and G. Armelles, “Highly sensitive detection of biomolecules with the magneto-optic surface-plasmon-resonance sensor,” Opt. Lett. 31(8), 1085–1087 (2006). [CrossRef]
38. J. Oh, Y. W. Chang, S. Yoo, D. J. Kim, S. Im, Y. J. Park, D. Kim, and K.-H. Yoo, “Carbon nanotube-based dual mode biosensor for electrical and surface plasmon resonance measurements,” Nano Lett. 10(8), 2755–2760 (2010). [CrossRef]
39. K. M. Byun, S. J. Yoon, D. Kim, and S. J. Kim, “Experimental study of sensitivity enhancement in surface plasmon resonance biosensors by use of periodic metallic nanowires,” Opt. Lett. 32(13), 1902–1904 (2007). [CrossRef]
40. K. Kim, D. J. Kim, S. Moon, D. Kim, and K. M. Byun, “Localized surface plasmon resonance detection of layered biointeractions on metallic subwavelength nanogratings,” Nanotechnology 20(31), 315501 (2009). [CrossRef]
41. W. Lee, Y. Kinosita, Y. Oh, N. Mikami, H. Yang, M. Miyata, T. Nishizaka, and D. Kim, “Three-dimensional superlocalization imaging of gliding Mycoplasma mobile by extraordinary light transmission through arrayed nanoholes,” ACS Nano 9(11), 10896–10908 (2015). [CrossRef]
42. Y. Oh, W. Lee, and D. Kim, “Colocalization of gold nanoparticle-conjugated DNA hybridization for enhanced surface plasmon detection using nanograting antennas,” Opt. Lett. 36(8), 1353–1355 (2011). [CrossRef]
43. Y. Kim, K. Chung, W. Lee, D. H. Kim, and D. Kim, “Nanogap-based dielectric-specific colocalization for highly sensitive surface plasmon resonance detection of biotin-streptavidin interactions,” Appl. Phys. Lett. 101(23), 233701 (2012). [CrossRef]
44. Y. Oh, W. Lee, Y. Kim, and D. Kim, “Self-aligned colocalization of 3D plasmonic nanogap arrays for ultra-sensitive surface plasmon resonance detection,” Biosens. Bioelectron. 51, 401–407 (2014). [CrossRef]
45. P. A. van der Merwe and A. N. Barclay, “Analysis of cell-adhesion molecule interactions using surface plasmon resonance,” Curr. Opin. Immunol. 8(2), 257–261 (1996). [CrossRef]
46. K. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, “Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy,” Biophys. J. 76(1), 509–516 (1999). [CrossRef]
47. S.-H. Kim, W. Chegal, J. Doh, H. M. Cho, and D. W. Moon, “Study of cell-matrix adhesion dynamics using surface plasmon resonance imaging ellipsometry,” Biophys. J. 100(7), 1819–1828 (2011). [CrossRef]
48. T. Son, J. Seo, I.-H. Choi, and D. Kim, “Label-free quantification of cell-to-substrate separation by surface plasmon resonance microscopy,” Opt. Commun. 422(422C), 64–68 (2018). [CrossRef]
49. T. Son, D. Lee, C. Lee, G. Moon, G. E. Ha, H. Lee, H. Kwak, E. Cheong, and D. Kim, “Superlocalized three-dimensional live imaging of mitochondrial dynamics in neurons using plasmonic nanohole arrays,” ACS Nano 13(3), 3063–3074 (2019). [CrossRef]
50. Y. Wang, C. Zhao, J. Wang, X. Luo, L. Xie, S. Zhan, J. Kim, X. Wang, X. Liu, and Y. Ying, “Wearable plasmonic-metasurface sensor for noninvasive and universal molecular fingerprint detection on biointerfaces,” Sci. Adv. 7(4), eabe4553 (2021). [CrossRef]
51. E. H. Koh, W.-C. Lee, Y.-J. Choi, J.-I. Moon, J. Jang, S.-G. Park, J. Choo, D.-H. Kim, and H. S. Jung, “A wearable surface-enhanced Raman scattering sensor for label-free molecular detection,” ACS Appl. Mater. Interfaces 13(2), 3024–3032 (2021). [CrossRef]
52. H. Lee and D. Kim, “Curvature effects on flexible surface plasmon resonance biosensing: segmented-wave analysis,” Opt. Express 24(11), 11994–12006 (2016). [CrossRef]
53. K. Hasegawa, J. U. Nöckel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75(6), 063816 (2007). [CrossRef]
54. A. Kolomenski, A. Kolomenskii, J. Noel, S. Peng, and H. Schuessler, “Propagation length of surface plasmons in a metal film with roughness,” Appl. Opt. 48(30), 5683–5691 (2009). [CrossRef]
55. T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nature Mater 9(12), 1015–1022 (2010). [CrossRef]
56. T. Yokota, P. Zalar, M. Kaltenbrunner, H. Jinno, N. Matsuhisa, H. Kitanosako, Y. Tachibana, W. Yukita, M. Koizumi, and T. Someya, “Ultraflexible organic photonic skin,” Sci. Adv. 2(4), e1501856 (2016). [CrossRef]
57. T. F. O’Connor, A. V. Zaretski, S. Savagatrup, A. D. Printz, C. D. Wilkes, M. I. Diaz, E. J. Sawyer, and D. J. Lipomi, “Wearable organic solar cells with high cyclic bending stability: Materials selection criteria,” Sol. Energy Mater. Sol. Cells 144, 438–444 (2016). [CrossRef]
58. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
59. S. Elhadj, G. Singh, and R. F. Saraf, “Optical properties of an immobilized DNA monolayer from 255 to 700 nm,” Langmuir 20(13), 5539–5543 (2004). [CrossRef]
60. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings111, (Springer-Verlag, 1988), Chap. 2 Surface Plasmons on Smooth Surfaces. [CrossRef]
61. T. Liebermann and W. Knoll, “Surface-plasmon field-enhanced fluorescence spectroscopy,” Colloids and Surfaces A: Physicochemical and Engineering Aspects 171(1-3), 115–130 (2000). [CrossRef]
62. K. Kim, Y. Oh, K. Ma, E. Sim, and D. Kim, “Plasmon-enhanced total-internal-reflection fluorescence by momentum-mismatched surface nanostructures,” Opt. Lett. 34(24), 3905–3907 (2009). [CrossRef]
63. L. Chen, H. Yu, M. Dirican, D. Fang, Y. Tian, C. Yan, J. Xie, D. Jia, H. Liu, J. Wang, F. Tang, X. Zhang, and J. Tao, “Highly transparent and colorless nanocellulose/polyimide substrates with enhanced thermal and mechanical properties for flexible OLED displays,” Adv. Mater. Interfaces 7(20), 2000928 (2020). [CrossRef]
64. J.-I. Park, J. H. Heo, S.-H. Park, K. I. Hong, H. G. Jeong, S. H. Im, and H.-K. Kim, “Highly flexible InSnO electrodes on thin colourless polyimide substrate for high-performance flexible CH3NH3PbI3 perovskite solar cells,” J. Power Sources 341, 340–347 (2017). [CrossRef]
65. H.-U. Kim, H. Y. Kim, H. Seok, V. Kanade, H. Yoo, K.-Y. Park, J.-H. Lee, M.-H. Lee, and T. Kim, “Flexible MoS2-polyimide electrode for electrochemical biosensors and their applications for the highly sensitive quantification of endocrine hormones: PTH, T3, and T4,” Anal. Chem. 92(9), 6327–6333 (2020). [CrossRef]
66. Q. Li, J. Zhang, X. Pan, Z. Zhang, J. Zhu, and X. Zhu, “Selenide-containing polyimides with an ultrahigh intrinsic refractive index,” Polymers 10(4), 417 (2018). [CrossRef]
