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Abstract
We demonstrate an elliptical gold nanodisk array (GNA) for engineering the spectral profile of surface lattice resonance (SLR). The nanodisk’s shape has a great impact on SLR. Small linewidth of 20 nm at an aspect ratio of 1.17, as well as large wavelength tuning of 64 nm within 4% strain via different orientations and polarizations, are achieved experimentally. The enhanced wavelength response of 6.93 nm per 1% strain variation for elliptical GNA is 2.4 times better than that for general circular GNA. Furthermore, the strain sensing for elliptical GNA approaches is 5.7 times greater than that for circular GNA.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Researches of nanophotonics on metallic nanoparticles have been developed prosperously owing to the advance of nanofabrication techniques [1–3]. Rich optical properties, such as localized surface plasmon resonance (LSPR) [4], surface lattice resonance (SLR) [5], and Fano resonance [6], can be realized by the geometry design or the arrangement of nanoparticles. Among these phenomenons, the high quality factor (Q) of plasmon resonance has been observed in Fano resonance and SLR, which can minimize the loss and realize narrow resonance peak in spectrum. This is beneficial to identify the signal variation and spectral profile shift for sensing applications, including refractive index sensing [7], coloration of imaging [8], biomolecular sensing, strain sensing [9], and so on.
For strain sensing, high Q and wide tunable range of resonance peak, and large sensitivity are crucial. The design of plasmonic nanostructures for narrow linewidth is essential to resolve signal variation under small strain. Many plasmonic nanostructures fabricated on flexible or plastic substrate are proposed to tune plasmon resonance in different manners [10–12]. However, there are challenges remained to be overcome, for example, critical condition for wavelength tuning [13], inefficient tuning of local gap distance [14], natural property of radiation damping for localized plasmonic mode [15], and cracks formed in the surrounding substrate [16]. Gold nanodisk array (GNA) embedded in polydimethylsiloxane (PDMS) for generating SLR can simultaneously solve the issues mentioned above. The SLR can be tuned by varying periodicity and high Young’s modulus of PDMS can avoid cracks during stretching.
The SLR-based strain sensors are demonstrated to exhibit better properties than the LSPR-based ones. Traditional LSPR-based devices show strain sensitivity ranging from 5 to 7 nm per 1% elongation and can approach large stretching percentage (around 100% strain) but with wide full width at half maximum (FWHM) in spectrum [17,18]. On the other hand, the SLR-based devices can reveal higher figure of merit (FOM) defined as the sensitivity divided by the FWHM of resonance peak. The FOM of aluminum-nanoparticle SLR-based stretchable device [8] can approach 0.295, much better than those (below 0.05) of traditional LSPR-based sensors [17,18]. However, more efficient methods to modulate the FWHM of SLR and increase the strain sensitivity are needed and the effect of shape of nanoparticles [16] on strain sensing performance has not been investigated and discussed. In this study, we propose to employ the elliptical GNA for tuning SLR by changing the aspect ratio. The shape effect on polarizability and fundamental optical properties of SLR under different polarizations are examined. Strain sensing is demonstrated with enhanced FOM.
2. Theory
The SLR can be described using the coupled dipole approximation (CDA) to intuitively calculate the extinction spectrum. The predicted collective resonance of nanoparticle array exhibits narrow linewidth of resonance profile when the separation of nanoparticle is close to the incident wavelength scale [19]. The optical properties of SLR have been investigated by a series of works, including different homogeneous environments, number effect, and different array arrangements [5,20]. These studies show that SLR can be understood as the interactions of individual dipoles with interference.
In this approximation, each particle is modeled with static polarizability and the particle is assumed to be ellipsoid with semiaxes a, b, and c. The corresponding polarizability can be expressed as [5]
where the and are relative permittivities of the metal and surrounding medium. The shape factor L can be calculated from the relation of integration [21]. When the size of nanoparticle cannot be neglected, the polarizability α can be treated by the modified long wavelength approximation (MLWA) [5] and expressed aswhere k is the wavevector in the homogeneous medium and is the semiaxis of the particle parallel to the incident electric field. The net dipole amplitude is the sum of the incident field plus the radiation from all other dipoles. Assuming a finite array, the effective polarizability can be written aswhere the array factor S is the total interference and scattering results contributed from all other dipoles. For the square array of dipoles, this factor iswhere θ is the in-plane angle between different located dipoles and r is the distance between any two selected nanoparticles. According to Eq. (3), the resonance of SLR is determined under the condition Re(1/α) = Re(S) to approach infinite polarizability.Once the effective polarizability is found, the absorption and scattering cross sections can be calculated using the following relations
and the extinction cross section can be obtained [5].3. Characteristics of SLR in elliptical GNA
Figure 1(a) shows the illustration of strain sensing for elliptical GNA. Applied strain (%) is defined by as indicated in Fig. 1(a). The shift of optical spectrum reflects the variation from the non-stretched state to stretched state. The transverse length a, longitudinal length b, and thickness c of the elliptical nanodisk is set as 60, 70, and 30 nm for simulation. The periodicity p is 550 nm in the arrangement of square lattice. The Lorentz-Drude model is utilized for the dielectric function of gold [22]. The refractive index of PDMS as homogeneous surrounding is 1.4. Figures 1(b) and 1(c) show the relations of S and 1/α to wavelength for the elliptical GNA under transverse and longitudinal polarizations. The intersections of curves Re(S) and Re(1/α) determine the resonance wavelengths according to the theory of CDA. The same results can be observed in the extinction spectra as shown in Figs. 1(d) and 1(e). For different polarizations, the orientation of dipole oscillation is changed resulting in different polarizabilities. A distinctive red-shift of resonance peak from transverse to longitudinal polarization is observed.
Fig. 1 (a) Illustration of strain sensing for elliptical GNA with spectral shift caused by an external stretching force. Theoretical spectra of S and 1/α under (b) transverse and (c) longitudinal polarizations. The resonance wavelength is indicated by the vertical dashed line. Theoretical extinction spectra under (d) transverse and (e) longitudinal polarizations.
Figures 2(a)-2(d) show the effect of aspect ratio R on polarizability and SLR under different polarizations. The aspect ratio R is defined as the longitudinal length divided by the transverse length (b/a). The b of nanodisk is varied from 60 to 90 nm while the a is fixed at 60 nm. Because of different polarizabilities for x, y, and z directions, the resonance peak can be tuned. A large wavelength shift of 150 nm is obtained for 30 nm variation of b. This geometry provides more design freedom for working wavelength. Figures 2(e) and 2(f) show the trends of FWHM and linewidth index (defined as 1/| ms |) with respect to different R. |ms| is the slope of tangential line of Re(S) at the intersection point with Re(1/α), as illustrated in Fig. 2(b) for b = 60 nm. The magnitude of |ms| indicates the degree of coherence. As |ms| increases, the intersected S also increases, meaning all dipoles resonating with the same phase condition for strong interference, resulting in narrower linewidth. Hence, the FWHM of SLR depends on the slope of Re(S) at the resonance condition [23]. For the transverse polarization, the smaller and less varied polarizability leads the slight linewidth variation between 10.7 and 17.7 nm. On the other hand, large variation of FWHM under longitudinal polarization is found owing to considerable change of polarizability.
Fig. 2 The spectra of Re(S) and Re(1/α), extinction spectra, and FWHM and 1/|ms| with different R under (a) (c) (e) transverse and (b) (d) (f) longitudinal polarizations.
Figure 3(a) shows the fabrication process of GNA embedded in PDMS. First, 240 nm A3 PMMA is spin-coated as electron beam resist on the InP substrate. The GNA pattern is defined on the PMMA by electron beam lithography. A 30 nm Au layer is deposited on the PMMA, followed by the lift-off process. Then, the sample is spin-coated a PDMS layer and bonded onto a PDMS substrate. The InP substrate is removed by wet etching (HCl:H2O = 3:1). Finally, the sample is sealed by spin-coating a PDMS layer on the top. Scanning electron microscope (SEM), microscope, and macroscopic images of the fabricated 100 × 100 μm2 elliptical GNA are shown in Fig. 3(b) with a = 60 nm, b = 70 nm, and p = 564 nm (examined by ImageJ software).
Fig. 3 (a) Fabrication process of elliptical GNA embedded in PDMS. (b) SEM, microscope, and macroscopic images of elliptical GNA. The red dashed region indicates the nanoparticle array.
The measured extinction spectra of circular and elliptical GNAs under different polarizations are shown in Fig. 4. The resonance peaks of circular GNA in Figs. 4(a) and 4(b) are the same (755 nm) for both polarizations as anticipated. For the elliptical GNA with R = 1.17 in Figs. 4(c) and 4(d), the resonance peaks show different behaviors under different polarizations. The measured resonance peak with smaller linewidth at 819 nm under transverse polarization is shorter than the one with broader linewidth at 851 nm under longitudinal polarization. This trend agrees with the simulation results. According to CDA, smaller polarizability under transverse polarization results in blue-shift of SLR resonance and narrower linewidth. Here, the periodicity of circular GNA is 480 nm instead of 550 nm for tuning SLR in the suitable range for latter strain sensing experiment.
Fig. 4 Measured and simulated extinction spectra of circular GNA under (a) transverse and (b) longitudinal polarizations. Measured and simulated extinction spectra of elliptical GNA under (c) transverse and (d) longitudinal polarizations.
4. Strain sensing
For strain sensing, Fig. 5(a) shows the six cases investigated for circular and elliptical GNAs under different polarizations and nanodisk orientations. Figure 5(b) presents the variation of spectra under stretching for cases A and E. The wavelength shifts Δλ1 and Δλ2 are 27.3 and 12 nm within 4% strain for elliptical and circular GNAs under transverse polarization. In Fig. 5(c), the resonance peaks slightly blue shift for cases B and F. This phenomenon can be explained by far-field coupling effect [24]. The collective radiative power is perpendicular to the induced electric field. The coherence strongly depends on the separation of each row. The corresponding variation of dipole-interactive strength is small when the stretching is perpendicular to the separation of each row, leading to slight change of resonance peak. Figures 5(d) and 5(e) show the variations of resonance peak and FWHM under different strains for cases A, E and B, F. The wavelength shift and FWHM of the elliptical GNA are larger and narrower than those of the circular GNA under transverse polarization. For the longitudinal polarization, the FWHM of SLR exhibits a dramatic decrease for case B owing to the nature characteristic of SLR beyond single nanoparticle dipolar resonance [7]. The SLRs for cases A, B, C, and D are observed with wavelengths ranging from 801 to 865 nm within 4% strain. As a result, wide wavelength tuning of 64 nm via an elliptical GNA is obtained under different orientations and polarizations.
Fig. 5 (a) Illustration of the six cases considered for strain sensing under different polarizations and nanodisk orientations for circular and elliptical GNAs. Measured extinction spectra before and after stretching for circular and elliptical GNAs under (b) transverse (cases A and E) and (c) longitudinal (cases B and F) polarizations. Measured resonance peak shift and FWHM variation under different strains for (d) cases A and E and (e) cases B and F. Comparisons of strain sensing properties for (f) SS and (g) FOM.
Figures 5(f) and 5(g) show the strain sensing performances for the six cases. Here, strain sensitivity (SS, nm/%) is defined as resonance peak shift per 1% strain variation by linear fitting and figure of merit (FOM, 1/%) is defined as SS divided by FWHM. The simulations are performed using COMSOL software. The Young’s modulus, Poisson’s ratio, and density are set according to [25]. Due to the positive Poisson’s ratio of PDMS, compressions in y- and z-directions appear when the stretching is applied in x direction. This deformation results in varied periodicity, which is taken into account for calculating SLR under different strains. Among these results, case A shows the best SS (6.93 nm/%) and FOM (0.34 1/%) experimentally. The simulated and measured FOM for elliptical GNA are 9.8 and 5.7 times larger than those for circular GNA. In addition, narrow FWHM of 15.8 nm and 92 nm wavelength tuning via different orientations and polarizations under 8% strain are observed.
The strain sensitivity is greatly influenced by the orientation of elliptical nanodisk, far-field coupling, and shape of nanodisk. The effects of these factors on SS are illustrated in Fig. 6, comparing different situations for better understanding. Since the collective radiative power is perpendicular to the induced electric field, the larger polarizability of elliptical nanodisk for the longitudinal direction results in better sensitivity (A > C, D > B). The relative directions of stretching and polarized field (parallel or perpendicular) affect the strength of far-field coupling. The induced field perpendicular to the stretching direction leads to larger wavelength shift compared to the parallel case (A > D, C > B, E > F). When these two factors are combined in a favorable way, best SS is obtained (A > B). Moreover, the effect of orientation of elliptical nanodisk is stronger than that of far-field coupling from D > C. For the effect of shape, SS of elliptical nanodisk is greater than that of circular one (A > E, C > E, B > F, D > F) owing to the larger polarizability and periodicity [7]. Hence, the thorough investigation of optical properties of the proposed elliptical GNA and the effects of orientation and shape of nanodisk under different polarizations enables us to optimize the strain sensing performance of SLR.
Fig. 6 Illustration of the effects of the orientation of nanodisk, far-field coupling, and shape of nanodisk on SS, comparing different situations for better understanding.
5. Conclusion
We investigate and demonstrate the SLR of elliptical GNA for enhanced strain sensing. The influence of shape on engineering the spectrum of SLR is studied by changing the aspect ratio. The CDA is used to calculate the shape effect on polarizability, resonance peak, and FWHM of SLR. Narrow FWHMs of SLR around 20 nm in experiment for elliptical GNA are obtained at aspect ratio R = 1.17 under 4% strain, better than those for circular GNA. For strain sensing, the optical properties of SLR under different polarizations are examined considering the effects of orientation of elliptical nanodisk, far-field coupling, and shape of nanodisk. An elliptical GNA can achieve a total of 64 nm variation of SLR within 4% strain via varying orientation and polarization. Furthermore, the elliptical GNA exhibits wavelength response of 6.93 nm per 1% strain variation, 2.4 times greater than that for circular GNA. Large FOM of 0.34 is observed experimentally, 5.7 times better than that for circular GNA. By engineering SLR of the proposed elliptical GNA, enhanced performance of strain sensing is realized. Our research provides insight for not only surface lattice resonance tuning but also strain sensing application, which is essential to the development of nanophotonics and optical sensors.
Funding
Ministry of Science and Technology (MOST) of Taiwan under contract numbers 103-2221-E-009-096-MY3, 106-2221-E-009-123-MY3, and 106-2221-E-009-124-MY2.
Acknowledgments
The authors acknowledge the financial support from Ministry of Science and Technology (MOST) and Research Team of Photonic Technologies and Intelligent Systems at NCTU within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. The fabrication of this work is supported by the facilities from Center for Nano Science and Technology (CNST) of National Chiao Tung University (NCTU) and National Nano Device Laboratories (NDL) in Taiwan.
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