We demonstrate successful integration of aperiodic arrays of metal nanoparticles with microfluidics technology for optical sensing using the spectral-colorimetric responses of nanostructured arrays to refractive index variations. Different aperiodic arrays of gold (Au) nanoparticles with varying interparticle separations and Fourier spectral properties are fabricated using Electron Beam Lithography (EBL) and integrated with polydimethylsiloxane (PDMS) microfluidics structures by soft-lithographic micro-imprint techniques. The spectral shifts of scattering spectra and the distinctive modifications of structural color patterns induced by refractive index variations were simultaneously measured inside microfluidic flow cells by dark-field spectroscopy and image correlation analysis in the visible spectral range. The integration of engineered aperiodic arrays of Au nanoparticles with microfluidics devices provides a novel sensing platform with multiplexed spatial-spectral responses for opto-fluidics applications and lab-on-a-chip optical biosensing.
© 2013 OSA
The integration of plasmon-enhanced biosensors with opto- and microfluidics technology is central to the development of optical sensing technology . Potentially, large-scale and high-density microfluidic networks with individually addressable pumps and valves analogous to electronic integrated circuits  can enable highly efficient sample preprocessing steps [3–5] with minimal human input. Small volumes of processed samples are combined with different assay reagents  and automatically delivered to integrated sensors for detection and diagnosis . In this context, biomolecular interactions, such as molecular binding, and biomass detection can be monitored by label-free and multiplexed optical biosensors [8–11]. Variations in the chemical composition of analyzed solutions or binding events of target molecules are currently probed by refractive-index detection [12–15]. Therefore, the integration of novel optical refractive index sensors with microfluidic platforms is important to the development of multifunctional compact devices  for biosensing, medical diagnosis, and drug discovery.
The integration of optical sensing devices with microfluidic channels has been demonstrated for high throughput biochemical sensors or “lab-on-a-chip” devices [16–19]. Detection of proteins and separation of small molecules have been achieved fluorescently using a photo-detector array embedded underneath polydimethylsiloxane (PDMS) microchannels  and using Surface-Enhanced Raman Scattering (SERS) substrates integrated in a metal-PDMS composite . Different chemicals, such as sodium chloride, urine and pesticides, have been detected by SERS either in a flowing condition or with stationary metal substrates inside microchannels [22, 23]. White and his associates  recently demonstrated 400pM sensitivity to Rodamine 6G by flowing both metal nanoclusters and target analytes through a liquid-core optical ring resonator.
Alternatively, label-free chemical sensing of ethanol solutions with various refractive indices has been recently demonstrated using one dimensional photonic crystal structures coupled to a planar waveguide , yielding a maximum refractive index sensitivity (Δλ/Δn) of 480nm/RIU. Erickson and his associates  further miniaturized opto-fluidic refractive index sensors by designing a single row of holes within a planar photonic crystal and demonstrated high refractive index sensitivity potentially leading to single molecule detection. Furthermore, two-dimensional gratings and photonic crystals for colorimetric sensing have been demonstrated  and the binding kinetics and biomolecular interactions of protein A with IgG molecules have been measured inside a plastic microfluidic device .
In a recent study, Lee and associates introduced a novel approach to optical biosensing based on the engineering of the spectral and the structural color responses of Deterministic Aperiodic Nanostructures (DANS), and experimentally demonstrated protein monolayer detection by dark-field scattering and autocorrelation analysis in the visible range with improved sensitivity compared to conventional periodic grating structures . DANS are multi-scale nanoparticle arrays with a degree of structural complexity that interpolates in a tunable fashion between disordered random systems and regular periodic structures. Differently from disordered random media, DANS are generated by deterministic algorithms rooted in discrete geometry, symbolic dynamics, and number theory . Deterministic aperiodic arrays of metal nanoparticles (i.e., plasmonic DANS) often manifest unique light localization and scattering properties enabled by the manipulation of aperiodic Fourier space [30–34], and can be fabricated using conventional nano-lithographic and imprint techniques [35–37]. Moreover, these structures sustain dense spectra of highly complex structural resonances, known as critical modes, which display characteristic electric field fluctuations in space with localization properties that are ideally suited for optical biosensing . Light scattering by DANS substrates with various incommensurate length scales produces characteristic structural colors, called “colorimetric fingerprints”, that are easily detected using white light dark-field microscopy [28, 39]. These complex chromatic patterns of scattered radiation provide exciting opportunities for multiplexed colorimetric detection where both spatial and spectral information are simultaneously utilized .
In this paper, we demonstrate DANS-integrated microfluidics devices and investigate optical refractive index sensing using different Au nanoparticle arrays with spectral measures ranging from singular to continuous Fourier space. The spectral character of the aperiodic Fourier space of DANS is a very important feature for the engineering of scattering-based sensors, since the intensity fluctuations of scattered radiation from aperiodic surfaces are directly determined by the spectral width (in spatial frequency space) of their Fourier spectral density . As a result, in order to engineer more efficient scattering-based DANS sensors with spectral and spatial (i.e., colorimetric) sensitivity to small perturbation of the refractive index, there is a need to design deterministic aperiodic nanostructures with broad spectral features and to characterize their spatial-spectral responses. In this study, we introduce plasmonic Galois arrays based on the pseudo-random one-dimensional (1D) Galois sequence, which features a continuous Fourier spectrum, and compare their performances against Gaussian Prime Arrays (GPA)  and quasi-periodic Fibonacci structures [33, 34, 40]. These structures feature very different Fourier spectral properties ranging from pure-point spectra with discrete scattering peaks to a broad continuum in reciprocal space. Aperiodic plasmonic arrays with different interparticle separations are fabricated by Electron Beam Lithography (EBL) and integrated with PDMS microfluidic channels by soft-lithographic micro-imprint techniques. Scattering spectra and colorimetric patterns are experimentally measured by conventional dark-field optical microscopy and autocorrelation image analysis, revealing how plasmonic Galois arrays exhibit the highest spectral and colorimetric sensitivity to refractive index variations. The successful integration of DANS scattering sensors with microfluidics devices provides a novel sensing platform with multiplexed spatial-spectral responses for opto-fluidics and lab-on-a-chip optical biosensing applications.
2. Generation of two-dimensional aperiodic surfaces
In the following, we will briefly discuss the methods followed to generate the aperiodic two-dimensional (2D) arrays of Au nanoparticles investigated in this paper. Galois nanoparticle arrays are based on the mathematical concept of a Galois field. In abstract algebra, a field is a set of elements with addition, subtraction, multiplication and division (except by 0) operations that satisfy the usual commutative, associative, and distributive laws . Galois fields, named after the French mathematician Évariste Galois, are fields with a finite number of elements (i.e., finite order fields) and have found numerous applications in physics, communication theory, error-correcting code, cryptography, and even artistic design . As an example, a residue system modulo a prime p forms a finite number field (i.e., a Galois field) of order p, which is indicated by GF(p). Of particular importance are finite number fields of order equal to a prime power pm (i.e., with pm elements), where p is a prime number and m is a positive integer. A Galois field of order pm is usually denoted as GF(pm) .
Galois sequences with binary values have been used in many engineering applications due to their distinctive spectral properties . In particular, Galois sequences derived from GF(2m) possess a flat Fourier spectrum but, in contrast to other pseudo-random binary sequences (e.g., Legendre sequences), are generated by a simple linear recursion. This provides a very convenient way to generate aperiodic systems starting from binary valued sequences. For instance, in the case of GF*(24), a binary Galois sequence looks as follow:42] that this sequence can be generated by the linear recursion:
By using the above recursion relation repeatedly, the first column of a 2D Galois array of nanoparticles can be generated with the 2m-1 elements of the 1D Galois sequence. Subsequent columns are generated by permutations of the elements determined by the binary values in the first column. The binary value of the n-th column will be the same as the ones in the first one if the n-th value in the first column is 1, otherwise the binary value of the n-th column will be “inverted” (1→0, 0→1) with respect to the first column, where 2 ≤ n ≤ 2m-1. By following this deterministic linear recursion and permutation scheme, a 2D aperiodic Galois array is easily obtained. Figure 1(a) illustrates that the 2D aperiodic Galois array is diagonally symmetric and the flat Fourier spectral property of a one-dimensional GP sequence is extended to two dimensions as shown in Fig. 1(d).
The second type of DANS array investigated in this study is the Gaussian prime array, which is generated based on the distribution of prime numbers in the complex plane . Based on this array structure, we have recently demonstrated  structural color sensitivity to molecular monolayers using inexpensive dark-field scattering microscopy in the visible spectral range. Therefore, it is important to compare the results obtained using alternative aperiodic structures with this important reference case. Gaussian primes (GP) are prime (i.e, irreducible) Gaussian integers. Gaussian integers are complex numbers whose real and imaginary parts are both integers and formally, they are the set:
We notice that not all ordinary primes in Z are still primes in the complex field C, for instance, 2 and primes of the form 4k + 1 can be factored in C (e.g., 2 = (1 + i)(1-i), 5 = (2 + i)(2-i), etc.) . Furthermore, it is known that when an ordinary prime number is written as a sum of two squares , for example, 5 = ( ± 1)2 + ( ± 2)2, there are exactly eight GPs corresponding to the prime 5, namely ± (1 ± 2i) and ± (2 ± i). Moreover, if one of a, b is zero and the other is a Gaussian prime, the four numbers ± a (or ± b), ± ia (or ± ib), called the associates of a (or b), are also Gaussian primes. As a result, Gaussian primes are symmetric about the real and imaginary axes. By plotting the real and imaginary components of GP numbers as horizontal and vertical coordinates, we can represent Gaussian primes geometrically in the complex plane, and produce a highly symmetric though aperiodic array shown in Fig. 1(b). Thus, in general the geometric structure of a 2D GP aperiodic array features an eightfold rotational symmetry.
However,despite the evident symmetry and apparent regularity of Gaussian prime arrays, a number of problems still remain open including the nature of their spectral Fourier properties [45,46]. The SEM image of a nanofabricated GP array of Au nanoparticles is shown in Fig. 1(h), while the calculated Fourier spectrum is displayed in Fig. 1(e). The spectrum features discrete peaks in the presence of a diffuse background. This is suggestive of a mixed Fourier spectrum composed of pure-point and continuous or singular continuous components for GP arrays.
The third aperiodic array investigated in this study consists of the well-known 2D Fibonacci quasi-periodic structure, which has been intensively studied in recent years [33, 34, 40, 47–50] and it is generated by the 2D generalization of the symbolic inflation rule: A→AB, B→A [33, 34], where A and B indicate the presence or the absence of a nanoparticle, as shown in Fig. 1(c). The Fourier spectrum of the Fibonacci array, shown in Fig. 1(f), consists of isolated Bragg peaks with incommensurate periods in the reciprocal space (i.e., pure-point spectrum). The SEM micrograph of a fabricated Fibonacci array is displayed in Fig. 1(i).
3. Sensitivity vs. Autocorrelation function
In this section we will briefly recall the main relations, originally discussed in Ref , that are useful to better understand correlation sensing with aperiodic arrays. The intensity fluctuations of the radiation scattered by a DANS array, which can be analyzed to extract valuable information on adsorbed analytes, are proportional to the area under the spectral density of colorimetric fingerprints. In particular, the mean square value of the fluctuating intensity, E[y2], is equal to the autocorrelation function (ACF) G(ξ) of the structural color patterns  evaluated at ξ = 0:52, 53]. Using this well-established approach, tiny variations of the optical image can be detected and quantified by simply measuring scattered radiation. We define the autocorrelation function of a spatially fluctuating signal in 1D, s(x), as :28]:33, 34]:Eq. (7), the normalized ACF can be calculated directly using Eq. (6). The normalized ACF profiles in 1D are then extracted from the 2D normalized ACF maps by cutting along the center-line of the images and normalizing with respect to the size of the arrays. This definition of the normalized ACF enables us to calculate the ACF variance of the spatial fluctuations of scattered intensity in a colorimetric fingerprint by evaluating the autocorrelation function in Eq. (6) in the limit when both ξ and η vanish :
4. Microfluidic integration
The aperiodic nanoparticle arrays for structural color sensing were fabricated using EBL on SiO2 substrates, as described in our previous work , and representative samples are demonstrated by the SEM pictures in Figs. 1(g)-1(i).
In this work, we extend the sensing capabilities of plasmonic DANS by integrating them with a PDMS microfluidics flow cell, using the process illustrated in Fig. 2(a) . In our integration process, the channel master is defined through photolithography in a 28μm-thick spin-coated layer of SU8 negative photoresist on a clean silicon wafer. With a second lithography step we add 1μm-thick 50μm-wide filling grooves on the top of channel mold to facilitate fluid filling [54, 55]. A thin layer of PDMS (10:1 base: curing agent) was subsequently cured on top of the SU8 master at 70°C for 2 hours. Half-through inlet and outlet holes were bored on the PDMS mold at both ends of the transferred microchannel patterns and the PDMS mold was oxygen-plasma treated along with the colorimetric DANS sensor on SiO2 substrate to form an optofluidic DANS device [56, 57]. The oxygen plasma process permanently bonded the device together and resulted in hydrophilic surfaces, which easily prime the microchannels with insensible amount of pressure. Epidermic syringe needles were then inserted from the sides of the PDMS mold into the holes as fluid access ports  and secured using silicone sealant, as shown in Fig. 2(b). Figure 2(c) clearly illustrates optofluidic DANS devices under a microscope, in which 2D aperiodic Galois, GP and Fibonacci arrays were positioned inside the microfluidic flow cell for colorimetric detection. The arrays are encapsulated inside a single 28 μm-thick 3.5mm by 2.5mm cell with 1μm-thick filling grooves on the ceiling, as shown in Fig. 2(c).
5. Experimental results
The colorimetric responses of the DANS arrays were measured in an oil immersion (noil = 1.518) microscope configuration under white light illumination using a dark-field condenser with NA 1.2-1.4, which resulted in an incident angle of approximately 20° with respect to the array plane. In Fig. 3 we show dark-field scattering images of different DANS surfaces of gold nanoparticles with various diameters and minimum interparticle separations designed for operation in the visible spectral range. All the images were collected in transmission using a 10x objective (NA 0.3) and imaged by a microscope-coupled CCD camera (DP21, Olympus). The main structural parameters of the fabricated structures, namely their number of particles (N), linear dimension of arrays (L), particle diameter (d), minimum (a) and average (<a>) first neighbor center-to-center separation, are summarized in Table 1 .
We can notice from Figs. 3(a)-3(c) that the colorimetric fingerprints of Galois arrays are more contrasted and easily detected. This follows from their broader spectrum of spatial frequencies, which results in larger intensity fluctuations as predicted by Eq. (4). As a result, the larger number of scattering resonances supported at multiple wavelengths by Galois arrays give rise to highly localized structural color patterns that offer larger sensitivity to local refractive index variations. In contrast, Fibonacci arrays possess isolated Bragg peaks with incommensurate periods that describe a more regular spatial structure with a smaller density of spatial frequencies. This implies smaller fluctuations in the scattered intensity and therefore more spatially uniform colorimetric fingerprints are formed, as demonstrated by dark-field data in Figs. 3(g)-3(i). We notice that the data in Figs. 3(g)-3(i) feature colorimetric responses similar to the monochromatic ones typically observed in regular periodic structures. On the other hand, the colorimetric responses of Gaussian prime arrays, shown in Figs. 3(d)-3(f), display intermediate characteristics in between the highly structured Galois fingerprints and the more uniform and almost monochromatic ones observed for quasi-periodic Fibonacci arrays, consistently with their mixed spectral character.
In order to demonstrate the sensing capability of different 2D DANS integrated in a microfluidic environment, the sensitivity to gradual refractive index changes induced by variations in the concentration of a glycerol solution were characterized by dark-field scattering spectra and ACF analysis. The microfluidics-integrated DANS were immersed in fluids with different index of refraction in the microchannel, and excited by white light illumination. The glycerol solution (diluted in Deionized water) of varying concentration ranging from 0% to 100% was sequentially pumped into the microfluidic flow cell (0%: n0 = 1.333, 25%: n25 = 1.364, 50%: n50 = 1.398, 75%: n75 = 1.435, 100%: n100 = 1.474 ).
The colorimetric fingerprints of each DANS array were first characterized by measuring the scattering spectra before the integration with the microfluidics flow cells. Figures 4(a) -4(c) shows the normalized scattering spectra of three representative Galois, GP, and Fibonacci arrays in air (black lines). The PDMS microfluidic flow cell was then integrated with the colorimetric DANS sensors and the peak resonant wavelengths of the DANS arrays shifted to shorter wavelengths due to the coupling with the PDMS slab that perturbed the optical path to the collection objective. The scattering spectra of the arrays as a function of different refractive indices of the microfluidics environment were then measured (Figs. 4(a)-4(c), color lines). A shift of the scattering peak to longer wavelengths was measured for all the structures as the refractive index in the channel was gradually increased, demonstrating the sensing capability of the integrated devices. Simultaneously, the fluctuations of the scattered intensity gave rise to distinctive modifications in the structural color patterns (i.e., colorimetric fingerprints) that were quantified by the normalized ACF variance. Figures 4(d)-4(f) illustrate the one-dimensional profiles of the normalized ACF collected from the perturbed colorimetric fingerprints of the investigated DANS sensors. The initial decay of the ACF profiles represents short-range spatial correlations in the colorimetric fingerprints, while periodic oscillations in the ACF profiles reflect long-range spatial correlations of the different components of the structural color patterns .
We can see in Figs. 4(d)-4(f) that Galois arrays exhibit the largest degree of structural disorder described by an intense and almost delta-like ACF profile. However, we also notice in Fig. 4(d) the presence of a complex substructure of oscillations superimposed to the central ACF peak of Galois arrays, which indicates a substantial degree of spatial correlations in the structure. Since the Fourier spectrum of Galois sequences has a continuous spectrum with constant amplitude in the limit of infinite-size structures, we attribute the presence of these residual correlations to the finite-size of the investigated Galois patterns. On the contrast, the fine oscillations in the ACF profile of Fibonacci arrays indicate the presence of the multiple spatial correlations typical of quasi-periodic structures . The ACF profiles observed for the colorimetric fingerprints of GP arrays feature smooth and slow decay traces that reflect the highly organized spatial and spectral structures shown in Figs. 1(b) and (e), respectively. It is also important to notice that the ACF profiles shown in Figs. 4(d)-4(f) feature decreasing peak intensity (i.e., variances) as the degree of structural disorder is decreased from Galois patterns to quasi-periodic Fibonacci ones.
In our experiments, we quantified the sensitivity of the colorimetric devices by plotting the normalized ACF variance (i.e., σ) of the structural color patterns measured at the peak scattering wavelength versus the glycerol solution index of refraction (i.e., n) in the microfluidics flow cell. In Figs. 4(d)-4(f), the measured ACF profiles indicate a substantial change in the variance of colorimetric fingerprints perturbed by an increasing concentration of glycerol solution.
Differently from conventional microfluidic sensors, the overall sensitivity of microfluidics integrated spatial-spectral DANS sensors is determined by both the normalized ACF variance of their structural color patterns, Δσ, and the shift Δλ in the peak wavelength (PWS) of their scattering spectra in response to variations of the refractive index Δn. Linear fits to the experimental data shown in Fig. 5(a) demonstrate the spectral sensitivities, ηs = Δλ/Δn, achieved with Galois, GP, and Fibonacci DANS arrays, which are approximately equal to 267nm/RIU, 145nm/RIU, 166nm/RIU, respectively. These values are comparable with what reported for photonic crystal structure biosensors based on the shift of their peak resonant wavelengths [13, 24, 60], but are here demonstrated using a multiplexed spatial-spectral approach that can inexpensively be implemented using standard microscopy in the visible spectral range. In Fig. 5(c), we demonstrate the high spectral sensitivity achieved with Galois arrays over the more uniform GP and Fibonacci ones. Furthermore, the autocorrelation analysis in Fig. 5(b) demonstrates the high colorimetric sensitivities, ηc = Δσ/Δn obtained for the three investigated DANS with approximately 0.0528RIU−1, 0.0575RIU−1, and 0.0090RIU−1 values, respectively.
In Fig. 5(d), we summarize our results by showing an overall “sensitivity matrix” where both the spectral and the colorimetric sensitivities to refractive index variations in the microfluidics environment are plotted for all the investigated DANS arrays. In particular, we can appreciate that although the colorimetric sensitivities of Galois and GP arrays are approximately the same (Fig. 5(b)) due to the presence of a substantial diffused background, Galois arrays feature improved spectral sensitivity and therefore provide both peak wavelength and colorimetric ACF variance shifts, as highlighted by the red circle in the sensitivity matrix. These results demonstrate that plasmonic Galois arrays featuring structural complexities with a high density of spatial frequencies similar to random systems (i.e., large integral area under their Fourier power spectra) result in large intensity fluctuations in their perturbed colorimetric fingerprints simultaneously accompanied by large wavelength shifts of their scattering spectra.
Our results have successfully demonstrated the integration of colorimetric DANS sensors with microfluidic devices for optical refractive index sensing. By making use of conventional dark-field microscopy and image autocorrelation analysis, distinctive spectral and colorimetric spatial modifications of the colorimetric fingerprints of DANS arrays were experimentally measured in the microfluidic flow cell and Galois arrays with high spectral and colorimetric sensitivities were demonstrated. The integration of plasmonic aperiodic arrays with microfluidics devices provides a novel approach with multiplexed spatial-spectral responses for opto-fluidics platforms and lab-on-a-chip optical biosensing.
The work was supported by the AFOSR program “Deterministic Aperiodic Structures for On-chip Nanophotonic and Nanoplasmonic Device Applications” under Award FA9550-10-1-0019, the U.S. Army NSRDEC PAO # U12-491, the SMART Scholarship Program, APIC Corporation and by PhotonIC Corporation.
References and links
3. J. Y. Zhang, J. Do, W. R. Premasiri, L. D. Ziegler, and C. M. Klapperich, “Rapid point-of-care concentration of bacteria in a disposable microfluidic device using meniscus dragging effect,” Lab Chip 10(23), 3265–3270 (2010). [CrossRef] [PubMed]
7. J. Y. Zhang, Q. Cao, M. Mahalanabis, and C. M. Klapperich, “Integrated microfluidic sample preparation for chip based molecular diagnostics.,” in Microfluidic Applications for Human Health, U. Demirci, A. Khademhosseini, R. Langer, and J. Blander, eds. (World Scientific Publishing Co., Hackensack, NJ, USA, 2012).
8. H. M. Hiep, T. Endo, K. Kermam, M. Chikae, D.-K. Kim, S. Yamamura, Y. Takamura, and E. Tamiya, “A localized surface plasmon resonance based immunosensor for the detection of casein in milk,” Sci. Technol. Adv. Mater. 8(4), 331–338 (2007). [CrossRef]
9. L. Yang, B. Yan, W. R. Premasiri, L. D. Ziegler, L. D. Negro, and B. M. Reinhard, “Engineering nanoparticle cluster arrays for bacterial biosensing: the role of the building block in multiscale SERS substrates,” Adv. Funct. Mater. 20(16), 2619–2628 (2010). [CrossRef]
10. F. Prieto, B. Sepúlveda, A. Calle, A. Llobera, C. Domínguez, A. Abad, A. Montoya, and L. M. Lechuga, “An integrated optical interferometric nanodevice based on silicon technology for biosensor applications,” Nanotechnology 14(8), 907–912 (2003). [CrossRef]
11. X. Fan, Advanced photonic structure for biological and chemical detection (Springer, 2009).
13. E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. 29(10), 1093–1095 (2004). [CrossRef] [PubMed]
14. B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85(3), 219–226 (2002). [CrossRef]
15. J. J. Amsden, H. Perry, S. V. Boriskina, A. Gopinath, D. L. Kaplan, L. Dal Negro, and F. G. Omenetto, “Spectral analysis of induced color change on periodically nanopatterned silk films,” Opt. Express 17(23), 21271–21279 (2009). [CrossRef] [PubMed]
16. C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: a new river of light,” Nat. Photonics 1(2), 106–114 (2007). [CrossRef]
18. J. Feng, V. S. Siu, A. Roelke, V. Mehta, S. Y. Rhieu, G. T. R. Palmore, and D. Pacifici, “Nanoscale plasmonic interferometers for multispectral, high-throughput biochemical sensing,” Nano Lett. 12(2), 602–609 (2012). [CrossRef] [PubMed]
19. T.-Y. Chang, M. Huang, A. A. Yanik, H.-Y. Tsai, P. Shi, S. Aksu, M. F. Yanik, and H. Altug, “Large-scale plasmonic microarrays for label-free high-throughput screening,” Lab Chip 11(21), 3596–3602 (2011). [CrossRef] [PubMed]
20. M. L. Chabinyc, D. T. Chiu, J. C. McDonald, A. D. Stroock, J. F. Christian, A. M. Karger, and G. M. Whitesides, “An integrated fluorescence detection system in poly(dimethylsiloxane) for microfluidic applications,” Anal. Chem. 73(18), 4491–4498 (2001). [CrossRef] [PubMed]
21. R. M. Connatser, L. A. Riddle, and M. J. Sepaniak, “Metal-polymer nanocomposites for integrated microfluidic separations and surface enhanced Raman spectroscopic detection,” J. Sep. Sci. 27(17-18), 1545–1550 (2004). [CrossRef] [PubMed]
22. K. W. Kho, K. Z. M. Qing, Z. X. Shen, I. B. Ahmad, S. S. C. Lim, S. Mhaisalkar, T. J. White, F. Watt, K. C. Soo, and M. Olivo, “Polymer-based microfluidics with surface-enhanced Raman-spectroscopy-active periodic metal nanostructures for biofluid analysis,” J. Biomed. Opt. 13(5), 054026 (2008). [CrossRef] [PubMed]
26. B. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B 81, 316–328 (2002).
27. C. J. Choi and B. T. Cunningham, “Single-step fabrication and characterization of photonic crystal biosensors with polymer microfluidic channels,” Lab Chip 6(10), 1373–1380 (2006). [CrossRef] [PubMed]
28. S. Y. Lee, J. J. Amsden, S. V. Boriskina, A. Gopinath, A. Mitropolous, D. L. Kaplan, F. G. Omenetto, and L. D. Negro, “Spatial and spectral detection of protein monolayers with deterministic aperiodic arrays of metal nanoparticles,” Proc. Natl. Acad. Sci. U.S.A. 107(27), 12086–12090 (2010). [CrossRef] [PubMed]
29. L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photonics Rev. 178–218 (2011).
31. S. Y. Lee, C. Forestiere, A. J. Pasquale, J. Trevino, G. Walsh, P. Galli, M. Romagnoli, and L. Dal Negro, “Plasmon-enhanced structural coloration of metal films with isotropic Pinwheel nanoparticle arrays,” Opt. Express 19(24), 23818–23830 (2011). [CrossRef] [PubMed]
32. J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express 20(3), 3015–3033 (2012). [CrossRef] [PubMed]
33. A. Gopinath, S. V. Boriskina, N.-N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8(8), 2423–2431 (2008). [CrossRef] [PubMed]
34. L. Dal Negro, N. N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A Pure Appl. Op. 10, 064013 (2008).
35. O. Dial, C. C. Cheng, and A. Scherer, “Fabrication of high-density nanostructures by electron beam lithography,” J. Vac. Sci. Technol. B 16(6), 3887–3890 (1998). [CrossRef]
36. D. Lin, H. Tao, J. Trevino, J. P. Mondia, D. L. Kaplan, F. G. Omenetto, and L. Dal Negro, “Direct transfer of sub-wavelength plasmonic nanostructures on bio-active silk films,” Adv. Mater. (Deerfield Beach Fla.) 24(45), 6088–6093 (2012). [CrossRef]
37. F. Carcenac, C. Vieu, A. Lebib, Y. Chen, L. Manin-Ferlazzo, and H. Launois, “Fabrication of high density nanostructures gratings (>500Gbit/in2) used as molds for nanoimprint lithography,” Microelectron. Eng. 53(1-4), 163–166 (2000). [CrossRef]
39. S. V. Boriskina, S. Y. K. Lee, J. J. Amsden, F. G. Omenetto, and L. Dal Negro, “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces,” Opt. Express 18(14), 14568–14576 (2010). [CrossRef] [PubMed]
40. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544–5555 (2008). [CrossRef] [PubMed]
41. D. A. Cox, Galois theory (Wiley-Interscience, 2004).
42. M. R. Schroeder, Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity (Springer, 2009).
43. E. W. Weisstein, “Gaussian prime” (From MathWorld–A Wolfram Web Resource), retrieved http://mathworld.wolfram.com/GaussianPrime.html.
44. J. Stillwell, Elements of number theory (Springer, 2003).
45. E. Gethner, S. Wagon, and B. Wick, “A stroll through the Gaussian primes,” Am. Math. Mon. 105(4), 327–337 (1998). [CrossRef]
46. G. Hardy and J. Littlewood, “Some problems of ‘Partitio numerorum’; III: on the expression of a number as a sum of primes,” Acta Math. 44(1), 1–70 (1923). [CrossRef]
48. S. Katsumoto, N. Sano, and S.-i. Kobayashi, “Electron propagation through a fibonacci lattice,” Solid State Commun. 85(3), 223–226 (1993). [CrossRef]
49. E. Maciá, “Physical nature of critical modes in Fibonacci quasicrystals,” Phys. Rev. B 60(14), 10032–10036 (1999). [CrossRef]
50. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003). [CrossRef] [PubMed]
51. D. E. Newland, AniIntroduction to random vibrations, spectral and wavelet analysis (John Wiley & Sons, Incorporated, 1993).
52. N. O. Petersen, P. L. Höddelius, P. W. Wiseman, O. Seger, and K. E. Magnusson, “Quantitation of membrane receptor distributions by image correlation spectroscopy: concept and application,” Biophys. J. 65(3), 1135–1146 (1993). [CrossRef] [PubMed]
53. P. W. Wiseman and N. O. Petersen, “Image correlation spectroscopy. II. optimization for ultrasensitive detection of preexisting platelet-derived growth factor-beta receptor oligomers on intact cells,” Biophys. J. 76(2), 963–977 (1999). [CrossRef] [PubMed]
54. C.-H. Choi, U. Ulmanella, J. Kim, C.-M. Ho, and C.-J. Kim, “Effective slip and friction reduction in nanograted superhydrophobic microchannels,” Phys. Fluids 18(8), 087105–087108 (2006). [CrossRef]
55. J. Davies, D. Maynes, B. W. Webb, and B. Woolford, “Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs,” Phys. Fluids 18(8), 087110–087111 (2006). [CrossRef]
56. S. Bhattacharya, A. Datta, J. M. Berg, and S. Gangopadhyay, “Studies on surface wettability of poly(dimethyl) siloxane (PDMS) and glass under oxygen-plasma treatment and correlation with bond strength,” J. Microelectromech. Syst. 14(3), 590–597 (2005). [CrossRef]
57. M. Morra, E. Occhiello, R. Marola, F. Garbassi, P. Humphrey, and D. Johnson, “On the aging of oxygen plasma-treated polydimethylsiloxane surfaces,” J. Colloid Interface Sci. 137(1), 11–24 (1990). [CrossRef]
58. A. M. Christensen, D. A. Chang-Yen, and B. K. Gale, “Characterization of interconnects used in PDMS microfluidic systems,” J. Micromech. Microeng. 15(5), 928–934 (2005). [CrossRef]
59. M. Polyanskiy, “RefractiveIndex.INFO” (2008), retrieved http://refractiveindex.info/?group=LIQUIDS&material=Glycerol.
60. B. Cunningham, J. Qiu, P. Li, and B. Lin, “Enhancing the surface sensitivity of colorimetric resonant optical biosensors,” Sens. Actuators B 87, 365–370 (2002).