We propose the attachment of a periodic array of gold nanoparticles (epitopes) to the equator of a Whispering Gallery Mode Biosensor for the purpose of plasmonically enhancing nanoparticle sensing in a self-referencing manner while increasing the capture rate of analyte to antibodies attached to these plasmonic epitopes. Our approach can be applied to a variety of whispering gallery mode resonators from silicon/silica rings and disks to capillaries. The interpretation of the signals is particularly simple since the optical phase difference between the epitopes is designed to be an integer multiple of π, allowing the wavelength shift from each binding event to add independently.
© 2012 OSA
The label-free detection of individual aqueous borne bio-nanoparticles (e.g., virus, protein) is an important goal in the bio-sensing field; analytes are detected at the ultimate sensitivity, in their native state (i.e., without interference from labels), without added preparation, and size/mass information can be extracted from the signals [1,2]. The first label-free single bio-nanoparticle detection by a microcavity was of InfluenzaA with a mass of 512 attograms (ag) from the reactive frequency shift  of a micro-spheroidal Whispering Gallery Mode (WGM) . In reactive sensing the evanescent field of the WGM polarizes the analyte, and as a reaction the cavity’s resonance shifts . The limit of detection (LOD) in the InfluenzaA experiment based on the reported signal/noise ratio (S/N) was ~170 ag. Since the reactive frequency shift is in proportion to mass, a virus such as Polio with a mass of 14 ag would not have been detected . Recently, the robust detection (S/N = 8) and sizing of a virus MS2 (6ag), which has less than half the mass of Polio was demonstrated by adding a nanoplasmonic structure (“plasmonic epitope”) to the sensing region, thereby forming a Whispering Gallery Mode Hybrid resonator (WGM-h) . This simple addition provided a frequency shift boost ~70x, and brought the LOD down to 0.4 ag (molecular weight equivalent of 240 kDa). This frequency shift enhancement occurs because the WGM drives the plasmonic epitope into resonance, thereby creating local hot spots that increase the reactive interaction between the virus and the cavity [7,8]. In addition, the virus is drawn to the plasmonic receptor due to enhanced light forces [6,7,9,10]. With this simple modification to the microcavity, viruses of all known sizes can be detected one at a time. This achievement means that a virus such as Polio can be detected with the ultimate sensitivity. In addition, the potential applications go well beyond virus to single protein. By projecting from the MS2 experiments, the authors of Ref.6 calculate that protein above an effective radius of 2 nm (molecular weight ~26 kDa) should be able to be detected and sized. However, an important problem still remains: a single nanoplasmonic epitope has little surface for analyte binding compared to the entire equator of the WGM microcavity. As a consequence, statistics are limited and the time between binding events is long. This paper is concerned with a cure for these problems.
In what follows we propose to attach a necklace of periodic plasmonic epitopes to the surface of a WGM microcavity (e.g., oblate spheroid in Fig. 1 ), thereby substantially increasing the available surface for analyte binding while still retaining the frequency shift boost. Beyond increasing the number of possible events and reducing the time between events the periodic configuration changes our measurement approach from “spectral step-sensing”  to “self-referencing” . Self-referencing can be understood in the following way. On a bare cavity a WGM circulating in a given direction or in the opposite direction will have the same frequency. A particle on the equator couples these modes and splits this degeneracy into two resonances. This splitting is apparent when the reactive shift exceeds the modal linewidth. For a single nanoparticle the splitting is twice the average shift, which enables the interaction to be measured from the frequency separation between the split modes. A large number of periodic epitopes are assured to produce mode splitting even if one epitope does not . The advantage of the “self-referencing” approach to measurement is that it is differential and can therefore eliminate common mode noise. A key advantage of the periodic epitopes is that this design builds a quasi-1D photonic band structure. Near the band edge where the inter-epitope optical phase difference is π, adsorption on any of the plasmonic epitopes produces additional splitting that is simply additive. In what follows we will discuss the enhancement by possible local plasmonic epitopes. Following this we will describe our theoretical approach, apply it to nonperiodic mode splitting problems in the past, and finally to periodic plasmonic epitope sensing of dielectric bio-nanoparticles.
2. Possible plasmonic epitopes
The space group of the 1D structure with its band edge inter-optical phase difference of π is apparent in Fig. 1, however the point group can be picked from a variety of local plasmonic structures. Solid gold spherical particles are a poor choice since they have a modest intensity enhancement at their hot spots of only ~37x , whereas a core shell structure can develop a tenfold further enhancement as shown from the results of quasi-static calculations  in Fig. 2(a) . The intensity enhancements in this figure were calculated at the top (T) or bottom (B) of the depicted shell and show the tunability of the core-shell structure as the shell thickness is changed. As in most local plasmonic structures, the field enhancements increase as one moves from the center of the visible spectrum toward the near infrared due to the reduced imaginary component of gold’s permitivity in this region . There are also nanoellipsoids [Fig. 2(b)] and nanorods [14,15] that can develop much larger enhancements, but will have to be carefully oriented; for TE excitation the orientation would be along a meridian with hotspots above and below the equator. This is experimentally possible by using micro/nanofabrication methods (such as photolithography and shadow masking, etc.) [16–19]. Specific detection can be achieved by functionalizing each epitope’s hot spot regions, i.e., with antibodies that exclusively allow binding of distinctive target analytes suspended in aqueous media.
3. Theory of one, two and N plasmonic epitopes
We start our analysis with a single plasmonic epitope on the equator of a microspheroid (Fig. 3 ) in an attempt to verify the results of Zhu et al.  before moving on to the necklace. We concentrate theoretically on the equatorial mode of an oblate axisymmetric microcavity. Without the plasmonic epitope or significant roughness we have a bare cavity in which two counterpropagating equatorial WGMs have identical resonance frequencies; they are degenerate. Since the microcavity is a high Q system, the modes of this system can be considered to be quasi-normal modes . This allows for quantum analogs such as photonic atoms . Such analogs compel one to borrow the theoretical machinery of quantum mechanics in order to describe high Q optical systems. For bound nanoplasmonic epitope perturbing a WGM, degenerate perturbation theory can be used to predict the eigenvalues and eigenstates of the system.
The oblateness of the spheroid lifts thedegeneracy of a sphere allowing the equitorial mode having a principle quantum number s, angular momentum quantum number , and a two fold degeneracy to be excited through guided wave excitation. This equitorial WGM (Fig. 3) provides the simplest model for which a first order perturbation will not couple radial modes having different principle quantum numbers since their frequencies do not cross. First, we start to interpret the signal in the presence of a plasmonic epitope attached to the WGM ring (equator) at an angle , as shown in Fig. 3; then we apply this theory to two epitopes and N epitopes, respectively. The N epitopes will first be considered as random and then periodic.
Our theory will apply the approach of Teraoka and Arnold . We will briefly review the results of this paper before applying it to the case of periodic epitopes.
First we start with the bare cavity framework. By separating the Helmholtz equation the azimuthal differential equation for the equitorial TE mode, which is almost entirely polarized along the polar direction () has the form:Eqs. (2) and (3), orthogonal basis vectors for these ccw and cw equatorial traveling waves are23]. The dielectric function of the cavity isand consequently the energy in either travelling wave , where the 2 accounts for an equal amount of electric and magnetic energy in the cavity, the matrix element requires integration over the cavity volume, and the usual conjugate relationship between the bra and ket is maintained. In order to return to a familiar result we will at first suppose that a perturbation to the cavity maintains the travelling wave. Having to polarize an added Rayleigh particle on the equator at position with excess polarizability having a real part corresponds to a dielectric perturbation . This perturbation costs the cavity energy , for which the frequency shift of either travelling wave isEq. (5) , this first order perturbation in frequency of either of the traveling wave states has become known in the literature as g [11,24]. Equation (5) is in a conveniently simple form by keeping Planck’s constant in place. However, because of our single photon approach is simply a place saver for the energy in the cavity divided by frequency. With converted in this way Eq. (5) can be evaluated using our basis states in Eq. (4) and the position of the plasmonic epitope with the result :Eq. (5) by including . However, our description currently is not complete since the dielectric perturbation can couple the traveling wave states.
Coupling requires interaction between the ccw and cw states through degenerate perturbation theory. For a two state system one has to solve an eigenvalue equation involving a 2x2 matrixEq. (5) as expected. Unique solutions are guaranteed by setting the determinant of the secular equations represented by Eq. (7) equal to zero.Eqs. (4) since and consequently the upper rt. element is , since the perturbation is at azimuthal angle . The matrix element on the lower left is just the conjugate of this, . With these inputs Eq. (9) becomes11]. The epitope clearly splits the degeneracy, and the magnitude of the splitting 2|g| depends only on the value of g. If g is a negative number, as it would be for an epitope having a polarizability which is positive, is larger than . As we go forward the subscript + on the frequency shift will be assigned to the state maintaining the higher frequency for g < 0 (i.e. ).
So far we have been assuming quasi-normal modes. Losses can cause the modal line widths to overlap, in which case one expects to measure a shift of a single feature that is the average of frequency shift of the two states. In order to look at the wave function amplitudes of the mixed states (i.e., the location of the nodes and anti-nodes in the relation to the epitopes), we need to determine the eigenvectors in Eq. (10). The solution of the following equation provides the required eigenvectors,Eq. (8) (using ), and normalizing the eigen vectors to a single photon (i.e., ), we obtain the following new states.
If another epitope is attached to the ring at an angle, then the eigenvalues in the presence of the two epitopes are derived from the determinant25] and reveals that splitting magnitude is not only determined by the size, shape, and refraction index of the epitope, but also is strongly dependent on the phase difference between them (i.e., relative positions). If the second epitope is identical to the first epitope (g1 = g2 = g), then the total magnitude of the splitting is 4|g|. This clearly indicates that the mode splitting disappears when the phase difference is π/2, which results from the fact that the two epitopes exert the same influence on each of the two standing wave modes. However, if the phase difference between any consecutive epitopes is π, this gives rise to the maximum splitting 4|g|. In order to get the eigenvectors, for simplicity and are assumed to be 0 and π. The solution of the following equation provides the required eigenvectors;Eq. (8), we get the same mutually orthogonal wave functions given by Eqs. (14) and 15.
Suppose now that N epitopes are attached to the surface in random positions on the ring. Then the eigenvalues of the system are derived from the determinant
Therefore the eigenvalues of the system are and the magnitude of splitting is 2|g|. This is consistent with the results of Chantada et al. .
Having shown that our theoretical approach is consistent with the results of others we now apply it to the case of periodic epitopes. If N identical epitopes are periodically attached to the ring in the presence of an optical phase difference ΔΦ between the consecutive epitopes, then the eigenvalues of the system are given by the following equation,Fig. 4 . It can be seen that maximum frequency shift is 20g at ΔФ = π due to the superposition of the travelling waves, and the average shift for all possible phase differences is 10g; this is exactly the shift produced by a travelling wave. The peak that appears at π is known as the first order peak and higher order peaks appear at integer multiples of π. The peak width is inversely proportional to N; in order to observe a significant splitting experimentally, the inter-epitopes phase difference must have an error of less than half of the peak width. If the position of each epitope is Gaussian-distributed around the correct position with standard deviation σ, the mode splitting will be reduced. For example, for an operating wavelength of 800 nm in silica, if σ/r = 0.025 μm/40 μm, the mode splitting becomes 90% of the full splitting. For a smaller σ of 0.015 μm, the fraction rises to 96%. If one epitope is missing, then the model fails. In practice, both issues can be dealt with by quality control and post-fabrication examination. We can also view the proposed periodic epitope system as a diffraction grating, with each epitope being the grating groove. It is well known that the grating sensitivity (resolution) depends upon the total number of grooves illuminated on the surface of the grating (N) and diffraction order (n). Although the sensitivity (magnitude of splitting) of the proposed epitope system is dependent on N like a diffraction grating, it is independent of n. In other words, the magnitude of the splitting is the same for all orders.
The eigenvectors for the N-periodic epitope system are given by the following Equation by assuming and .Eqs. (14) and 15. The eigenstates of a system with 10 epitopes on a ring are shown in Fig. 5 . It is clear that all epitopes are at anti-nodes (symmetric mode) in one case, resulting in the maximum frequency shift 20g. In the other case, all epitopes are at nodes (asymmetric mode), which results in no shift.
For N periodic epitopes, that are different from each other, then the eigenvalues of the system are given by the following equation,
In the case of mode splitting, along with the physical properties of the epitopes, the quality factor of the microcavity also plays a major role. To observe the mode splitting requires that its magnitude be considerably larger than the line width of WGM resonance. In the case of a cavity containing N particles, the observation of mode splitting requires
In single dielectric nanoparticle (e.g., bio-nanoparticle) detection, if a dielectric analyte is adsorbed to any one of the epitopes then (N-1) epitopes have the same interaction with the microcavity (g) and one epitope which adsorbs the analyte particle has the different interaction (). Here it is to be noted that is the combined interaction of single dielectric analyte particle and epitope with the microcavity. The splitting now changes to . Similarly, if two dielectric nanoparticles are adsorbed each at the center of the dipole hot spots of two different epitopes (i.e., position T or B in Fig. 2), then the splitting becomes, and so on. Here, the interactionis proportional to the real part of effective polarizability of a plasmonic-dielectric cluster. Suppose that an extremely small dielectric particle (i.e., a protein molecule) is attached to a spherical plasmonic epitope, thenfor transverse electric (TE) mode excitation can be estimated using the following analytical expression:Fig. 2), excess polarizability of the adsorbed dielectric particle, and the squared quantity on the right hand side is the intensity enhancement factor at the center of the hot spot.
Figure 6 demonstrates the advantage of the periodic plasmonic epitope system with respect to sensing. We first start the discussion of this figure by considering adsorption onto the equator of a bare resonator (lower trace), and take our analyte to be the smallest aqueous borne RNA virus [MS2, 13.6 nm radius]. This discussion will be an extension of Ref  for which the radius of the silica resonator was and the unperturbed resonance occurs near a free space wavelength of 780nm . To be consistent with Ref , we will talk of shifts, splittings and linewidth changes in wavelength terms. Adsorption on the silica equator is calculated to produce a resonance shift of 0.25 fm, well below the 2fm r.m.s. noise in Ref.6. The calculated splitting would be 2x0.25 = 0.5 fm, much smaller than the experimental linewidth of 1950fm (Q = 4x105). If 8 copies of the virus adsorbed randomly on the equator of this resonator the mean splitting for a statistical number of trials would be , still much smaller than the line width. Even if we increase the Q to ~4x106 (line width = 195 fm), a value easily within reach of such a microcavity , the splitting due to these 8 viruses would still not be seen. It is to be noted that the mode splitting due to the imperfections (Rayleigh scattering) will be present when the ring resonator has an extremely high Q-factor (~108). With a low Q-factor, such a splitting does not occur. In our case, because of the lossy metallic nanoparticles, our Q-factor is much lower (~106).
Next we consider the effect of including functionalized periodic plasmonic epitopes as binding sites. We suppose now that epitopes are added to the microcavity described in the preceding paragraph. These epitopes will be of the same type as the single epitope used in Ref.6 (gold nanoshell with silica core). We will need some additional experimental information concerning the shift |g| and linewidth change produced by the addition of the single nanoshell; in wavelength terms we have measured these to be 175 fm and 230 fm, respectively, during our experiments described in Ref. 6. From the inequality Eq. (27) no splitting would be seen for the adsorption of a single nanoshell for a Q ~4x106 ; 2|g| = 350 fm does not exceed the overall linewidth of 425 fm (195 fm from the original linewidth plus 230 fm caused by the deposition of a nanoparticle). Splitting would be seen for two epitopes separated by an optical phase of nπ since the magnitude of the splitting is 700 fm compared with an overall linewidth of 635 fm. With 4 optimally placed epitopes the splitting magnitude would be 2x4x175 = 1400 fm, well in excess of the overall linewidth. The key question now is: how much additional splitting will be produced by the binding of viruses onto the functionalized hotspots of these epitopes? From Ref.6, which only used one epitope, the maximum shift for a virus was 17 fm, which would have produced an unseen increase in splitting of 34 fm. With 8 viruses adsorbing on our 4 epitopes for which a splitting should be apparent, an increase in splitting of ~280 fm is calculated (upper trace in Fig. 6). So we have gone from an invisible splitting of 3.5 fm by random adsorption on silica to a visible increase of 280 fm for adsorption onto a phased array of 4 epitopes.
Ultimately the above ideas are envisioned to be used for ring or disk resonators formed by micro and nanofabrication. The use of plasmonic epitopes easily recovers sensitivity lost due to lower Q factors, by providing a boost in signal strength . Such a ring configuration is shown in Fig. 7 .
Recent research shows that silicon nitride rings on silicon can provide a Q-factor above 106 in the near infrared [28–30], where modes of plasmonic epitopes are easily excited, and single virus and protein detection can be anticipated. By using an array of ring resonators with each carrying different selective receptors, a lab on a chip multiplexed to the same laser can be anticipated.
S.A. and B.G. thank the NSF for supporting this work (Grant No. CBET 0933531). X.F. thanks the NSF for supporting this work (Grant No. CBET 1037097 and 1158638).
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