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Abstract
In spite of tremendous advancements in modern diagnostics, there is a dire need for reliable, label-free detection of highly contagious pathogens like viruses. In view of the limitations of existing diagnostic techniques, the present theoretical study proposes a novel scheme of detecting virus-like particles employing whispering gallery and quasi-whispering gallery resonant modes of a composite optical system. Whereas whispering gallery mode (WGM) resonators are conventionally realized using micro-disk, -ring, -toroid or spherical structures, the present study utilizes a rotationally symmetric array of silicon nanowires which offers higher sensitivity compared to the conventional WGM resonator while detecting virus-like particles. Notwithstanding the relatively low quality factor of the system, the underlying multiple-scattering mediated photon entrapment, coupled with peripheral total-internal reflection, results in high fidelity of the system against low signal-to-noise ratio. Finite difference time domain based numerical analysis has been performed to correlate resonant modes of the array with spatial location of the virus. The correlation has been subsequently utilized for statistical analysis of simulated test cases. Assuming detection to be limited by resolution of the measurement system, results of the analysis suggest that for only about 5% of the simulate test cases the resonant wavelength shift lies within the minimum detection range of 0.001-0.01 nm. For a single virus of 160 nm diameter, more than 8 nm shift of the resonant mode and nearly 100% change of quality factor are attained with the proposed nanowire array based photonic structure.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The field of photonics and optics offers in itself ingenious techniques of detecting and analyzing microorganisms based on the rich physics of light-matter interaction. Consequently, over recent years, optical biosensors and detectors have emerged as a major area of research and innovation not only for diagnostic purposes, but also for therapeutic, imaging and monitoring applications. Different embodiments of micro- and nano-scale photonic structures, such as photonic crystals [1,2], optical fibers [3,4], whispering gallery mode resonators [5,6], ring resonators [7] and plasmonic nanostructures [8,9] have been utilized to design and realize biophotonic sensors and detectors for a variety of applications. Among these variegated systems, the whispering gallery mode (WGM) resonator is one of the most promising ones owing to its unique characteristics as a low-volume, high-Q microcavity. As light is tightly confined within a WGM structure because of total internal reflection, it is highly susceptible to the change of local refractive index - an aspect that has been extensively utilized for biosensing [5,10], ultra-low threshold lasing [11–13], quantum information processing [14], optical modulation [15], and light filtering applications [16,17].
Ever since the seminal work by Vollmer et al. [18] on the detection of proteins by a spherical WGM resonator, WGM resonators have been reportedly utilized for detecting a variety of bacteria [19,20], cancer cells [21] and viruses [22–24]. A silicon-on-insulator biosensing platform commercialized by Genalyte has reportedly been utilized by a number of research groups for detecting clinically-relevant protein molecules [21,25–27]. Based on the early proposition of He et al. [24], self-referencing has been demonstrated as a possible means of biosensing by Acharrya et al. [28] and DeGoede et al. [29]. This technique holds promise for realizing inexpensive and portable WGM biosensors as it does not require bulky, expensive and high-precision spectrometers, or tunable laser-sources. In regards to the materials system, though silicon has remained to be the most widely used one, silicon nitride [30,31] and silicon oxynitiride [32] have also shown promise for realizing high-performance WGM resonators.
In spite of such significant advancements with WGM resonator based biosensing schemes, there is plenty of scope for further research and development so that these resonators can be reliably and cost-effectively utilized in true medical environment. Efficient excitation of WGM-resonators has remained to be an active area of exploration [33–39]. Surface-functionalization with target-specific capture agents is also one of the major challenges to ensure bio-specificity of these biosensors in complex testing environment [40–42]. Another prospective area of exploration is the design of novel WGM-resonators which can be conveniently realized for reliable, label-free detection of biomolecules or pathogens. Whispering gallery mode based biosensing schemes have conventionally relied on photonic structures like micro-rings, microtoroids, micro-disks and micro-spheres. However, as far as label-free detection of pathogens is concerned, the prospect of utilizing a rotationally symmetric nanowire array in the form of WGM resonator has remained entirely untapped, in spite of the remarkable potential of such systems for realizing tunable, high-density, defect-free photonic structures [43,44].
In the present study, we theoretically propose and investigate a rotationally symmetric nanowire array based WGM resonator for label-free detection of virus-like particles (VLPs). Finite difference time domain (FDTD) based numerical analysis suggests that the proposed silicon-nanowire based WGM resonator is significantly more sensitive to minute changes of the optical field caused by a single virus, when compared with the sensitivity of a conventional solid-cylindrical WGM resonator. Detailed numerical simulations have been performed to correlate resonant wavelengths of the structure with spatial location of a single virus onto the array, and the resultant mapping has been utilized for statistically analyzing random test cases having a large number of samples. Statistical simulation results suggest than in the presence of a suitable functionalizing binding material, only 5% of the samples containing a single virus fall within the minimum detection range of resonant wavelength shift or change of quality-factor. The proposed technique is also found to have robustness against low signal-to-noise ratio. As no fluorescent, colorimetric or enzymatic tag is needed in the proposed scheme, it should be applicable for rapid diagnosis of VLPs with a high degree of accuracy at low-cost. In fact the proposed technique offers a means of label-free detection of not only viruses, but also numerous other pathogens, biomolecules and extraneous particles relevant to biosensing, clinical diagnostics and environmental monitoring.
2. Structure optimization
Whereas previous studies on WGM-based pathogen detection relied upon conventional solid-cylindrical, spherical, ring, or toroid shaped photonic-structures, the present study considers a rotationally symmetric array of nanowires for designing the high-Q resonant modes necessary for VLP detection. A schematic illustration of the nanowire (NW) based composite structure, along with a spherical VLP on top, is shown in Fig. 1(a). It may be noted that the spherical shape of the VLP is representative of several common viruses, including the Coronavirus-2 (SARS-CoV-2) or the novel Coronavirus which is the cause of the COVID-19 pandemic as declared by World Health Organization [45]. According to experimental studies, the diameter of such spherical shaped viruses vary approximately from 60-200 nm [1,46–48]. In our two-dimensional (2D) finite difference time domain (FDTD) based numerical analysis, diameter of the VLP is taken to be 100 nm. An enlarged top view of the array (shown as an inset of Fig. 1(a)) shows that the NWs form among themselves a hexagonal lattice, which not only results in rotational symmetry of the structure, but also hexagonal close packing of the NWs. Similar hexagonal close packed (HCP) arrays have been utilized for directionality control of GaN-based WGM resonators [49]. Considering materials availability and ease of implementation, we propose here silicon nanowire based WGM resonators which can be conveniently implemented employing a low-cost, high-throughput lithography technique like nanoimprint [50,51]. Unless otherwise specified, refractive index of the background medium is taken to be 1.335, which is approximately equal to the refractive index of blood or serum containing the virus [52]. It is to be noted that if the virus sample constitutes a different background medium, such as saliva, the designed structure will work just as fine as the refractive index of saliva is very close to that of serum [53]. In accordance with previous experimental reports [1,54], refractive index of the VLP is taken to be 1.48.
Fig. 1. (a) Schematic of the proposed nanowire array based photonic structure with a virus-like particle attached to it (inset shows a closeup view of the array showing diameter, $d$ of an individual nanowire and distance, $a$ between two uniaxial nanowires); (b) Resonant wavelength and Q-factor of the resonant mode as the overall diameter (D) of the NW array is varied; here inset shows near-field profiles on the top surface along x-y plane, and also along diameter of the array (x-z plane) obtained from 3D simulations (see Visualization 1 and Visualization 2); (c) Change in resonant wavelength and Q-factor with the change of distance between nanowires, while keeping the diameter of individual nanowires fixed at $d$=70 nm; (d) Change in resonant wavelength and Q-factor with the change of diameter of the nanowires, while maintaining a fixed nanowire spacing of $a$=15 nm. Here mode numbers are shown as (r,m), where $r$ and $m$ represent radial and azimuthal mode numbers respectively.
To optimize the diameter and spacing of the nanowire array to be utilized for VLP detection, FDTD analysis is performed employing open-source software package MEEP [55]. Throughout the study, resonant modes of the nanowire array are calculated by first positioning a Gaussian pulse source at the center, and then by applying filter diagonalization method to decompose the magnetic and electric field components of the light wave [56]. Only transverse magnetic (TM) modes have been considered for virus detection as transverse electric (TE) modes are found to be weakly confined in such structures [49,57]. In our 2D simulation, the VLP is considered in-plane with the nanowires. This is representative of a practical scenario where a VLP will be attached to the top-surface of the array having nanowires of finite length along the z-direction. Light circulates azimuthally along 2D cross-sectional plane of the array and forms the WGM near the periphery just as it would in a circularly symmetric solid structure. To check the validity of the 2D simulation model, 3D simulations with and without VLP at the top surface of the nanowire array have also been performed where the nanowires were considered to be of finite height. The quality factor and resonant wavelength obtained for the 3D cases are very similar to the values obtained for the equivalent 2D case. The corresponding near field images (shown as insets of Fig. 1(b)) at the top surface along x-y plane, and along the diameter of the array (x-z plane) clearly suggests the formation of WGM in the 3D structure. The radial and azimuthal mode numbers of the WGM for the 2D and 3D cases are also found to be identical. Moreover, for both 2D and 3D simulations maximum shift of resonant wavelength are attained for same locations of the virus when they are considered on the top-surface of the NW-array. These results suggest that the 2D simulations reasonably represent the 3D cases to be realized during actual experiments. It is to be noted that in the 3D case, in addition to the circulation along azimuthal direction, light propagates along the length of nanowires in the array. From near-field distribution, it is observed that the WGM sustains throughout the length of the array, except at half the height of the nanowires - where a node is created because of longitudinal mode created by Fabry-Perot resonance along the vertical direction. This node however does not influence VLP detection as the VLP is positioned at the top surface of the array and interacts with the modes located therein.
Resonant wavelength ($\lambda _{r}$) and quality (Q)-factor obtained for different diameters ($D$) of the array are shown in Fig. 1(b). The nanowire diameter ($d$), and the minimum spacing ($a$) between adjacent NWs considered in these arrays are 70 nm and 15 nm respectively. For these dimensions, a maximum Q-factor of $10^{5}$ is obtained at $\lambda _{r}=1.2 \mu m$ when the overall diameter of the array is 2.5 $\mu$m. However this mode is found to be less sensitive to the presence of a VLP onto the array as the resonant wavelength in this case is significantly larger than the particle diameter. To overcome this limitation, a NW-array of 2 $\mu$m diameter is considered throughout this study. This not only ensures high detection sensitivity of the array because of the corresponding smaller resonant wavelength, but also reasonably high Q-factors in the order of $10^{4}$.
To obtain optimum dimensionalities of the NWs in the array for VLP detection, resonant wavelengths and quality factors of the most strongly confined WGM modes are plotted in Figs. 1(c)-(d) for varying diameter (d) and spacing (a) of nanowires in the array. Radial (r) and azimuthal (m) mode numbers, denoted as (r,m) in these plots, have been obtained from near-field distribution of the corresponding resonant modes. As can be observed, the Q-factor of the array decreases monotonically with the increase of both nanowire spacing and diameter, whereas the resonant wavelength varies from 750 -1000 nm depending on mode number (r,m). It is noteworthy that resonant wavelength of the structure closely matches with theoretical calculations performed based on the following relation:
Here $D_{R}$ is effective diameter of optical path for different radial modes and $n_{eff}$ is effective refractive index of the array calculated from its effective dielectric constant ($\epsilon _{eff}$) obtained from the following Maxwell-Garnett equation-based approximation of effective medium theory [58]:According to Figs. 1(c)–1(d), high-Q whispering gallery resonant modes are expected for smaller spacing and diameter of nanowires in the array. However, whereas too small a spacing may give rise to processing and fabrication related complexities, loss of detection sensitivity is observed when NW diameters are less than 70nm. This is illustrated in Fig. 2(a), where shift of resonant wavelength caused by a VLP placed 100 nm inwards from the periphery of an array is shown for four different diameters of the NWs. From this plot it is evident that having small sized NWs does not necessarily result in a high detection sensitivity of the array. The highest shift of resonant wavelength, and also a significant change of Q-factor are obtained for a NW-array of $d$=70 nm and $a$=15 nm. Henceforth, this NW-array is taken as the optimized structure for VLP detection in this study. To understand resonant characteristics of this optimized structure in further details, its resonant modes residing within ultra-violet (UV) to near-infrared (NIR) regime of the spectra are calculated and plotted in Fig. 2(b). Whispering gallery resonant modes having azimuthal mode numbers of 14, 18 and 22 are obtained with Q-factors in the order of $\sim$ $10^{4}$. Among these WGMs, the (14,1) mode localized at 993.273 nm represents the most tightly confined one with Q$\sim$16621. Near field image of this mode is shown in Fig. 2(c). To have an estimate of the effect of heterogeneity resulting from process variation in the array, the (14,1) mode is reevaluated considering 2% uniform random variation of nanowire diameter and position. As shown in the inset of Fig. 2(b), the resonant wavelength shifts from about 0.6 nm to 1.6 nm for such variations. For both spatial and structural variations of this order, the Q-factors remain high enough for the NW-array to be able to detect the presence of a VLP.
Fig. 2. (a) Shift of resonant wavelength and percentage change of Q-factor of the most strongly confined resonant mode in the presence of a single virus, while NW-diameter of the array is varied (resonant wavelength of each mode in the absence of virus is shown within parentheses); (b) Q-factors for different modes of the optimized structure (inset shows resonant modes for 2% uniform random variation of nanowire diameter and position); Near field images of the resonant modes located at (c) 993.273 nm and (d) 357.465 nm; (e) Shift in resonant wavelength for change of background medium of the nanowire array and conventional solid cylindrical WGM resonator (inset shows schematic of the solid cylindrical resonator and its near-field profiles for the highest Q-factor mode).
As can be observed from Fig. 2(b), while the most strongly confined mode of the optimized NW-array resides in the NIR regime, another strongly localized mode resides at 357.465 nm with a Q-factor of $\sim$9500. Near field image of this mode, shown in Fig. 2(d), represents multiple-scattering mediated weak localization of light. Consequently a more distributed field profile is obtained with a relatively smaller Q-factor in this case. Such field distributions are characteristics of microrod- or nanowire-array based optical resonators and the corresponding modes are often referred to as quasi- or triangular-WGMs [12]. In spite of having relatively smaller Q-factors, quasi-WGMs offer significant advantages over the conventional whispering gallery resonant modes in terms of virus detection. As resonant characteristics of WGMs are dominated by standing-wave formed near periphery of the structure, the WGM-resonator is found to be less sensitive to disturbance caused by a virus located near the center of the structure. On the contrary, as will be shown in subsequent section, the referred quasi-WGMs of the NW array are highly sensitive to any perturbation caused by a virus located around the center of the array.
The near-field images of Figs. 2(c)-(d) indicate that two different underlying mechanisms contribute to the emergence of WGM and quasi-WGM resonant modes in the NW-array. In previous theoretical and experimental works based on similar structures [49,59], it has been shown that nanowires in such an array constitute an effective medium where peripheral total internal reflection of light results in the emergence of WGMs- just as it would in a rotationally symmetric solid medium. These modes can be predicted by the effective-medium theory, as has been shown by the Maxwell-Garnett based approximation in this work. By varying the size and density of the nanowires, the effective refractive index of the array can be varied and thus the resonant mode can be tuned as required. These modes are more suitable for detecting VLPs located near the periphery of the array as the optical field for such modes are more or less confined in the peripheral region and therefore are easily perturbed by a VLP attached in there.
The mechanism of multiple-scattering mediated quasi-WGM, on the other hand, is more in line with the random scattering of light in an array of dielectric scatterers. It has been shown both theoretically and experimentally that random scattering of light leads to photon entrapment in a disordered array of nanowires [44,57]. Whereas a weakly disordered system results in weak localization of light with moderately high Q-factors (in the order of $10^{3}-10^{4}$), a strongly disordered system leads to strong-localization of light with Q-factors in the order of $10^{4}-10^{5}$ or higher. Though the rotationally symmetric nanowire array in this study does not constitute a disordered medium, it is likely that the underlying multiple-scattering events of light in the array results in the resonant mode observed at 357.465 nm. It is also to be noted that resonant wavelength of these scattering mediated modes cannot be predicted by the effective medium based approximations of Eqs. (1)–(2). Instead, the scattering t-matrix based formulation reported by Arya et al. applies well in this case [60]. This theoretical framework defines a characteristic scattering length $l_{c}=\sqrt {3 / \pi }(c / \omega )$ for a random medium of circular scatterers. Here $\omega$ is angular frequency and $c$ is velocity of the electromagnetic (EM) wave in the background medium. According to this formalism, the EM wave undergoes a transition from extended state to localized state when the condition $l / l_{c} \leq 1$ is satisfied. Here $l=c /(2 \gamma )$ is the diffusion length and
To have a quantitative estimate of the advantage of the optimized NW array based resonator, its sensitivity to change of refractive index of the background medium is calculated and compared with the results obtained for a conventional solid cylindrical structure of equal diameter. Here refractive index of the background medium of both the resonators is varied from 1.31 to 1.40 and the resultant wavelength shift of the most strongly localized mode are calculated and plotted in Fig. 2(e). For the NW-array based structure, wavelength shift per unit change of refractive index unit (RIU) is calculated to be 97.34 nm/RIU, which is about five times of the value obtained for the solid cylindrical structure. Such difference arises from the field-distributions of the two structures. Near-field profile for the highest Q-factor mode of the solid structure is shown as an inset of Fig. 2(e). As can be observed, unlike the case of the nanowire array, this highest Q-factor mode has a radial mode number of $r=2$. Consequently the structure becomes less sensitive to the presence of a VLP when it is positioned in between the two radial field profiles. As discussed later in the text, detection sensitivity of the proposed scheme is obtained to be even higher when resonant wavelength shift resulting from the presence of a VLP is compared. Furthermore, as WGMs of higher radial modes are more difficult to excite and detect [36], the NW-array based WGM having $r=1$ appears to be more suitable for VLP detection.
3. Spatial and spectral mapping for virus detection
The biosensing scheme proposed in this work is based on the change of resonant wavelength and Q-factor resulting from the presence of a VLP in the resonator. It has been assumed that during experiments, a suitable functionalizing material will be utilized to attach the VLP onto the resonator- a practice that is quite standard for detecting pathogens employing biosensors [1,2,61,62]. This will ensure that the virus will remain attached to the resonator and therefore will remain immobile during measurements. The presence of the virus in the array will not only perturb field distribution of the resonant mode, but will also change the effective refractive index of the medium, thereby resulting in an overall shift of the resonant wavelength. This shift of resonant wavelength, denoted as $\Delta \lambda _{r}$, is shown in Fig. 3(a) for different radial positions of the virus onto the array. For the whispering gallery resonant mode, $\Delta \lambda _{r}$ is maximum when the virus is located near periphery of the structure. The maximum value of $\Delta \lambda _{r}$ is obtained to be $\sim$4.3 nm when the virus is located $\sim$100 nm inwards from the periphery. At this location, the virus is essentially positioned at the maxima of the WGM standing wave and consequently the field distribution (shown as an inset of Fig. 3(a)) is perturbed significantly. To understand how this perturbation is related to the angular position of the virus, $\Delta \lambda _{r}$ is calculated and plotted by positioning the VLP at different polar angles ($\theta$) of the resonator (Fig. 3(b)). Here three different diameters of the VLP are considered, and for each case the radial distance of the VLP is kept fixed at 900 nm. As expected, $\Delta \lambda _{r}$ increases as the size of the perturbing particle increases, and for a VLP of 140 nm diameter the shift can be as much as 8 nm. As the NW-array is not perfectly radially symmetric due to the presence of truncated NWs at the cavity boundary, $\Delta \lambda _{r}$ changes periodically with angular position of the array. For the array under consideration, the minima of $\Delta \lambda _{r}$ consistently appears at integer multiples of $\theta =\pi /4$, thereby ensuring that $\lambda _{r}$ remains predictable enough. Though $\Delta \lambda _{r}$ becomes significantly small ($\sim$0.2 nm) when the diameter of the VLP is 60 nm, it is $3.2\pm 1.1$ nm or higher when the diameter of the VLP is $\geq$100 nm.
Fig. 3. (a) Shift of resonant wavelength with respect to the 993.273 nm and 357.465 nm modes for different positions of the virus on the array (inset shows near field image for a virus positioned near the periphery of the array); (b) Shift of resonant wavelength of the 993.273 nm mode for different diameters and angular positions of the virus-like particle located near periphery of the array; False color map illustrating shift in resonant wavelength with respect to the (c) 357.465 nm and (d) 993.273 nm modes for different positions of the virus on the array; (e) Shift of the 993.273 nm resonant mode obtained for different diameters of the virus (the solid line represents linear fit of the results and inset shows resonant modes calculated over a narrow spectral range).
From Fig. 3(a), it becomes quite obvious that the whispering gallery resonant mode is significantly perturbed when the VLP is located near periphery of the NW-array. However when the VLP is located towards the center of the array, it has minimal impact on the field distribution of the WGM. To overcome this shortcoming, the quasi-WGM of $\lambda _{r}$ = 357.465 nm is considered for detecting a virus positioned within $\sim$600 nm radius of the array. The corresponding shift of resonant wavelength for this case is shown in Fig. 3(a). Though the $\Delta \lambda _{r}$ values obtained for this case are smaller than what is attained with the dominant mode, it offers significantly higher sensitivity for detecting a virus positioned around the center of the array. To have a better overview of the dependence of resonance wavelength on spatial location of the virus, a spatial mapping of $\Delta \lambda _{r}$ is performed. As can be observed from the false color plots of Figs. 3(c)-(d), the quasi-WGM mode of $\lambda _{r}$ = 357.465 nm is better suited for detecting a virus positioned near the center, whereas better sensitivity is attained with the $\lambda _{r}$ = 993.273 nm mode for virus positioned elsewhere. It is noteworthy that for both the modes, the shift of resonant wavelength is radially asymmetric in nature. As mentioned earlier, this is related to the asymmetry arising from the presence of truncated NWs at the cavity boundary. Whereas $\Delta \lambda _{r}$ values ranging from 0.01 nm to 4.3 nm readily offers a significant figure of merit, it is quite promising that even greater shift of $\lambda _{r}$ is expected with viruses of larger diameters. The effect of virus-size variation on resonant wavelength is shown in greater detail in Fig. 3(e), where the maximum value of $\Delta \lambda _{r}$ is plotted for different diameters of the VLP positioned at 900 nm radial distance and $0^{\circ }$ angular position of the array. The inset of this plot shows the resonant wavelengths over a very narrow spectral range, obtained for viruses having diameters of 60 nm and 160 nm. As can be observed, for viruses of small size, the resultant shift of $\lambda _{r}$ is relatively small. This is owing to the fact that field distribution of the resonant mode is less perturbed by particles of smaller size. As the size of the VLP increases, $\Delta \lambda _{r}$ increases almost linearly and it can exceed 8 nm when the diameter of the VLP is 160 nm. It may be noted that spherical shaped viruses, such as the Coronavirus, are usually 80 nm - 200 nm in size. Hence the resonant wavelength shifts obtained here offer significant detection sensitivity for practical applications.
While the change of resonant wavelength offers one of the most effective means of detecting a VLP, it cannot be denied that the sensitivity of this approach is rather limited if a virus is positioned within 550-650 nm radial distance of the array- an area highlighted as ‘blind-spot’ in Fig. 3(a). To overcome this limitation, we have explored the possibility of utilizing Q-factor as another means of detecting the presence of a virus in the test sample. The Q-factor of the resonant mode obtained for different spatial locations of the virus is shown in Figs. 4(a). Depending on spatial location of the virus, significant change in Q-factor is observed for both the modes. Similar to the trends obtained for $\Delta \lambda _{r}$, the change of Q-factor (referred as $\Delta Q_{r}$) is significantly high for the dominant resonant mode when the virus is positioned near the periphery. The quasi-WGM mode on the other hand exhibits more than 90% change of $\Delta Q_{r}$ when the virus is positioned around center of the array. Therefore the two modes in a way complement each other to enhance overall sensitivity of the array. It is also noteworthy that Q-factor can change by as much as 28% over the region referred to as blind-spot in Fig. 3(a). This suggests that Q-factor variation offers an effective means of VLP detection, specially over the spatial region where sensitivity is relatively low for detection based on resonant wavelength shift. To have an estimate of how Q-factor would change depending on angular position and size of the virus, $\Delta Q_{r}$ is calculated and plotted as a function of $\theta$ for three different diameters of the VLP. The results shown in Fig. 4(b) suggest that similar to the dependence of $\Delta \lambda _{r}$ on $\theta$ (shown in Fig. 3(b)), Q-factor changes periodically with angular position of the virus. The minima of $\Delta Q_{r}$ consistently appears at integer multiples of $\theta =\pi /4$, just as it was observed in the $\Delta \lambda _{r}$ vs. $\theta$ plot. Though $\Delta Q_{r}$ varies between $7\times 10^{3}-1.3\times 10^4$ for the VLP of 60 nm diameter, the value consistently remains in the order of $10^{4}$ for particles of larger diameters, thereby offering significant detection sensitivity in terms of change of Q-factor.
Fig. 4. (a) Change in Q-factors of the considered resonant modes, obtained for different positions of the virus on the array; (b) Change in Q-factor of the 993.273 nm mode obtained for different diameters and angular positions of the virus-like particle located near periphery of the array; False color map illustrating Q-factor variation of the (c) 357.465 nm and (d) 993.273 nm resonant modes for different positions of the virus on the array; (e) Change in Q-factor obtained for different diameters of the virus (the solid line represents exponential fit of the results and the inset shows near field image obtained for a virus of 160 nm diameter positioned near periphery of the array).
Similar to the case of resonant wavelength shift, mapping has been performed for Q-factor variation considering different spatial locations of a single virus onto the array. The resultant false color plots of Figs. 4(c)-(d) suggest that the variation is radially asymmetric. As mentioned earlier, such asymmetry possibly arises from the non-uniformity at the periphery of the array resulting from truncated nanowires. Dependence of Q-factor on the size of a single virus, located at $\theta =0^\circ$ near the periphery of the array, is shown in Fig. 4(e). As expected, the Q-factor decreases, and consequently $\Delta Q_{r}$ increases, with increasing diameter of the virus as it significantly perturbs symmetry of the system and thus offers the optical field a path to leak out of the resonator. This is confirmed by the near field image shown as an inset of Fig. 4(e), which shows that the standing wave is significantly perturbed by a single virus of 160 nm diameter in size, placed near periphery of the array. It is noteworthy that a linear fit of the relation between $\Delta \lambda _{r}$ and VLP-diameter ($s$) is obtained from Fig. 3(e), whereas the $\Delta Q_{r}$ vs. $s$ plot of Fig. 4(e) shows an exponential fit. These results suggest that size of the virus can be uniquely determined by assessing the change of resonant wavelength and Q-factor of the array.
4. Performance analysis
The results presented in this work so far suggest that the proposed nanowire array based optical resonator holds significant promise for label-free detection of virus-like particles. To have a quantitative estimate, detection sensitivity of the NW array is compared with the results obtained for a solid-cylindrical WGM resonator. It has already been shown in Fig. 2(e) that compared to a conventional WGM resonator, the proposed sensor offers about 5 times higher sensitivity to the change of refractive index of background medium. However, it remains to be seen whether such high sensitivity can also be attained when the change of refractive index is caused by the attachment of a single virus onto the array. To this end, sensitivity of the two structures are compared when VLPs are located in the central and peripheral regions of the structures. As shown in Table 1, the sensitivity of the NW array based resonator is about 8 times higher than the value attained with an equivalent solid-cylindrical structure when the virus is positioned near the periphery. The detection wavelength in this case is in the NIR regime. For the case when the virus is positioned at the center of the array, the UV resonant mode offers the best sensitivity for both the structures. However, whereas the solid-cylindrical structure utilizes WGM in this case, the resonant mode utilized for the NW array is quasi-WGM in nature, as has been discussed in detail in Section 2. It is remarkable that the sensitivity attained for the NW array with such a radially asymmetric mode is $\sim$742 times higher than the sensitivity attained with the conventional structure. Therefore irrespective of the position of the virus, the proposed NW array based optical resonator is expected to offer significantly higher sensitivity compared to the solid-WGM structure.
Table 1. Comparison of performance characteristics of nanowire array based resonator and solid cylindrical WGM resonator
In spite of the enhanced performance characteristics, it cannot be denied that Q-factor of the NW array is about 1-2 orders lower than the Q-factors obtained for regular WGM resonators. This raises the concern whether the NW array based detector is more susceptible to noise compared to the conventional WGM resonator. To address this point, figure of merit (FOM) defined in Eq. (2) is calculated and compared for the two structures [63].
Here FWHM is obtained from Q-factor of resonant mode of the corresponding structure. The obtained results are shown in Table 1, which suggests that when VLPs are located near the periphery, FOM of the solid structure is about 1.6 times higher than the FOM obtained for the NW array. However for a virus positioned near the center, FOM of the NW array is about 7 times higher. Therefore, the high sensitivity of the NW array more than compensates for its comparatively low Q-factor, thereby making it a better alternative to the conventional WGM resonator.To further evaluate performance characteristics of the NW array based detector, statistical analysis is performed considering three different simulated test cases comprising of 500 samples each. For each sample, a single VLP is randomly placed onto the NW array and the corresponding shift of resonant wavelength and change of quality factor are recorded based on the mapping presented in Section 3. It is to be noted that uniform random distribution has been considered to randomly place any virus onto the array- one at a time, for each test case. The results of the three simulated test cases are shown in Figs. 5(a)-(c), where position of the dot represents spatial location of the virus, diameter of the dot represents percentage change of Q-factor, and color bar indicates the corresponding shift of resonant wavelength. For all three test cases, the outer circle shown by a red dashed line represents the periphery of the array. The inner circle having 600 nm radius (shown by brown dashed line in Figs. 5(a)-(c)) encompasses the region where viruses are detected based on UV spectral analysis, whereas the region outside corresponds to the NIR wavelength based detection regime.
Fig. 5. (a)-(c) Shift in resonant wavelength and percentage change in Q-factor for three different simulated test cases having 500 samples each, where for each sample a virus is randomly positioned on the nanowire array (the diameter and color of a dot represents percentage change in Q-factor and shift in the resonant wavelength respectively, whereas position of the dot shows spatial location of the virus); Histograms representing detection summary in terms of (d) resonant wavelength shift and (e) percentage change of Q-factor (the bars indicate variation obtained for the three different simulated test cases)
A summary of the results obtained for the simulated test cases is shown by the histograms of Figs. 5(d)-(e), which represent statistical variability of resonant wavelength shift and percentage change of Q-factor respectively. The bars shown in these plot correspond to the range of variation obtained for the three test cases. It is noteworthy that more than 50% samples are expected to result in a shift of resonant wavelength of 0.1 nm or larger. Such a value of $\Delta \lambda _{r}$ is significantly higher than the values reported elsewhere for resonant wavelength shift based biosensing schemes [9,64]. It is also noteworthy that only 5% samples fall within the minimum $\Delta \lambda _{r}$ range of 0.001-0.01 nm. Promising characteristics are obtained for percentage change of Q-factor as well. As can be observed, more than 60% samples correspond to $\Delta Q_{r}$ values of 40% or higher. Only about 4.6% of the test samples lie within the minimum range of $\Delta Q_{r}$. It is noteworthy that the $\Delta \lambda _{r}$ values for these 4.6% samples lie within the minimum wavelength shift range of 0.001-0.01 nm. Therefore detection sensitivity for these samples will be essentially limited by spectral resolution of the measurement system. The present study does not discuss specificity of the proposed technique as it has been considered that appropriate functionalization material is used to bind the virus with the NW array.
5. Experimental considerations
The feasibility of implementation of the proposed technique will depend on experimental considerations related to processing or fabrication of the nanowire array, resolution of the measurement system, and binding of the virus particles to the nanowire array. The proposed nanowire based photonic structure can be experimentally realized employing different fabrication and processing techniques. In fact several groups have already reported fabrication of nanostructures of similar dimensionalities using techniques like nanoimprint [50,51], electron beam lithography (EBL) [65], reactive ion etching (RIE) [66], focused ion beam (FIB) milling [67] and metal assisted chemical etching (MACE) [68]. As has been shown in Fig. 2(b), 2% random variation of nanowire diameter or position decreases both resonant wavelength and Q-factors of the nanowire array. Nevertheless, both Q-factor and $\Delta \lambda _{r}$ remain high enough for VLP detection. This offers some degree of flexibility in the design and implementation of the array. Resonant wavelength and Q-factor of the nanowire array can be measured using micro-photoluminescence spectroscopy, which is well established as a reliable technique for experimentally characterizing WGM resonators [33,35].
Efficient excitation of WGM-resonators is still an active area of research and different techniques have been suggested to this end. To efficiently excite a mode with high angular number, light can be incident onto the NW-array at an angle through a focusing lens [34,35,37]. Also other experimentally reported techniques like tapered-fiber coupling [35,36], prism-coupling [35,38,39] or fiber-tip coupling [35] can be adopted to efficiently excite the array. Upon excitation, light can be collected out of plane using an objective lens of suitable magnification and numerical aperture, or using optical fibers [33]. Considering two different excitation wavelengths of these scheme, i.e. 993.273 nm for WGM and 357.465 nm for quasi-WGM modes, the near infra-red resonant mode needs to be collected from peripheral region of the structure (Fig. 6(a)), whereas the UV resonant mode has to be collected from the central region of the structure, as shown in Fig. 6(b). According to the statistical analysis presented in the previous section, about 5% of the simulated test samples fall within the minimum wavelength shift range of 0.001-0.01 nm. This suggests that in case the spectral measurement system is resolution limited for wavelength shifts of 0.001 – 0.01 nm or lower, statistically the VLP will remain undetected for about 5% of the test cases because of random positioning of the particle onto the array. Hence this should be the minimum resolution of the spectroscopic measurement system for being able to detect the presence of a virus with a high degree of accuracy.
Fig. 6. Schematic of the possible experimental setup of the proposed biosensor; for the 993.273nm and 357.465nm resonant mode out of plane light will be collected (a) from the peripheral region and (b) from the central region respectively.
In our study we have considered the virus particle to be immobile because of its binding with the nanowires. In case the virus remains loosely attached to the array because of weak binding, the resonance wavelength and quality factor of the system will change in accordance with the mapping shown Section 3. In such scenarios, detection performance will not only depend on spectral resolution of the measurement system, but possibly also on how fast the measurement system can respond to the temporal change of resonant condition of the photonic structure. To have a quantitative estimate of how a slight change in the orientation of the virus would influence detection performance, rates of change of $\Delta \lambda _{r}$ and $\Delta Q_{r}$ are calculated with respect to radial position of the virus. From our analysis, the maximum rate of change of $\Delta \lambda _{r}$ and $\Delta Q_{r}$ are obtained to be 0.015 nm/nm and 40/nm respectively. Such small rates of change suggest that detection performance will not be remarkably affected by a slight change in the orientation of the virus-like particle.
An important concern with label-free pathogen detection is the choice of suitable functionalizing binding material, as non-specific binding interactions in such schemes can often result in background signals which may interfere with signals specific to virus detection. To reduce non-specific binding, suitable functionalizing binding materials have to be utilized so that only the virus of desired type is bound onto the array. As an example, anti-S or anti-N biomarkers can be utilized to specifically bind S-spike proteins of Coronavirus onto the nanowires [69]. Selective surface functionalization of silicon nanowires based on nanoscale joule heating can also be adopted to reduce non-specific binding [70]. In the presence of suitable functionalizing material and also upon appropriate pre-processing and cleaning - possibly in accordance with the experimental works on WGM-based biosensing [40,41]- only the desired VLP will be bound to the array. Moreover, by utilizing antibodies specific to different viruses, the proposed technique can be extended for detecting numerous pathogens in a label-free, rapid manner. Notwithstanding such prospects, it goes without saying that practical effectiveness of the proposed scheme can only be assessed by means of detailed experiments, where experimental implementation of a suitable functionalizing technique for binding viruses onto the silicon nanowire based resonator array is expected to play the governing role. Such a detailed study was beyond the scope of the present theoretical work. It is envisaged that the proposed novel sensing scheme presented herein would stimulate further research and experimental investigation in this regard.
6. Conclusions
In summary, a novel photonic technique of detecting virus-like particles has been proposed based on a rotationally symmetric array of silicon nanowires, where the presence of a single virus significantly perturbs the whispering gallery and quasi-whispering gallery resonant modes of the system. Finite difference time domain based numerical analysis has been performed to estimate the shift of resonant wavelength and change of quality factor of resonant modes in the UV and NIR regimes of the spectra and the corresponding changes have been correlated with spatial location of the virus. Statistical analysis has been performed by randomly positioning a single virus onto the NW array for test cases having a large number of samples. The results of our analysis suggest that the proposed structure offers significantly higher sensitivity in comparison to conventional WGM resonators. The resonant wavelength shift has a linear dependence on the diameter of the virus whereas the quality factor of the mode appears to vary exponentially. For a single virus of 160 nm diameter, the resonant wavelength can shift by more than 8 nm, whereas the change of quality factor can be as much as 100%. Statistical analysis suggests that for simulated test cases comprising 500 samples each, only about 5% of the samples fall within the minimum wavelength shift range of 0.001-0.01 nm and the minimum quality factor variation range of $\sim$14-20%. Because of the underlying silicon platform, the proposed photonic structure should be realizable in a low-cost manner employing high-throughput lithography techniques. Moreover, by using suitable functionalizing binding material, the scheme should be applicable for detecting numerous pathogens in a label-free, rapid manner, and may also be utilized for environmental monitoring purposes.
Acknowledgments
M.M.H. and M.Z.B. acknowledge the support and facilities received from the Department of Electrical and Electronic Engineering and the Institute of Information and Communication Technology of Bangladesh University of Engineering and Technology.
Disclosures
The authors declare no conflicts of interest.
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.
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