Physical modeling of interference enhanced imaging and characterization of single nanoparticles

Interferometric imaging schemes have gained significant interest due to their superior sensitivity over imaging techniques that are solely based on scattered signal. In this study, we outline the theoretical foundations of imaging and characterization of single nanoparticles in an interferometric microscopy scheme, examine key parameters that influence the signal, and benchmark the model against experimental findings. © 2016 Optical Society of America OCIS codes: (180.3170) Interference microscopy; (070.0070) Fourier optics and signal processing; (260.1960) Diffraction theory. References and links 1. A. Yurt, G. Daaboul, J. H. Connor, B. B. Goldberg, and M. S. Ünlü, “Single nanoparticle detectors for biological applications,” Nanoscale 4(3), 715–726 (2012). 2. D. R. Walt, “Optical methods for single molecule detection and analysis,” Anal. Chem. 85(3), 1258–1263 (2013). 3. R. K. Jain, and T. Stylianopoulos, “Delivering nanomedicine to solid tumors,” Nat. Rev. Clin. Oncol. 7(11), 653—664 (2010). 4. World Health Organization, “World health statistics 2015” (WHO, 2015). http://www.who.int/gho/ publications/world_health_statistics/2015/en/ 5. E. Kelly and S. J. Russell, “History of oncolytic viruses: genesis to genetic engineering,” Molecular Therapy 15(4), 651—659 (2007). 6. G. Steiner, “Surface plasmon resonance imaging,” Anal. Bioanal. Chem. 379(3), 328–331 (2004). 7. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A. 105(52), 20701–20704 (2008). 8. S. H. Gage, “Modern dark-field microscopy and the history of Its development,” Trans. Am. Microsc. Soc. 39, 95–141 (1920). 9. M. R. Monroe, G. G. Daaboul, A. Tuysuzoglu, C. A. Lopez, F. F. Little, and M. S. Unlu, “Single nanoparticle detection for multiplexed protein diagnostics with attomolar sensitivity in serum and unprocessed whole blood,” Anal. Chem. 85(7), 3698–3706 (2013). 10. O. Avci, N. L. Ünlü, A. Y. Ozkumur, and M. S. Ünlü, “Interferometric reflectance imaging sensor (IRIS)–a platform technology for multiplexed diagnostics and digital detection,” Sensors 15(7), 17649–17665 (2015). 11. D. D. Nolte, Optical Interferometry for Biology and Medicine (Springer, 2012). 12. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006). 13. K. L. Kelly , E. Coronado , L. L. Zhao , and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). 14. E. Seymour, G. Daaboul, X. Zhang, S. M. Scherr, N. S. Ünlü, J. H. Connor, and M. S. Ünlü, “DNA-directed antibody immobilization for enhanced detection of single viral pathogens,” Anal. Chem 87(20), 10505–10512 (2015). #258126 Received 26 Jan 2016; accepted 7 Mar 2016; published 10 Mar 2016 (C) 2016 OSA 21 Mar 2016 | Vol. 24, No. 6 | DOI:10.1364/OE.24.006094 | OPTICS EXPRESS 6094


Introduction
With the advent of nanotechnology, there is a rapidly growing impact of nanoparticles in biotechnology research necessitating new tools for detection and accurate characterization of individual particles [1].Artificial nanoparticles can be utilized as labels for molecular detection [2] and for drug delivery [3].While viruses (nature's nanoparticles) cause over a million fatalities every year [4], they can also be engineered as promising agents for cancer therapy [5].Optical microscopy has been an essential tool in biological sciences for visualization of micron size particles.However, optical imaging techniques demonstrated inadequate success in visualization of natural and artificial nanoparticles as a consequence of fundamental visibility and resolution limitations due to diffraction.Various optical techniques such as Surface Plasmon Resonance Microscopy (SPRM) [6], Whispering Gallery Mode (WGM) sensors [7], Total Internal Reflection Fluorescence [2], and Dark-field Microscopy [8] have been employed to detect nanoparticles.The challenge in detecting nanoparticles using those aforementioned techniques often lies in the complexity of the system, as well as the lack of means to discriminate between the target and unspecifically bound impurities, such as dust particles, cellular debris, and molecular aggregates, which further decrease the sensitivity of the technique.Labeling the target analytes with metallic nanoparticles to generate an easily differentiated signal can also contribute to discriminating target analytes from impurities [9,10].
Studying biomolecular dynamics especially in confined environments have necessitated the use of smaller nanoparticles (tens of nanometers in diameter) as labels.However, some of the aforementioned imaging techniques that are solely based on the scattered light fall short of meeting this need.This is mainly due to the scaling of the scattered electric field with the third power of the nanoparticle radius (r 3 ) -hence the intensity scales with r 6 .This leads to a rapid decrease in signal for small nanoparticles, making them indistinguishable from the background.The development of the interferometric microscopy schemes have rendered the r 3 scaling in the detected signal, so they have gained significant attention for their highly sensitive nanoparticle detection capabilities, and their simple, cost-effective, and high-throughput setups.The detailed comparisons of interferometry based techniques with other commonly used methods can be found in [1,10], along with their physical principles explained in [11].
Optical interferometry provides enhanced visibility of nanoparticles and it can be implemented in a simple common path modality [10].However, interferometry is prone to errors due to small phase variations in high numerical aperture optical systems resulting in errors in detection and characterization accuracy.Hence, it is crucial to have a robust physical model that accounts for various parameters including the phase between the target and reference signals.
In this study, we develop a rigorous numerical model for single particle optical response in a common-path wide-field interferometric optical system.While the basic elements of the model can be utilized for analysis of generic wide-field imaging systems, we concentrate on imaging on a layered sensor where specular reflection of illumination provides the reference light for interferometry.We analyze both low-index contrast dielectric nanoparticles (similar to natural nanoparticles) and metallic nanoparticles and validate the model with experimental results.We demonstrate that specific signatures for nanoparticles can be acquired by varying the focus of the optical system with respect to the physical structure, thus scanning the relative phase.
The paper is organized as follows.In the remainder of the introduction, we present an overview of the interferometric imaging schemes.In section 2, we provide a detailed description on the theoretical model of the single particle response in a common-path wide-field interferomteric imaging scheme.Section 2.1 provides a discussion on Köhler illumination in the context of wide-field imaging.The incident and reflected fields assuming Köhler illumination are formulated in section 2.2, and the image of the reflected field is formulated in section 2.3.The Green's function formulations for a point dipole near a planar interface is given in section 2.4, and the image of the electric field due to a point dipole near a layered sensor is provided in section 2.5 along with the overall response of a nanoparticle in a common-path wide-field interferomteric imaging scheme.In section 3, we provide a detailed analysis of the main parameters that affect the interferometric nanoparticle signal using the theoretical model simulations, which are also benchmarked against experimental findings in section 3.4, and final remarks are provided in section 4.

Interferometric microscopy
In general, signal in an interferometric microscopy scheme is primarily affected by the polarizability of the particle (α), amplitude of the reference field (E re f ) and the phase lag between them (θ ) as given in the following equation: where Note that α is the polarizability of the spherical nanoparticle, r is its radius, and ε p & ε m are the permittivities of the particle and its surrounding medium, respectively; and the particle is assumed to be a Rayleigh scatterer, for its size is much smaller than the illumination wavelength.
In Equation 1, for small particles, |E sca | 2 can be neglected, thus the cross term ( |E re f ||E sca |) that scales with r 3 dominates the signal, and the phase lag (θ ) can be adjusted such that the signal is maximized.
Mainly, the reference field can be obtained in a double-path interferometer configuration as shown in Fig. 1(a), or it can be obtained using a reflective sensor in a common-path interferometer scheme as shown in Fig. 1(b) and 1(c).The latter has the prominent advantage of simplicity and robustness.Furthermore, the signal in a typical interferometric microscopy is also affected by the particle geometry, illumination wavelength, and defocus.In this study, we combine rigorous numerical modeling with experimental measurements to show how these aforemtioned factors affect the signals from the nanoparticles.We present our model for the interferometric imaging scheme shown in Fig. 1(c), yet it can easily be adapted to the other interferometric nanoparticle imaging schemes as well.

Theoretical model
In this section, we provide a detailed description of the physical model for the single particle response in a wide-field common-path interferometric microscopy.Our model makes use of the Angular Spectrum Representation (ASR) approach outlined in [12].We start with a short discussion on Köhler illumination, and then define the field representations of the incident and reflected light in a Köhler illumination scheme.

Köhler illumination
In Köhler illumination, the light source is in a Fourier plane with respect to the sample plane so as not to contain the image of the source in the final image.Therefore, the light source is placed at the back focal plane of a lens which essentially performs a Fourier transform.Hence, the field at the field stop (A f s ) is in the Fourier plane with respect to the plane of the light source, and is imaged onto the object plane by two lens configuration that is a 4f system with nonunity magnification as shown in Fig. 2. To model the Köhler illumination, we use incoherent plane waves of angles that span the numerical aperture (NA) of the objective.This consideration can be justified by segmenting the light source into small regions, where each segment can be represented by a point source with E s (x, y) = δ (x, y).Since the first lens (f 1 ) performs a Fourier transform, we can write the field at the field stop (A f s ) as follows: where λ is the wavelength of the illumination light, and n is the refractive index of the surrounding medium.At (x,y) = (0,0) E f s becomes as follows: which is a plane wave propagating along the optical axis.Similarly, for a segment at (x,y) = (x o , y o ) E f s is as follows: which is another plane wave with (k x , k y ) = (k 1 . Therefore, for a spatially incoherent light source, each point source produces a plane wave that is incoherent with others.As a result, each plane wave can be propagated and treated separately, and in the end at the detector, their intensities are summed, as opposed to their fields, which would be the case for a coherent light source.

Incident & reflected fields in Köhler illumination
We first start with the mapping of incident field components from the light source onto the sample plane that is the sensor surface.We follow the ASR approach, as detailed in [12], to map the input electric field components to the corresponding total driving fields at the sample plane as illustrated in Fig. 3 below.Fig. 3. Geometric illustration of the focusing of an incident field by a condenser lens (adapted from [12]).
Since each plane wave is treated separately in Köhler illumination, the focal incident fields in spherical coordinates due to each plane wave component E i,m can be expressed as given below.Note that the reader is referred to Appendix A for the details of this derivation.
Moreover, with an interface at z = z d (z d denotes the defocus -the distance between the sensor surface and the focal plane of the objective as illustrated in Fig. 13 in Appendix B), we can find the reflected field components by invoking the boundary conditions as follows: The formulations for the Fresnel coefficients, r p , r s , can be found in Appendix B.
Hence the total driving field (E t,m ) due to the plane wave corresponding to index m at (ρ, ψ, z) = (ρ o , ψ o , z o ) can be found as the sum of incident and reflected electric fields:

Image of the reflected light
Akin to the our approach in Section 2.2, to model the imaging of the reflected field onto detector (image) plane, we follow the same ASR approach.In this case, the field that is reflected off of the sensor is imaged by a two lens system with magnification, M = n 1 f 2 /n 2 f 1 , as shown in the Fig. 4 below.
As a series of incoherent plane waves cover the angular spectrum range in Köhler illumination, we can write the reflected fields in the image plane due to each plane wave component, E r,m , as given in the equation below.The reader is referred to Appendix C for the details of this derivation.

Far-field Green's function for point dipoles near a planar interface
The electric field at a point, r, due to a dipole (µ µ µ) located at r 0 in a homogeneous and isotropic medium can be expressed as follows: where ↔ G 0 (r, r 0 ) denotes the Green's function.When the electric dipole is near a planar interface, apart from the primary Green's function, ↔ G 0 (r, r 0 ), we also have a reflected Green's function, ↔ G r (r, r 0 ), due to the reflections off the layered sensor.Hence the overall Green's function that defines the field in the upper half space is the sum of these two functions such that: where far-field Green's functions in spherical coordinates are given in Appendix D. Hence, the far-field electric field in spherical coordinates can be expressed as follows: For the sake of brevity, we skipped the intermediate steps of the derivation.The reader is referred to reference [12] for the details.
Note that the dipole moment can be formulated as follows: where E t is the total electric field incident upon the nanoparticle (see Section 2.2), and the polarizability tensor, ↔ α α α, for a spherical nanoparticle is as follows: As the sub-wavelength spherical particles scatter as electric dipoles [12], we can use the quasi-static approximation given in Equation 2 for particle polarizability (α).However, for a more accurate model, it is imperative to correct this quasi-static approximation with electrodynamic corrections, which are also known as radiative damping correction (c rad ), and dynamic depolarization corrections (c dep ) [13].The radiative damping stems from the spontaneous emission from the induced dipole moment, which results in broadening in the resonance spectrum; whereas the dynamic depolarization is due to finite r/λ ratio, leading to a red shift in the plasmon resonance with increasing size, which is more pronounced in resonant metallic nanoparticles [13].Therefore, the corrected polarizabilities used in the model for the dielectric and metallic particles are as follows: where k = 2π/λ , and α in Equation 14 is replaced by α d and α m are for non-resonant dielectric nanoparticles and resonant metallic nanoparticles, respectively.Note that within the context of this study, we limit our discussion to modeling of spherical nanoparticles.However, the polarizability calculations can also be extended to model scattered field components from asymmetric particles such as spheroids within the limits electrostatic approximations.

Image of a point dipole near a planar interface
In this section, we formulate the image of the dipole field in the detector plane through a two lens system as shown in Fig. 5 below.We follow the same steps as in section 2.3.
The image of a point dipole in the spherical coordinates can be expressed as given below.Note that the details of this formulations can be found in Appendix E.

Ẽu
p (θ , φ )e jη p e jk 2 w e − jζ p dφ dθ (16 where Next, we calculate these integrals for x-, y-and z-oriented dipole moments.Note that the integration over φ can be calculated analytically using Bessel function closure relations, which, along with the electric field formulations in the image plane for x-, y-and z-oriented dipole moments, can be found in Appendix F. For brevity, we skip the intermediate steps of the derivations.For more details on the derivations, the reader is referred to reference [12].
Finally, we write the overall response observed at the detector as the interference of the reflected and scattered fields as follows:

Results and discussion
In this section, we first study the effects of sensor characteristics, dipole height, and nanoparticle type (dielectric or metallic) on the interferometric signal using the model outlined in the previous section.Next, we provide a comparison between the physical model simulations and experimental results for dielectric and gold nanoparticles.Of particular interest is the effect of the axial difference between the physical structure (nanoparticle on the sensor surface) and the minimum beam waist position (or focus) of the optical imaging system.In the following discussion, we define defocus as the axial shift of the top sensor surface from the focal plane of the objective lens.Experimental and theoretical studies demonstrate that the additional phase introduced by the axial shift manifest itself in a particle-specific signature of nanoparticle response.Thus defocus responses allow for distinguishing dielectric and metallic nanoparticles which may otherwise have identical response at a particular axial position.In addition, experimentally, defocus scans are of great importance to maximize visibility for all the particles, as they can be at different heights, and/or they can be of different type or size.Below in Fig. 6, we present an illustration of two scenarios: i) the focal plane of the objective corresponds to sensor surface, ii) the sensor is moved by an offset (z d ).In the simulations and experiments, the following parameters are used: NA = 0.8, Note that the nanoparticles are assumed to be resting on the sensor surface except for the case in Section 3.2.In addition, the wavelength selection is based upon the fact that the spherical gold nanoparticles that range from 30 nm to 60 nm in diameter exhibit surface plasmon resonances between 523 nm and 536 nm.

The effects of the nanoparticle size on its interferometric response
Nanoparticle size directly affects the interferometric signal as can be observed from Equation 2-the increase in the radius of the spherical nanoparticle results in scattered field enhancement.In the light of this observation, we simulate the nanoparticle responses for various sizes for both dielectric and metallic particles in Fig. 7.
The nanoparticle response is defined in terms of the normalized intensity (contrast) that is calculated by dividing the total intensity response by the reference signal (I t /I r ).The nanoparticles can exhibit strong negative or positive peaks in their defocus responses depending on their type and axial location.As can be seen from Fig. 7 above, for a given range of defocus, the contrast of the nanoparticle response increases with increasing size, and for a given type of particle, our model can be used as a forward model to discriminate particles based on size.

The effects of the layered sensor on the nanoparticle response
As the phase between reference field and the scattered field is controlled by the thickness of the first layer of the sensor as illustrated in Fig. 6, and also in Fig. 13 in Appendix B, (the layer characterized with ε 2 , µ 2 ), the interferometric signal of a nanoparticle depends upon this thickness (see Equation 1).In Fig. 8, we present the simulated defocus curves for both a dielectric nanoparticle with 60 nm radius and a gold nanoparticle with 30 nm radius for a range of thicknesses (d).As can be seen from Fig. 8, for a given range of d, the nanoparticle response (contrast) is enhanced the most for d = 100 nm in the case of a non-resonant dielectric particle, whereas for a metallic particle, the increase in d first enhances the negative contrast up to d = 80 nm and after that this response starts to get reversed, and the positive contrast is enhanced the most around d = 120 nm.

The effects of the nanoparticle height with respect to the sensor surface on its interferometric response
The axial position of the nanoparticle (h) affects the observed interferometric signal, where it factors in as additional phase term as can be seen in Equation D.2.The nanoparticle responses for different axial positions are given for dielectric and metallic particles in Fig. 9.
#258126 The discussion on dipole height becomes particularly relevant when carrying out interference enhanced nanoparticle imaging in solution, since nanoparticles exhibit Brownian motion in both axial and lateral directions.The elevation of the nanoparticles from the surface of the sensor alters their interferometric signatures as can be closely observed from Fig. 10 below.While the 60 nm spherical gold nanoparticle resting on the surface exhibits a strong negative contrast, when it is elevated by 40 nm, its response changes to a strong positive peak observed in a different defocus plane.This phenomenon necessitates the data acquisition in stacks of various defocus planes (z-stack images) so as to detect all of the gold nanoparticles in a field of view.In other words, the z-stack image acquisition renders the digital detection of nanoparticles that are at different heights with respect to sensor surface.In Fig. 11, we show the responses of gold nanoparticles for three different defocus planes.Regions 1 and 2 indicate signal levels that are within ±%1 and ±%2 contrast regions, respectively.When the signal from the gold nanoparticle labels fall in these regions, labeled particle #258126 detection becomes challenging, as the impurities such as dust particles in the background could also be the source of this level of signal depending on the assay quality, which validates the need for z-stack image acquisition.For instance, the blue dashed curve lying in Region 2 corresponds to ~37% of 60 nm gold nanoparticles that would go undetected when only imaged at z d = −500 nm, assuming a uniform height distribution between 0 and 100 nm for the nanoparticles.Fig. 11.Spherical gold nanoparticle (r = 30 nm) responses with respect to h for three defocus planes: z d = -500 nm, 0 nm, and 500 nm.Region 1 and Region 2 correspond to signal levels that fall between ±%1 and ±%2 contrast regions, respectively.Furthermore, it has been shown that the capture efficiency of the probes can be enhanced when they are elevated from the layered surface, and when these probes are used in conjunction with secondary probes tagged with metallic particles for signal enhancement, the variation in the nanoparticle heights, which can vary between 10 -30 nm [14], have to be taken into consideration.Therefore, in this case, z-stack image acquisition becomes even more crucial for accurate digital detection and counting of gold nanoparticle labels.

Comparison between the physical model and experimental findings
In this section, we benchmark our physical model against the experimental data.For a dielectric particle, we use spherical silica beads with 60 nm radius, and for a metallic particle, we use spherical gold beads with 30 nm radius.As for the sensor, we use thermally grown silicon dioxide atop silicon wafer.The wafers with 100 nm oxide thickness were purchased from Silicon Valley Microelectronics.The sensors are plasma-ashed to negatively charge the surface, and then, the silica and gold beads in DI solution are spin-coated on two different sensor surfaces.The electrostatic attraction immobilizes the beads onto the surface.For the metallic beads, we etch the oxide layer of sensor down to 60 nm for enhanced negative contrast (see Fig. 8), however, since the dielectric nanoparticle response is enhanced the most when d = 100 nm (see Fig. 8), for the silica beads, we directly use the sensor without etching.The experimental results along with the theoretical model simulations are shown in Fig. 12 below.As can be seen from Fig. 12, the experimental data follows the defocus trends that our model predicted for both dielectric and metallic spherical nanoparticles.The slight deviations of the experimental data from the theory can be attributed to the size variations of the nanoparticles, potential misalignments and vibrations in the experimental setup, and the spectral width (∆λ ≈ 40 nm) of the illumination source.

Conclusion
In this paper, we presented a detailed theoretical study of interference enhanced nanoparticle imaging and characterization for a common-path interferometer.Our theoretical study can be adapted to model other types of interferometry based systems as well.Furthermore, we provided a comprehensive discussion on the key factors that affect the interferometric responses of the spherical nanoparticles, which can be of dielectric or metallic type.We show that the interferometric defocus response of a nanoparticle depends upon sensor characteristics, nanoparticle size and location of the particle with respect to the sensor surface, and is material-specific.These dependencies, when taken into account in the light of a theoretical model as presented here, can be an asset for nanoparticle characterization, as well as improved nanoparticle visibility, especially for low-index particles.Additionally, the physical model outlined in this study rigorously accounts for the factors that determine the interferometric signal, and is of great utility for accurately characterizing nanoparticles based on their interferometric signals.Furthermore, there are implications for practical systems for interferometric detection.Specifically, for common-path interferometric systems utilizing a layered reference reflector as described here, focal plane is not well-defined and accurate autofocusing maybe a significant challenge.As we have shown in this theoretical treatment, image acquisition at multiple focal planes provides unique signatures for nanoparticles on the sensor surface without precise focusing, thus relaxing the instrumentation requirements.
Then taking the Fourier transform of (A.2) will yield the classical angular spectrum representation of fields as follows: Ẽi,∞ (k x , k y )e j(k x x+k y y) e − jk z1 z dk x dk y k z1 (A.3) Expressing the field in spherical coordinates using the following transformations: θ )sin(θ )e jη x e − jk 1 zcosθ dφ dθ (A.5) where θ max = sin −1 (NA/n 1 ).
Since each plane wave is treated separately in Köhler illumination, we can also write the incident focal fields due to each plane wave component E i,m as follows: ,m e − jk 1 zcosθ m (A.6) where As can be seen from Fig. 4, the reflected field first gets refracted by the objective lens ( f 1 ) then it propagates to the tube lens ( f 2 ) and gets refracted again.Following the geometrical optics, the field at the far-field image plane can be written as follows: Now we can find the reflected field in the near field of image space by carrying out a nearto-far field transformation (as in Equation A.2) as follows: Thus in the image plane the reflected field is: From geometrical optics, as illustrated in Fig. 4, we observe that θ and θ u are related as follows: where M denotes the magnification.Furthermore, for typical imaging systems with high magnification (M 1), we can further make the following approximations based on trigonometric identities and Taylor series expansion: Note that when cosθ u is in the phase term, we use its approximation as given in Equation (C.8), and when it is in the amplitude term, we simply use cosθ u = 1, and sinθ u = 0.In addition, to further simplify the equations, we carry out the following spherical coordinate transformations (similar to Equation A.4): For z-oriented dipole, its electric field in the image plane is: where 1 + Φ 1 ]J 0 (η p )e − jζ p dθ (F.8a) 1 sin 2 θ cosθ J 1 (η p )e − jζ p dθ (F.8b) 1 ]J 2 (η p )e − jζ p dθ (F.8c)

Fig. 6 .
Fig. 6.Defocus scan: (a) when the sensor surface is in the same plane as the focal plane of the objective (z d = 0 case), (b) when the sensor is displaced by z d in the negative direction.

Fig. 7 .
Fig. 7. Spherical nanoparticle responses with respect to radius for silica and gold particles: (a) Simulation parameter (shown in red), (b) Silica particle responses, and (c) Gold particle responses.