We numerically analyze PiFM’s lateral and vertical (subsurface) imaging performance in the visible and IR regimes. The lateral spatial resolution and subsurface imaging capabilities are limited by the field spatial confinement near the tip apex, which is directly proportional to the excitation wavelength. In addition, we show that near-field optical force exerted on the tip due to sample molecular resonance is indeed in the detectable range. Moreover, driving sample on (off) resonance reveals high (low) contrast. The strength of the optical forces is assessed for metal (gold), polymers (Polystyrene and Polymethylmethacrylate), and solid (SiC). By increasing tip-coating thickness from 5 nm to 35 nm, the gap-field enhancement decreases to about 40%. In IR, force spectrum over an absorption band is predominantly following the real part of the polarizability, as predicted by dipole-dipole approximation.
© 2017 Optical Society of America
Atomic Force Microscopy (AFM) [1, 2] is a powerful technique for profiling surface features with atomic resolution. However aside form chemical force microscopy , Kelvin Probe Force Microscopy  and Magnetic Resonance Force Microscopy , AFM’s are unable to identify chemical species at the nanoscale. Recently, a new high resolution nanoscopy principle was introduced that converts an AFM into a chemical imaging and spectroscopy device based on measuring the local optical dipole-dipole interaction forces between tip and sample . This technique – image force microscopy or photo induced force microscopy (PiFM) – has evolved to the point where dipole-dipole force interactions have been used to measure the Raman effect, perform pump-probe excited state spectroscopy, map electromagnetic fields and plasmon oscillations all on the nano scale [7–10].
AFM’s also cannot detect buried features. While, surface chemical information has been explored with nanoscale resolution using scanning near-field optical microscopy (SNOM, Apertuless SNOM – or sSNOM) [11–13], depth investigation (or subsurface imaging) of chemical composition remains mostly unexplored. Ideally, subsurface imaging techniques should not only be able to image features at high resolution, but also be able to identify those features through their spectral signatures.
In this paper, the contrast and imaging performance of PiFM for surface and sub-surface imaging is investigated using numerical analysis techniques. In section 2, we introduce our basic simulation set up and parameters used. Our simulations are first validated against standard, closed form solutions for optical forces on an isolated nano sphere obtained from Mie theory. We then validate our modeling for electromagnetic field calculations by comparing against published work. Following these validations, in section 3, we calculate the forces between tip and sample in PiFM under different conditions: the effect of tip-coating thickness and sample thickness are investigated. The force spectrum over absorption bands in the IR and visible region are shown in section 4. Subsurface imaging and contrast performance are studied in section 5. In section 6, we discuss our results. Finally, in section 7, we provide some brief concluding remarks.
2. Numerical simulation
2.1 Brief theory of near field optical forces in PiFM
In SNOM and sSNOM optical interactions in the near field are detected in the far field. In PiFM, near field optical interactions are detected as local forces in the near field. Near fields are typically many orders of magnitude stronger than the far fields. Also, we know that force detection can be accomplished with extreme sensitivity using an AFM . We therefore envision PiFM to become a very powerful tool for optical nanoscopy/spectroscopy. The detected force in PiFM is related to the polarizability of both tip and sample; if the tip polarizability response is known, then chemical information can be extracted about the sample. Furthermore, because the probed forces are spatially confined to the nanoscale, PiFM is capable of providing spectroscopic information with extremely high spatial resolution - well below 10nm.
In earlier work, a dipole-dipole coupling approximation was used to analyze the tip-sample mutual coupling forces in PiFM upon optical excitation [9, 15, 16]. While this approximation is likely to breakdown at tip-sample separations less than 5 Å, it can still be useful for studying the underlying physics and trends of the near-field tip-sample force interaction. To recap this work in brief, we considered the tip and probed sample to be two optical dipolar spheroids with dipole moments pt, and ps respectively. In the electrostatic approximation, the time averaged optical gradient force calculated at the tip or sample particle is proportional to the strength of pt and ps, and can be written asEq. (1). If we consider the tip dipole to be at the origin of our coordinate system, then the sample dipole is located at a radial distance r from this origin. The optically induced dipole moment of the tip or the sample is proportional to the local electric field at pt or psAppendix A). Therefore, chemical information can be extracted by studying the force spectrum behavior over the optical absorption band of the sample.
In this paper, we will be using detailed 3D electromagnetic simulations to understand the contrast and imaging performance in PiFM. Maxwell stress tensor () formulation will be employed to calculate <Fopt> on the tip under different conditions, where <Fopt> can be written as10]. The can be expressed as17]. In Appendix A, we compare our numerical results with those obtained using the simple closed form dipole-dipole analysis described earlier.
2.2 Problem setup
Finite Element Method (FEM) with frequency domain approach is utilized to analyze the tip-sample field components using a three-dimensional mesh. A non-uniform meshing based on free tetrahedral elements is employed, where dense mesh elements are in the vicinity of tip-sample region with minimum element size of 1.5 Å; the maximum element size is one tenth of the wavelength of the incident beam, which is required for converged solutions. The dielectric constants are assumed to be uniform within the boundary of the tip, sample, particle and substrate. The far-field boundary condition is modeled using a perfectly matched layer (PML). The setup is simulated using COMSOL Multiphysics software. See Appendix B for more details about simulation setup and meshing distribution.
In Fig. 1, the silicon (Si) tip with 10° half angle is terminated by a spherical section of radius 100 Å and separated from the sample-coating film by 5 Å gap. The particle, shown in Fig. 1(b), has radius 50 Å. The substrate, (not shown in the figure) is Si and is assumed to be semi-infinite. The setup is irradiated by a z-polarized standing wave composed of two counter propagating monochromatic plane waves traveling in + x and -x directions each with electric field amplitude (E0) of 106 [V/m]. The surrounding medium is air. These simulation parameters are fixed throughout this paper.
First, we simulate the configuration shown in Fig. 1(a) to investigate the effect of tip-coating thickness (Tth) and sample-coating thickness (Sth) on the gap electric field enhancement (Eenh = |Eloc/Einc|) and the time averaged force <Fopt> exerted on the tip. Then, the electric field penetration along b- - b’ segment and the electric field spatial confinement along a- - a’ segment is studied. Second, by setting Tth and Sth to 50 Å and 300 Å respectively, the force spectrum over an absorption band is investigated for both IR and visible regions. Finally, vertical and lateral subsurface imaging is investigated in Fig. 1(b). The particle is placed within the sample at a depth d, and is swept either along direction 1 or 2 as indicated in the figure. In Fig. 1(b), the deposited film is Polymethylmethacrylate (PMMA) with 300 Å thickness; it is the host medium for the particle. The particle material has been considered as Polystyrene (PS), gold (Au), and hexagonal silicon carbide (SiC). The dielectric functions have been taken from [18–22].
2.3 Bench marking against known closed form expressions
Before simulating any problem using Finite Element Method, our model must be validated. Our simulation will be validated by reproducing published data in prior publications for field distributions near tips and by implementing Mie scattering analysis simulations of optical forces and fields on particles and comparing the results to closed form analytical solutions.
Mie theory describes the scattering and absorption processes of an incident electromagnetic plane wave by a homogenous sphere. A sphere of 1000 Å and refractive index surrounded by air has been modeled to extract the extinction and absorption cross sections. The extinction cross section () has been analyzed and compared to the analytical Mie solution done by Matzler code . Our COMSOL simulation gave excellent agreement with Mie solutions. Since Maxwell stress tensor (MST) formalism was employed for all force calculations, validating MST equations used by COMSOL was essential. Particles exposed to an incident field experience a radiation force along the incident direction. This force is proportional to the pressure cross section which in turn is proportional to the extinction and scattering cross sections. Once the pressure cross section is determined, the radiation force can be calculated. Radiation force must equal the integral of MST over the particle surface. Figure 2 shows a comparison between the Mie solution and our COMSOL results. The MST integral results are identical to the radiation force calculated by COMSOL and the Matzler code. These validations enhanced the confidence in our COMSOL analysis.
To add another level of confidence to our simulations, the simulated configuration done by  was repeated (Fig. 3). In this configuration, solid silver tip is irradiated by a monochromatic plane wave of wavelength 632.8nm, at an incident angle of π/3. The tip is surrounded by air, and modeled as a solid metal cone terminated by sphere of radius 250 Å. Dielectric constants are the same as in . The resulting field distribution near tip apex is normalized to the incident field (Einc) to yield the field enhancement Eenh shown in Fig. 3. The results are in excellent agreement with published data . Having validated our simulation geometry using Mie analysis and previously published data, we are now ready to simulate the various configurations mentioned in section 2.
3. Effect of coating materials and thickness
In this section, we study the effect of Tth and Sth on Eenh in the tip-sample gap and on <Fopt> exerted on the tip. Simulation setup is as shown in Fig. 1(a); this simulation is done in the IR regime where the wavelength is 10 μm, and the Si substrate is coated with PMMA of different thicknesses. Figure 4 (left column) shows Eenh as a function of Tth, where each curve represents specific Sth. Note that the thinner the Sth, the higher the Eenh, which is ascribed to the tip-substrate optical interaction. Note that, no further field enhancement at Sth beyond 60 nm can be seen, which suggests that the tip-substrate optical interaction saturates at this thickness, but Eenh is still present. We also note that the smaller the tip radius, (which corresponds to thinner Tth), the higher the Eenh. One should not jump to the conclusion that since Eenh is decreasing, <Fopt> should also decreases. In fact, Fig. 4 (right column) shows the opposite situation; although Eenh decreases, <Fopt> increases with tip size, corresponding to thicker Tth. We will elaborate on these points in the discussion section. Three tip-coating metals have been studied: silver (Ag), platinum (Pt), and gold (Au). In all three cases, Eenh is found to be roughly the same in the mid infrared.
The lateral resolution of PiFM, can be ascribed to the steep electric field gradient near tip apex. In the next simulation, we fix Sth to 300 Å, assign tip-coating material as Au, and simulate Fig. 1(a) at different Tth. Figure 5(c) shows the electric field calculated along b - - b’ segment; it decreases at a faster rate for smaller Tth. Electric field spatial confinement near the tip apex has been extracted along the a - - a’ segment and plotted in Fig. 5(b). Clearly, small Tth results in higher spatial confinement. We see that thicker Tth enhances the depth imaging capabilities while thinner Tth increases the surface imaging resolution.
4. Optical force spectrum at an absorption band
The <Fopt> spectrum over an absorption band, (in the case of the electrostatic approximation), is dependent on both tip and sample polarizabilities. Since the imaginary part of Au polarizability is negligible in the IR region (the tip being assumed to be spheroid), one may attribute the force behavior over an IR absorption band to predominantly follow the real part of the sample polarizability. In the visible region, however, we cannot attribute <Fopt> to either the real or imaginary part of tip or sample polarizability because both contribute to <Fopt>. To validate the former statements, Fig. 1(a) is simulated to investigate the <Fopt> spectrum in IR absorption band. The sample materials are chosen as PMMA in one simulation and as Polystyrene (PS) in another. The incident wavelengths cover the absorption band of PMMA at 1730 cm−1 and of PS at 700 cm−1. Figure 6(a) and 6(b) show that the force spectrum takes a distorted S shape, indicating that the <Fopt> predominantly follows the real part of sample polarizability—as expected from dipole-dipole electrostatic theory. In the visible, <Fopt> spectrum is governed by the dielectric functions due to the plasmon effect. Figure 1(c) is simulated where an Au particle is located at the surface i.e. d is zero. Since the particle radius is 50 Å, we used the nanoparticle dielectric function for the Au particle and bulk dielectric function for the tip. Interestingly, the force spectrum shows three plasmon peaks, which can be explained by hybridization effects [25–28]. Nanoshell plasmonic spectrum is governed by the interaction of the inner and outer induced charges at the interfaces of the metal shell (tip coating layer). These induced charges oscillate symmetrically (bonding) at lower energy and anti-symmetrically (ant-bonding) at higher energy. Under weak interaction, these oscillation energies overlap showing one plasmon peak. However, under strong interaction, the energy difference between these oscillations increases showing two plasmon peaks. Depending on the thickness of the shell (coating thickness), the plasmon peaks can be tuned . The third plasmon peak is attributed to the particle plasmon which oscillates at higher energy due to damping (the averaged path of the electron before colliding is smaller than the mean free path). Figure 6(c), proves that the force spectrum in this case cannot be simply attributed to either real or imaginary part of αs.
5. Subsurface imaging and contrast performance
5.1 Vertical imaging
To assess the vertical imaging capabilities of the PiFM, Fig. 1(b) is simulated. The particle is scanned along direction 2 (cf. Fig. 1(b)) by varying d within the sample (PMMA). The excitation wavelength is at the max resonance of the Au particle (based on Fig. 6(c)). In a separate simulation, a PS particle is excited at the absorption band of 700 cm−1. Equation (4) is employed to calculate <Fopt> exerted on the tip as a function of particle position. Because the tip-sample dipole interaction is dominated by the dipole of the particle (PS or Au), the force exerted on the tip varies with the particle location. The closer the particle is to the tip, the higher the force. We note that the force in Figs. 7(a) and 7(b) decreases exponentially as the particle is swept along direction 2. For a given tip radius, the rate of decrease in force with particle depth depends on the wavelength of the exciting beam - the force decreases more rapidly with depth in the visible than in the IR. In both cases (Au and PS), the force effect of the buried particle on tip is easily detectible and in the pN range. Lastly, the particle was swept along direction 1 at d = 0, 2, 5 nm, and 10 nm to study the contrast defined as Fcont = <Fopt > - <Fopt, min>. Figure 7(c) shows that at depth d = 5 nm, the Au particle contrast disappears, which is attributed to the rapid electric field gradient shown in (a). For PS particle, the field gradient is smaller leading to a measurable contrast even near d = 5 [nm] (Fig. 7(d)). Therefore, IR-PiFM subsurface imaging capability is improved. Note that in both cases the line width of the Lorentzian-like lines is expanding as the particle dives into the sample.
5.2 Lateral imaging performance
By driving the particle on (or off) resonance we can estimate spectral contrast high (or low) as the tip laterally scans the sample. This has been demonstrated for PS, Au, and SiC particles in Fig. 8 and Table 1. The figure shows that at resonance the tip feels a stronger force than off resonance. Therefore, if the noncontact (or tapping mode ) is used in PiFM, the cantilever will experience stronger attractive forces on resonance than off resonance as the tip is scanned across the particles. This will result in a detectable change in the second frequency resonance (responsible for PiFM signal), and the particles will appear as a bright spot in the PiFM image. The three particles were chosen so that PiFM lateral imaging performance could be assessed in the visible, where the plasmon effect is excited, and in different IR regions. We note that in the IR, high contrast is observed in the case where the particle has refractive index (k) less than one in PS and higher than one in SiC. The full width half maximum of the lorenztian-like curves is not simply the convolution of the tip profile with that of the particle. Because the effective field is confined laterally to a region much smaller than the tip apex, a much sharper image is obtained.
Modeling the tip-sample interaction by spheres  excludes substrate and lightening rod effects. The gap-field enhancement will be significantly underestimated, which in turn lowers the calculated optical forces by orders of magnitude. Figure 4 clearly shows the substrate effect (left column) on the gap-field enhancement. In addition, Bohn et al. numerically analyzed the antenna effect and emphasized that gap-field enhancement is strongly dependent on the lightening rod effect . Therefore, care must be exercised to include all the aforementioned effects in order to get a valid result out of any numerical modeling of near-field optical forces. In Fig. 4, as Tth increases, the tip apex becomes blunter. Since the incident field is polarized along the tip axis, the maximum charge density will be at the tip apex. Therefore, the blunter the tip apex the lesser the density, which lowers the Eenh in the gap . Although Eenh decreases as Tth increases, <Fopt> increases. This is due to the increase in the self-consistent fields (incident plus scattered fields), which are captured in the MST expression (Eq. (5).
F. Huang et al. experimentally mapped the electromagnetic near-field distribution on a glass substrate by measuring the tip-substrate optical force interaction, leading to the conclusion that optical forces are clearly detectable . A. Ambrosio et. al. were able to obtain pure refractive index contrast imaging in the visible regime on a dielectric nanostructure that contains a 2D array of titanium dioxide (TiO2) with refractive index of 2.428 embedded in a matrix of electron beam resist with refractive index of 1.56 . In this paper, we show by numerical simulation that optical forces are indeed in the detectable pN range. In the near-field regime, the gradient force dominates – it depends on the real part of the product of tip and sample polarizabilities, namely Re [αt α*s]. In section 4, the force behavior over an absorption band in the IR region for PMMA and PS shows that the optical forces are predominantly dispersive. This can be attributed to the negligible-imaginary part of the tip polarizability. Dipole-dipole approximation qualitatively agrees with the numerical analysis. However, tip sample interaction can be described more quantitatively by the extended dipole method . For example, A. Cvitkovic et. al. were able to implement material-specific imaging by sSNOM in IR regime, where Au and PS nanoparticle are dispersed and immobilized on Si substrate . In addition, dyadic green functions have been incorporated to quantitatively describe tip sample interaction, and PIFM signals have been studied with isotropic and anisotropic polarizability of the tip . The analysis presented in this letter is the first full wave numerical simulation of photo induced forces as applied to microscopy. We hope it will guide further experiments in this field.
We have shown for the first time that PiFM subsurface imaging is a viable possibility. The resolution of PiFM is not limited by tip radius but rather by the spatially confined electric field enhancement localized to few nanometers near the tip apex. Because this confinement is wavelength dependent, the full width at half maximum (FHWL) of the field intensity near the apex shrinks in the visible region while it expands in the IR region. PiFM reveals higher subsurface depth imaging capability in the IR since the field penetrates deeper due to the intensity expansion near the apex. Figure 7 illustrates force gradients, in both visible and IR regions, calculated as a function of particle position, by sweeping the particle vertically within the sample coating. Note that the force gradient decays faster in the visible than in the IR region, which implies that the particle image contrast will disappear faster in the visible (Fig. 7(c)) than the IR (Fig. 7(d)). This can be explained using a simple Hertzian dipole (HD) model (Fig. 9). The force gradient decay rate is essentially due to the electric field gradient. The near-field electric field equation of HD in spherical coordinates can be approximated as:Eq. (6), we immediately see that the near electric field decays faster at smaller wavelengths matching the results shown in Fig. 7; subsurface imaging performance is, therefore, improved using IR excitation. To further enhance the subsurface imaging capability, tip apex radius should be increased by increasing its coating thickness. Figure 5 shows the electric field gradient measured vertically through b - - b’ segment. With larger tip radius, electric field penetrates deeper into the PMMA, allowing for deeper subsurface imaging.
High image contrast is achieved by driving the particle at resonance while low contrast is achieved by driving it off resonance as shown in Fig. 8. For Au particle, the contrast is due to the plasmon effect. In IR, high contrast has been observed for particle with extinction coefficient less than one (i.e. PS), and higher than one (i.e. SiC). The real part of the refractive index (n) of SiC is fixed for on and off resonance; thus, contrast is purely due to imaginary part (k). Note that in all cases the calculated forces are in the pico newton range.
Finally, we compare PiFM and sSNOM. The contrast signal in both PiFM and sSNOM originates from the same local electric field distributions. However, there are fundamental differences that distinguish PiFM from sSNOM. PiFM is a force sensing technique with sensitivity limited only by the cantilever sensitivity, which reaches 0.1 pN in ambient condition and 10−18 N at low temperatures in UHV conditions. Note that, in PiFM, near field forces are sensed in the near field leading to a background free signal while the sSNOM technique detects the scattered signal from the tip apex, the cantilever and the sample. The sensitivity of sSNOM is limited by the intensity detection scheme, and the wanted signal is typically overwhelmed by the unwanted signal (or background). This background signal has been largely reduced by means of heterodyne and pseudoheterodyne detection methods [36, 37]. Since sSNOM measures intensity, local polarization is difficult to recover. This polarization is perfectly recovered by PiFM technique i.e. sensed forces directly report the dipole orientation. Furthermore, the optical force detection technique is sensitive to local field gradients and exhibits a z−4 dependence while the local scattered field exhibits a z−3 dependence (where z is the tip-sample distance). This gives PiFM a higher resolution in comparison to sSNOM. While phase information on the tip- sample interaction is missing in PiFM technique, the sSNOM technique can perfectly recover phase information. Both techniques are capable of single molecule detection although neither has demonstrated this so far.
Modelling the tip and sample by two interacting spheres excludes the effect of the substrate as well as the lightening rod effect. The absence of these effects underestimates the gap-field enhancement which in turn reduces optical forces by orders of magnitudes. However, after a certain sample thickness, gap-field enhancement saturates and is dominated by the sample thickness and not by the substrate. Moreover, the tip-coating thickness effect must also be considered; the enhancement reduction as coating thickness increases to 35 nm is roughly 40%. In the IR, coating the tip with Pt, Au, or Ag does not make a significant difference on the gap-field enhancement. Force behavior has been assessed over absorption bands, and the results show dispersive behavior at PS and PMMA absorption bands, namely 700 cm−1 and 1730 cm−1 respectively. However, force behavior is absorptive when Au particle is interacting with the tip over the visible region. Numerically, near-field optical forces developed in the vicinity of the tip apex due to tip-sample optical interactions is shown to be detectable and in the pN range. PiFM allows embedded particles to be imaged laterally with higher spatial resolution in the visible region than in the IR, ascribed to the spatial confinement of the gap-field enhancement at the tip. Vertical imaging range is increased using IR excitation, as the force gradient is less steep than in the visible. In addition, increasing tip coating thickness allows electric fields to penetrate deeper into the sample, which further enhances the subsurface imaging range.
Appendix A dipole approximation
Assuming linear relationship between dipole moment and the incident beam, tip and sample have been replaced by point dipoles with dipole moment having the same time dependency as the incident beam. The z-component, averaged optical force can be written as in Eq. (7) and COMSOL-built-in equations are normalized and compared. Figure 10 shows excellent agreement.
Appendix B COMSOL simulation
Truncating the simulation domain and preventing incident beam from reflecting can be achieved by terminating the domain by PML with suggested thickness (based on COMSOL user guide) of λ/2, where λ is the wavelength of the incident beam, and assigning the outer boundary of PML to scattering boundary condition (SBC). However, PML costs memory. Therefore, PML thickness should be optimized to cause sufficient attenuation, which is possible with thickness less than λ/2. Figure 11 shows the domain in detail.
A nonuniform tetrahedral mesh has been implemented over the domain with dense mesh elements near the regions of interest. We have maintained 4 meshing points, as shown in Fig. 12(d) in the tip-sample gap to enhance the accuracy of the results. The minimum element mesh size is 0.15nm. For Au coating layer, boundary layer mesh must be used to resolve skin depth, this is shown in Fig. 12(c).
NSF Center for Chemical Innovation, Chemistry at the Space-Time Limit (CaSTL) under Grant No. CHE-1414466; Saudi Arabian Cultural Mission (SACM).
We would like to acknowledge support of the NSF Center for Chemical Innovation, Chemistry at the Space-Time Limit (CaSTL) under Grant No. CHE-1414466, and the Saudi Arabian Cultural Mission (SACM) for financial support
References and links
1. G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Phys. Rev. Lett. 56(9), 930–933 (1986). [PubMed]
2. Y. Martin, C. C. Williams, and H. K. Wickramasinghe, “Atomic force microscope–force mapping and profiling on a sub 100‐Å scale,” J. Appl. Phys. 61, 4723–4729 (1987).
3. C. D. Frisbie, L. F. Rozsnyai, A. Noy, M. S. Wrighton, and C. M. Lieber, “Functional Group Imaging by Chemical Force Microscopy,” Science 265(5181), 2071–2074 (1994). [PubMed]
4. M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramasinghe, “Kelvin probe force microscopy,” Appl. Phys. Lett. 58, 2921–2923 (1991).
5. J. A. Sidles, “Noninductive detection of single‐proton magnetic resonance,” Appl. Phys. Lett. 58, 2854–2856 (1991).
6. I. Rajapaksa, K. Uenal, and H. K. Wickramasinghe, “Image force microscopy of molecular resonance: A microscope principle,” Appl. Phys. Lett. 97(7), 073121 (2010). [PubMed]
7. I. Rajapaksa and H. Kumar Wickramasinghe, “Raman spectroscopy and microscopy based on mechanical force detection,” Appl. Phys. Lett. 99(16), 161103 (2011). [PubMed]
8. J. Jahng, J. Brocious, D. A. Fishman, S. Yampolsky, D. Nowak, F. Huang, V. A. Apkarian, H. K. Wickramasinghe, and E. O. Potma, “Ultrafast pump-probe force microscopy with nanoscale resolution,” Appl. Phys. Lett. 106, 083113 (2015).
9. F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. K. Wickramasinghe, “Imaging Nanoscale Electromagnetic Near-Field Distributions Using Optical Forces,” Sci. Rep. 5, 10610 (2015). [PubMed]
10. T. U. Tumkur, X. Yang, B. Cerjan, N. J. Halas, P. Nordlander, and I. Thomann, “Photoinduced Force Mapping of Plasmonic Nanostructures,” Nano Lett. 16(12), 7942–7949 (2016). [PubMed]
11. B. Hecht, B. Sick, and U. P. Wild, “Scanning near-field optical microscopy with aperture probes: Fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
12. F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution,” Science 269(5227), 1083–1085 (1995). [PubMed]
13. H. K. Wickramasinghe and C. C. Williams, “Apertureless Near-Field Optical Microscope,” U.S. Patent No. 4 947 034, August 1990.
14. K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, and D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” J. Microelectromech. Syst. 9, 117–125 (2000).
15. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos Trans A Math Phys Eng Sci 362(1817), 719–737 (2004). [PubMed]
16. J. Jahng, J. Brocious, D. A. Fishman, F. Huang, X. Li, V. A. Tamma, H. K. Wickramasinghe, and E. O. Potma, “Gradient and scattering forces in photoinduced force microscopy,” Phys. Rev. B 90, 155417 (2014).
17. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University 2006)
18. W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared Properties of Hexagonal Silicon Carbide,” Phys. Rev. 113, 127–132 (1959).
19. O. N. Tretinnikov, “IR spectroscopic study of the effect of polymer nanofilm thickness on its surface density,” J. Appl. Spectrosc. 75, 64–68 (2008).
20. V. M. Zolotarev, B. Z. Volchek, and E. N. Vlasova, “Optical constants of industrial polymers in the IR region,” Opt. Spectrosc. 101, 716–723 (2006).
21. R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
22. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [PubMed]
23. C. Mätzler, “MATLAB Functions for Mie Scattering and Absorption,” (Institut für Angewandte Physik, 2002)
24. Z. Yang, Q. Li, F. Ruan, Z. Li, B. Ren, H. Xu, and Z. Tian, “FDTD for plasmonics: Applications in enhanced Raman spectroscopy,” Chin. Sci. Bull. 55, 2635–2642 (2010).
25. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization Model for the Plasmon Response of Complex Nanostructures,” Science 302(5644), 419–422 (2003). [PubMed]
26. D. Östling, P. Apell, and A. Rosén, “Theory for Collective Resonances of the C 60 Molecule,” Europhys. Lett. 21, 539 (1993).
27. J. W. Liaw, M. K. Kuo, and C. N. Liao, “Plasmon Resonances of Spherical and Ellipsoidal Nanoparticles,” J. Electromagn. Waves Appl. 19, 1787–1794 (2005).
28. Z. Jian, D. Xing-chun, L. Jian-jun, and Z. Jun-wu, “Tuning the number of plasmon band in silver ellipsoidal nanoshell: refractive index sensing based on plasmon blending and splitting,” J. Nanopart. Res. 13, 953–958 (2011).
29. E. Prodan and P. Nordlander, “Structural Tunability of the Plasmon Resonances in Metallic Nanoshells,” Nano Lett. 3, 543–547 (2003).
30. H. U. Yang and M. B. Raschke, “Resonant optical gradient force interaction for nano-imaging and -spectroscopy,” New J. Phys. 18, 053042 (2016).
31. J. L. Bohn, D. J. Nesbitt, and A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A 18(12), 2998–3006 (2001). [PubMed]
32. A. Ambrosio, R. C. Devlin, F. Capasso, and W. L. Wilson, “Observation of Nanoscale Refractive Index Contrast via Photoinduced Force Microscopy,” ACS Photonics 4, 846–851 (2017).
33. A. Cvitkovic, N. Ocelic, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 15(14), 8550–8565 (2007). [PubMed]
34. A. Cvitkovic, N. Ocelic, and R. Hillenbrand, “Material-Specific Infrared Recognition of Single Sub-10 nm Particles by Substrate-Enhanced Scattering-Type Near-Field Microscopy,” Nano Lett. 7(10), 3177–3181 (2007). [PubMed]
35. F. T. Ladani and E. O. Potma, “Dyadic Green’s function formalism for photoinduced forces in tip-sample nanojunctions,” Phys. Rev. B 95, 205440 (2017).
36. R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000). [PubMed]
37. N. Ocelic, A. Huber, and R. Hillenbrand, “Pseudoheterodyne detection for background-free near-field spectroscopy,” Appl. Phys. Lett. 89, 101124 (2006).