## Abstract

We demonstrate analytically and numerically that the detection of the spectral response of a single spherical Au nanoantenna allows one to map very small (down to 5·10^{−4} RIU) variations of the refractive index of an optically transparent sample. Spectral shift of the dipole local plasmon resonance wavelength of the nanoantenna and the spectral sensitivity of the method developed was estimated by using simple analytical quasi-static model. A pointed scanning probe based on fiber microaxicon with the Au spherical nanoantenna attached to its tip was proposed to realize the RI mapping method. Finite-difference time-domain numerical simulations of the spectral properties of the proposed probe are in good agreement with the theoretical quasi-electrostatic estimations for a radius of the nanoantenna not exceeding the skin depth of Au.

© 2014 Optical Society of America

## 1. Introduction

The state-of-the-art development and fabrication level of various nanodevices imposes stringent requirements on the microscopic analysis methods of their critical dimensions, chemical composition, topography and local optical properties [1]. The analysis of the structural and topographical sample properties can be efficiently implemented by using atomic force (AFM) and scanning electron microscopy (SEM) methods, while detecting small changes in refractive index (RI), which fully determines the local chemical composition and optical properties of the sample, is usually based on the interaction of the sample surface with a light field localized at nanoscale [2].

Light radiation is difficult to focus on the sample under study by means of far-field optics due to the fundamental diffraction limit. Optical nanoantennas exhibit higher performance in light control and localization at nanoscale [2], thus giving opportunities for practical applications of optical manipulation for nano-objects [3], nanolithography [4], excitation and detection of single-molecule fluorescence at 20-nm resolution [5], as well as subwavelength RI microscopy [6]. In the letter case the development of such a superresolution technique for refractometric studying of the optically transparent samples opens up broad prospective for novel practical applications in integrated optics for the characterization of various nanophotonic devices recorded in the photosensitive materials, in microbiology for nonfluorescent study of the biological samples, etc.

To control the “nanoantenna - sample” distance as well as to simultaneously map the local RI changes, the nanoantenna should be placed at the extremity of the scanning probe. With the aperture-type scanning near-field optical microscopy such a nanoantenna typically represents a localized light source, fabricated in the form of the through nanosized aperture on the end of a tapered optical fiber coated with an opaque metal film [7]. The described approach is widely used in subwavelength optical microscopy of different samples with a lateral resolution down to 50 nm, however, the low throughput of the nanoaperture greatly limits the sensitivity of a-SNOM in RI mapping at nanoscale. The use of resonant apertures with higher throughput (bowtie apertures [4,8], C-shaped apertures [9], and apertures surrounded by concentric grooves [10]) can improve the lateral resolution of the a-SNOM (down to 20 nm), however, is not able to provide the significant sensitivity increase in detecting small RI changes (12-% intensity signal change at extremely high RI variation Δn~2 refractive index units (RIU) [6]).

With the scattering-type near-field optical microscopy (s-SNOM) the metal tip of the AFM cantilever placed in the focal spot of the laser source and focused the radiation due to the “lighting rod effect” serves us such nanoantenna [11]. The radiation localized near the tip of such a “probe-like” nanoantenna is scattered owing to its interaction with the sample surface. The scattered signal intensity varies with the topographic and local optical properties of the surface [11] or even sub-surface [12] of the samples, which provides to s-SNOM with the possibility to detect sample chemical composition. However, the extended focal spot irradiated the tip adds the background contribution to the nanoantenna signal. This problem is solved by detection of nonlinear processes or by modulation techniques [13], which greatly complicate the practical realization of high-resolution refractometers based on the s-SNOM. Furthermore, in some applications the presence of the broad focal spot on the sample surface is highly undesirable [5]. It also should be noted that the metalized cantilever tip can be approximately considered us pointed nanoantenna owing to its infinite size does not support geometrical resonances [5]. Recently to avoid this drawback resonant probes (also referred to as “pointed probes”) based on the dipole or monopole nanoantennas were proposed and mapping and enhancement of the single-molecule fluorescence with a lateral resolution down to 20 nm were demonstrated [14,15]. However, the scattered radiation intensity from the resonant nanoantenna, as for the case of a-SNOM with resonant apertures, weakly dependents on the sample RI local changes preventing the usage of the nonfluorescent methods in studying the optically transparent samples.

It is known that the use of spectrally-based signal processing techniques in SNOM systems, instead of amplitude ones, can increase the sensitivity of these systems [16]. Similar approach based on the detection of the nanoantenna’s spectral response rather than the scattered signal intensity seems to increase the sensitivity of the SNOM methods as well as provides them with the possibility to detect small RI changes of the optically transparent samples. Such an approach could be based on the fact that the metal nanoparticles have a pronounced spectral dependence of the local plasmon resonance (LPR) on the surrounding media RI [17]. Therefore, the metal nanoparticle placed at the extremity of the scanning probe can act as the pointed nanoantenna. The detection of the spectral response of this nanoantenna can provide the high-precision mapping of RI changes of the optically transparent samples. In this paper, by detecting the spectral response of the simplest nanoantenna - spherical Au nanoparticle placed at the extremity of the transparent dielectric probe fabricated in the form of a fiber microaxicon (FMA), we will theoretically demonstrate the possibility of mapping the extremely small RI variations (down to 5·10^{−4} RIU under ideal conditions) with the lateral resolution comparable to the nanoantenna diameter. We will show that the use of the FMA as a scanning probe provides an efficient excitation of dipole LPR in the nanoantenna by the diffraction-limited focal spot as well as the collection of the nanoantenna signal. The results of numerical simulations based on the 3D finite-difference time-domain (FDTD) method are in good agreement with the theoretical quasi-electrostatic estimations for the radius of the nanoantenna not exceeding the skin depth of Au.

## 2. Theoretical estimations

To estimate the LPR spectral response of the spherical nanoantenna placed in the close proximity with the optically transparent sample surface we use a quasi-electrostatic theory. It is known that the impact of an electromagnetic wave with an electric field amplitude *E* on the spherical Au nanoparticles with a radius *a* and a dielectric permittivity *ε _{Au}* and a permeability

*μ*= 1, surrounded by a transparent homogeneous dielectric medium with the dielectric permittivity

*ε*= 1 leads to its polarization with a dipole moment

_{m}*ε*and permeability the

_{m}*μ*= 1 placed at a distance

*d = z*from the nanoantenna’s center (Fig. 1(a)). The incident electromagnetic field is assumed to be polarized in the direction perpendicular (s-polarization) to the medium surface (Fig. 1(a)).

_{0}–aIn this case, the polarized nanoantenna will induce surface charges in the medium. The electromagnetic field of these surface charges can be described by an equivalent field of an image dipole [18] with dipole moment *p _{ekv}*

*= α*

**,**

*p**α =*(

*ε*1)/(

_{m}-*ε*+ 1), located inside the medium at a distance

_{m}*2z*from the nanoantenna’s center (Fig. 1(a)). This field will affect the nanoantenna in turn. In this case, we can show that the dipole moment of the nanoantenna will affect only the longitudinal component (directed along the z-axis in Fig. 1(a)) generated by the reflected component of the dipole electric field, which can be written as:

_{0}**can be expressed in term of the field-related charges in the equivalent medium (Fig. 1(b)) polarized by the**

*E*_{ekv}**field and surrounded the nanoantenna from all sides [18]where**

*E**ε*is the dielectric permittivity of the equivalent medium. By equating the Eqs. (2) and (3) we can found that the relationship between the

_{ekv}*ε*and the dielectric permittivity

_{ekv}*ε*of the semi-infinite medium located at the distance

_{m}*z*can be expressed by the following equation

_{0}*ε*is a high-frequency limit dielectric constant,

_{∞}*λ*- plasma wavelength, and

_{p}*γ*- damping parameter. By neglecting the

_{r}*ε*dispersion, assuming (

_{ekv}*ε*- 1) <<1 and using the known equation for the dipole LPR of small spherical nanoparticles [7]we can obtain from Eq. (4) the simple equation describing the relationship between the dipole LPR wavelength

_{m}*λ*of the spherical Au nanoantenna and the dielectric permittivity of semi-infinite medium

_{SP}*ε*:

_{m}*λ*and

_{media}*λ*are the dipole LPR wavelengths of the spherical Au nanoparticle in the homogeneous dielectric medium with

_{vak}*ε*and

_{m}*in vacuo*respectively.

The *λ _{media}* and

*λ*values can be obtained by applying the Eq. (6) to the Eq. (5). Note that in general Eq. (5) gives quite good approximation for Au dispersion curve in the visible spectral range. However, the use of such expression to determine exact

_{vak}*λ*and

_{media}*λ*values are quite approximate, which leads to the incorrect estimates of

_{vak}*λ*value in turn. Therefore, to carefully estimate the

_{SP}*λ*and

_{media}*λ*values we used a modified Drude-Lorentz model [19]:

_{vak}*λ*- interband transition wavelength,

_{m}*γ*- interband transition broadening expressed in wavelength,

_{m}*A*and

_{m}*Φ*- dimensionless critical points amplitude and phase [19], respectively). The following equation allows one to describe the spectral dependence of the dielectric permittivity of Au, that best fits the experimentally measured values for bulk material [20], thereby gives more accurate estimations for ${\lambda}_{media}$and ${\lambda}_{vak}$. By repeating the aforementioned calculations for p-polarized exiting field (Fig. 1(a)), we can obtain

_{m}*λ*(λ

_{SP}/λ_{SP0}_{SP0}- dipole LPR wavelength of the nanoantenna

*in vacuo*) of the nanoantenna on the semi-infinite medium RI n

_{m}(curves 1-5), calculated for different “nanoantenna-medium” distances

*d*and incident field polarizations in accordance with the Eqs. (7) and (9) respectively.

The *λ _{SP}/λ_{SP0}*(n

_{m}) dependencies calculated for the nanoantenna fully surrounded by the homogeneous dielectric medium [21] are also presented in the Fig. 2(a) and 2(b) (dashed curves). As seen, the change in the RI leads to a detectable red-shift of the dipole LPR wavelength in comparison with the

*λ*, with the shift value being strongly depended on the incident field polarization as well as the “nanoantenna-media” distance

_{SP0}*d*. Nevertheless, even for nanoantenna located in a close proximity to the surface the spectral shift is significantly lower than for the nanoantenna fully surrounded by this dielectric media (dashed curves in Figs. 2(a) and 2(b)). Figure 2(c) shows the dependence of a linear section slope

*S*=

_{λ}*dλ*/dn

_{sp}_{m}of the

*λ*(n

_{SP}_{m}) curve calculated near n

_{m}= 1.35 on the “nanoantenna-media” distance

*d*. As seen, the maximum spectral sensitivity

*S*~19 nm/RIU is expectedly achieved at

_{λ}*d*= 0 and s-polarized excitation field owing to the distribution character of the energy density maxima in the nanoantenna near-field [22,23]. At the same time the

*S*value is only 4 times lesser than the sensitivity of the nanoantenna fully surrounded by the medium. In accordance with the resolution of modern optical spectrum analyzers estimated by using the Rayleigh criterion (no worse than 0.02 nm) [24], the method proposed can detect the RI changes ~10

_{λ}^{−3}RIU. However, by taking into account the existing state-of-the-art methods of optical spectra processing based on detection of the spectral peak “center of gravity”, we can expect an increase in the resolution of the method at least by an order of magnitude (down to 10

^{−4}RIU).

## 3. FDTD analysis

As it was mentioned above at real experimental conditions the nanoantenna should be placed at the extremity of the scanning probe, which can excite the nanoantenna with minimized sample background illumination. Such a probe must be integrated into the standard feedback system of a scanning probe microscope, and thus provide an opportunity to move nanoantenna with high precision in a close proximity to the sample surface. We assume that the nanoantenna is located on tip of the fiber microaxicon (FMA) (Fig. 1(c)) fabricated at the flat endface of a standard step-index optical fiber (OF). The semi-infinite homogeneous media (referred to as “sample”) with a refractive index n_{m} ranging in 1 - 1.7 RIU is placed at a distance *d* from the nanoantenna. Note that the FMA presence causes some difference of the geometry under consideration from the conditions described in the analytical estimations. Also it should be noted that the abovementioned quasi-static approximation holds true only for relatively small nanoantenna radius (*i.e.*, at *a*<*σ*, where *σ* - the skin depth of Au).

In order to take into account the retardation effects inside the nanoantenna [23] at *a*>*σ*, and to assess the influence of the FMA on the spectral sensitivity of the developed method we use numerical simulation based on the 3D-FDTD method [25]. The FMA is assumed to have a full taper angle θ = 90°, the base diameter equals to the OF core diameter and axial symmetry along the OF optical axis (z axis in Fig. 1(c)). Such a microlens can focus the laser radiation into the diffraction-limited spot with a lateral size ~*λ/*2 and a focal depth ~0.3*λ* [26], providing a high excitation efficiency of the dipole LPR nanoantenna with the minimized background illumination as well as a possibility to collect the nanoantenna spectral signal. The following parameters were used to model the OF properties: fiber core diameter *D _{core}* = 4.5 μm, cladding diameter

*D*= 125 μm and the numerical aperture NA = 0.14. The FMA is excited by a broadband p-polarized Gaussian source (its position is marked in Fig. 1(c)) with a central wavelength

_{clad}*λ*= 532 nm and spectral FWHM

_{s}*Δλ*= 100 nm, providing the single-mode propagation regime in this spectral range at the abovementioned OF parameters. Perfectly matched layers are used as the boundary conditions. Mesh size of the computational cell equals to 1x1x1 nm

_{FWHM}^{3}. Moreover, to increase the calculation accuracy an additional mesh covering the nanoantenna area with the cell size 0.2x0.2x0.2 nm

^{3}is used.

Figure 3(a) shows the dependence of the relative dipole LPR wavelength *λ _{SP}/λ_{SP0}* of the nanoantenna on the sample RI n

_{m}calculated for small nanoantenna radii (

*a*<

*σ*) and “nanoantenna-sample” distance

*d*= 0. Note that the dipole LPR wavelength λ

_{SP0}is not constant and rises linearly [27] with the nanoantenna radius

*a*(Fig. 3(b)), which is also consistent with the data presented in [22,23]. Therefore, to present the calculated and analytical curves on one plot the

*λ*value in each case is normalized to the correspondent

_{SP}*λ*value determined at n

_{SP0}_{m}= 1. Figure 3(a) also shows the

*λ*(n

_{SP}/λ_{SP0}_{m}) dependencies obtained using the analytical estimations for the s- (Eq. (7)) and p-polarized (Eq. (9)) radiation, as well as

*λ*(n

_{SP}/λ_{SP0}_{m}) dependence, which corresponds to the well-known case of the nanoantenna fully surrounded by a homogeneous dielectric medium (black solid line in Fig. 3(a)). As seen, for

*a*<25 nm the relative spectral shift

*λ*obtained using numerical simulations is in good agreement with the analytical estimations. Moreover, at low RI (1<n

_{SP}/λ_{SP0}_{m}<1.3) calculated and analytical curves rises almost linearly with n

_{m}. However, the

*λ*value for all calculated curves exceeds the analytical value estimated by using the Eq. (7) for the p-polarized radiation. This fact can be partly explained by the presence of the longitudinal component of the electromagnetic field directed along z axis (Fig. 1(c)) in the FMA focal spot excited the nanoantenna. Furthermore, the analysis of the data in Fig. 3(b) shows, than all calculated curves tend to increase the slope

_{SP}/λ_{SP0}*S*with the n

_{λ}_{m}growth (in the range 1.3<n

_{m}<1.7), while the slope of the

*λ*(n

_{SP}/λ_{SP0}_{m}) dependencies estimated by using the analytical Eqs. (7) and (9) demonstrates the decrease with n

_{m}growth. This mismatch between the calculated and analytical results can be explained by the fact that the assumption (

*ε*- 1)<<1 made in analytical estimations does not hold true at the specified RI range. Moreover, to provide more accurate analytical estimation one should take into account the multiple reflections of the image dipole and nanoantenna that contribution increases with n

_{m}_{m}.

Figure 2(c) shows the dependencies of a linear section slope of the *λ _{SP}*(n

_{m}) curve estimated near n

_{m}= 1.35

*S*/dn

_{λ}= dλ_{sp}_{m}on the “nanoantenna-sample” distance

*d*calculated for different nanoantenna radii

*a*. As seen, despite some mismatch in the maximum values estimates at

*d*= 0, the behavior of the calculated curves shows good agreement with the analytical results. It should be noted that the

*S*dependence for the nanoantennas with the radius larger than the skin depth of Au has a similar behavior with both numerical calculations for the

_{λ}(d)*a*<

*σ*and the analytical estimations. However, with nanoantenna radii growth the slope of the

*λ*(n

_{SP}/λ_{SP0}_{m}) dependence increases, exceeding the estimated value (Eq. (9)) at

*a*>25 nm and reaching the maximum value

*S*~21 nm/RIU at

_{λ}*a*= 50 nm. Such behavior apparently can be explained by the increasing influence of the field retardation effect with the nanoantenna radius growth [32].

It also should be noted that the spectral FWHM of the dipole LPR is strongly increased with the nanoantenna radius [22] due to the higher-order (nondipole) plasmon modes excitation. Potentially it significantly complicates the spectral shift registration. To estimate the lateral resolution of the method developed, we simulated the case when the dipole LPR wavelength varies with a step-like spatial change of the sample surface RI. The results of the correspondent numerical calculations for nanoantennas with the radii *a* = 50 nm and *a* = 25 nm are shown in Fig. 3(c). As seen, the lateral resolution estimated as the difference between the 10-% and 90%-levels of the maximum signal improves with the nanoantenna radius decreasing and reaches ~55 nm, which approximately corresponds to the doubled nanoantenna radius. Summing up these facts, in spite of the spectral sensitivity increase with *a,* both the spectral LPR peak broadening and the lateral resolution worsening can reduce the positive effect of the large-radius nanoantennas. Thus, in our opinion, the use of the nanoantennas with *a*<50 nm is optimal in terms of the achieved lateral resolution and provided sensitivity (down to 5·10^{−4} RIU under ideal conditions). Obviously, the change in LPR spectral position associated with the “nanoantenna-sample” distance or the microroughness of the sample surface could lead to an error in evaluating the actual RI change, thus the stated spectral sensitivity can be lower. This error as well as possible artifacts in the RI map seems to be minimized by simultaneously scanning the topography and the RI of the sample surface using SNOM feedback system based on a standard tuning fork.

It should be noted that the spectral shift registration can be performed in the far-field by using the focusing optics as well as directly through the FMA. Figure 3(d) shows the calculated reflection spectra of the scattered nanoantenna radiation collected by the FMA. As seen, all the local maxima of the reflection spectrum (solid curves) are slightly blue-shifted [23,28] in comparison with the spectra calculated in the nanoantenna’s near-field (dashed curves). Nevertheless, the “far-field” *λ _{SP}/λ_{SP0}*(n

_{m}) dependence (not presented here) is similar to that of presented in the Fig. 3(a).

Finally, the use of the nanoantennas with an elongated geometrical shape (higher aspect ratio [29],) or specially designed resonant nanoantennas [30,31] can substantially increase the sensitivity of the developed method as well as provide better lateral resolution. These features as well as the experimental realization of the proposed probe will be discussed in our forthcoming paper.

## 4. Conclusions

In summary, in the paper we showed analytically and numerically that the use of spectral response detection of a single spherical Au nanoantenna allows one to map very small (down to 5·10^{−4} RIU under ideal conditions) variations of the RI of the optically transparent sample. Simple analytical quasi-static model allowed estimating the spectral shift of the nanoantenna dipole LPR wavelength and the spectral sensitivity of the method developed was presented. The pointed scanning probe based on fiber microaxicon with the Au spherical nanoantenna attached to its tip was proposed to realize the RI mapping. The finite-difference time-domain numerical simulations of the spectral properties of the proposed probe are in agreement with the theoretical quasi-electrostatic estimations for the radius of the nanoantenna, not exceeding the skin depth of Au.

The fabrication technique and the focusing properties of the FMAs are carefully studied [26,32]. The nanoantenna on the extremity of the microaxicon can be fabricated by using several developed experimental techniques: ion-beam milling, laser-induced femtosecond transfer [33], single Au nanoparticles scanning [34], etc. To precisely control the distance between the nanoantenna and the sample surface a standard SNOM feedback system based on a tuning fork can be used providing the possibility to simultaneously map the local RI changes and the topography of the sample under study. Spectral signal detection can be performed in the far field by using focusing lens as well as directly through the fiber probe, which significantly simplifies the optical signal system. The experimental realization of the probe proposed will be presented in our forthcoming paper.

## Acknowledgments

The authors acknowledge partial support from Russian Foundation for Basic Research (Projects nos. 14-02-31323-mol_a, 14-02-00205-a).

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