Abstract

We report experimental observations of optical hot-spots associated with surface phonon polaritons in boron nitride nanotubes. As revealed by near-field optical microscopy, the hot-spots have mode volumes as small as 2.7×106λ030 is the wavelength of the exciting light in vacuum), which are in the deep subwavelength regime. Such strong light-trapping leads to ultrahigh field enhancement with a Purcell factor of ≃1.8 × 106. Remarkably, the hot-spots are not induced by designed structures, but by random scatterings with the rough gold substrate. The ultrahigh field enhancement can be used to improve nonlinear infrared spectroscopy, thermal emitters and detectors, and label-free molecule sensing at nanoscales.

© 2017 Optical Society of America

1. Introduction

Plasmonic subwavelength light-trapping has become a very important discipline in nanophotonics [1–3] for visible and near infrared frequencies. At mid infrared frequencies, noble-metal plasmonic structures become less effective, giving their roles to graphene plasmonic devices [4–6] and phonon polaritonic devices [7–11]. Surface phonon polaritons (SPhPs) offer opportunities for trapping light at mid infrared frequencies [12,13]. Studies have revealed strong light-matter interaction of SPhPs in SiO2 [7], SiC [8], and boron nitride (BN) thin films [9, 10]. Recently, one-dimensional (1D) guided surface phonon polaritons in boron nitride nanotubes (BNNTs) were reported by our group [11, 14]. SPhPs in BNNTs exhibit very high effective refractive indices; up to neff ≃70 was observed in a 19 nm-diameter BNNT on gold substrate [11]. Here the effective refractive index is defined as neff ≡ λ0SPhP where λ0 and λSPhP are the photonic wavelength in vacuum and the SPhP wavelength in the BNNTs, respectively. The effective refractive index of SPhPs in BNNTs is much greater than the effective refractive index of most plasmonic devices made of noble metals [3,15] and graphene [4,5]. BNNTs even show much stronger light-trapping than BN thin films [11]. Moreover, the effective refractive indices of the BNNTs are found to be inversely proportional to the nanotube diameter, which allows considerable tunability of the effective refractive index [11]. Thus, BNNTs can serve as a tunable deep-subwavelength light-trapping system which is of tremendous application values (such as nano-sensors, nano-laser, and nonlinear optical devices) at mid infrared frequencies.

For many applications in nanophotonics, such as optical biosensing, light-trapping into a zero-dimensional hot-spot is necessary. Here we report observation of such hot-spots in BNNTs and reveal a mechanism for the formation of such hot-spots due to random coherent scattering. This mechanism is known as Anderson localization, which is first studied in electronic systems and then for classical waves. Anderson localization leads to the suppression of diffusion and formation of confined states in disordered media [16,17]. In photonics, field localization has been theoretically studied and experimentally observed in various disordered or periodic dielectrics [18–23] and metals [24–26] where coherent multiple scattering opens a photonic band gap. Strong localization in disordered media have been shown as a tool to achieve highly localized states [16]. For light and polariton waves, such strongly localized states are highly desirable in achieving strong light-matter interaction which are indispensable for the study of quantum electrodynamics, low-threshold lasing, enhanced nonlinear spectroscopy, biological and molecular sensing and energy applications [18–27]. Here, we report direct experimental observations of localization of 1D SPhPs in BNNTs. By utilization of high spatial resolution (≃10 nm) infrared scattering type scanning near-field optical microscopy (s-SNOM) technique, the local field intensity of the SPhP waves at different locations along the BNNTs is measured. It is found that the presence of quasi-randomly distributed, nanometer-sized gold grains from a rough gold substrate induce localization of 1D SPhPs in BNNT. Both weak and strong localization regimes are observed where the dispersions of SPhPs are weakly or strongly modified. Weak and strong localization of SPhPs are tuned by the frequencies of SPhPs as controlled by the incident light. The observed phenomena are explained by theory and calculations based on the transfer matrix method.

In s-SNOM, a sharp metallic probe atomic force microscopy (AFM) tip scatters the near field of electromagnetic field generated by the sample, and induces a far-field detectable optical signal. There are two passages of exciting the sample. The first way is to excite the sample with the far-field propagating electromagnetic wave (i.e., the wave directly from the laser). The second way is to excite the sample through the probe AFM tip itself. The metallic tip acts as a nano-antenna that concentrates the electromagnetic energy into the near-field of the probe tip with high field energy density, when exposed to far-field electromagnetic field. Polarization of the sample in proximity to the metallic tip is induced efficiently and effectively [4, 5]. In BNNTs, SPhPs are excited by an incident laser field through the metallic tip above the BNNT. The excited SPhP wave then propagates along the nanotube and is reflected back by the terminals or disordered regions where gold nano-grains scatter the SPhP wave. The back-reflected waves are collected right below the AFM tip and converted into far-field photons through the near-field coupling between the SPhPs in the BNNT and the polarizable metallic tip. The amplitude and phase of the near-field signal depend on the amplitude and phase of the reflected SPhP wave in the BNNT at the positions beneath the tip. A reference optical field from the same source of the incident laser field is used for homodyne amplification of the near-field signal at a selected phase [11].

Figure 1(a) shows the scheme of the custom-made s-SNOM with infrared light sources. The custom-made experimental setup consists of a tunable infrared laser source, an atomic force microscope, and an interferometer. A quantum cascade laser (QCL, Daylight Solutions) was used to provide a tunable, continuous wave, mid infrared laser of 100 mW average power. An asymmetric Michelson interferometer was built with a 50:50 IR beam splitter. One half of the infrared laser was focused at the probe tip apex of a commercial available atomic force microscope (Multimode AFM Digital Instrument) by a 90° off-axis gold coated parabolic mirror (f = 25 mm). The AFM operated in tapping mode with platinum coated tips (DPE 14 HQ Mikromash) driven at Ω ≃ 137 kHz. The tip-scattered infrared light is collimated by the same parabolic mirror, homodyned by a phase controlled reference beam, and detected by a mercury cadmium telluride (MCT) detector (J15D12 Teledyne-Judson) with a homodyne technique described in literature [28]. The near-field signal is extracted by a lock-in amplifier (HF2Li Zurich Instrument) demodulated at the third harmonic of the tip oscillation frequency (3Ω). π/2 phase method is used to map the SPhPs in Figs. 1(c), 1(e), and Fig. 2. Both in-phase and π/2 phase homodyne near-field signal are registered, and the total amplitude calculated from the vector sum of two homodyne signals is used to form the near-field image in Fig. 3. A near-field image is formed when the tip is scanned over the sample and substrate, together with the topography image.

 figure: Fig. 1

Fig. 1 (a) A scheme of s-SNOM setup. (b) AFM topography of a BNNT on top of another BNNT. The scale bar is 300 nm. (c) Near-field images taken at 1410 cm−1 of the BNNTs in (b). (d) AFM topography of a rough gold surface, the gold nano-grains have diameter from 10nm to 30 nm. The scale bar is 200 nm. (e) s-SNOM IR image of the same substrate area.

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 figure: Fig. 2

Fig. 2 BNNT near-field images at different infrared frequencies. (a) Topography of a BNNT. (b) AFM phase image of the BNNT. (c–g) Near-field images of the BNNT at 1380 cm−1, 1398 cm−1,1410 cm−1 1422 cm−1, and 1435 cm−1, respectively, taken with π/2 homodyne technique [28]. (h) Fourier transformation of the SPhPs of the BNNT showing a splitting of spatial frequency peaks. (i) Scheme of back reflections of SPhPs by nanotube terminal and gold nano-grains.

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 figure: Fig. 3

Fig. 3 BNNT near-field images at different infrared frequencies. (a) Topography of a BNNT. (b–f) Near-field images of the BNNT at 1380 cm−1, 1415 cm−1,1420 cm−1 1430 cm−1, and 1590 cm−1, respectively. The color scale is tuned to maintain the same color for near-field response of the gold substrate.

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When a BNNT is on top of another BNNT [see Fig. 1(b)], the effect of gold substrate is much weakened, as proved by a previous numerical investigation on the properties of SPhPs in BNNTs on gold substrates [29]. In this limit, clear coherent beating patterns of the SPhPs are observed [see Fig. 1(c)]. As revealed in Ref. [11], these beating patterns are due to the coherent propagation of SPhP waves launched by the AFM tip and reflected back by the near-by terminal of the BNNT. In the other limit, when the BNNT lies directly on the substrate, the effect of the substrate on the SPhPs in the BNNT is pronounced. It was revealed in Refs. [11, 29] that the effective refractive index of SPhP can be considerably modified by tuning the distance between the BNNT and the gold substrate. In previous simulations [29], only smooth gold substrate is considered. However, in reality the gold substrates have rough surfaces which are comprised of nanometer-sized grains. These gold nano-grains have random sizes and positions. Fortunately, there geometry can be probed by the AFM topography [see Fig. 1(d)]. We find that these gold nano-grains have a quasi-random distribution with an approximate periodicity of 55 nm. Their typical sizes are 10–30 nm. The mid infrared s-SNOM can also reveal their random distributions on the surface of the gold substrate [see Fig. 1(e)]. The strongly inhomogeneous s-SNOM signals on the surface of the gold substrate reflects that the mid infrared optical waves are propagating in a non-percolating medium where strong local scattering enhances the s-SNOM signal at some random positions. The localization of surface plasmon polaritons in such a non-percolating gold surface is a known phenomena, which is essentially similar to the surface plasmon polaritons on individual gold nano-grains [3]. The focus of this work is to study the properties of the SPhP waves in BNNTs on rough gold substrate. When a BNNT is placed on a rough gold substrate, the distance between the gold substrate surface and the BNNT surface (termed as the “height”) is now a function of position (a local property). According to the random nature of the gold nano-grains, the height should also follow a quasi-random distribution with a periodicity about 55 nm. Since the effective refractive index is strongly influenced by the height, such a system can be regarded as SPhP waves propagating along BNNTs with random refractive index, if the effective refractive index can be defined locally. Such position-dependent refractive index is recently confirmed by other experimental results [30]. We shall show that such a random model can capture the essential physics of SPhPs on the rough gold substrates. Moreover, it reflects the most relevant processes regarding the s-SNOM signal, the random scattering by the gold nano-grains. Our speculation is also confirmed by the fact that for BNNTs on top of other BNNTs the coherent beating of SPhP waves are clearly visible, whereas for BNNT that directly lie on the gold substrate the coherent beating is usually distorted and the spatial periodicity of the beating is often obscure. We believe that in the former regime, the large height of the BNNTs makes the random scattering by the gold nano-grains rather weak, whereas in the latter regime, the small height of the BNNTs considerably enhances the random scattering effect. In fact, almost all coherent quantum beating of SPhP waves are observed on in the former regime.

Figures 2(a) and 2(b) show the topography and phase image of a BNNT on a rough gold substrate, respectively, obtained through the mechanical imaging capabilities of the AFM instrument. Figures 2(c)2(g) show the corresponding near-field response of the BNNT and substrate for a series of frequencies. From high frequency to low frequency, the nodal patterns of the s-SNOM signals extend, from the termini to the middle of the BNNT. Such reduced spatial decay of the near-field signals in the BNNT, in contrast with the much stronger spatial decay for BNNT on top of another BNNT in Fig. 1(c). We argue that the reduced decay of the s-SNOM signal is due to the enhanced scattering by the gold nano-grains. This argument is supported by the one-dimensional Fourier transformation of the s-SNOM signals along the BNNT direction. Compared with the case of a BNNT on top of another BNNT with clear peaks in the Fourier-transformed s-SNOM signals (a single peak for each frequency), the BNNT on top of the rough gold substrate has smeared Fourier peaks or even multiple peaks, as shown in Fig. 2(h). For simplicity, we shall term the situation with one BNNT on top of another as “smooth substrate”, since the observed clear coherent patterns of s-SNOM signal is qualitatively similar to the s-SNOM signals of BNNTs on smooth gold substrate. As schematically shown in Fig. 2(i), scattering of the 1D SPhPs by the random gold nano-grains can enhance the back-reflection and hence increases the s-SNOM signals.

The AFM and s-SNOM measurements of another sample is shown in Fig. 3. The AFM topography in Fig. 3(a) shows the BNNT on the rough gold substrate. The s-SNOM signals at various frequencies are presented in Figs. 3(b)3(f). Here the s-SNOM signals in the figures are the total amplitude which manifests the local optical field intensity right beneath the AFM tip [4]. The spatial distribution of the s-SNOM signal changes from almost uniform at 1380 cm−1 to non-uniform at 1415 cm−1, and then to a strong peak in the BNNT at 1430 cm−1. In the middle of the BNNT the s-SNOM signal is significantly enhanced at the frequency of 1430 cm−1. This very strong s-SNOM signal over a region of 0.5μm can only be explained by the emergence of strongly localized SPhPs. In Fig. 3(f), the s-SNOM signal on the BNNT at the frequency 1590 cm−1 is suppressed, which is probably because this frequency is off-resonance with the SPhPs. Such strong frequency-dependent localization of SPhPs is one of the main results in this work. The profiles of SPhPs along the BNNT for frequencies 1380 cm−1, 1415 cm−1, 1430 cm−1, and 1590 cm−1 are extracted and shown in Fig. 4(b), which directly shows the strong localization of the SPhPs at 1590 cm−1.

 figure: Fig. 4

Fig. 4 (a) Averaged wavevector q as a function of frequency for three samples. One sample is placed on smooth gold substrate, while the other two samples (1 and 2) are placed on rough gold substrates. The method for averaging the wavevector when there are multiple peaks in the spatial frequency is explained in the main text. (b) Profiles of near-field signal of the BNNT at different infrared frequencies. The intensities are normalized assuming the same level of near-field (NF) signal on the gold substrate. The total amplitudes of near-field signals are plotted in the figure, by taking the vector sum of near-field amplitudes collected at two quadrature homodyne phase conditions [31]. Strong localization of SPhPs leads to an increase of total near-field signal that even greatly surpass the strong near-field signal from the rough gold substrate for the frequency 1430 cm−1.

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An alternative way to manifest the effects of the random scattering of the SPhPs is to plot the dispersion relations using the averaged wavevector. Despite the fact that the spatial Fourier transformation of the s-SNOM signals does not have a clear single peak, one can always obtain an average wavevector of the SPhPs from these Fourier transformed data. In Fig. 4(a) the averaged wavevector versus frequency curves are plotted for three samples. One of the sample is a BNNT on top of another BNNT (simplified as “smooth substrate”), while the other two samples are on rough gold substrates. For the first case, the curve in Fig. 4(a) gives the dispersion relation. For the latter two cases, the curves give the approximate dispersion relation in the two BNNTs. We note that the light-line is well below these curves, consistent with the large effective refractive index and deep subwavelength light-trapping. The dispersion relation for the BNNT for the first deviate from the linear behavior. Particularly for Sample 2, the dispersion curve has a down-turn near 1420cm−1. Such nonlinear dispersions are another sign of strong random back-scattering the SPhP waves.

To explain the observed phenomena, we apply the theoretical formalism of the transfer matrix method. The s-SNOM signal is proportional to the local electric field beneath the metallic tip which consists of two parts: the injected electromagnetic wave and the waves reflected by the gold nano-grains and BNNT terminals. In the limit of smooth gold substrate the waves are reflected back only BNNT terminals. Therefore clear quantum beats of SPhP oscillations are indicated in the s-SNOM signal from which one can extract the wavelength and dispersion relation of the SPhPs. The dispersion relation of SPhP can be written as ω = qc/neff where q is the wave-vector, c is the velocity of light and neff is the effective refractive index which is generally complex and q-dependent. The wavelength of SPhP at given frequency varies with BNNT diameters and the distance between BNNT and the gold substrate [11]. For a BNNT lying on a rough gold substrate the distance between the BNNT and the gold substrate varies randomly. This distance depends on the distribution of gold nano-grains. The packing of the nano-grains is not entirely a random configuration. In fact, the Fourier transformation of the real-space profile of the s-SNOM signal on the rough gold substrate has a broadened peak [11]. This observation indicates that the gold nano-grains are quasi-randomly distributed with an average spatial periodicity.

The metallic AFM tip separates the wire into two parts: the left part and the right part. Each part is characterized by a reflection coefficient ri (i = L, R), calculated by the transfer matrix method. When the two parts combine together, the total s-SNOM signal is determined by the total reflection coefficient

rtot=rL+rR+2rLrR1rLrR.
Using the homodyne technique the real or imaginary part of the total wave amplitude ∼ (2+rtot) (the number 2 accounts the injected wave to the left and right) can be measured. The coefficients rL and rR, can be calculated by the transfer matrix method. The electric field at each point can be determined by the left-going wave amplitude El and the right-going wave amplitude Er. The total electric field is E(z) = Er(z) + El(z) and its spatial derivative is ∂zE(z) = iq(Er(z)−El(z)) where z is the coordinate along the BNNT. Both E(z) and ∂zE(z) are continuous along the BNNT.

If SPhPs are excited in the middle of a piece of filament with length a = 1 nm and labeled as j, then

(Ej,r+rREj,riqj(Ej,rrREj,r))=(M11(R)M12(R)M21(R)M22(R))(02iqNEN,r),
where Ej,r is the right-going wave launched by the incoming electromagnetic wave. At the BNNT right terminal boundary (labeled as N), the right-going wave is EN,r, while the left-going wave is EN,l=EN,r by assuming a hard-wall boundary at the BNNT terminals. This hard-wall boundary is consistent with the optical phonon boundary conditions in a nanotube as revealed in a recent work [32]. The above equation gives
rR=iqjM12(R)M22(R)iqjM12(R)+M22(R).

The transfer matrix M(R) is given by

M^(R)= m=j+1NM^m1
where
M^m=(cos(qma)sin(qma)/qmsin(qma)qmcos(qma)),
where qm = (nm,eff + ikeff)ω/c is the complex wavevector in filament m which is of length a. Here nm,eff describes the effect of the quasi-random distribution of the gold nano-grains. As revealed by the s-SNOM signal of the rough gold substrate in Fig. 1(e), the optical properties of the substrate shows quasi-random nature. Fourier transformation of the real-space distribution of the s-SNOM signal gives broadened peak [11, 14, 29], i.e., it is a quasi-random distribution with an approximate periodicity of 55 nm and the AFM topography reveals that the gold nano-grains at the surface of the substrate is of size 10–30 nm. We require that the effective refractive index follows the same distribution nature as the s-SNOM signal of the rough gold substrate. Thus, we assume the form nm,eff = nm,period + nm,rand where the nm,period describes a periodic profile of the refractive index with the periodicity of 55 nm and nm,rand follows a uniform distribution. The nm,period is taken to be a periodic step distribution between 0.5neff0 and neff0 where neff0 is extracted from the SPhP dispersions in the regions where regular quantum beat of SPhP is still visible. In our calculations, we actually work with quasi-random distributions of the permittivity which is a uniform distribution with a width of 0.15(neff0)2. Previous study reveals that spatial damping coefficient is almost constant for all frequencies measured as κ = keffω/c = 0.45 × 106 m−1 [11].

Similarly, for the left part of the BNNT,

M^(L)= m=j10M^m.
and
(rLEj,l+Ej,liqj(rLEj,lEj,l))=(M11(L)M12(L)M21(L)M22(L))(02iq0E0,l),
where Ej,l is the right-going wave launched by the incoming electromagnetic wave. At the BNNT left terminal boundary (labeled as 0), the left-going wave is E0,l, while the right-going wave is E0,r=E0,l. Accordingly, we have
rL=iqjM12(L)+M22(L)iqjM12(L)M22(L).

We calculate the amplitude of the reflection coefficient rtot for a BNNT on the rough gold substrate. The rough gold substrate is modeled by quasi-randomly distributed gold nano-grains that modify locally the effective permittivity (refractive index) of the SPhPs along the BNNT. The calculation results from transfer matrix method are shown in Fig. 5 where the modular square of the total wave amplitude |2 + rtot|2 (which is proportional to the local optical field intensity) is plotted. These results qualitatively repeat the observations via s-SNOM measurement for different frequencies in Fig. 4(b). To demonstrate the effect of disorders, we also plot the same quantity for smooth gold substrate in Fig. 5 (the blue curves). For all four frequencies disorders modulate the optical wave intensity distribution along the BNNT. For frequencies other than 1430 cm−1, the effect of the rough gold substrate is not significant, agreeing with the measurement. Both the experiments and the calculations indicate that the strong localization is highly sensitive to the frequency. In the calculations, we observed that such frequency-sensitivity is due to the interference between the multiple back-scattering waves. When the frequencies ≃1430 cm−1, the quasi-randomly distributed scattering centers give approximately constructive interference between the multiple back-scattering waves, leading to strong localization. Indeed the quarter wavelength of the SPhPs at this frequency is close to the spatial periodicity of the gold nano-grains (≃ 55 nm), confirming the Anderson localization mechanism [18–21] for the 1D SPhPs.

 figure: Fig. 5

Fig. 5 Local optical field intensity (in arbitrary units) |2+rtot|2 of SPhPs as a function of the exciting position which mimics the near-field signal as function of AFM tip position along a BNNT of 1.6 μm length for frequencies (a) 1400 cm−1, (b) 1410 cm−1, (c) 1420 cm−1 and (d) 1430 cm−1. The blue curves are for a BNNT on the smooth gold substrate, while the red curves are for a BNNT on the rough gold substrate.

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AFM scan of local mechanical properties along the BNNT confirms that the appearance of peaks in the s-SNOM signal is not due to a structural defect on the BNNT (the 0.5μm size is also too large for a mechanically stable structural defect). The observed hot-spot can be regarded as a highly confined “cavity mode” in the BNNT of length 0.5μm. The quality factor of this mode as determined from the wavevector q = 5.8 × 107 m−1 and the damping κ = 0.45 × 106 m−1 is Q ≃ 64. The mode volume is V=2πξzξ2 where ξz = 0.5 μm and ξ = 0.017 μm are the characteristic decaying length of the field amplitude along z and perpendicular directions, respectively. ξ is determined by the fact that in deep sub-wavelength regime q2ξ2. We hence estimate the mode volume as V=2.7×106λ03 and the Purcell factor as 3Qλ03/(4π2V)=1.8×106. In comparison, the Purcell factor is ∼ 104 for photonic crystal nanocavities [33], ∼ 10 for hybrid semiconductor-metal plasmonic cavities [34], ∼ 103 for semiconductor microdisk [35], and ∼ 104 for plasmonic cavities [36]. Such ultrahigh Purcell factor can be useful for ultra-sensitive molecule sensing and very low-threshold lasing [12,13]. It can also greatly enhances thermal emission and absorption [37] (detection [38]), serving as perfect nano-sized local thermal emitters, absorbers and detectors. Molding the flow of mid infrared light could enable the control of thermal conduction in real life systems which is important for advanced energy technologies [39]. Comparing with other strong-coupling systems in mid infrared region, graphene plasmon (∼ 8 meV [6]) usually has larger linewidth than phonon-polaritons (∼ 3 meV) in BNNT [11]. Coherent coupling between phonon polariton in hexagonal boron nitride sheet and plasmon in graphene have been revealed in a planar heterostructure [40]. In previous studies we have demonstrated transfer of energy between BNNT SPhPs and graphene plasmons in a 1D/2D geometry. Our studies open the possibility for photonic circuits based on patterned boron nitride and graphene nanostructures in the mid infrared regime.

2. Conclusions

We have observed localization of SPhPs in a one dimensional system provided by BNNTs in rough gold substrates where the gold nano-grains on the interfaces of the substrate scatters the SPhPs randomly. The s-SNOM technique enables a high spatial resolution (10 nm) scan of the local field and hence facilitates a direct, local measurement of reflection and localization properties of SPhPs with high precision. Strongly localized waves are detected in the BNNT, which provides possibility for nano-scale sensing and imaging. The highly confined local modes in the deep sub-wavelength regime have very strong field and energy concentration which could be useful for nanoscale mid infrared optics which includes potential enhance nonlinear optics (e.g., lasing and Raman spectroscopy), thermal emitter and detector and biological and molecular sensing at nanoscale with the synergy of metal plasmonics and phonon polaritonics.

Appendix: numerical methods

The total field amplitude at the position where energy is injected via the AFM tip consists of several parts. For continuous excitation as in the case of experiments in this work, the total amplitude is given by

n=0(2+rL+rR)(rLrR)n=2+rtot
with rtot = (rL + rR + 2rLrR)/(1 − rLrR). Here the amplitude of the injected SPhP wave going along the left or right is taken as 1. The first reflected SPhP wave has amplitude rL and rR from the left and right sides, respectively. The second reflected wave have amplitude of rLrR, and so on. The total field amplitude is the summation of all those contributions. rL and rR is calculated via the transfer matrix method when the refractive index neff is given.

In the calculation the BNNT is modeled by a chain of grid points. Each point is of length 1 nm while the total length is 1.6 μm. The effective permittivity at the m-th grid point is generated in the following way,

εm,eff=12(εeff0+εeffg)j=0Ntot/Npαjδεexp[(mjNp)22σ2]
where εeff0 and εeffg are the effective permittivity for the SPhP waves without and with proximity to a gold nano-grain. Ntot is the total number of grid points. Physically, Np is the average distance between the randomly distributed gold nano-grains. The number of gold nano-grains is Ntot/Np+1.δε12(εeff0εeffg) is half of the difference between the two effective permittivity. αj is a random variable uniformly distributed in the region [−0.5, 0.5]. σ is the range that a gold nano-grain can modify the effective permittivity of the SPhP waves considerably, which is approximately the size of the gold nano-grain.

According to previous study [11], the periodicity of the random distribution of nano grains is about 55 nm. AFM height images reveals that the gold nano-grains at the surface of the substrate is of size 10–30 nm. Particularly for frequency 1405 cm−1 the wavelength is 184 nm, while for 1415 cm−1 the wavelength is 142 nm. From these data we extract a linear dispersion relation which should be applicable around 1410 cm−1.

Funding

National Natural Science Foundation of China (grant no.: 11675116), Natural Sciences and Engineering Research Council of Canada, and the Soochow university.

Acknowledgments

We thank Pierre Berini and Behnood G. Ghamsari for helpful discussions, Gregory O. Andreev for support with the instrumentation, Chunyi Zhi, Yoshio Bando and Dmitri Golberg for providing boron-nitride nanotubes.

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12. S. Law, V. Podolskiy, and D. Wasserman, “Towards nano-scale photonics with micro-scale photons: the opportunities and challenges of mid-infrared plasmonics,” Nanophotonics 2, 103–130 (2013). [CrossRef]  

13. R. Stanley, “Plasmonics in the mid-infrared,” Nat. Photon. 6, 409–411 (2012). [CrossRef]  

14. X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014). [CrossRef]   [PubMed]  

15. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008). [CrossRef]  

16. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]  

17. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena: Localization and Mesoscopic Phenomena, vol. 88 (Springer, 2006).

18. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984). [CrossRef]  

19. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987). [CrossRef]   [PubMed]  

20. K. Arya, Z. B. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985). [CrossRef]   [PubMed]  

21. J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989). [CrossRef]   [PubMed]  

22. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef]   [PubMed]  

23. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989). [CrossRef]   [PubMed]  

24. D. J. Bergman and Y. Imry, “Critical behavior of the complex dielectric constant near the percolation threshold of a heterogeneous material,” Phys. Rev. Lett. 39, 1222–1225 (1977). [CrossRef]  

25. S. I. Bozhevolnyi, “Localization phenomena in elastic surface-polariton scattering caused by surface roughness,” Phys. Rev. B 54, 8177–8185 (1996). [CrossRef]  

26. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: Surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16 (2003). [CrossRef]  

27. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990). [CrossRef]   [PubMed]  

28. X. G. Xu, A. E. Tanur, and G. C. Walker, “Phase controlled homodyne infrared near-field microscopy and spectroscopy reveal inhomogeneity within and among individual boron nitride nanotubes,” J. Phys. Chem. A 117, 3348–3354 (2013). [CrossRef]   [PubMed]  

29. B. G. Ghamsari, X. G. Xu, L. Gilburd, G. C. Walker, and P. Berini, “Mid-infrared surface phonon polaritons in boron-nitride nanotubes,” J. Opt. 16, 114008 (2014). [CrossRef]  

30. X.-G. Xu, unpublished (2017).

31. Z. H. Kim, B. Liu, and S. R. Leone, “Nanometer-scale optical imaging of epitaxially grown gan and inn islands using apertureless near-field microscopy,” J. Phys. Chem. B 109, 8503–8508 (2005). [CrossRef]  

32. T. Thonhauser and G. D. Mahan, “Phonon modes in si [111] nanowires,” Phys. Rev. B 69, 075213 (2004). [CrossRef]  

33. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003). [CrossRef]  

34. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature (London) 461, 629–632 (2009). [CrossRef]  

35. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003). [CrossRef]  

36. E. J. R. Vesseur, F. J. G. de Abajo, and A. Polman, “Broadband purcell enhancement in plasmonic ring cavities,” Phys. Rev. B 82, 165419 (2010). [CrossRef]  

37. E. Rephaeli, A. Raman, and S. Fan, “Ultrabroadband photonic structures to achieve high-performance daytime radiative cooling,” Nano Lett. 13, 1457–1461 (2013). [PubMed]  

38. P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, and J. H. Shim, “High-precision nanoscale temperature sensing using single defects in diamond,” Nano Lett. 13, 2738–2742 (2013). [CrossRef]   [PubMed]  

39. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009). [CrossRef]   [PubMed]  

40. V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014). [CrossRef]   [PubMed]  

References

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  1. K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (sers),” Phys. Rev. Lett. 78, 1667 (1997).
    [Crossref]
  2. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1997).
    [Crossref] [PubMed]
  3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  4. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (London) 487, 77–81 (2012).
  5. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (London) 487, 5 (2012).
  6. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photon. 7, 394–399 (2013).
    [Crossref]
  7. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light–matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
    [Crossref]
  8. J. Le Gall, M. Olivier, and J.-J. Greffet, “Experimental and theoretical study of reflection and coherent thermal emission by a sic grating supporting a surface-phonon polariton,” Phys. Rev. B 55, 10105 (1997).
    [Crossref]
  9. S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride,” Science 343, 1125–1129 (2014).
    [Crossref] [PubMed]
  10. J. D. Caldwell, A. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, and A. J. Giles, “Sub-diffraction, volume-confined polaritons in the natural hyperbolic material, hexagonal boron nitride,” Nat. Commun. 5, 7507 (2014).
    [Crossref]
  11. X. G. Xu, B. G. Ghamsari, J. Jiang, L. Gilburd, G. O. Andreev, C. Y. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
    [Crossref] [PubMed]
  12. S. Law, V. Podolskiy, and D. Wasserman, “Towards nano-scale photonics with micro-scale photons: the opportunities and challenges of mid-infrared plasmonics,” Nanophotonics 2, 103–130 (2013).
    [Crossref]
  13. R. Stanley, “Plasmonics in the mid-infrared,” Nat. Photon. 6, 409–411 (2012).
    [Crossref]
  14. X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014).
    [Crossref] [PubMed]
  15. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008).
    [Crossref]
  16. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
    [Crossref]
  17. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena: Localization and Mesoscopic Phenomena, vol. 88 (Springer, 2006).
  18. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
    [Crossref]
  19. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
    [Crossref] [PubMed]
  20. K. Arya, Z. B. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
    [Crossref] [PubMed]
  21. J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989).
    [Crossref] [PubMed]
  22. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref] [PubMed]
  23. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
    [Crossref] [PubMed]
  24. D. J. Bergman and Y. Imry, “Critical behavior of the complex dielectric constant near the percolation threshold of a heterogeneous material,” Phys. Rev. Lett. 39, 1222–1225 (1977).
    [Crossref]
  25. S. I. Bozhevolnyi, “Localization phenomena in elastic surface-polariton scattering caused by surface roughness,” Phys. Rev. B 54, 8177–8185 (1996).
    [Crossref]
  26. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: Surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16 (2003).
    [Crossref]
  27. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
    [Crossref] [PubMed]
  28. X. G. Xu, A. E. Tanur, and G. C. Walker, “Phase controlled homodyne infrared near-field microscopy and spectroscopy reveal inhomogeneity within and among individual boron nitride nanotubes,” J. Phys. Chem. A 117, 3348–3354 (2013).
    [Crossref] [PubMed]
  29. B. G. Ghamsari, X. G. Xu, L. Gilburd, G. C. Walker, and P. Berini, “Mid-infrared surface phonon polaritons in boron-nitride nanotubes,” J. Opt. 16, 114008 (2014).
    [Crossref]
  30. X.-G. Xu, unpublished (2017).
  31. Z. H. Kim, B. Liu, and S. R. Leone, “Nanometer-scale optical imaging of epitaxially grown gan and inn islands using apertureless near-field microscopy,” J. Phys. Chem. B 109, 8503–8508 (2005).
    [Crossref]
  32. T. Thonhauser and G. D. Mahan, “Phonon modes in si [111] nanowires,” Phys. Rev. B 69, 075213 (2004).
    [Crossref]
  33. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003).
    [Crossref]
  34. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature (London) 461, 629–632 (2009).
    [Crossref]
  35. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
    [Crossref]
  36. E. J. R. Vesseur, F. J. G. de Abajo, and A. Polman, “Broadband purcell enhancement in plasmonic ring cavities,” Phys. Rev. B 82, 165419 (2010).
    [Crossref]
  37. E. Rephaeli, A. Raman, and S. Fan, “Ultrabroadband photonic structures to achieve high-performance daytime radiative cooling,” Nano Lett. 13, 1457–1461 (2013).
    [PubMed]
  38. P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, and J. H. Shim, “High-precision nanoscale temperature sensing using single defects in diamond,” Nano Lett. 13, 2738–2742 (2013).
    [Crossref] [PubMed]
  39. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009).
    [Crossref] [PubMed]
  40. V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014).
    [Crossref] [PubMed]

2014 (6)

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride,” Science 343, 1125–1129 (2014).
[Crossref] [PubMed]

J. D. Caldwell, A. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, and A. J. Giles, “Sub-diffraction, volume-confined polaritons in the natural hyperbolic material, hexagonal boron nitride,” Nat. Commun. 5, 7507 (2014).
[Crossref]

X. G. Xu, B. G. Ghamsari, J. Jiang, L. Gilburd, G. O. Andreev, C. Y. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014).
[Crossref] [PubMed]

B. G. Ghamsari, X. G. Xu, L. Gilburd, G. C. Walker, and P. Berini, “Mid-infrared surface phonon polaritons in boron-nitride nanotubes,” J. Opt. 16, 114008 (2014).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014).
[Crossref] [PubMed]

2013 (5)

X. G. Xu, A. E. Tanur, and G. C. Walker, “Phase controlled homodyne infrared near-field microscopy and spectroscopy reveal inhomogeneity within and among individual boron nitride nanotubes,” J. Phys. Chem. A 117, 3348–3354 (2013).
[Crossref] [PubMed]

E. Rephaeli, A. Raman, and S. Fan, “Ultrabroadband photonic structures to achieve high-performance daytime radiative cooling,” Nano Lett. 13, 1457–1461 (2013).
[PubMed]

P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, and J. H. Shim, “High-precision nanoscale temperature sensing using single defects in diamond,” Nano Lett. 13, 2738–2742 (2013).
[Crossref] [PubMed]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photon. 7, 394–399 (2013).
[Crossref]

S. Law, V. Podolskiy, and D. Wasserman, “Towards nano-scale photonics with micro-scale photons: the opportunities and challenges of mid-infrared plasmonics,” Nanophotonics 2, 103–130 (2013).
[Crossref]

2012 (3)

R. Stanley, “Plasmonics in the mid-infrared,” Nat. Photon. 6, 409–411 (2012).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (London) 487, 77–81 (2012).

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (London) 487, 5 (2012).

2010 (1)

E. J. R. Vesseur, F. J. G. de Abajo, and A. Polman, “Broadband purcell enhancement in plasmonic ring cavities,” Phys. Rev. B 82, 165419 (2010).
[Crossref]

2009 (2)

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature (London) 461, 629–632 (2009).
[Crossref]

2008 (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008).
[Crossref]

2005 (1)

Z. H. Kim, B. Liu, and S. R. Leone, “Nanometer-scale optical imaging of epitaxially grown gan and inn islands using apertureless near-field microscopy,” J. Phys. Chem. B 109, 8503–8508 (2005).
[Crossref]

2004 (1)

T. Thonhauser and G. D. Mahan, “Phonon modes in si [111] nanowires,” Phys. Rev. B 69, 075213 (2004).
[Crossref]

2003 (3)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[Crossref]

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: Surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16 (2003).
[Crossref]

2002 (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light–matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
[Crossref]

1997 (3)

J. Le Gall, M. Olivier, and J.-J. Greffet, “Experimental and theoretical study of reflection and coherent thermal emission by a sic grating supporting a surface-phonon polariton,” Phys. Rev. B 55, 10105 (1997).
[Crossref]

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (sers),” Phys. Rev. Lett. 78, 1667 (1997).
[Crossref]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

1996 (1)

S. I. Bozhevolnyi, “Localization phenomena in elastic surface-polariton scattering caused by surface roughness,” Phys. Rev. B 54, 8177–8185 (1996).
[Crossref]

1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[Crossref] [PubMed]

1989 (2)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
[Crossref] [PubMed]

J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989).
[Crossref] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

1985 (1)

K. Arya, Z. B. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[Crossref] [PubMed]

1984 (1)

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[Crossref]

1977 (1)

D. J. Bergman and Y. Imry, “Critical behavior of the complex dielectric constant near the percolation threshold of a heterogeneous material,” Phys. Rev. Lett. 39, 1222–1225 (1977).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003).
[Crossref]

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (London) 487, 77–81 (2012).

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Andreev, G. O.

X. G. Xu, B. G. Ghamsari, J. Jiang, L. Gilburd, G. O. Andreev, C. Y. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (London) 487, 5 (2012).

Arya, K.

K. Arya, Z. B. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[Crossref] [PubMed]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003).
[Crossref]

Atwater, H.

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-bn heterostructures,” Nano Lett. 14, 3876–3880 (2014).
[Crossref] [PubMed]

Avouris, P.

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photon. 7, 394–399 (2013).
[Crossref]

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Xia, F.

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Xu, X. G.

X. G. Xu, B. G. Ghamsari, J. Jiang, L. Gilburd, G. O. Andreev, C. Y. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
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X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014).
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B. G. Ghamsari, X. G. Xu, L. Gilburd, G. C. Walker, and P. Berini, “Mid-infrared surface phonon polaritons in boron-nitride nanotubes,” J. Opt. 16, 114008 (2014).
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X. G. Xu, A. E. Tanur, and G. C. Walker, “Phase controlled homodyne infrared near-field microscopy and spectroscopy reveal inhomogeneity within and among individual boron nitride nanotubes,” J. Phys. Chem. A 117, 3348–3354 (2013).
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H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photon. 7, 394–399 (2013).
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A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: Surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16 (2003).
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S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride,” Science 343, 1125–1129 (2014).
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Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature (London) 461, 629–632 (2009).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (London) 487, 5 (2012).

Zhi, C.

X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014).
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X. G. Xu, B. G. Ghamsari, J. Jiang, L. Gilburd, G. O. Andreev, C. Y. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
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H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photon. 7, 394–399 (2013).
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ACS Nano (1)

X. G. Xu, J.-H. Jiang, L. Gilburd, R. G. Rensing, K. S. Burch, C. Zhi, Y. Bando, D. Golberg, and G. C. Walker, “Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene,” ACS Nano 8, 11305–11312 (2014).
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B. G. Ghamsari, X. G. Xu, L. Gilburd, G. C. Walker, and P. Berini, “Mid-infrared surface phonon polaritons in boron-nitride nanotubes,” J. Opt. 16, 114008 (2014).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (London) 487, 5 (2012).

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Figures (5)

Fig. 1
Fig. 1 (a) A scheme of s-SNOM setup. (b) AFM topography of a BNNT on top of another BNNT. The scale bar is 300 nm. (c) Near-field images taken at 1410 cm−1 of the BNNTs in (b). (d) AFM topography of a rough gold surface, the gold nano-grains have diameter from 10nm to 30 nm. The scale bar is 200 nm. (e) s-SNOM IR image of the same substrate area.
Fig. 2
Fig. 2 BNNT near-field images at different infrared frequencies. (a) Topography of a BNNT. (b) AFM phase image of the BNNT. (c–g) Near-field images of the BNNT at 1380 cm−1, 1398 cm−1,1410 cm−1 1422 cm−1, and 1435 cm−1, respectively, taken with π/2 homodyne technique [28]. (h) Fourier transformation of the SPhPs of the BNNT showing a splitting of spatial frequency peaks. (i) Scheme of back reflections of SPhPs by nanotube terminal and gold nano-grains.
Fig. 3
Fig. 3 BNNT near-field images at different infrared frequencies. (a) Topography of a BNNT. (b–f) Near-field images of the BNNT at 1380 cm−1, 1415 cm−1,1420 cm−1 1430 cm−1, and 1590 cm−1, respectively. The color scale is tuned to maintain the same color for near-field response of the gold substrate.
Fig. 4
Fig. 4 (a) Averaged wavevector q as a function of frequency for three samples. One sample is placed on smooth gold substrate, while the other two samples (1 and 2) are placed on rough gold substrates. The method for averaging the wavevector when there are multiple peaks in the spatial frequency is explained in the main text. (b) Profiles of near-field signal of the BNNT at different infrared frequencies. The intensities are normalized assuming the same level of near-field (NF) signal on the gold substrate. The total amplitudes of near-field signals are plotted in the figure, by taking the vector sum of near-field amplitudes collected at two quadrature homodyne phase conditions [31]. Strong localization of SPhPs leads to an increase of total near-field signal that even greatly surpass the strong near-field signal from the rough gold substrate for the frequency 1430 cm−1.
Fig. 5
Fig. 5 Local optical field intensity (in arbitrary units) |2+rtot|2 of SPhPs as a function of the exciting position which mimics the near-field signal as function of AFM tip position along a BNNT of 1.6 μm length for frequencies (a) 1400 cm−1, (b) 1410 cm−1, (c) 1420 cm−1 and (d) 1430 cm−1. The blue curves are for a BNNT on the smooth gold substrate, while the red curves are for a BNNT on the rough gold substrate.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

r t o t = r L + r R + 2 r L r R 1 r L r R .
( E j , r + r R E j , r i q j ( E j , r r R E j , r ) ) = ( M 11 ( R ) M 12 ( R ) M 21 ( R ) M 22 ( R ) ) ( 0 2 i q N E N , r ) ,
r R = i q j M 12 ( R ) M 22 ( R ) i q j M 12 ( R ) + M 22 ( R ) .
M ^ ( R ) =   m = j + 1 N M ^ m 1
M ^ m = ( cos ( q m a ) sin ( q m a ) / q m sin ( q m a ) q m cos ( q m a ) ) ,
M ^ ( L ) =   m = j 1 0 M ^ m .
( r L E j , l + E j , l i q j ( r L E j , l E j , l ) ) = ( M 11 ( L ) M 12 ( L ) M 21 ( L ) M 22 ( L ) ) ( 0 2 i q 0 E 0 , l ) ,
r L = i q j M 12 ( L ) + M 22 ( L ) i q j M 12 ( L ) M 22 ( L ) .
n = 0 ( 2 + r L + r R ) ( r L r R ) n = 2 + r t o t
ε m , eff = 1 2 ( ε eff 0 + ε eff g ) j = 0 N t o t / N p α j δ ε exp [ ( m j N p ) 2 2 σ 2 ]

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