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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 22268–22279

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Multi-scheme approach for efficient surface plasmon polariton generation in metallic conical tips on AFM-based cantilevers

F. De Angelis, R. Proietti Zaccaria, M. Francardi, C. Liberale, and E. Di Fabrizio  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 22268-22279 (2011)
http://dx.doi.org/10.1364/OE.19.022268


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Abstract

We report on the possibility of realizing adiabatic surface plasmon polaritons compression on metallic conical tips built-in on AFM cantilevers by means of different approaches. The problem is faced considering the role of the source, when linear and radial polarizations are assumed, associated to different fabrication schemes. Nano-patterned devices properly combined with metallic conical tips can affect the adiabatic characteristic of the surface electric field. The results are analyzed in terms of tradeoff between fabrication difficulties and device performances. Suggestions on the best possible scheme are provided.

© 2011 OSA

1. Introduction

In the last couple of decades we have been witnessing an enormous advance in the field of nanotechnology. Faster computers, LCD based mobiles, nanoparticles for UV absorption in suntan lotions are just few of many examples where nanotechnology plays a fundamental role. The merit of this is mainly due to the common advance of design and fabrication methods. Present fabrication techniques, such as Focused Ion Beam (FIB) lithography, guarantee a resolution of less than 10 nanometers which is about five times more precise than ten years ago [1

S. Cabrini, A. Carpentiero, R. Kumar, L. Businaro, P. Candeloro, M. Prasciolu, A. Gosparini, C. Andreani, M. De Vittorio, T. Stomeo, and E. Di Fabrizio, “Focused ion beam lithography for two dimensional array structures for photonic applications,” Microelectron. Eng. 78–79, 11–15 (2005). [CrossRef]

]. This strong improvement in the fabrication has determined a new youth for some well-established topics such as plasmonics. In fact, already at the beginning of the 20th century, R. W. Wood had observed unknown optical features in the reflection spectrum of metallic gratings [2

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396 (1902).

]. Again, in 1956 R. Fano introduced the term polariton to define the coupling between bound electrons and incoming photons [3

U. Fano, “Atomic theory of electromagnetic interactions in dense materials,” Phys. Rev. 103(5), 1202–1218 (1956). [CrossRef]

] and in 1968 the scientists H. Raether, E. Kretschmann and A. Otto presented methods for the coupling of photons on a flat metallic surface [4

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216(4), 398–410 (1968). [CrossRef]

,5

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturf. 23A, 2135–2136 (1968).

]. At that point, many theoretical important aspects related to plasmonics were well known but strong limitations in the fabrication techniques could not allow their accurate experimentation at the nano-scale level. Nowadays, we are however in a situation where nanostructures can be fabricated and accurately compared with theory [6

F. De Angelis, C. Liberale, M. L. Coluccio, G. Cojoc, and E. Di Fabrizio, “Emerging fabrication techniques for 3D nano-structuring in plasmonics and single molecule studies,” Nanoscale 3(7), 2689–2696 (2011), doi:. [CrossRef] [PubMed]

] which explains the vast interest for plasmonics more than 100 years after the work of W. Wood.

In this paper we will focus on a peculiar aspect of energy transfer at the nanoscale, known as adiabatic compression of plasmon polaritons [7

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

], particularly applied to AFM-based devices. This effect, which manifests itself on metallic structures such as conical tips, is characterized by the strong concentration of electric energy density on the apex of the nanostructure, associated to a negligible back reflection during the propagation of the surface plasmon polaritons (SPP). During the propagation, both group velocity and phase velocity of the SPP tend to zero at the extreme tip of the cone with a contemporary increase of the surface field. Hence, adiabatic compression can play a very important role in any field requiring high radiation confined in a nanometric spot. For example, metallic conical-like structures with nanometer apex were demonstrated to be useful for applications such as near field scanning optical spectroscopy and microscopy [8

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008). [CrossRef] [PubMed]

–13

A. Weber-Bargioni, A. Schwartzberg, M. Cornaglia, A. Ismach, J. J. Urban, Y. Pang, R. Gordon, J. Bokor, M. B. Salmeron, D. F. Ogletree, P. Ashby, S. Cabrini, and P. J. Schuck, “Hyperspectral nanoscale imaging on dielectric substrates with coaxial optical antenna scan probes,” Nano Lett. 11(3), 1201–1207 (2011). [CrossRef] [PubMed]

].

One of the open issues in adiabatic nanostructures fabricated on AFM cantilever for spectroscopy applications is how to efficiently couple the external laser radiation with the conical waveguide. A plasmonic conical waveguide, such as a silver nanocone, supports efficiently a TM0 mode [7

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

,14

A. J. Babadjanyan, N. L. Margaryan, and Kh. V. Nerkararyan, “Superfocusing of surface polaritons in the conical structure,” J. Appl. Phys. 87(8), 3785 (2000). [CrossRef]

,15

F. De Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, M. L. Coluccio, G. Cojoc, A. Accardo, C. Liberale, R. Proietti Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. Di Fabrizio, “Breaking the diffusion limit with super hydrophobic delivery of few molecules to plasmonic nanofocusing structures,” Nat. Photonics (accepted for publication).

], also known as a radial mode, i.e. a transverse magnetic mode with electric radial symmetry with respect the cone axis. However, this kind of mode cannot be easily generated. A recent work [16

M. Agio, X.-W. Chen, and V. Sandoghdar, “Nanofocusing radially-polarized beams for high-throughput funneling of optical energy to the near field,” Opt. Express 18(10), 10878–10887 (2010). [CrossRef] [PubMed]

] showed that a laser beam radially polarized can efficiently be coupled to a cylindrical waveguide when the laser is properly focused onto the cone base through a high numerical aperture lens (NA~1). In fact, optimal coupling requires a laser spot of the same size of the nanocone base (typically in the range of 300 nm). Moreover, the laser beam must be well aligned with the nanocone axis, with accuracy far below the spot size. In particular, this last requirement represents practical difficulties to satisfy stability during AFM and spectroscopy experiments. Furthermore, due to design and architectural choices, many commercial AFM systems do not allow the employment of additional lenses necessary to focus the laser on to the nanocone base. Therefore, there is a need for developing alternative approaches for the generation of SPP with TM0 symmetry that can be adiabatically compressed at the tip end. In this paper we propose different approaches that allow an efficient and practical coupling through the use, alternatively, of both radially and linearly polarized laser beams.

In the following calculations we have employed two different simulation packages [17

C. S. T. Microwave Studio, 2010, www.cst.com.

,18

Lumerical Solutions, www.lumerical.com.

] based on different numerical algorithms which have provided strong agreement for all the simulated geometrical configurations.

2. Adiabatic compression through radially polarized beam

In this section we shall show the role played, on the same device, by two different kinds of sources such as a radially polarized beam and a linearly polarized plane wave. The device is formed by an ideal conical structure made of silver surrounded by air. The dimensions of the cone are: base 300 nm and height 2.5 μm, which corresponds to an apex angle of 0.12rad (6.88 degrees). The chosen wavelength is λ = 633 nm corresponding to a dielectric function of silver ε = εreal + i εimag = −14.469 + i 1.094 [19

A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef] [PubMed]

]. Figure 1 shows the value of the electric field on the cone when, either a radially polarized source or a plane wave, impinges on the base of the cone. For both cases the direction of propagation of the illumination source is assumed parallel to the axis of the metallic cone, and the laser spot size is comparable to the nanocone base (laser spot 450 nm). As expected, the SPP mode field strongly depends on the source symmetry. In particular, in Fig. 1(A) and 1(B), when a radial source is used, an adiabatic compression of the SPP mode can be observed. On the contrary, as in Fig. 1(C) and 1(D), when the source is a plane wave, no adiabatic compression is produced even though SPP were generated. This can be easily seen by looking at the distance between two adjacent maxima of the field. It confirms that a mode with radial symmetry is associated to adiabatic compression as stated in the pioneering work of Stockman [7

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

]. Hence, a radial source, differently by a plane wave, can generate a TM0 mode in the cone and the SPP energy is delivered to a strongly localized region comparable to the radius of curvature of the cone apex [10

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010). [CrossRef] [PubMed]

]. For completeness, we have also included the vectorial profiles for both the electric field E and the magnetic field H when either radial or linear polarizations were considered.

Fig. 1 Calculated electric field amplitude of an ideal silver nanocone when radially (A, B, and E) and linearly (C, D, and F) polarized laser beams are employed (λ = 633 nm). The cone is 2.5 μm long with 150 nm base radius. The corresponding electric and magnetic field vectors are also calculated in the XY plane starting from Z = 2000nm. (A), (C). Amplitude of Ex component of the electric field showing the progressive reduction of the effective wavelength in A (adiabatic compression) and constant wavelength in B (no compression). Red and blue colors represent the phase of the field. (B), (D). Field lines of the total electric field are reported. (E), (F). Plot of the surface electric field along the nanocone CST package was used with a resolution at the apex of 0.3 nm.

When a radial source (Fig. 1(B)) is considered, the electric field calculated at the tip apex results 400 times higher than at the base of the cone, whereas analytical solution of Maxwell equations suggested a value of 1000 [7

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

]. This difference relies mainly on the role of the absorption that in the present calculations was included by considering the imaginary part of the permittivity.

We notice that these calculations represent the ideal case where SPP are directly excited on the cone. Hereafter, we report FDTD simulations [18

Lumerical Solutions, www.lumerical.com.

] of devices and configurations as close as possible to the experimental situations where both the cantilever and the substrate (the typical sample support) are included. The cone has height 2.0μm, base 300nm and vertex angle 9°. Three mesh parameters have been chosen for the cone. In particular, from the base to 150nm from the apex it shows 5nm mesh, which becomes 1nm up to the last few nanometers of the cone where it turns to be 0.5nm (unless otherwise specified). The simulative region is roughly 1μmx1μmx3μm with perfectly matched layer boundary conditions. In Fig. 2(A) is shown the electric field intensity on the nanocone. The source is a radially polarized beam perfectly aligned with the cone axis and focused on to the cone base (lens NA = 0.7, λ = 630 nm, beam spot ~450 nm, focus depth ~600 nm). In proximity of the tip we placed a dielectric slab (n = 1.5, mesh = 200nm) representing the substrate under investigation (see sketch in the figure). The slab is located at 0.5 nm from the tip end, corresponding to the minimum possible gap when operating in “tapping mode” during Raman-AFM measurements. In Fig. 2 the electric field is normalized with respect to the source amplitude, differently from Fig. 1 where it was normalized with respect to the field at the cone base. To better compare the presented results, we set the electric field scale bars of the Figs. 2–4 to the same limits. In Fig. 2(B) is reported a detail of the tip-substrate region with tip-end mesh equal to 0.2nm and in Fig. 2(C) the electric field induced onto the substrate by the plasmonic tip (XY plane). The latter consists of a spot of about 5 nm in diameter with a maximum field about 110 times higher than the laser source. This number gives a clear and direct evaluation of the advantage of using plasmonic tip and adiabatic compression in terms of localization of the exciting field. The electric field shows also a strong contribute along the z-axis (not shown) which, together with the radial shape as in Fig. 2(C), defines a typical TM0 mode.

Fig. 2 Calculated electric field intensity for a real silver nanocone (radius of curvature 5 nm, λ = 633 nm) interacting with a dielectric (n = 1.5, gap 0.5 nm) when radially polarized source is employed. (A). Overall device including silicon nitride cantilever (n = 2.0327 + i 0.0118 [20

A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt. 37(22), 5291–5297 (1998). [CrossRef] [PubMed]

]). (B). Detail of intensity map at the tip end. (C). Electric field intensity induced by the tip on to the substrate. Black arrows are added to show the direction of the electric field: intensity distribution and direction correspond to the same radial polarization used to excite plasmons at the cone base. It is interesting to notice how the 0.5nm mesh utilized in Fig. 2(A) produces two hot spots close to the tip end which are removed by the finer 0.2nm mesh of Fig. 2(B). FDTD Lumerical package was used.
Fig. 4 Electric field intensity for three different devices illuminated with linearly polarized laser beam (λ = 633 nm). On the left, the sketch of the setup. (A). tilted laser beam (40°). The vectorial representations of both electric and magnetic fields are calculated at z = 0 nm (base of the cone) and z = 1900 nm (100 nm from the tip end). Notice how the vectorial field becomes TM0 in the vicinity of the cone apex. (B). A phase shifter step patterned on the silicon nitride cantilever (315 nm thick. The total cantilever thickness is 500 nm) induces an optimal phase shift that enables a TM0-like mode. (C). A silicon based photonic crystal cavity L3 is used to couple the incident linearly polarized laser beam (with a tilt angle°) with the nanocone. Both FDTD Lumerical and CST packages were used.

As previously mentioned, the direct employment of a radially polarized beam can present some experimental difficulties. In fact, in AFM-Raman experiments the presence of many sources of misalignment between the laser beam and the nanocone prevents from an efficient generation of SPP-TM0 wave. In particular, the main sources of errors are: X-Y translational misalignments, focusing errors, and tilting of the cantilever. The efficacy of using radial polarization relies on the fact that the SPP on the cone and the laser beam have the same radial symmetry therefore, when the laser intensity profile symmetrically overlaps the base of the nanocone, the condition of optimal coupling is achieved. Unfortunately, this “free error” overlap doesn’t occur stably during experiments. Furthermore, the requirement of high NA lenses causes additional difficulties. For instance, in the example of Fig. 2, a beam misalignment of 150 nm with respect of the cone axis and an out of focus of 300 nm implies a decrease of the electric field, at the cone apex, of a factor 2. Hence, misalignment is a source of uncertainty that can easily cause misinterpretation in spectroscopy measurements. A more stable situation could be obtained by fabricating larger nanocones which, having larger base size, would allow the employment of lower NA lenses and then easier alignment. We performed calculations to evaluate this approach (data not reported for brevity) but, although it represents an improvement in term of ease of use, it does not overcome the misalignment problem. In addition, nanocones with wider base and same height are more difficult to fabricate, and they do not satisfy the requirements for adiabaticity (small vertex angle). A better approach consists in fabricating on the cantilever additional micro-nano structures that acting as microlenses are able to focus and to align the laser beam on to the nanocone. An example of this structure is a Fresnel micro zone plate.

The zone plate, compatible with radial symmetry, weakens these constrains and, ought to its lens properties with low working distance, allows the coupling with the nanocone. We simulated a zone plate with focal length of 1 μm, 8 zones, wavelength λ = 630 nm and silver thickness of 80 nm. The result is reported in Fig. 3 together with a sketch of the device. The zone plate is illuminated through a lens of NA = 0.1 (spot size ~3.5 μm) with a radially polarized beam. By comparing Fig. 2 (isolated cone) with Fig. 3 (cone + zone plate) the electric field enhancement changes from da 110 to 71, but for the latter case the difficulties related to misalignments are strongly reduced. In other words, the coupling with zone plate is less sensitive to misalignment and defocusing errors compared to a cone with 300 nm base directly illuminated with a radial source. Considering further improvements, such as, high efficient blazed zone plate [21

Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express 18(4), 3438–3443 (2010). [CrossRef] [PubMed]

], our calculations suggest that this device can be exploited to overcome the mentioned difficulties associated to radially polarized beams.

Fig. 3 Calculated electric field intensity for a real nanocone (same as in Fig. 2) when a micro zone plate (λ = 633 nm, focal length 1 μm) is used to focus the laser beam on the nanocone base. FDTD Lumerical software was used.

3. Adiabatic compression through linearly polarized beam

In the previous section we have considered the employment of radially polarized beams for the generation of pure TM0 modes able to propagate along a silver nanocone and reach the tip end where adiabatic compression occurs. We also discussed some technical difficulties that make this approach an experimental challenge. In this section we show how linearly polarized laser beams can be used to generate modes which are still able to be adiabatically compressed. We call these modes TM0-like modes. The intention is to provide a simple experimental method accessible to a wider class of instruments.

We will consider three different configurations as a function of increasing complexity and efficiency. The simplest way for the creation of a TM0-like mode on a metallic cone by means of a linearly polarized wave is by impinging on the cone along a direction not strictly parallel to its axis. This method does not require, for the laser coupling, any additional patterned structure at the base of the metallic cone. Figure 4(A) shows the results of the simulations. The cone has a base of 300 nm, height equal to 2.0 μm corresponding to an apex angle of 0.16 rad (9 degrees). The calculations are tailored on fabrication results that can be routinely reached by the current technology. The apex of the cone was chosen with a radius of curvature of 5 nm. The wavelength λ is 633 nm which corresponds to a silver dielectric function ε = −14.469 + i 1.094. The impinging plane wave is tilted by an angle between 10 and 40 degrees respect to the axis of the cone. The working principle is based on a phase shift at the base of the cone due to the impinging angle. In fact, due to different path lengths, the plane wave will reach opposite sides of the cone base with a phase difference which will excite the SPP at different times. The calculated electric field (Fig. 4(A)), after the propagation along the cone surface, confirms this behavior by showing the radial symmetry of the field at the cone apex. The typical dipole structure related to a linearly polarized plane wave is lost during the propagation and only the mode with radial symmetry prevails close to the cone apex. This behavior is clearly shown in the vectorial plot on the right side of Fig. 4(A). Finally, it is important to notice that the field profile along the cone depends on the impinging angle. In fact, while a tilted wave always realizes a TM0-like mode close to the cone-end, the mode intensity is strictly related to the angle between the incoming source and the axis of the cone. The reason can be understood by the phase-shift mechanism behind the creation of a TM0 mode.

As shown on the left of Fig. 4(A), under the dielectric slab (the cantilever) is placed a metal layer (silver, 80 nm thick) with an aperture of 600 nm aligned with the nanocone. Its purpose is to stop the fraction of the laser beam that does not impinge on the nanocone. This aperture enables the exploitation of lager laser beam without significant drawback. The problem of the focus and misalignment in now completely removed. For instance, a lens with NA~0.1 (generating a beam spot and focus of few microns) can be easily used to deliver the laser beam on to the metal aperture.

The next approach to create a TM0-like mode on a metallic cone involves a minimal increase of fabrication difficulties. In fact, the additional fabrication regards an asymmetric dielectric slab located at the base of the cone. The slab, made of Si3N4 (refractive index equal 2) with a thickness step of 315nm, is partially aligned with the cone axis, as shown in Fig. 4(B). Its role is to provide a phase shift to the impinging light. In fact, when an axially aligned plane wave is shined on the base of the cone and to the “phase shifter”, an SPP is generated and it propagates along the cone to the apex with the intensity profile as shown in Fig. 4(B). The working mechanism is similar to the situation described in the previous section: the source field undergoes a phase shift due to the asymmetric slab which creates an anti-phase field at the base of the cone. This phase shift generates an SPP mode that, as in the previous case, during the propagation along the cone will become a TM0-like mode when close to the tip apex (right side of Fig. 4(A)) producing an intense radially symmetric field.

In Fig. 4(C) is reported a L3 Cavity [10

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010). [CrossRef] [PubMed]

] illuminated by a tilted beam. The generated plasmonic mode has a similar structure and behavior as in the two previous cases. In this last case, when optimized at 633 nm wavelength, the SPP field at the tip apex is the highest, as shown in the comparison calculations of Fig. 6 .

Fig. 6 Electric field amplitudes with respect to the wavelength for all the considered devices. C_RP: isolated nanocone, radial polarization. CT_LP: isolated nanocone, tilted linear polarization. CPS_LP: nanocone and phase shift step, linear polarization. CZP_RP: nanocone and zone plate, radial polarization. CPhCL3_LP: nanocone and photonic crystal cavity, tilted linear polarization.

4. Fully radial symmetry devices

The final device we present relies on a photonic crystal (PhC) slab cavity [22

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000). [CrossRef] [PubMed]

] able to support radial modes. Many are the applications involving PhC [23

A. J. Turberfield, M. Campbell, D. N. Sharp, M. T. Harrison, and R. G. Denning, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef] [PubMed]

–25

R. Proietti Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, and S. Kawata, “Manipulating full photonic band gaps in two dimensional birefringent photonic crystals,” Opt. Express 16(19), 14812–14820 (2008). [CrossRef] [PubMed]

], such as non-linear phenomena [26

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. K. De Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68(16), 161306 (2003). [CrossRef]

–28

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15(24), 16161–16176 (2007). [CrossRef] [PubMed]

], SERS analysis [29

G. Das, F. Mecarini, F. Gentile, F. De Angelis, H. Mohan Kumar, P. Candeloro, C. Liberale, G. Cuda, and E. Di Fabrizio, “Nano-patterned SERS substrate: application for protein analysis vs. temperature,” Biosens. Bioelectron. 24(6), 1693–1699 (2009). [CrossRef] [PubMed]

], light guiding [30

M. Galli, M. Agio, L. C. Andreani, L. Atzeni, D. Bajoni, G. Guizzetti, L. Businaro, E. Di Fabrizio, F. Romanato, and A. Passaseo, “Optical properties and photonic bands of GaAs photonic crystal waveguides with tilted square lattice,” Eur. Phys. J. B 27(1), 79–87 (2002). [CrossRef]

–32

H. Zhao, R. Proietti Zaccaria, P. Verma, J. Song, and H. Sun, “Single-mode operation regime for 12-fold index-guiding quasicrystal optical fibers,” Appl. Phys. B 100(3), 499–503 (2010). [CrossRef]

] or for employment in chemical and biological sensor technology [33

J. F. Song and R. Proietti Zaccaria, “Manipulation of light transmission through sub-wavelength hole array,” J. Opt. A, Pure Appl. Opt. 9(9), S450–S457 (2007). [CrossRef]

], however to the best of our knowledge this is the first time that a photonic crystal is proposed for supporting radial modes. In particular, in Fig. 5(A) and 5(B) is reported the calculation of a photonic crystal cavity named H1 which selects a radial mode at λ = 633 nm wavelength. This radial mode has the complete symmetry to generate a pure TM0 SPP mode as calculated in Fig. 1(A) and 1(B). The photonic crystal utilized in this configuration follows a hexagonal periodicity of air holes in dielectric. The chosen material is silicon showing a refractive index n = 3.917 + i 0.0122 at λ = 633 nm [34

S. Adachi, “Model dielectric constants of Si and Ge,” Phys. Rev. B Condens. Matter 38(18), 12966–12976 (1988). [CrossRef] [PubMed]

]. The crystal periodicity is P = 290 nm with holes radius R = 0.4P and slab thickness t = 162 nm. The cavity is H1-like, namely it is defined by a single missing hole. This device has the property of well supporting a range of radial modes. In particular, the electric field distribution of the 633nm TM0 mode is shown in Fig. 5(A) and its vectorial structure in Fig. 5(B). Details of the band analysis necessary for the design of the present photonic crystal slab will follow in a next publication. In Fig. 5(C) is reported the nanocone on H1 radial mode cavity. As can be seen in the right side of the Fig. 5(C) the SPP mode has the structure of a pure TM0 mode, adiabatically compresses during the propagation. From practical point of view, this device should be less sensitive to alignment and defocusing experimental errors.

Fig. 5 Photonic crystal slab supporting radial modes at λ = 633 nm. (A) Total electric field at z = 162 nm (slab thickness). (B) Vectorial plot of the electric field at z = 162 nm. Its radial TM0 structure is well evidenced by the figure. (C) Full SPP propagation and field calculation for a silver nanocone on H1 radial cavity. Notice that in this case the field has a pure TM0 structure, both at the base of the cone where the SPP is launched, and at the apex proximity (dashed line at top of (C)). CST package was used.

5. Discussion: a comparison between the proposed approaches

In Fig. 6 are reported the amplitude of the electric fields around the tip apex for five configurations. In particular, for the isolated nanocone shined by a radial beam (C_RP), the cone on a zone plate (CZP_RP) with radial source, and the isolated nanocone with tilted linearly polarized laser beam (CT_LP), we calculated the absolute value of the electric field in the range of wavelengths from 530 nm to 830 nm. These configurations show an oscillating behavior of the field amplitude as a function of the source wavelength. Similar behavior was already observed in [16

M. Agio, X.-W. Chen, and V. Sandoghdar, “Nanofocusing radially-polarized beams for high-throughput funneling of optical energy to the near field,” Opt. Express 18(10), 10878–10887 (2010). [CrossRef] [PubMed]

] and its explanation is due to the setting up of a stationary field at the base of the cone that depends both on its radius of curvature and the exciting wavelength. We point out that the launching field profile at the cone base is different, as a function of the wavelength. Furthermore, for the two configurations CZP_RP and the cone coupled to a L3 photonic crystal cavity [10

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010). [CrossRef] [PubMed]

] with tilted linearly polarized source (CPhCL3_LP), due to the strong dependence of the optimization conditions on the wavelength, we have calculated the value of the electric field only at λ = 633nm. The results show that the use of tilted beam on a L3 cavity is the most efficient device for the generation of TM0-like modes on a metallic nanocone. However, from fabrication point of view, this is also the most complicated configuration; hence a tradeoff between efficiency and fabrication has to be considered. Furthermore, considering the simpler use and fabrication, the cone combined with phase shifter (Fig. 4(B)) deserves attention. We have chosen to exclude from Fig. 6 the cone on H1 photonic crystal cavity because the complete optimizations have to be considered. However, preliminary results show that the field is even higher than that in the L3 cavity. This is not surprising considering its intrinsic capability of supporting pure TM0 modes.

Finally, we include examples of fabricated devices, whose architecture characteristics can be considered as a general demonstration of the lithographic feasibility of a wider category photonic devices. In Figs. 7(A) –7(C) it is shown an isolated cone (A), a cone on L3 Si3N4 cavity (B) and a H1 silicon cavity (C).

Fig. 7 Example of nanocone and cavities fabricated on AFM cantilever. (A) isolated cone on AFM Si3N4 cantilever 100 nm thick. (B) nanocone on L3 cavity fabricated on AFM Si3N4 cantilever 100 nm thick. (C) H1 cavity on AFM Si Cantilever 1 micron thick.

6. Conclusion

We have introduced different devices for the realization of adiabatic compression on noble metal nanocones. We have faced the problem from a simulation point of view but, at the same time, we have taken into account the most important experimental issues, such as, fabrication difficulties, illumination source properties and beam/device alignment. Our results show that for a realistic device all the reported structures can be used for the creation of TM0-like or pure TM0 modes on noble metal nanocones with relevant enhancement. We believe that adiabatic compression will play an important role for future applications, such as in single molecule detection, where AFM microscopy will be combined with enhanced spectroscopy, particularly for measurements where a strong signal to noise ratio is needed.

Acknowledgments

The authors gratefully acknowledge support from European Projects SMD FP7-NMP-2008-SMALL-2 proposal No. CP-FP 229375-2 and Nanoantenna FP7-HEALTH-2009, Grant No. 241818. FOCUS project proposal #270483- ICT-2009 8.7 - FET proactive 7: Molecular Scale Devices and Systems.

References and links

1.

S. Cabrini, A. Carpentiero, R. Kumar, L. Businaro, P. Candeloro, M. Prasciolu, A. Gosparini, C. Andreani, M. De Vittorio, T. Stomeo, and E. Di Fabrizio, “Focused ion beam lithography for two dimensional array structures for photonic applications,” Microelectron. Eng. 78–79, 11–15 (2005). [CrossRef]

2.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396 (1902).

3.

U. Fano, “Atomic theory of electromagnetic interactions in dense materials,” Phys. Rev. 103(5), 1202–1218 (1956). [CrossRef]

4.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216(4), 398–410 (1968). [CrossRef]

5.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturf. 23A, 2135–2136 (1968).

6.

F. De Angelis, C. Liberale, M. L. Coluccio, G. Cojoc, and E. Di Fabrizio, “Emerging fabrication techniques for 3D nano-structuring in plasmonics and single molecule studies,” Nanoscale 3(7), 2689–2696 (2011), doi:. [CrossRef] [PubMed]

7.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

8.

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008). [CrossRef] [PubMed]

9.

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46(2), 20101 (2009). [CrossRef]

10.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010). [CrossRef] [PubMed]

11.

N. C. Lindquist, P. Nagpal, A. Lesuffleur, D. J. Norris, and S. H. Oh, “Three-dimensional plasmonic nanofocusing,” Nano Lett. 10(4), 1369–1373 (2010). [CrossRef] [PubMed]

12.

C. C. Neacsu, S. Berweger, R. L. Olmon, L. V. Saraf, C. Ropers, and M. B. Raschke, “Near-field localization in plasmonic superfocusing: a nanoemitter on a tip,” Nano Lett. 10(2), 592–596 (2010). [CrossRef] [PubMed]

13.

A. Weber-Bargioni, A. Schwartzberg, M. Cornaglia, A. Ismach, J. J. Urban, Y. Pang, R. Gordon, J. Bokor, M. B. Salmeron, D. F. Ogletree, P. Ashby, S. Cabrini, and P. J. Schuck, “Hyperspectral nanoscale imaging on dielectric substrates with coaxial optical antenna scan probes,” Nano Lett. 11(3), 1201–1207 (2011). [CrossRef] [PubMed]

14.

A. J. Babadjanyan, N. L. Margaryan, and Kh. V. Nerkararyan, “Superfocusing of surface polaritons in the conical structure,” J. Appl. Phys. 87(8), 3785 (2000). [CrossRef]

15.

F. De Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, M. L. Coluccio, G. Cojoc, A. Accardo, C. Liberale, R. Proietti Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. Di Fabrizio, “Breaking the diffusion limit with super hydrophobic delivery of few molecules to plasmonic nanofocusing structures,” Nat. Photonics (accepted for publication).

16.

M. Agio, X.-W. Chen, and V. Sandoghdar, “Nanofocusing radially-polarized beams for high-throughput funneling of optical energy to the near field,” Opt. Express 18(10), 10878–10887 (2010). [CrossRef] [PubMed]

17.

C. S. T. Microwave Studio, 2010, www.cst.com.

18.

Lumerical Solutions, www.lumerical.com.

19.

A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef] [PubMed]

20.

A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt. 37(22), 5291–5297 (1998). [CrossRef] [PubMed]

21.

Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express 18(4), 3438–3443 (2010). [CrossRef] [PubMed]

22.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000). [CrossRef] [PubMed]

23.

A. J. Turberfield, M. Campbell, D. N. Sharp, M. T. Harrison, and R. G. Denning, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef] [PubMed]

24.

R. Proietti Zaccaria, S. Shoji, H. B. Sun, and S. Kawata, “Multi-shot interference approach for any kind of Bravais lattice,” Appl. Phys. B 93(1), 251–256 (2008). [CrossRef]

25.

R. Proietti Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, and S. Kawata, “Manipulating full photonic band gaps in two dimensional birefringent photonic crystals,” Opt. Express 16(19), 14812–14820 (2008). [CrossRef] [PubMed]

26.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. K. De Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68(16), 161306 (2003). [CrossRef]

27.

A. M. Malvezzi, F. Cattaneo, G. Vecchi, M. Falasconi, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. De Vittorio, “Second-harmonic generation in reflection and diffraction by a GaAs photonic-crystal waveguide,” J. Opt. Soc. Am. B 19(9), 2122 (2002). [CrossRef]

28.

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15(24), 16161–16176 (2007). [CrossRef] [PubMed]

29.

G. Das, F. Mecarini, F. Gentile, F. De Angelis, H. Mohan Kumar, P. Candeloro, C. Liberale, G. Cuda, and E. Di Fabrizio, “Nano-patterned SERS substrate: application for protein analysis vs. temperature,” Biosens. Bioelectron. 24(6), 1693–1699 (2009). [CrossRef] [PubMed]

30.

M. Galli, M. Agio, L. C. Andreani, L. Atzeni, D. Bajoni, G. Guizzetti, L. Businaro, E. Di Fabrizio, F. Romanato, and A. Passaseo, “Optical properties and photonic bands of GaAs photonic crystal waveguides with tilted square lattice,” Eur. Phys. J. B 27(1), 79–87 (2002). [CrossRef]

31.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010). [CrossRef] [PubMed]

32.

H. Zhao, R. Proietti Zaccaria, P. Verma, J. Song, and H. Sun, “Single-mode operation regime for 12-fold index-guiding quasicrystal optical fibers,” Appl. Phys. B 100(3), 499–503 (2010). [CrossRef]

33.

J. F. Song and R. Proietti Zaccaria, “Manipulation of light transmission through sub-wavelength hole array,” J. Opt. A, Pure Appl. Opt. 9(9), S450–S457 (2007). [CrossRef]

34.

S. Adachi, “Model dielectric constants of Si and Ge,” Phys. Rev. B Condens. Matter 38(18), 12966–12976 (1988). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Integrated Optics Devices

History
Original Manuscript: July 8, 2011
Revised Manuscript: August 25, 2011
Manuscript Accepted: August 26, 2011
Published: October 24, 2011

Virtual Issues
Collective Phenomena (2011) Optics Express

Citation
F. De Angelis, R. Proietti Zaccaria, M. Francardi, C. Liberale, and E. Di Fabrizio, "Multi-scheme approach for efficient surface plasmon polariton generation in metallic conical tips on AFM-based cantilevers," Opt. Express 19, 22268-22279 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22268


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References

  1. S. Cabrini, A. Carpentiero, R. Kumar, L. Businaro, P. Candeloro, M. Prasciolu, A. Gosparini, C. Andreani, M. De Vittorio, T. Stomeo, and E. Di Fabrizio, “Focused ion beam lithography for two dimensional array structures for photonic applications,” Microelectron. Eng.78–79, 11–15 (2005). [CrossRef]
  2. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag.4, 396 (1902).
  3. U. Fano, “Atomic theory of electromagnetic interactions in dense materials,” Phys. Rev.103(5), 1202–1218 (1956). [CrossRef]
  4. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys.216(4), 398–410 (1968). [CrossRef]
  5. E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturf.23A, 2135–2136 (1968).
  6. F. De Angelis, C. Liberale, M. L. Coluccio, G. Cojoc, and E. Di Fabrizio, “Emerging fabrication techniques for 3D nano-structuring in plasmonics and single molecule studies,” Nanoscale3(7), 2689–2696 (2011), doi:. [CrossRef] [PubMed]
  7. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93(13), 137404 (2004). [CrossRef] [PubMed]
  8. F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett.8(8), 2321–2327 (2008). [CrossRef] [PubMed]
  9. Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys.46(2), 20101 (2009). [CrossRef]
  10. F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol.5(1), 67–72 (2010). [CrossRef] [PubMed]
  11. N. C. Lindquist, P. Nagpal, A. Lesuffleur, D. J. Norris, and S. H. Oh, “Three-dimensional plasmonic nanofocusing,” Nano Lett.10(4), 1369–1373 (2010). [CrossRef] [PubMed]
  12. C. C. Neacsu, S. Berweger, R. L. Olmon, L. V. Saraf, C. Ropers, and M. B. Raschke, “Near-field localization in plasmonic superfocusing: a nanoemitter on a tip,” Nano Lett.10(2), 592–596 (2010). [CrossRef] [PubMed]
  13. A. Weber-Bargioni, A. Schwartzberg, M. Cornaglia, A. Ismach, J. J. Urban, Y. Pang, R. Gordon, J. Bokor, M. B. Salmeron, D. F. Ogletree, P. Ashby, S. Cabrini, and P. J. Schuck, “Hyperspectral nanoscale imaging on dielectric substrates with coaxial optical antenna scan probes,” Nano Lett.11(3), 1201–1207 (2011). [CrossRef] [PubMed]
  14. A. J. Babadjanyan, N. L. Margaryan, and Kh. V. Nerkararyan, “Superfocusing of surface polaritons in the conical structure,” J. Appl. Phys.87(8), 3785 (2000). [CrossRef]
  15. F. De Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, M. L. Coluccio, G. Cojoc, A. Accardo, C. Liberale, R. Proietti Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. Di Fabrizio, “Breaking the diffusion limit with super hydrophobic delivery of few molecules to plasmonic nanofocusing structures,” Nat. Photonics (accepted for publication).
  16. M. Agio, X.-W. Chen, and V. Sandoghdar, “Nanofocusing radially-polarized beams for high-throughput funneling of optical energy to the near field,” Opt. Express18(10), 10878–10887 (2010). [CrossRef] [PubMed]
  17. C. S. T. Microwave Studio, 2010, www.cst.com .
  18. Lumerical Solutions, www.lumerical.com .
  19. A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37(22), 5271–5283 (1998). [CrossRef] [PubMed]
  20. A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt.37(22), 5291–5297 (1998). [CrossRef] [PubMed]
  21. Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express18(4), 3438–3443 (2010). [CrossRef] [PubMed]
  22. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature407(6804), 608–610 (2000). [CrossRef] [PubMed]
  23. A. J. Turberfield, M. Campbell, D. N. Sharp, M. T. Harrison, and R. G. Denning, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000). [CrossRef] [PubMed]
  24. R. Proietti Zaccaria, S. Shoji, H. B. Sun, and S. Kawata, “Multi-shot interference approach for any kind of Bravais lattice,” Appl. Phys. B93(1), 251–256 (2008). [CrossRef]
  25. R. Proietti Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, and S. Kawata, “Manipulating full photonic band gaps in two dimensional birefringent photonic crystals,” Opt. Express16(19), 14812–14820 (2008). [CrossRef] [PubMed]
  26. A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. K. De Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B68(16), 161306 (2003). [CrossRef]
  27. A. M. Malvezzi, F. Cattaneo, G. Vecchi, M. Falasconi, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. De Vittorio, “Second-harmonic generation in reflection and diffraction by a GaAs photonic-crystal waveguide,” J. Opt. Soc. Am. B19(9), 2122 (2002). [CrossRef]
  28. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express15(24), 16161–16176 (2007). [CrossRef] [PubMed]
  29. G. Das, F. Mecarini, F. Gentile, F. De Angelis, H. Mohan Kumar, P. Candeloro, C. Liberale, G. Cuda, and E. Di Fabrizio, “Nano-patterned SERS substrate: application for protein analysis vs. temperature,” Biosens. Bioelectron.24(6), 1693–1699 (2009). [CrossRef] [PubMed]
  30. M. Galli, M. Agio, L. C. Andreani, L. Atzeni, D. Bajoni, G. Guizzetti, L. Businaro, E. Di Fabrizio, F. Romanato, and A. Passaseo, “Optical properties and photonic bands of GaAs photonic crystal waveguides with tilted square lattice,” Eur. Phys. J. B27(1), 79–87 (2002). [CrossRef]
  31. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett.10(3), 891–895 (2010). [CrossRef] [PubMed]
  32. H. Zhao, R. Proietti Zaccaria, P. Verma, J. Song, and H. Sun, “Single-mode operation regime for 12-fold index-guiding quasicrystal optical fibers,” Appl. Phys. B100(3), 499–503 (2010). [CrossRef]
  33. J. F. Song and R. Proietti Zaccaria, “Manipulation of light transmission through sub-wavelength hole array,” J. Opt. A, Pure Appl. Opt.9(9), S450–S457 (2007). [CrossRef]
  34. S. Adachi, “Model dielectric constants of Si and Ge,” Phys. Rev. B Condens. Matter38(18), 12966–12976 (1988). [CrossRef] [PubMed]

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