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Transversely Divergent Second Harmonic Generation by Surface Plasmon Polaritons on Single Metallic Nanowires Yang Li, Meng Kang, Junjun Shi, Ke Wu, Shunping Zhang, and Hongxing Xu Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04016 • Publication Date (Web): 15 Nov 2017 Downloaded from http://pubs.acs.org on November 16, 2017

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Transversely Divergent Second Harmonic Generation by Surface Plasmon Polaritons on Single Metallic Nanowires Yang Li, #, † Meng Kang, #, † Junjun Shi,‡ Ke Wu,§ Shunping Zhang, *, † and Hongxing Xu*, †, ‡ †

School of Physics and Technology, Center for Nanoscience and Nanotechnology,

and Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, Wuhan University, Wuhan 430072, China. ‡

The Institute for Advanced Studies, Wuhan University, Wuhan 430072, China.

§

School of Physics, Huazhong University of Science and Technology, Wuhan,

430074, China.

Corresponding Author *E-mail: [email protected]. Phone: +8627 68752219. *E-mail: [email protected]. Phone: +8627 68752253.

ACS Paragon Plus Environment

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ABSTRACT: Coherently adding up signal wave from different locations are a prerequisite for realizing efficient nonlinear optical processes in traditional optical configurations. While nonlinear optical processes in plasmonic waveguides with subwavelength light confinement are in principle desirable for enhancing nonlinear effects, so far it has been difficult to improve the efficiency due to the large momentum mismatch. Here we demonstrate, using remotely excited surface plasmon polaritons (SPPs), axial collimated but transversely divengent second harmonic (SH) generation in a single silver nanowire - monolayer molybdenum disulfide hybrid system. Fourier imaging of the generated SH signal confirms the momentum conservation conditions between the incident and reflected SPPs and reveals distinct features inherent to the 1D plasmonic waveguides: (i) the SH photons are collimated perpendicular to the nanowire axis but are divergent within the perpendicular plane; (ii) the collimation (divergence) is inversely proportional to the length of the active region (lateral confinement of the SPPs); and (iii) the SH emission pattern resembles that of an aligned dipole chain on top of the substrate, with an emission peak at the critical angle. Our results pave the way to generate and manipulate SH emission around subwavelength waveguides and open up new possibilities for realizing high efficiency on-chip nonlinear optics.

KEYWORDS: Nonlinear waveguide, surface plasmons, light-matter interaction, second harmonic generation, transition metal dichalcogenides.


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Surface plasmon polaritons (SPPs) – electromagnetic wave-coupled charge density oscillation confined on metal-dielectric interfaces – have attracted significant interests in the past two decades because they enable on-demand light manipulation on the nanoscale or on-chip photonic circuits

1, 2

. Nonlinear processes involving SPPs are

one of the key functionalities to this end

3, 4

, on account of (i) the local field

enhancement caused by the tight confinement of the SPPs the basic requirement for all-optical logic or switching

5-7

, and (ii) nonlinearity is

8, 9

. Reaching the efficient

nonlinear processes is a crucial step for realizing the potential of SPPs in optical systems - this would allow the nonlinear responses from different positions to be added up coherently

10, 11

. However, the demonstrations of momentum conserved

nonlinear processes in plasmonic systems have so far been limited.

The difficulty to realize phase matching in plasmonic systems originates in the highly dispersive nature of SPPs and that their momentums are larger than that of photons in air. Therefore, harmonic processes – such as two SPPs annihilating and generating a second harmonic (SH) photon or a SH SPPs – are impossible because of phase mismatch

12

. Addressing this would require well-designed configurations for

nonlinear momentum compensation

13-15

. Successful attempts to achieve phase

matched second harmonic generation (SHG) rely on the use of the Kretschmann configuration for copropagating SPPs

16

. For counter-propagating SPPs

17, 18

, the

momentum conservation condition is fulfilled automatically so that directional SHG beaming can be observed. However, these demonstrations are limited to 2D metal films where SPPs are only confined in the normal direction. Achieving collimated


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beaming in lower dimension systems are more appealing to take full advantage of the field confinement19.

On the other hand, in 0D plasmonic nanostructures where field enhancement is significant, the concept of phase matching is invalid 20-27. SHG from metal films with random nanoscale roughness is proven to be depolarized and incoherent

20, 21

; For a

well-defined but centrosymmetric nanoparticle, SHG is generally very weak

22

. A

symmetry breaking together with modal overlapping mechanism is essential for efficient second order nonlinearity in 0D systems

23, 24, 26

. For 1D plasmonic

waveguides, the translational symmetry along the guide axis is recovered so that the phase matching conditions can in principle be available. Although SHG from a nanowaveguide has been elegantly demonstrated by counter-propagating guided pulses in a photonic waveguide or by copropagating SPPs in a plasmonic waveguide 28, 29

, the momentum conversed coherent SHG from 1D plasmonic systems has not

been realized. In this Letter, we present the first experiment observation of momentum conversed coherent SHG in a 1D plasmonic nanowire (NW) waveguide and identify distinctive features that are inherent to the 1D system. Collimated SH emission perpendicular to the NW was observed and featured as those from an aligned dipole chain on top of the substrate, with an emission peak at the critical angle.

The success of the demonstration presented in this Letter relies on the combination of a chemically synthesized crystallized silver (Ag) NW and a single


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atomic layer of molybdenum disulfide (MoS2) with large second order susceptibility 30-32

(Figure 1a). The Ag NWs with atomic smooth surfaces were synthesized by a

self-seeding polyol process

33

, which resulted in NWs with pentagonal crossection,

with a typical diameter of 120 nm and length of 10 µm. Single crystal MoS2 was grown on silica substrate via solid-source chemical vapor deposition (S1 in Supporting Information). The choice of transparent silica substrate (thickness 0.17 mm) ensures the low propagation loss of the fundamental mode of SPPs on Ag NW 34, and enables the collection of signals from the substrate sides of the sample. The interaction with the substrate pulls the near-field of the fundamental mode of SPPs close to the NW-substrate interface (Figure 1c). The use of monolayer MoS2 with non-centrosymmetry provides a thin, but intense source of SH polarization, right at the position where the near-field of SPPs is maximized (Figure 1b,c).

To realize and characterize our setup we drop casted and dried Ag NWs onto the monolayer MoS2 and used a femtosecond laser beam (central wavelength 796 nm, pulse duration ~ 20 fs) focused on one end of the Ag NW through an objective. When the incident laser light shone onto the NW has polarization that is parallel to the NW axis, fundamental SPPs are excited and propagate along the NW. A fraction of the guided SPPs on the NWs are scattered off at the other end, and the rest partly reflected back to the Ag NW. The SH photons emitted from the hybrids were collected by the same objective responsible for focusing the laser beam, either from the air side (100X, NA = 0.9, Olympus) or through the silica substrate (100X oil immersion, NA = 1.3, Olympus). The SH signal was then sent to a spectrometer or to an imaging


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CCD camera (S2 in Supporting Information). A bandpass filter (380 nm - 420 nm) was placed in the collection path to remove any unwanted light except the signal at the SH frequency.

Figure 1. (a) Schematic of the remotely excited SHG from a Ag NW deposited on a monolayer MoS2 sheet. (b) The cross-section of the sample at the overlapping region (not to scale). (c) Electric field profile of the fundamental SPPs modes on the hybrid system. (d) Schematic drawing of momentum conservation in SHG by the copropagating (Path I) or counter-propagating SPPs (Path II).

To describe the optical response and SHG in our system, a nonlinear source term can be incorporated into the waves equation at the SH frequency as 10:

∇ 2E ( 2ω ) +

4ω 2 4ω 2 ( 2) ε 2 ω ⋅ E 2 ω = − P ( 2ω ) , ( ) ( ) c2 ε0c 2

(1)

2 where 2 ω is the SH frequency, ε ( 2ω ) is the relative permittivity and P ( ) ( 2ω )

is the nonlinear polarization at 2ω . Note that the monolayer MoS2 not only provides


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the nonlinear source term at the SH frequency as indicated by Eq. (1), but also serves as a compositive dielectric that’s part of the plasmonic waveguide at the fundamental frequency. Thus the second harmonic signal from the overlap region inherits the property of SPPs.

There are two distinct processes of coherent SHG by SPPs, which can be understood based on the energy and momentum conservation (Figure 1d). In path I, the SH photons are generated by copropagating SPPs and leak away from the Ag NW. This process is inefficient since the phase-matched conditions cannot be satisfied because the momentums of bound SPPs are larger than those of photons in air or in the substrate 12. In path II, the SH photons are generated by counter-propagating SPPs, so that the relationship k (/ / ) ( 2ω ) = k spp (ω ) − k spp (ω ) = 0 holds. The vanishing 2

parallel momentum indicates that the generated coherent SH signals will emit only in the direction perpendicular to the Ag NW (Figure 1d). However, this momentum conservation relation has no restriction on the directionality of the SHG in the perpendicular plane. Therefore, the SH photons are collimated perpendicular to the nanowire axis but are divergent within the perpendicular plane. The emission pattern of SHG and its dependence on the structural parameters are the major subject in the following.

To confirm that the remotely excited SPPs induce SHG, we first excited the Ag NW-monolayer MoS2 hybrids from the air side with a 1.3 mW incident power (Figure 2a,b). SPPs are launched in the Ag NW when the femtosecond laser is focused on the


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bottom end of the NW (Figure 2c), with a spot size of 1 µm. In order to avoid the direct excitation of SHG in MoS2 through the peripheral part of the beam, we only chose those samples with a distance over 3 µm between the incident end of the Ag NW and the edge of the MoS2 sheet. By switching the filters into the optical path, we can also perform direct imaging of the SPPs generated SH on the Ag NW (Figure 2d). We found that the SH emission only occurs from the overlap area of the Ag NW and the MoS2 but not from any other region of the sample. To confirm this, we compared the spectra of the signal at three representative positions: an overlap region (marked A), the output terminal (marked B) and the input terminal (marked C) of the NW, as indicated by the dash squares in Figure 2d. To that end a slit in front of the spectrometer was used to select the collecting areas, and the integration time was 5 s. Figure 2e shows that there is no SH emission from the regions B and C (very weak signal with a long enough integral time). Only in region A, the spectrum has a clear peak at 398 nm, which is consistent with the fundamental wavelength centered at 796 nm. This confirms that the SH emission was not from the input end of the Ag NW, but is generated locally at the overlap region. Also, the absence of SH emission at the NW terminals indicates that the recoupling efficiency of SHG into the SPPs is very low or the propagation loss is high at the SH frequency. The narrow spectral width of the peak (Figure 2d) and its incident power dependence (Figure 2d) confirm that the emission light originates from SHG. Additionally, no SH emission was observed if we rotated the incident polarization from parallel to perpendicular to the NW axis – a situation in which the fundamental SPPs cannot be effectively launched


35, 36

. This

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further indicates that the SH signal from the overlap region is generated by the propagating SPPs.

Figure 2. (a) Optical image of the hybrid system. (b) SEM image of the sample. (c) Fault color imaging of SPPs propagation under the 796 nm laser excitation at the bottom end of the Ag NW. (d) Fault color image at the SH frequency. The dash white lines in (a) - (d) delimit the MoS2 sheet. The scale bar in (a) is also applied to (b) - (d). (e) The spectra of three special regions marked in (d). Region A, B and C correspond to an overlap region, the output terminal and the excitation terminal, respectively. (f) Power dependence of SHG from point A. A slope of 1.96 indicates the expected quadratic dependence.

To increase the collection efficiency of the SH signal, we turned the sample over and excited the Ag NW using a 100X oil immersion objective. The refractive index of the oil (n = 1.5) was matched to that of the silica substrate. In this configuration, the SH light was collected from the substrate side within a collection angle of ± 60


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degrees. The incident power of the laser beam was 5 mW. We observed a distinguishable spatial pattern of the SHG with local bright and dim areas (a representative sample shown in Figure 3a). A line cut of the intensity profile along the Ag NW (Figure 3b) confirms the periodic modulation of the SH signal. The Ag NW behaves as a 1D Fabry-Pérot cavity for SPPs

37, 38

. As shown in Supporting

Information S3, the reflectivity at the NW terminal is about 25% and the 1/e propagation length of SPPs is 28 µm, meaning that there is only one reflection of SPPs at the NW terminal after it is excited. The interference of the incident and the reflected SPPs forms an interference pattern that modulates the SH polarization in the monolayer MoS2. As a result, a regular beating pattern is observed in the far field SH image at the overlap region. By exchanging the input and output ends of the NW, we observe a stronger SH intensity when the MoS2 sheet is closer to the output end (S3 in Supporting Information), which indicates that the intensity of reflected SPPs determines the intensity of SH emission.

To further confirm that the observed spatially modulated SHG pattern originates from the counter-propagating SPPs, we modeled the hybrid structure using full wave finite element method (FEM, COMSOL Multiphysics, V5.2a) (S4 in Supporting Information). Structural parameters used in the calculations were obtained according to the experiments. In order to avoid calculating the whole NW which was too large, we used mode analysis at the cross section of the NW and selectively launched the fundamental SPPs into the system. Symmetric boundary condition was used at the other end to model the reflection of SPPs at the NW terminal. We first calculated the


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counter-propogating SPPs field distribution at the fundamental frequency and evaluated the nonlinear polarization in the monolayer MoS2. We then calculated the SH field distribution using the nonlinear polarization as the source term10. And the far field SH Fourier imaging was transferred from the SH near-field using the Stratton-Chu formula 39. To compare with the SHG image observed in the experiment, the calculated SH near-field distribution at a cut plane 1.3 µm from MoS2 monolayer are shown in Figure 3c. The beating period of the simulations agrees well with the one in the measurements (Figure 3d), which demonstrates that the coherent SHG at the overlap region is caused by the counter-propagating SPPs. A slight derivation between simulation and experiment can be further eliminated if the calculation domain can be large enough to better take account of the diffraction. Further calculations also shows a large-scale modulation on the local SHG distribution originating from the effect of dispersion of SPP since a relative broad spectrum laser was used. Fortunately, it affects only the visibility but not the period of the interference pattern from counter-propagating SPPs.

Figure 3. (a) SEM image of a Ag NW-monolayer MoS2 hybrids. (b) Close-up SHG


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image at the overlap region when the NW is excited through the quartz substrate at the left end. (c) SH intensity calculated in quartz substrate at a plane located 1.3 µm from the monolayer MoS2. The scale bar is 2 µm in (a), and 500 nm in (b) and (c). (d) The intensity profiles along the Ag NW, corresponding to the dash lines in (b) and (c), respectively.

Further, to obtain information about the wavevector distribution hidden in the real plane SHG image, we recorded the Fourier/real image plane by the same CCD camera by flipping up/down a lens (S2 in Supporting Information). The pattern on the Fourier plane is related directly to the wavevector distribution of the SH intensity I(θ, ϕ) 40, where θ and ϕ are the spherical coordinates indicating the direction of the radiation, as defined in the Figure 4a. Here the outer circle on the Fourier image represents the maximum collecting angle of the objective and the inner circle is determined by the critical angle of the air-glass interface

40, 41

. Furthermore, α is the angle between the

Ag NW and the armchair direction of the MoS2 sheet. While the perpendicular bisectors (edges) of the MoS2 triangles are aligned with the armchair (zigzag) directions42. We also confirmed this in the present work, and take the perpendicular bisectors of the MoS2 triangles as the armchair. The SH Fourier plane image obtained with the different α are shown in Figure 4b-d. In the Fourier image, we see the angular distribution of the SH emission is dominant for ϕ = 90°, which means that the direction of the SHG is perpendicular to the Ag NW. This is a direct consequence of the momentum conservation condition between counter-propagating SPPs (Figure 1d). It is also different from the case where the SPPs are confined only in one


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dimension

29

, because, in contrast, the SHG from NW with field confined in two

dimension is divergent in the ϕ = 90° plane. Here the maximum value appears at θ = 0° and at angles larger than the critical angle.

Figure 4. (a) Schematic of our Fourier imaging microscopy collecting signals with an oil objective. (b)-(d) Normalized experimental SH Fourier plane images obtained for different angle between the Ag NW and the monolayer MoS2. (e)-(g) Normalized simulated Fourier image with the different angle between the Ag NW and the monolayer MoS2, corresponding to the (b)-(d). (h)-(j) Comparison between the angular distribution of the SH emission in experiments (b)-(d) and simulations (e)-(g) at ϕ = 90°.

The experimental measurements (Figure 4b-d) also show that the emission pattern of the SHG depends on α. This orientation dependence can be understood by looking at the susceptibility of the MoS2 crystal 30-32. In our experiment, the CVD synthesized


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monolayer MoS2 belongs to the D3h point group. Thus, the second order susceptibility tensor has a single non-zero element χ ( 2 ) ≡ χ ( 2 ) = − χ ( 2) = − χ ( 2) = − χ ( 2) , where (a, b, bbb

aab

aba

baa

c) belong to the crystal coordinates. The corresponding nonlinear polarization is ( 2)

( 2)

P = 2ε0 χ Eip2 [cos(3α - 2θc )xˆ +sin(3α - 2θc ) yˆ ] ,

(2)

where ε 0 is the permittivity of vacuum, Eip is the module of the in-plane electric field and θ c is the angle between the in-plane electric field and the Ag NW. So the 2

far field SH Fourier images I (θ ,ϕ) ∝ cos (ϕ − 3α + 2θc ) present a periodical change of 60° to the angle α (S4 in Supporting Information). Taking into account the complex near-field of SPPs, our full wave simulation results with the α = 10°, 15° and 20° qualitatively agree with the experiment results (Figure 4e-g). A line profile of the emission pattern shown in the Figure 4h-j further confirms the overall agreement between the experimental and the simulation results. The SH emission has two strong side lobes within the plane perpendicular to the nanowire axis, as shown in Figure 4b-j. The two side lobes are both out of critical angle, because they originate from evanescent wave of SH polarization. Evanescent waves interact with the surface of substrate and then are converted into propagating wave, so that they can be detected in the far field.

To capture and describe further the main mechanisms that govern the SHG, we further simplify the model by describing the system in terms of an aligned dipole chain at 2ω ; this because the nonlinear polarization is the source term in Eq. (1). Here the dipoles are in phase but with a position dependent amplitude that mimic the


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nonlinear polarization generated by the interference pattern of the counter-propagating SPPs. The length of the dipoles is determined by the length Lo of the overlap region in the experiments. They locate in air at a distance of 0.62 nm (which corresponds to the thickness of monolayer MoS2) above the quartz surface. To account for the non-zero in-plane (x and y) polarization of MoS2, the experimental SH emission can be regarded as a linear superposition of these two orthogonal in-plane dipole orientations. As a proof of principle, we have shown that the superposition of the in-plane dipole chains can reproduce the experiment results very well (S5 in Supporting Information).

The length of the dipole chain determines how many dipoles add up coherently, which determines the collimation of the SH emission. The divergence of the SH beam in x direction ∆kx in Fourier image is proportional inversely to the length Lo of the overlap region，i.e. ∆kx ∝1 Lo . As shown in Figure 5, experimental results confirm this conclusion. When the length extends to infinity, divergence ∆kx approaches zero (S6 in Supporting Information), implying perfectly collimated SH emission. Taking into account the loss of SPPs, the collimation is eventually restricted by the propagation length of SPPs ∆kx,min ∝1 Lspp (S7 in Supporting Information), if

Lo >> Lspp . In contrast, when Lo approaches zero, the Fourier image of SHG shows a distribution similar to the Fourier image of a point dipole source above a dielectric substrate (S6 in Supporting Information). The divergence of SHG in the transverse plane, i.e., yz plane, is inversely proportional to the confinement of the SPPs. In the current case of the fundamental SPPs on a metal NW, it is proportional to the diameter of the NW. The smaller the diameter is, the more the SHG diverges (S8 in Supporting


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Information). If the NW is suspended in homogeneous background, the SHG should be collimated perpendicular to the NW but radially divergent.

Figure 5. The beam divergence of the SH radiation pattern ∆kx as a function of the length of the overlap region.

A close up examination at the Fourier image reveals weak interference fringes (side lobes) - a feature in agreement with those observed in Ref. 29. This can attribute to the contribution of copropagating SPPs, as evidenced by the dipole chain model. This process suffers from phase mismatch so that the SH intensity is much weaker than the case for counter-propagating SPPs mainly discussed in this Letter. As revealed by the dipole chain model, the interference fringe spacing is proportional inversely to the length of the structures. If the length is long enough, the SHG emission is concentrated to the glancing angle (S9 in Supporting Information).

In summary, we have demonstrated the transversely divergent coherent SHG from counter-propagating SPPs on 1D plasmonic waveguide. As a consequence of the translation symmetry along the waveguide, SH photons are collimated perpendicular to the waveguide axis due to the coherent superposition. Different from SHG


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processes in 2D plasmonic systems, the SH photons are divergent in other directions due to the confinement in the lateral dimension of the waveguide. The collimation (corresponding to the translation axis) of the SHG beam can be controlled by the length of the active region or the propagation length of the SPPs, while the divergence can be tuned by the spatial confinement of SPPs within the cross section of the waveguide. What’s more, Fourier imaging also reveals a SH emission peak at the critical angle of the air-substrate interface, resembling those patterns from an aligned dipole chain. Control over the SH near-fields or the emission patterns can be accomplished by rotating the MoS2 sheet with respect to the waveguide axis. Our finding opens new possibility in generating new type of coherent light beams and serves as guide for designing compact, high efficiency nonlinear elements. Finally, the combination of high quality plasmonic NW and monolayer MoS2 with strong second order susceptibility also provides an ideal test-bed for plasmonic spontaneous parametric down-conversion process, which is one of the key functionality for integrated quantum optic circuits.

ASSOCIATED CONTENT

Supporting Information Detailed description of the sample fabrication and characterization, second harmonic microscopy, second harmonic Fourier microscopy and simulations. This material is available via the Internet at http://pubs.acs.org.


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Author Contributions S.P.Z. conceived the idea. Y.L. and J.J.S. prepared the samples and performed the experiments. M.K. performed the theoretical simulations. K.W. provided the CVD grown MoS2 samples. Y.L., M.K., S.P.Z. and H.X.X. analyzed the data. S.P.Z., Y.L., M.K., and H.X.X. wrote the paper. #Y. L. and M. K. contributed equally to this work. Notes The authors declare no competing financial interests.

ACKNOWLEDGMENTS

This work was supported by the Ministry of Science and Technology (Grant No. 2015CB932400), the National Natural Science Foundation of China (Grants Nos. 11674256 and 11674255).

REFERENCES (1) Ebbesen, T. W.; Genet, C.; Bozhevolnyi, S. I. Phys. Today 2008, 61, 44-50. (2) Gramotnev, D. K.; Bozhevolnyi, S. I. Nat. Photonics 2010, 4, 83-91. (3) Kauranen, M.; Zayats, A. V. Nat. Photon. 2012, 6, 737-748. (4) Cai, W.; Vasudev, A. P.; Brongersma, M. L. Science 2011, 333, 1720-1723. (5) Lu, F. F.; Li, T.; Hu, X. P.; Cheng, Q. Q.; Zhu, S. N.; Zhu, Y. Y. Opt. Lett. 2011, 36, 3371-3373. (6) Utikal, T.; Zentgraf, T.; Paul, T.; Rockstuhl, C.; Lederer, F.; Lippitz, M.; Giessen, H. Phys. Rev. Lett. 2011, 106, 133901.


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(7) Li, G. Y.; de Sterke, C. M.; Palomba, S. Laser & Photonics Reviews 2016, 10, 639-646. (8) Chang, D. E.; Sørensen, A. S.; Demler, E. A.; Lukin, M. D. Nat. Phys. 2007, 3, 807-812. (9) MacDonald, K. F.; Sámson, Z. L.; Stockman, M. I.; Zheludev, N. I. Nat. Photonics 2008, 3, 55-58. (10) Boyd, R. W., Nonlinear Optics (third edition). Academic Press: 2008. (11) Zhu, S.; Zhu, Y. Y.; Ming, N. B. Science 1997, 278, 843-846. (12) de Hoogh, A.; Opheij, A.; Wulf, M.; Rotenberg, N.; Kuipers, L. ACS Photonics 2016, 3, 1446-1452. (13) Renger, J.; Quidant, R.; van Hulst, N.; Palomba, S.; Novotny, L. Phys. Rev. Lett. 2009, 103, 266802. (14) Constant, T. J.; Hornett, S. M.; Chang, D. E.; Hendry, E. Nat. Phys. 2015, 12, 124-127. (15) Raschke, M. B.; Berweger, S.; Atkin, J. M., Ultrafast and Nonlinear Plasmon Dynamics. In Plasmonics: Theory and Applications, Springer: 2013. (16) Grosse, N. B.; Heckmann, J.; Woggon, U. Phys. Rev. Lett. 2012, 108, 136802. (17) Chen, C. K.; de Castro, A. R.; Shen, Y. R. Opt. Lett. 1979, 4, 393. (18) Fukui, M.; Sipe, J. E.; So, V. C. Y.; Stegeman, G. I. Solid State Commun. 1978, 27, 1265-1267. (19) Rodrigo, S. G.; Harutyunyan, H.; Novotny, L. Phys. Rev. Lett. 2013, 110, 177405. (20) Dadap, J. I.; Shan, J.; Eisenthal, K. B.; Heinz, T. F. Phys. Rev. Lett. 1999, 83, 4045-4048. (21) Anceau, C.; Brasselet, S.; Zyss, J.; Gadenne, P. Opt. Lett. 2003, 28, 713. (22) Butet, J.; Bachelier, G.; Russier-Antoine, I.; Jonin, C.; Benichou, E.; Brevet, P. F. Phys. Rev. Lett. 2010, 105, 077401. (23) Zhang, Y.; Grady, N. K.; Ayala-Orozco, C.; Halas, N. J. Nano Lett. 2011, 11, 5519-5523. (24) Thyagarajan, K.; Rivier, S.; Lovera, A.; Martin, O. J. Opt. Express 2012, 20, 12860-12865. (25) Butet, J.; Brevet, P. F.; Martin, O. J. ACS Nano 2015, 9, 10545-10562. (26) Celebrano, M.; Wu, X.; Baselli, M.; Grossmann, S.; Biagioni, P.; Locatelli, A.; De Angelis, C.; Cerullo, G.; Osellame, R.; Hecht, B.; Duo, L.; Ciccacci, F.; Finazzi, M. Nat. Nanotechnol 2015, 10, 412-417. (27) Nappa, J.; Russier-Antoine, I.; Benichou, E.; Jonin, C.; Brevet, P. F. J. Chem. Phys. 2006, 125, 184712. (28) Yu, H.; Fang, W.; Wu, X.; Lin, X.; Tong, L.; Liu, W.; Wang, A.; Shen, Y. R. Nano Lett. 2014, 14, 3487-3490. (29) Viarbitskaya, S.; Demichel, O.; Cluzel, B.; Colas des Francs, G.; Bouhelier, A. Phys. Rev. Lett. 2015, 115, 197401. (30) Kumar, N.; Najmaei, S.; Cui, Q.; Ceballos, F.; Ajayan, P. M.; Lou, J.; Zhao, H. Phys. Rev. B 2013, 87, 161403. (31) Malard, L. M.; Alencar, T. V.; Barboza, A. P. M.; Mak, K. F.; de Paula, A. M.


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Phys. Rev. B 2013, 87, 201401. (32) Li, Y.; Rao, Y.; Mak, K. F.; You, Y.; Wang, S.; Dean, C. R.; Heinz, T. F. Nano Lett. 2013, 13, 3329-3333. (33) Sun, Y. G.; Xia, Y. N. Adv. Mater. 2002, 14, 833-837. (34) Zhang, S.; Xu, H. ACS Nano 2012, 6, 8128-8135. (35) Zhang, S.; Wei, H.; Bao, K.; Håkanson, U.; Halas, N. J.; Nordlander, P.; Xu, H. Phys. Rev. Lett. 2011, 107. (36) Wei, H.; Zhang, S.; Tian, X.; Xu, H. Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 4494-4499. (37) Ditlbacher, H.; Hohenau, A.; Wagner, D.; Kreibig, U.; Rogers, M.; Hofer, F.; Aussenegg, F. R.; Krenn, J. R. Phys. Rev. Lett. 2005, 95, 257403. (38) Lee, H. S.; Kim, M. S.; Jin, Y.; Han, G. H.; Lee, Y. H.; Kim, J. Phys. Rev. Lett. 2015, 115, 226801. (39) Jin, J., The Finite Element Method in Electromagnetics. Wiley: 2002. (40) Lieb, M. A.; Zavislan, J. M.; Novotny, L. J. Opt. Soc. Am. B 2004, 21, 1210. (41) Wolf, D.; Schumacher, T.; Lippitz, M. Nat. Commun. 2016, 7, 10361. (42) Hsu, W. T.; Zhao, Z. A.; Li, L. J.; Chen, C. H.; Chiu, M. H.; Chang, P. S.; Chou, Y. C.; Chang, W. H. ACS Nano 2014, 8, 2951-2958.


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Nano Letters

(a)

796 nm

(b)

398 nm

kspp

AgNW

SiO2 substrate (c)

kspp (d)

Path I

2kspp(ω)

Path II

k 0(2ω)

k 0(2ω)

k ( 2 )(2ω) kspp(ω) k s u b(2ω)


kspp(ω) k s u b(2ω)

Nano Letters

(a)

(b)

(c)

(d)

B

796 nm

A

C

3 µm

(e)

(f)

Intensity (a.u.)

20 counts Intensity (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Wavelength (nm)


Slope = 1.96

Power (mW)

398 nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

(b)

Nano Letters

(d)

Intensity (a.u.)

Page 23 of 25

Exp. Sim. 1 µm

(c)

Distance (µm)


Nano Letters

1.5

z (c-axis)

1.0

b

α θ

ky/k0

y

α = 11° (c)

(b)

α = 15.5° (d)

α = 20°

Max

0.5 0

-0.5 -1.0

x

a

-1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.0 -0.5 0 0.5 1.0 1.5

kx/k0

1.5

kx/k0

kx/k0

α = 15° (g)

α = 10° (f)

(e)

α = 20°

1.0

ky/k0

(a)

Intensity(a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.5 0

-0.5 -1.0

0

-1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.0 -0.5 0 0.5 1.0 1.5

φ

(h) -60

-90

-30

0

kx/k0

(i)

30 60

-30

-60

-90


0

kx/k0

(j)

30 60

-60

-90

kx/k0

-30

0

30 60

90

Page 25 of 25

0.15

0.12

'kx

'kx/k0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

0.09

0.06 2

3

4

Lo ( P


5

6

Transversely Divergent Second Harmonic ... - ACS Publications

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