## Abstract

We propose an approach of steering the second harmonic (SH) emission from a single plasmonic structure, through local excitations of plasmon. The proposed idea is confirmed experimentally, by adjusting the incident beam position at the fundamental frequency, on a single plasmonic antenna. A significant directivity change ( ± 52°) for the SH emission is observed with submicrometer adjustment ( ± 250 nm) of the excitation beam position, over broadband SH frequencies. Providing a simple method of controlling the directivity of frequency-converted light, our approach paves the way to new design strategy for nonlinear optical devices with various nonlinear wavefronts.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The steering of wavefront is a general topic of wave-mechanics, and has been intensively studied especially for electromagnetic waves over a wide frequency range from microwave [1] to optics [2]. The controlled radiation directivity could provide efficient energy conversion between localized electromagnetic energy and free-space radiation [3], novel functionalities for single photon sources [4], wireless interconnects [5,6], molecular spectroscopy [7], plasmonics [8], and optical microscopy [9].

Recent efforts on the wavefront steering has been extended to nonlinear optics (*e.g.*, second or third harmonic generation) [10–12], to achieve beam steering functions for frequency-converted light and to be prepared for the increasing demands of multi-functionalities in a single optoelectronic circuit [11]. Focusing on the nonlinear optical process of second harmonic generation (SHG) [13–19], while this process is forbidden in centrosymmetric materials (*e.g.*, metals) within the electric dipole approximation [20], it is well known that second-order nonlinear polarizations can be generated from broken symmetry at the surface or from higher multipole interactions in the bulk of centrosymmetric materials [21,22]. Recent study has also revealed that these nonlinear conversions are not restricted to the excitation area, but rather occur along the entire structure by the propagating surface plasmon (SP) mode [23]. Quite a number of demonstrations on the steering of SH light have been made so far: by engineering effective nonlinear polarizability tensor of particles, in periodic array [24–28] or in molecules [29], or by utilizing the interference of exited multipole modes from the electric and magnetic resonances in a single particle [30–33]. However, to our best knowledge, controlling the wavefront of SH light from a single plasmonic structure under single beam excitation has not been demonstrated.

In this paper, using the nonlinear polarizations generated from the propagating SP mode, we propose an approach steering the SH radiation from a thin single plasmonic antenna structure. This approach is based on the nonlinear optical process of SHG in a thin plasmonic structure [23], where the SH emission directivity is determined by the wavevector of the fundamental excitation fields, especially inside the metallic structure. The directionality control of SH emission is achieved by fine-adjusting the position of the incident beam impinging on a single plasmonic antenna, as well as changing the resonance condition of antenna geometry, which leads to different local plasmon excitations and wave-vector momentums at the fundamental frequency. The measured SH emission patterns from different antenna geometries are confirmed with numerical results of excellent fit. Over 13.4dB, highly directional switching of SH emission with large angle sweep (52° ~-52°) is demonstrated.

## 2. SH emission directivity from a thin single plasmonic structure

Far-field emission intensity from dipole sources near a flat dielectric interface can be obtained through far-field Green’s function [34]. When these dipole sources (*p*) are confined to a deep subwavelength (<< *λ*) thickness region parallel to the interface, the retardation along the perpendicular direction to the interface can be neglected, and the far-field emission intensity *I _{i}* (

*k*,

_{x}*k*) can be approximated as (see Appendix A):

_{y}*k*and

_{x}*k*are the wavevector for each axis,

_{y}*p*denotes

_{i}*i*component dipole sources, and

**N**

*(*

_{i}*k*,

_{x}*k*) is derived from the far-field Green’s function with single

_{y}*i*-polarized dipole source at the origin. These equations show that the far-field emission intensity

*I*(

_{i}*k*,

_{x}*k*) is determined by the distribution of the dipole sources

_{y}*p*in the reciprocal space

_{i}*S*(

_{i}*k*,

_{x}*k*) with fixed contribution from far-field Green’s function by

_{y}**N**

*. It is noted that, if all dipole components are considered, there will be constructive or destructive interferences in the far-field. Especially, if all dipole components share the same wavevector distribution as in the case of dipoles in the propagating SP mode, the overall tendency will maintain, as in Ref [35].*

_{i}This well-known relation between far-field emission directivity and distribution of the dipole sources in the reciprocal space can be directly applied to the control of SH emission directivity. In the process of SHG, second-harmonic generated nonlinear polarizations *P*^{2}* ^{ω}* inside the plasmonic structure act as dipole sources at 2

*ω*, and the wavevector of these nonlinear polarizations is related to that of the fundamental field (

*E*) via

^{ω}*P*

^{2}

*~*

^{ω}*E*. Especially, in case these nonlinear polarizations exist in a thin plasmonic antenna, the directivity of SH emission is thus governed by the wavevector of the fundamental field inside the structure via Eqs. (1) and (2) [Fig. 1(a)], which can be controlled, for example, by adjusting the position of local plasmon excitation [Fig. 1(b)]. This background theory and the use of thin-metal antenna thus together constitutes the first basis idea of our proposal.

^{ω}E^{ω}## 3. Result and discussion

In the experiment, we adjusted the position of the incident beam on a gold antenna [Fig. 2(a)], which was fabricated on a glass substrate through an electron beam lithography, lift-off, and annealing process [Fig. 2(b)]. The incident beam was *x*-polarized and tightly focused in an 893 nm full-width at half-maximum (FWHM) spot on the antenna, and the relative position of the incident beam to the antenna center [*x*_{0} in Fig. 2(a)] was adjusted along the *x*-axis with piezo stage. Depth of the antenna (*d* = 5 μm) was set much larger than the focusing spot (FWHM = 893 nm) and the wavelength (*λ*_{1} = 888 nm) of the incident beam, to confine the SH emission in a narrow *k _{y}* region (at

*k*= 0) and thus obtain a highly directional SH emission (see Appendix B for detailed analysis). SH signals in transmission direction were measured at back-focal plane of the oil-immersion objective lens (NA = 1.65), which provides Fourier transformed far-field patterns [36]. For numerical analysis, first, the fields at the fundamental frequency were calculated from the finite element method (FEM) simulation, and then, using these fundamental fields, hydrodynamic model based second-order nonlinear polarizations (

_{y}*P*) ([37], see Appendix C) were calculated. Finally, the SH fields were FEM-calculated using these nonlinear polarizations as sources at the SH wavelength (

^{2ω}*λ*

_{2}= 444 nm). It is noted that the computed SH fields were converted to the far-field using Lorentz reciprocity theorem [38].

Figure 2(c) shows the measured SH emission patterns from an antenna of *w* = 870 nm, for different incident beam positions of *x*_{0} = −250, 0, and 250 nm. In good agreement with FEM-calculated results [Fig. 2(d)], these results clearly demonstrate that the directivity of SH emission can be changed by adjusting the incident beam position on a single plasmonic structure. A significant directivity change, of maximum intensity angle from 39° to −39° [Fig. 2(e)], is obtained with only submicrometer adjustment ( ± 250 nm) of the incident beam position. It is noted that this change of SHG emission directivity is well explained by the change of fundamental fields’ dominant wavevector, which depends on the position of the incident beam [Fig. 2(f)]. Convolution of these fundamental fields in a reciprocal space (*E ^{ω}**

*E*) dictates the distribution of

^{ω}*P*in a reciprocal space [Fig. 2(g)], and thus the SH emission directivity as well via Eqs. (1) and (2) [see Appendix D for detailed calculation and resultant SH emission patterns obtained via Eqs. (1) and (2)]. It is important to note that in Fig. 2(g), we only show the surface-parallel nonlinear polarization component (${P}_{\text{surf,}\parallel}^{2\omega}$ ~ ${E}_{x}^{\omega}$${E}_{z}^{\omega}$, detailed expression for each nonlinear polarization component is presented in Appendix C), which provides dominant contribution to the total SH emission patterns for this antenna (

^{2ω}*w*= 870 nm, see Appendix E).

We also studied the variation of the controllable directivity by modulating the antenna geometry (*i.e.*, resonance condition). Figure 3(a) shows the measured SH emission patterns as a function of incident beam position (*x*_{0}), for the antennas of different widths (*w* = 870, 645, 420, and 205 nm, from left to right). It is noted that because the emitted SH signals from these antennas are highly confined at *k _{y}* = 0 [

*e.g.*, Figs. 2(c) and 2(d)] due to the quasi-1D nature of the antenna (see Appendix B), here we plot the

*k*-projected SH emission patterns in Fig. 3(a), for the ease of comparison. The measured SH emission patterns agree well with numerically obtained results [Fig. 3(b)], except minor discrepancies near

_{x}*k*= 0, which is attributed to the leakage of the incident beam (see Appendix F for detailed analysis). For all antenna geometry, it can be seen that the adjustment of the incident beam position

_{x}*x*

_{0}provides variation of the SH emission directivity, as shown in experimental [Fig. 3(a)] and numerical [Fig. 3(b)] results. As shown in Fig. 2(f), the relative magnitude between counter propagating SP modes depends on the incident beam position, which results in changes of the relative magnitudes of SH emission peaks and thus the variation of the SH emission directivity. Within this scheme, the most dramatic change of the maximum emission angle occurs near

*x*

_{0}= 0, where the dominant wavevector component of the fundamental fields changes. We further note that the peak positions and widths of the SH emission can be further adjusted by changing the antenna width [Figs. 3(a) and 3(b)], since the peaks of the fundamental fields are influenced by the width of the antenna (see Appendix G for a detailed analysis).

These results imply that additional degree of freedom in the SH emission pattern can be achieved by modifying the geometry of particle, not limited to a simple linear antenna, when combined with the position-dependent excitation.

It is worth to note that the antennas of *w* = 870 and 420 nm exhibit clearly directional, asymmetric switching (from *k _{x}* > 0 to

*k*< 0) near

_{x}*x*

_{0}= 0, when compared to the symmetric case of

*w*= 645 and 205 nm. This observed behavior is related to the resonance condition of the fundamental fields [Fig. 3(c)]. When the antennas are in resonance [

*w*= 645 and 205 nm in Fig. 3(c)], standing-wave inside the antenna, formed by counter-propagating SP waves, leads strong surface-normal electric fields ${E}_{\perp}^{\omega}$ at both ends of the antenna [Fig. 3(d)]. Such strong surface-normal electric fields ${E}_{\perp}^{\omega}$ at the antenna ends significantly enhance symmetric SH emissions from surface-normal nonlinear polarization component [${P}_{\text{surf,}\perp}^{2\omega}$~ ${E}_{\perp}^{\omega}{E}_{\perp}^{\omega}$, see Fig. 3(e)], which hinder asymmetric switching near

*x*

_{0}= 0 [see also Fig. 9 in Appendix E]. On the other hand, the antennas in off-resonance mode (

*w*= 870 and 420 nm), having no strong surface-normal electric fields ${E}_{\perp}^{\omega}$ at the antenna ends due to absence of standing-wave [Fig. 3(d)], exhibit no strong symmetric SH emissions from ${P}_{\text{surf,}\perp}^{2\omega}$ [Fig. 3(e)], which lead highly asymmetric switching near

*x*

_{0}= 0.

For the smallest off-resonance antenna (*w* = 420 nm), we now assess its directional switching performance. To this end, we define the left-to-right ratio I_{L}/I_{R} as the integrated SH emission intensity ratio of I_{SHG}(*k _{x}*/

*k*

_{2}< −1) to I

_{SHG}(

*k*/

_{x}*k*

_{2}> 1). The measured I

_{L}/I

_{R}varies largely from −6.5 dB to 7 dB in the experiment and from −9 to 9 dB in the calculation, with submicrometer adjustment of the incident beam position [Fig. 4(a)]. The maximum intensity angle reaches −52° for

*x*

_{0}= 250 nm [Fig. 4(b), see Fig. 13 in Appendix H for polar plotted data], and stays between −52° ~-59° for larger

*x*

_{0}. It is also found that for this directional transition, high maximum directivity

*D*

_{max}(

*D*

_{max}= (2π/

*P*

_{rad}) Max [

*p*(

*θ*,

*ϕ*)], where

*ϕ*is the azimuthal angle in the

*x*-

*y*plane,

*p*(

*θ*,

*ϕ*) is the angular power density, and

*P*

_{rad}is the total radiated power in transmission direction [39]), is maintained: greater than 13.4 and 14.5 dB in the experiments and calculations, respectively. Numerical assessment also shows that the large I

_{L}/I

_{R}(> 5 dB) and high directivity (> 15 dB) at

*x*

_{0}= 250 nm can be maintained over a wide wavelength range [from 780 nm to 980 nm, Fig. 4(c)] and over a wide range of antenna widths [from 370 nm to 510 nm Fig. 4(d)]. These broadband characteristic and robustness against structural modulation are attributed to the fact that the asymmetric directional emission ( = large I

_{L}/I

_{R}) is originated from the non-resonant (broadband) propagating SP waves that lead asymmetric wavevector distribution of the fundamental fields [

*e.g.*, Fig. 2(f)]. It is also noted that for this antenna (

*w*= 420 nm), a large amount of SHG photons (~4.9 × 10

^{6}SHG counts/s) were measured at

*x*

_{0}= 250 nm, under the average pump power of 9.5 mW (peak power

*P̂*

_{1}

*= 9.9 kW with a repetition rate of 80 MHz and pulse length of 12 fs). Considering the losses introduced by all optical elements in the collection pathway, the emitted photons from the antenna can be roughly estimated as ~1.23 × 10*

_{ω}^{8}SHG photons/s, corresponding to an SHG peak power

*P̂*

_{2}

*= 5.71 × 10*

_{ω}^{−5}W. The calculated nonlinear coefficient (

*γ*

_{SHG}) is 5.83 × 10

^{−13}W/W

^{2}(where

*γ*

_{SHG}=

*P̂*

_{2}

*/(*

_{ω}*P̂*

_{1}

*)*

_{ω}^{2}[19]), which indicates that still a large amount of SHG photons (10

^{5}photons/s) can be obtained from this directional SH antenna with a smaller pump power of 0.27 mW (peak power = 0.28 kW). The obtained broadband and robust directional switching with high maximum directivity could be utilized for switching devices such as optical transistors.

## 4. Conclusion

In summary, we proposed a new approach of steering the SH emission through the local plasmon excitations on a thin single plasmonic antenna. The proposed approach is based on a relation between SH emission directivity and wavevector of the fundamental fields inside the thin plasmonic structure, which is related through the nonlinear polarizability and Green’s function. By employing a single plasmonic antenna, we experimentally and theoretically confirmed the control of SH emission directivity over large angle, from 52° to −52°, with submicrometer adjustment of the incident beam position ( ± 250 nm). It is further shown that the SH emission directivities can be as well manipulated by simply modulating the width of the antenna, and is related to the resonance condition as well as the position of the local plasmon excitation. High left-to-right ratio I_{L}/I_{R} (> 5dB) and maximum directivity (> 15dB) are demonstrated over broadband frequency and large tolerances in antenna width. Combining the result of this study to other plasmonic structures of various geometries, we expect extended controllability in the nonlinear wavefront.

## Appendix A Far-field intensity from dipoles above a flat dielectric interface

When dipoles are located above a flat dielectric interface [Fig. 5(a)], the far-field **E**_{∞} observed at **r** = (*x*, *y*, *z*) in the far-zone, *i.e*. *r* >> *λ* where *r* = (*x*^{2} + *y*^{2} + *z*^{2})^{1/2}, can be determined by [34]:

*μ*

_{0}and

*μ*

_{+}are the vacuum and relative permeability of the upper half space (

*z*> 0) respectively, and $\overleftrightarrow{G}(r,r\text{'})$ is the asymptotic far-field form of the dyadic Green’s function [34] which defines the far-field

**E**

_{∞}(

**r**) of an electric dipole

**p**located at

**r**

*’*= (

*x’*,

*y’*,

*z’*). This Green’s function has the following form in the upper half space (

*z*> 0)

*z*< 0)Here, ${\overleftrightarrow{G}}_{\text{0}}$, ${\overleftrightarrow{G}}_{\text{ref}}$, and ${\overleftrightarrow{G}}_{\text{tr}}$ are the Green’s functions for primary, reflected, and transmitted fields [34].

In each of these Green’s functions, source position **r***’* = (*x’*, *y’*, *z’*) dependency can be separated in the following forms

*k*and

_{x}*k*are the wavevector along the

_{y}*x*and

*y*axis, and

*k*

_{z}_{+}= (

*k*

_{+}

^{2}–

*k*

_{x}^{2}–

*k*

_{y}^{2})

^{1/2}and

*k*

_{+}=

*n*

_{+}(2

*ω*/

*c*) is the wavenumber for the upper half space. Substituting Eqs. (6)-(8) in Eq. (3) and using the far-field relation (

*x*/

*r*,

*y*/

*r*,

*z*/

*r*) = (

*k*/

_{x}*k*,

_{m}*k*/

_{y}*k*,

_{m}*k*/

_{zm}*k*) [34] (

_{m}*m*= + or -, depending on the sign of

*z*), we have

*z’*=

*z*

_{0}(that is |

*z’*-

*z*

_{0}| <<

*λ*) [Fig. 5(a)], the term ${e}^{\mp i{k}_{z+}z\text{'}}$ can be considered as ${e}^{\mp i{k}_{z+}{z}_{0}}$, and the far-field intensity can be approximated as

**r**

*’*= (0, 0, 0) [see Eqs. (6)-(8)], and can be determined irrespective of distribution of the dipoles. Therefore, Eqs. (10) and (11) prove that the far-field intensity in the reciprocal space

*I*(

*k*,

_{x}*k*) (that is, directivity of the emission) can be determined by the distribution of the dipoles in the reciprocal space

_{y}**S**(

*k*,

_{x}*k*).

_{y}A more direct connection between distribution of the dipoles in reciprocal space and the emission directivity can be obtained when a single component of the dipoles (*i.e. x*-polarized dipoles) is considered. The far-field intensity from *x*-polarized dipole sources is

**N**

*is the first column vector of $\overleftrightarrow{N}$ and*

_{x}*I*(

_{x}*k*,

_{x}*k*) is directly determined by the distribution of the dipole sources in the reciprocal space with fixed contribution from |

_{y}**N**

*|*

_{x}^{2}.

The emission directivity from *y*- or z-polarized components also can be expressed:

In Figs. 5(b) and 5(c), the |**N*** _{x}*|

^{2}and |

**N**

*|*

_{z}^{2}for glass substrate (

*n*= 1.79) are plotted (|

**N**

*|*

_{y}^{2}is just 90° rotated image of |

**N**

*|*

_{x}^{2}).

## Appendix B Effect of the antenna depth on the SH emission directivity

In this appendix, we investigate the effect of the antenna depth on the SH emission directivity. Figure 6(a) shows the FEM-calculated fundamental electric fields, for different antenna depths under the same incidence condition of our work. This result shows that when *d* > 3*λ*_{1}, the effect of the increasing depth on the fundamental fields becomes negligible [Fig. 6(a)], and the fields are converged to result in same *y*-dependency with the incident Gaussian beam, as proven in Fig. 6(b). This indicates that the *x*-dependency of the induced field is invariant along the *y*-direction for the antenna of *d* > 3*λ*_{1}, and thus, it is possible to express the fundamental fields inside the antenna as follows:

*b*= 758 nm is the waist of the incident Gaussian beam. This equation also shows that for the same incident condition (

*b*= 758 nm), the

*y*-dependency of the field can be fixed, and thus the antenna can be considered as quasi-one-dimensional (Quasi-1D) along the

*x*axis. From this electric field distribution, second order nonlinear polarizations can be approximated as:

*I*(

_{i}*k*,

_{x}*k*) from the antenna of

_{y}*d*> 3

*λ*

_{1}should be confined near

*k*= 0 due to |

_{y}*Y*(

_{i}*k*)|

_{y}^{2}, of which distribution is plotted in Fig. 6(c).

Indeed, the FEM-calculated SH emission patterns for the same set of the antennas show highly confined SH emissions for *d* > 3*λ*_{1} [Fig. 7(a)]. In addition, significant improvement of the maximum directivity *D*_{max} is observed [from 7 dB to 14.5 dB in Fig. 7(b)] when the depth of the antenna is increased from *λ*_{1}/4 (*d* = 0.2 μm) to 4*λ*_{1} (*d* = 3.6 μm). Based on these, we employ antennas of *d* = 5 μm (> 5*λ*_{1}) in our work to obtain highly directional SH emission.

## Appendix C Hydrodynamic model based second-order nonlinear polarizations

In centrosymmetric materials such as metals, the second-order nonlinear polarizations, which is a source of second harmonic generation, originate from the field gradient in the bulk and symmetry breaking at the surface [22]. These nonlinear polarizations and corresponding nonlinear susceptibilities can be analytically obtained from the hydrodynamic model (HDM) with some approximations as follows [37]:

*e*is the charge of an electron, ${m}_{e}^{*}$ is the effective electron mass,

*ν*is the electron collision rate, and

*n*

_{0}is the background electron charge density. In our work, we used the following values for all the calculations: ${m}_{e}^{*}$ = 1.06 ×

*m*,

_{e}*ν*= 1.07 × 10

^{14}s

^{−1},

*n*

_{0}= 5.9 × 10

^{22}cm

^{−3}. The superscripts $\omega $ and $2\omega $ denote the values at the fundamental (1

*ω*) and SH (2

*ω*) frequency respectively, and the subscripts ⊥ and ∥ denote the surface-normal and surface-parallel components, respectively. In deriving Eqs. (23)-(31), a harmonic time dependence (

*e*) has been assumed.

^{iωt}## Appendix D Far-field SH emission calculated with Eqs. (1) and (2)

In Eqs. (1) and (2), we show that SH emission directivity *I _{i}* (

*k*,

_{x}*k*) is approximately proportional to the wavevector distribution of the nonlinear polarization |

_{y}*S*(

_{i}*k*,

_{x}*k*)| through

_{y}*I*(

_{i}*k*,

_{x}*k*) ~ |

_{y}**N**

*(*

_{i}*k*,

_{x}*k*)|

_{y}^{2}|

*S*(

_{i}*k*,

_{x}*k*)|

_{y}^{2}, where |

**N**

*(*

_{i}*k*,

_{x}*k*)|

_{y}^{2}is the far-field intensity from the

*i*-polarized dipole source at the origin. In this appendix, we show how the SH emission directivity is determined by Eq. (1), for the case of

*w*= 870 nm. Figure 8(a) shows the far-field intensity from an

*x*-polarized dipole source at the origin (|

**N**

*|*

_{x}^{2}) and Fig. 8(b) shows the wavevector distribution of the surface-parallel nonlinear polarization component (${P}_{\text{surf,}\parallel}^{2\omega}$,

*x*-polarized in our excitation scheme) at the bottom surface (|

*S*|

_{x}^{2}). With these data, the SH emission directivity

*I*can be calculated with Eq. (1) by multiplying |

_{x}**N**

*|*

_{x}^{2}and |

*S*|

_{x}^{2}. As can be seen, the SH emission directivity

*I*is determined by the distribution of the nonlinear polarization |

_{x}*S*|

_{x}^{2}, with fixed contribution from |

**N**

*|*

_{x}^{2}. The calculated emission patterns [Fig. 8(c)] well agree with the measured results [Fig. 2(c)] as well as the numerically obtained results [Fig. 2(d)].

## Appendix E SH emission patterns from individual nonlinear polarization components

Figure 9 shows *k _{x}*-projected SH emission patterns, which are FEM-calculated including all of the nonlinear polarization components (1st row), with only the surface-parallel (${P}_{\text{surf,}\parallel}^{2\omega}$, 2nd row), with only the surface-normal (${P}_{\text{surf,}\perp}^{2\omega}$ 3rd row), and with only the bulk (${P}_{\text{bulk}}^{2\omega}$, 4th row) component, respectively. As can be seen, surface-parallel nonlinear polarization component (${P}_{\text{surf,}\parallel}^{2\omega}$) provide the most similar radiation pattern when compared to the SH emission including all of the nonlinear polarization components, or experimentally obtained emission pattern [Fig. 3(a)]. Furthermore, we note that for the antenna widths of off-resonance mode (

*w*= 870 nm and 420 nm) which exhibit clear directional switching around

*x*

_{0}= 0 [see Fig. 3 and corresponding descriptions], ${P}_{\text{surf,}\parallel}^{2\omega}$ provides about twice as much contribution as other components.

## Appendix F Analysis of measured signals near *k*_{x} = 0

_{x}

The background SHG signals around *k _{x}* = 0 [Fig. 3(a)] were also observed from the bare glass substrate without gold antenna [Fig. 10(a)], which indicates that these signals are not generated from the antenna. Most significantly, the background SHG signals clearly exhibiting ring shape, being zero close to the center [Fig. 10(a)], resemble the shape of the secondary mirror of the reflective objective [Fig. 10(b)]. Taken together, we conclude that these background SHG signals around

*k*= 0 is not from the sample, but generated at the metallic reflective objective lens and focused onto the sample along with the incident beam at the fundamental frequency.

_{x}To examine the effect of this background SHG signal on the measured SHG signals from the antenna [Fig. 3(a)], we performed the numerical analysis with Gaussian beam excitation at the SH frequency [Fig. 11(a)], to obtain its far-field pattern. Figure 11(b) shows the resultant far-field patterns at the SH frequency in transmission direction, for the same set of beam positions and antenna widths as in Fig. 3(a). These results show that the far-field patterns obtained from the Gaussian beam excitation at the SH frequency are well confined near *k _{x}* = 0 and being smallest at

*x*

_{0}= 0, in good agreement to the measured signals near

*k*= 0 [Fig. 3(a)].

_{x}## Appendix G SH emission dependence on antenna width

Since the smaller width of an antenna will provide a larger spatial momentum component in the reciprocal space and thus wider wavevector distribution at the fundamental frequency, the SHG directivity will be dependent upon the antenna width, especially when the width of an antenna is smaller than the incident beam width. In Fig. 12 we plot the amplitudes of the Fourier transformed fields of the antenna [|${E}_{x}^{\omega}$| in Fig. 12(a), |${E}_{z}^{\omega}$| in Fig. 12(b), and |${P}_{\text{surf,}\parallel}^{2\omega}$| in Fig. 12(c)] for the incident condition of *x*_{0} = 0 as a function of the antenna width, obtained from FEM. As can be seen, antennas of smaller width (*w* < 600 nm) show much broader *k*-space distribution than those of larger antennas, in their fundamental fields *${E}_{x}^{\omega}$* and ${E}_{z}^{\omega}$. As a result, peak positions and width of the nonlinear polarizations are changed [Fig. 12(c)], leading different peak directions for SH emissions [also see Figs. 3(a) and 3(b)].

## Appendix H SH emissions for the antenna of *w* = 420 nm

## Funding

National Research Foundation of Korea (NRF) through the Global Frontier Program (2014M3A6B3063708); Basic Science Research Program from the NRF (2016R1A6A3A04009723); Deutsche Forschungsgemeinschaft (SPP1839, grant LI 580/12).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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