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Modal symmetry controlled secondharmonic generation by propagating plasmons Tzu-Yu Chen, Julian Obermeier, Thorsten Schumacher, FanCheng Lin, Jer-Shing Huang, Markus Lippitz, and Chen-Bin Huang Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b02630 • Publication Date (Web): 23 Aug 2019 Downloaded from pubs.acs.org on August 26, 2019

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Nano Letters

Modal symmetry controlled second-harmonic generation by propagating plasmons Tzu-Yu Chen,† Julian Obermeier,¶ Thorsten Schumacher,¶ Fan-Cheng Lin,§ Jer-Shing Huang,k Markus Lippitz,∗,¶ and Chen-Bin Huang∗,† †Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan ‡International Intercollegiate PhD Program, National Tsing Hua University, Hsinchu 30013, Taiwan ¶Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany §Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan kLeibniz Institute of Photonic Technology, 07745 Jena, Germany ⊥Research Center for Applied Sciences, Academia Sinica, Taipei 115-29, Taiwan E-mail: [email protected]; [email protected]

Abstract A new concept for second-harmonic generation (SHG) in an optical nanocircuit is proposed. We demonstrate both theoretically and experimentally that the symmetry of an optical mode alone is sufficient to allow SHG even in centro-symmetric structures made of centro-symmetric material. The concept is realized using a plasmonic twowire transmission-line (TWTL), which simultaneously supports a symmetric and an anti-symmetric mode. We first confirm that emission of second-harmonic light into the symmetric mode of the waveguide is symmetry-allowed when the fundamental excited waveguide modes are either purely symmetric or anti-symmetric. We further switch the

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emission into the anti-symmetric mode when a controlled mixture of the fundamental modes is excited simultaneously. Our results open up a new degree of freedom into the designs of nonlinear optical components, and should pave a new avenue towards multi-functional nanophotonic circuitry. E(ω) tion

nera SH ge

fundamental

E(2ω)

second harmonic

symmetric

anti-symmetric

Keywords Surface plasmon polaritons, second-harmonic generation, surface nonlinear optics, optical nanocircuit, polarization-selective control

Manuscript As documented in all nonlinear optics textbooks, 1 second-harmonic generation (SHG) from a bulk material requires a non-centro-symmetric crystal structure. This poses a problem, e.g., for nonlinear plasmonics, as typical noble metals, such as gold, silver and aluminum, are centro-symmetric and do not allow SHG in their bulk form. A strategy to circumvent this limitation is to make use of the non-vanishing second-order susceptibility at the surfaces, where the symmetry is automatically broken. 2–5 However, when measured in the far-field, despite local SHG is permitted at the interface, contributions of two opposing surface elements cancel out in the dipole approximation, provided the nanostructure is exhibiting centrosymmetry. 6 Past efforts in plasmonic-assisted SHG circumvented this cancellation through designing asymmetric nanostructures. 7–13 In these cases, the asymmetry of the structure

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a

y

2D cut plane

z x

b

symmetric mode

anti-symmetric mode

E(ω) local generation (2)

P (2ω) emission

E(2ω)

Figure 1: Schematics of second-harmonic generation within a fully symmetric waveguide. A two-wire transmission line (a) supports two optical modes of different symmetry as depicted by the electric field vectors E(ω) in the 2D cross section of the waveguide (b). We show the real part of the in-plane components along the surface of the wires. The orientation of the fields is color-coded with respect to the local surface normal. The nonlinear polarization P(2) (2ω) responsible for second-harmonic generation is proportional to the square of the field along the surface normal. This polarization launches an optical wave in case a suitable second-harmonic mode E(2ω) is present. In the case of the waveguide, the symmetric mode is symmetry-allowed for both fundamental modes.

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causes asymmetric field distributions, so that the amplitude of one surface contribution overwhelms that of the opposing surface. 7–13 In this letter we will demonstrate that secondorder nonlinear light-matter interaction is possible also by appropriate symmetry of the light alone, without adjusting the symmetry of the matter. A key ingredient of our approach is a propagating waveguide mode. The particular structure we use is a plasmonic two-wire transmission line (TWTL, shown in Fig. 1a) and consists of two parallel gold nanowires of identical size and shape. 14–16 Gold has a sizable second-order surface nonlinearity, 5 but any other surface nonlinearity would lead to the same effect. Although the waveguide is fully centro-symmetric, it supports a symmetric and an anti-symmetric mode with the surface fields E(ω) shown in Fig. 1b, similar to the bonding and anti-bonding orbital in a hydrogen molecule or the two eigenmodes of a coupled pendulum. However, due to the absence of any resonance condition, both modes are broadband and cover large parts of the optical and near-infrared spectrum. These two modes offer an important new degree of freedom over other approaches of second-harmonic generation by propagating plasmons. 13,17–23 Most importantly, as compared to past SHG demonstrations using propagating plasmons, our approach ensures that the resulting second-harmonic signals are still bounded waveguide modes of the TWTL. We label the two modes by the symmetry of the normal components of the electric field, which agrees with the symmetry of the charge distribution. During the propagation of the fundamental field, a nonlinear polarization P(2) (2ω) is generated locally (Fig. 1b). It can emit into the waveguide modes available at the second-harmonic frequency 2ω. While the fundamental (λ = 1560 nm) and second harmonic fields (λ = 780 nm) are computed with a finite element solver (Comsol), the nonlinear polarization P (2) is calculated as 5

2 Pn(2) = 0 χ(2) nnn En

,

(1)

where E is the driving fundamental field and n denotes the vector component normal to

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the local surface patch under consideration. In second-harmonic generation at gold surfaces (2)

the tensor component χnnn of the nonlinear susceptibility dominates, i.e., only the vector components along the surface normal n enter. 5 The coupling efficiency η of the nonlinear polarization P(2) to the waveguide modes at the second harmonic E(2ω) can be calculated as 24,25 Z η =

∂A

2 P (2ω) · E(2ω) ds ∗

(2)

(2)

where the overlap integral runs over the surface of the waveguide. The integral reduces to a line integral along the circumference ∂A in case phase matching and damping is neglected, as possible for waveguides of a fixed length. In case of symmetric and anti-symmetric field distributions the nonlinear polarization is always fully symmetric. In consequence, only emission into the symmetric mode is allowed, while the integral for the anti-symmetric mode vanishes. In analogy, this is what forbids second-harmonic emission from a small plasmonic sphere, as free space modes are anti-symmetric upon point inversion. This argument remains valid even when taking all tensor components of the nonlinear susceptibility χ(2) into account (see Supporting Information, section S3). We set out to demonstrate these predictions in the experiment sketched in Fig. 2a. The waveguide consists of two identical single-crystalline gold wires of about 100 nm width and distance (Fig. 2b). A plasmonic nanoantenna that is connected to the transmission line (left side of structure) allows to independently excite the two eigenmodes of the waveguide by controlling the polarization direction of the incoming laser beam. Here, the excitation polarization along the antenna leads to a dipolar charge distribution and launches the antisymmetric mode (inset Fig. 2b). In case of excitation polarization parallel to the waveguide, the antenna part acts as scattering edge launching the symmetric mode. For detection, we make use of the different modal distribution of the electric field-amplitude. The antisymmetric mode is confined between the wires while the symmetric mode has a higher field

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a

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mode detector

E(ω)

excitation

on

nerati SH ge

E(2ω) antenna

SH emission

b antenna

TWTL

mode detector

sym.

1 µm

polarization controlled mode excitation

mode dependent emission position and polarization

anti-sym.

��nm

c +

+

+

��nm

d

+

-

symmetric mode

-

+

+

anti-symmetric mode max.

0 Etot (arb. u.)

Figure 2: Experimental realization of a plasmonic two-wire transmission line supporting two waveguide modes. (a) In the setup, the fundamental laser beam is focused on the antenna part of the waveguide. During propagation in the waveguide, light at the second harmonic frequency is generated and then scattered from the other end of the waveguide. (b) SEM micrograph of the structure, fabricated by focused ion-beam milling of a crystalline gold flake. The dashed circles depict the excitation and detection foci. A thin connection between the wires serves as incoupling antenna. A region without gap between the wires acts as mode detector. Depending on the polarization direction of the laser focused in the antenna part, either the symmetric (c) or the anti-symmetric mode (d) is launched. The latter is localized between the two wires and scattered out on the left end of the mode detector. The symmetric mode travels until the right end of the mode detector. The emission has the same polarization direction as the excitation beam.

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amplitude outside the wire pair (Fig. 2c and 2d). This allows us to use a two-wire/single-wire interface (right side of structure) as mode detector. 14,16,26 The emitted light has the identical polarization that was necessary to excite the mode. second harmonic

symmetric mode

fundamental

anti-sym. mode

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a

c

excitation

emission

software filtered (intensity x0.4)

analyzer

d

b

software filtered (intensity x0.4)

0

analyzer

max. intensity (arb. u.)

Figure 3: Emission from the mode detector signals generation of second-harmonic in the symmetric mode, independent of the launched fundamental mode. (a-b) Imaging the sample at the fundamental wavelength shows next to the strong reflection of the incoming laser beam a second spot either at the left or at the right end of the mode detector. Depending on the input polarization (black arrows) either the symmetric (a) or anti-symmetric mode (b) is launched. (c-d) Imaging at the second harmonic shows also emission from the antenna itself, but independent of the launched fundamental mode always emission from the symmetric waveguide mode at the right end of the mode detector. The weak spot at the left end of the mode detector is a background signal from locally generated second harmonic (see Supporting Information). For better visibility, the displayed intensity of excitation sides is reduced by 60%. The experimental results for linear and nonlinear propagation are shown in Figs. 3a-b and 3c-d, respectively. When exciting the antenna with parallel polarization, the symmetric mode is launched leading to fundamental emission at the right end of the mode detector (Fig. 3a). Rotating the polarization by π/2, the excited mode changes and consequently also the emission point (Fig. 3b). Figures 3a-b show the two distinct modes of the TWTL can be deterministically excited at the fundamental frequency, therefore providing the unique ability to clearly investigate their respective SH emission properties independently. In the second harmonic images Figs. 3c-d the analyzer selects the emission from the symmetric mode at the right end of the mode detector. A discussion of the other polarization direction is presented in the Supporting Information (section S5). We find locally generated second-harmonic 7


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emission from the input antenna, independent on the fundamental excitation polarization. In contrast to other publications, the waveguide itself appears dark, demonstrating high surface quality. 19 Most important, we find second-harmonic emission for both excitation polarization directions at the right end of the mode detector, i.e., stemming from the symmetric mode only. The excitation of the anti-symmetric fundamental mode (Fig. 3d) leads to a slightly lower emission intensity, in agreement with our numerical simulations, due to a reduced mode overlap. Regardless of which pure fundamental mode is excited, we always generate second-harmonic during propagation in the waveguide that is emitted into the symmetric mode at frequency 2ω. This process is symmetry-allowed and takes place although both the material as well as the structure is centro-symmetric. We exclude sizable SHG at antenna and mode-detector as well as substrate-induced effects (see Supporting Information, sections S5 and S6). A natural question that arises at this stage is whether second harmonic could also be emitted into the anti-symmetric mode, and if possible, how? Obviously the fundamental harmonic excitation and subsequently the second harmonic emission into purely symmetric mode is only a special case of a more general behavior, that is what we want to pursue further. In general, each superposition of fundamental modes may be launched, leading to a nonlinear mode mixing during propagation and emission from different second harmonic modes. In case of our waveguide, the fundamental field can be written as a coherent superposition of the symmetric (Es ) and anti-symmetric (Eas ) waveguide mode:

Etot = ck Es + c⊥ r exp(iφ) Eas

,

(3)

with ck = cos θ, c⊥ = sin θ being the amplitude of the two polarization components of the fundamental laser beam. The parameters r and φ take into account that the antenna coupling efficiency differs in amplitude and phase between the two modes. When observing the fundamental wavelength, the mode detector projects the total field Etot again on the two

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a data points

fundamental

c

E in

� mode detector

antenna

model fit

b second harmonic

c

E in

�

analyzer orientation

c phase difference (rad)

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Nano Letters

anti-symmetric mode

/2

0

fit

0

symmetric mode

fit

/4 excitation angle

0

/2 0

/4 excitation angle

coupling efficiency

/2

max.

Figure 4: Coupling efficiency of the second-order nonlinear polarization to the symmetric and anti-symmetric mode of the waveguide. (a) Fundamental emission intensity at the two modedetector spots, as a function of the excitation beam polarization angle θ. The linear response follows the expected behavior. (b) Observing the nonlinear emission at the mode detector in the expected emission polarization direction shows a more complex behavior. It can be modeled taking the coherent superposition of symmetric and anti-symmetric fundamental mode into account. From the fit parameters (r = 0.83, φ = 73.3 deg) we can predict the full excitation polarization and phase dependence of the second harmonic emission intensity of the symmetric and anti-symmetric mode (c). eigenmodes. We observe the expected cos2 θ (sin2 θ) dependence for the symmetric (antisymmetric) mode intensity, as depicted by the polar representation in Fig. 4a. The behavior gets more complicated when observing the second harmonic emission (Fig. 4b). For the symmetric mode we find a deep minimum near 45◦ between the differing peak values at 0◦ 9


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and 90◦ that were already discussed in Fig. 3b. Moreover, the anti-symmetric mode shows emission with a peak at 45◦ linear input polarization of the fundamental mode. These results can be fully recovered by the model that we already introduced, taking only the dominating normal components of the fields into account. The total fundamental field (2)

Etot gives rise to a nonlinear polarization Pn (eq. 1) that is coupled with certain efficiency η (eq. 2) into the target mode at second harmonic frequency. The nonlinear polarization can be separated into a symmetric and anti-symmetric part,

2 + 2ck c⊥ r exp(iφ) Es Eas Pn(2) ∝ c2k Es2 + c2⊥ r2 exp(2iφ) Eas {z } {z } | |

.

(4)

anti-symmetric

symmetric

2 describe symmetric field distributions, only the cross term contributes As both Es2 and Eas

to second-harmonic generation in the anti-symmetric mode. The model is fitted to the full dataset using r and φ as free parameters as well as scaling parameters for the absolute intensity. The unknown exact spatial shape of the modes Es and Eas for the coupling efficiency integral (eq. 2), as well as the influence of the second harmonic phase matching can be absorbed in the free parameters r and φ (see Supporting Information, section S4). Altogether this leads for our structure to the fitting parameters r = 0.83, φ = 73.3◦ . In Fig. 4c we explore the dependence of the mode intensities on the relative phase φ and the polarization direction θ. For each eigenmode we calculate the coupling efficiency integral (2)

(eq. 2) using the the nonlinear polarization Pn

from eq. 4. The anti-symmetric mode is

excited at the second harmonic if both modes are simultaneously present in the fundamental field. It scales with the product of both mode amplitudes, but it does not depend on the relative phase φ. The symmetric target mode in contrast shows interference between the nonlinear polarizations that are generated by each fundamental mode. For a relative phase φ around π/2 this interference is destructive, causing the minimum around θ = 50.1◦ . The position and depth of this minimum depends on the value of r. This makes it possible to switch the second harmonic emission fully from the symmetric to the anti-symmetric mode

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and back again by turning the fundamental excitation polarization direction. To conclude, the symmetry of plasmonic waveguide modes offers new opportunities not available in far-field optics. This allows us to demonstrate a novel scheme for nanoscale second-harmonic generation. Although both material and structure are fully symmetric, the propagating plasmon in a smooth two-wire transmission-line generates second-harmonic in the waveguide modes. When the traveling fundamental field is either purely symmetric or anti-symmetric, only the symmetric second-harmonic mode is excited. We show a coherent superposition of the fundamental modes allows to control the amplitude ratio in the emitting modes, up to fully switching to the anti-symmetric second-harmonic mode. This allows a new route to deterministically tailor the propagating second-harmonic signals. Optimized phase-matching conditions can further enhance the efficiency of our structure. Moreover, our finding could be extended to any generalized waveguide systems where modes of suitable symmetry are supported. Our results opens up a completely new degree of freedom into the designs of nonlinear optical components, and should path a new avenue towards multifunctional nanophotonic circuitry. We anticipate impacts in performing nonlinear frequency mixings, polarization control, waveform shaping, selective routing, as well as potentials in quantum processing using our proposed scheme.

Supporting Information Available All data needed to evaluate the conclusions in the paper are present in the paper and/or the supporting information. Additional data related to this paper may be requested from the authors. This material is available free of charge via the Internet at http://pubs.acs.org.

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Author information Corresponding authors Email: C.B.H [email protected]; M.L. [email protected]

ORCID Markus Lippitz: 0000-0003-1218-6511 Chen-Bin Huang: 0000-0002-5824-3318 Jer-Shing Huang: 0000-0002-7027-3042

Author contributions T.Y.C. and J.O. contributed equally. C.B.H. conceived the concept. T.Y.C. and C.B.H. designed and performed the experiments. J.O., T.S. and M.L. performed numerical simulations and data processing. F.C.L. and J.S.H. prepared the samples. C.B.H., J.O., T.S., and M.L. wrote the manuscript. All authors commented on the manuscript.

Notes The authors declare no competing financial interest

Acknowledgement The authors thank the support from the Ministry of Science and Technology, Taiwan, under Grant MoST 106-2112-M-007-004-MY3.

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