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Mode coupling based on split-ring resonators and waveguide and second harmonic enhancement

Hao Chen, Menglai Zhang, Zhaofu Qin, Taozheng Hu, Zhuo Chen, and Zhenlin Wang

Open Access Open Access

Abstract

Nonlinear optical processes are promising for many applications, and recently great attention has been paid to improve the nonlinear efficiency of plasmonic metasurfaces. Here, we propose a hybrid structure consisting of a gold split-ring resonator (SRR) based nonlinear metasurface on top of a dielectric waveguide layer. By adjusting the periodicity of the SRR array, we demonstrate that the coupling between the magnetic surface plasmons (MSPs) of the SRRs and the waveguide modes could greatly enhance the second harmonic generation (SHG) intensity. Compared with the conventional SRR arrays, the hybrid structures could provide an extra enhancement in the SHG intensity of more than one order of magnitude.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Plasmonic metasurfaces, artificial optical two-dimensional plane composed of subwavelength elements, has attracted many researchers’ interest in recent years. Under the irradiation of incident electromagnetic wave, the phenomenon of collective oscillation of free electrons in metal is called localized surface plasmons (LSP), which can make the electric field localized in the range of sub-wavelength and significantly enhance the electric field intensity near the metal particles [1–3]. Metasurfaces based on LSP resonance are widely used in enhancing Raman scattering, [4–7] plasmon nanolasers, [8–10] biosensor [11–14] etc. In addition, nonlinear optical processes could be boosted effectively by LSP resonances in nanostructures, which greatly reduce the interaction length required to produce strong nonlinear effects [15–18]. The occurrence of nonlinear metasurfaces result in the significant reduction in the overall size of nonlinear devices, enabling them to be integrated into nanophotonic circuits, thus opening the way for new nanoscale nonlinear optical devices [19–21].

In order to enhance the localized electric field and improve the efficiency of nonlinear optical processes in plasmonic metasurfaces, significant efforts have been devoted. The nonlinear conversion efficiency of the metasurfaces can be significantly improved when the incident wavelength is near the position of LSP resonance [22–24]. Recently, researchers show that introducing surface lattice resonance at the position of LSP resonance could enhance nonlinear optical processes [25–27]. The principle is to introduce diffraction mode near the resonance of metal element structure by changing the periodicity and surface lattice resonance is generated by mode coupling. With increasing the period, the SHG can be enhanced several times under the combined action of mode coupling and dilution effect [28,29]. However, strong coupling of the diffraction modes and the LSP resonances requires a symmetric dielectric environment and the electric field direction matching between the incident wave and the diffraction mode. That is to say only the diffraction coupling in one particular axis could enhance the SHG intensity.

In this paper, we propose a hybrid structure consisting of gold SRR array on top of a dielectric waveguide layer to enhance the second harmonic generation (SHG) of plasmonic metasurfaces. The periodicity variation of the SRR array along the x-axis and y-axis would result in the strong coupling between the MSPs of SRRs and the waveguide modes, which could enhance the SHG intensity by 9-15 times compared with the conventional SRR array with the same periodicity.

2. Result and discussion

The geometry of the hybrid structure under study is schematically depicted in Fig. 1. It consists of an array of gold SRRs on top of an epoxy resin layer (nw = 1.6) with a thickness of t sandwiched between a glass substrate (ns = 1.46) and an air superstrate (na = 1.0). Each SRR is assumed to have a fixed height, base length, arm width and arm gap of h = 30 nm, l = 200 nm, w = 80 nm, and g = 100 nm, respectively. The coordinate system is introduced to make that the x-axis points along the SRR base, the y-axis is parallel to the SRR arms, and the z-axis is defined by the surface normal of the sample. The periods of the rectangular array along the x- and y-axes are ax and ay, respectively. The SRRs have been employed here, because it has been demonstrated that they could exhibit a relatively high SHG efficiency due to the broken central symmetry [30,31]. Numerical simulations were preformed based ${\varepsilon _{Au}}(\omega ) = 1 - [\omega _p^2/({\omega ^2} + i\omega \gamma )]$ on a commercial finite element method (Comsol Multiphysics). In the calculations, the relative permittivity of gold is described by Drude model: , where ω is the angular frequency of the incident electromagnetic wave, ${\omega _p} = 1.38 \times {10^{16}}{s^{ - 1}}$ and $\gamma = 1.075 \times {10^{14}}{s^{ - 1}}$ are the plasma frequency and damping rate, respectively [32,33]. Periodic boundary conditions are applied to the four side boundaries located in the xz and yz planes. The top and bottom boundaries in the xy plane are terminated with perfectly matched layers to absorb reflected and transmitted light in z-axis. To excite the MSP resonance, the incident wave is polarized along the x direction and normally incident on the SRR array along the z direction.

 figure: Fig. 1.

Fig. 1. Schematic diagram illustrating the geometry of the hybrid structure, the coordinate system, and the polarization configuration. The hybrid structure consists of a rectangular array (ax and ay: periods along the x- and y-axes) of gold SRRs on top of a slab waveguide layer with a thickness of t. The height (h), base length (l), the arm width (w) and the arm gap (g) of the SRRs are fixed to h = 30 nm, l = 200 nm, w= 80 nm, and g = 100 nm, respectively.

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In such a hybrid structure, the epoxy resin film, sandwiched between a glass substrate and an air superstrate with refractive indices smaller than that of the resin film, can act as an asymmetrical planar optical waveguide to support transverse electric (TE) guided modes (the electric field oscillates parallel to the interfaces) and transverse magnetic (TM) guide modes (the magnetic field oscillates parallel to the interfaces) below their corresponding cut-off wavelengths:[34]

$${\lambda _{TE\_cut}} = \frac{{2\pi t\sqrt {{\varepsilon _{wg}} - {\varepsilon _{sub}}} }}{{j\pi + {{\tan }^{ - 1}}\left( {\sqrt {\frac{{{\varepsilon_{sub}} - {\varepsilon_{air}}}}{{{\varepsilon_{wg}} - {\varepsilon_{sub}}}}} } \right)}}$$
$${\lambda _{TM\_cut}} = \frac{{2\pi t\sqrt {{\varepsilon _{wg}} - {\varepsilon _{sub}}} }}{{j\pi + {{\tan }^{ - 1}}\left( {\frac{{{\varepsilon_{wg}}}}{{{\varepsilon_{air}}}}\sqrt {\frac{{{\varepsilon_{sub}} - {\varepsilon_{air}}}}{{{\varepsilon_{wg}} - {\varepsilon_{sub}}}}} } \right)}}$$
where j is the order of the waveguide modes, t is the thickness of the waveguide layer, and ɛair, ɛwg and ɛsub are the relative dielectric constants of air, waveguide layer and substrate layer, respectively. As shown in Fig. 2(a), the cut-off wavelengths for fundamental (j = 0) and first-order (j = 1) waveguide modes are plotted against the thickness of the epoxy resin layer. It should be noted that the SRR array has a dual role. On the one hand, it provides the MSP resonance related to the individual SRRs. According to our calculations, the SRR array with a square lattice of ax = ay = 400 nm could support a MSP resonance at 1670 nm. Therefore, the spectral wavelength range is restricted to 1300–2300 nm. As seen from Fig. 2(a), when the waveguide thickness is chosen to be t = 650 nm, such a wavelength range locates well within the single-mode operation region for TE0 and TM0 waveguide modes. On the other hand, the SRR array could act as a grating coupler and couple the incident wave into the guided waves when the momentum conservation condition:
$${\vec{k}_{wg}} = {\vec{k}_{/{/}}} + m{G_x}\hat{x} + n{G_y}\hat{y}$$
is satisfied, where m and n are integers related to the diffraction order, kwg is the waveguide wave vector, k// is the component of the wave vector of the incident light in the x-y plane, Gx = 2π/ax and Gy = 2π/ay are reciprocal lattice vectors. For normal incidence, k// vanishes and therefore kwg is solely determined by the period ax and ay. According to Eq. (3), the spectral position of the waveguide modes (λwg) is given by:
$$\frac{{{n_{eff}}}}{{{\lambda _{wg}}}} = \frac{m}{{{a_x}}} + \frac{n}{{{a_y}}}$$
where neff is the effective refractive index of the waveguides. In the following discussions, either ax or ay is fixed to 400 nm while varying the other one. Under such circumstances, it could be easily deduced from Eq. (4) that the spectral position of the waveguide mode λwg possibly locates within the spectral range of interest from 1300 nm to 2300 nm only when (m = 0, n = 1) for the fixed ax = 400 nm and (m = 1, n = 0) for the fixed ay = 400 nm, since the effective refractive index neff is smaller than the refractive index (nw = 1.6) of the dielectric waveguide layer. This also means that when ax (or ay) is fixed to 400 nm, only the first order diffraction along the y-axis (or x-axis) could contribute to the excitation of the waveguide mode. Since the incident wave in our case is polarized along the x-axis, the first order diffraction along the y-axis and x-axis correspond to the excitation of TE0 and TM0 waveguide modes, respectively. For example, Fig. 2(b) shows the transmission spectrum of a rectangular array of SRRs with ax = 400 nm and ay = 900 nm, in which only the first order diffraction along the y-axis is expected to excite the TE0 mode. A broad transmission dip and a relatively narrow dip could be observed at 1750nm and 1345 nm in the spectrum, respectively. An induced magnetic dipole is presented in the Hz component of the magnetic field distribution on the xy-plane at 1750nm [right inset in Fig. 2(b)], which indicates that the broad dip corresponds to the MSP resonance of the SRR. To demonstrate that the narrow dip is attributed to the excitation of TE0 mode, the Ex component of the electric field distributions at 1345 nm is plotted in the left inset in Fig. 3(b) as a color map on the yz-plane. As expected, the Ex field is mainly concentrated within the waveguide layer and forms a standing wave pattern as a result of interference between waveguide modes traveling along opposite ± y directions.

 figure: Fig. 2.

Fig. 2. (a) Cut-off wavelengths for fundamental and first-order waveguide modes as a function of the waveguide thickness. Black vertical arrow line indicates a single-mode operation region for TE0 and TM0 waveguide modes supported in a t = 650 nm thick waveguide. (b) Normal incidence transmission spectrum of a rectangular array of SRRs with ax = 400 nm and ay = 900 nm on top of the waveguide layer. Left inset shows the Ex component of electric field distribution on the yz-plane at 1345 nm. Right inset shows the Hz component of magnetic field distribution on the xy-plane at 1750nm.

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 figure: Fig. 3.

Fig. 3. Transmission dips extracted from spectral positions with different ax (a) and ay (b). The red and blue solid circles represent the longer and the shorter wavelength transmission dips of the hybrid structure, respectively. The black dashed line represents the undisturbed waveguide modes, and the black dotted line represents the MSP positions of the SRR array based on the semi-infinite epoxy resin layer. In Fig.3a, the period of SRR array in y direction is fixed to ay = 400 nm; and the period in x direction is fixed to ax = 400 nm in Fig. 3(b).

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In order to study the coupling between the MSP resonance and the TM0 guided modes, normal incidence transmission spectra are calculated for a series of rectangular arrays of SRRs on top of the 650-nm-thick waveguide with the periodicity ay being fixed to ay = 400 nm while varying ax from 900 nm to 1300 nm in a step of 20 nm. Figure 3(a) shows the extracted spectral positions of the two transmission dips as a function of the periodicity of ax. As mentioned above, for the fixed ay = 400 nm, only the first order diffraction along the x-axis contributes to the excitation of the TM0 mode within the spectral range of interest from 1300 nm to 2300 nm, i.e., TE modes and higher order TM modes in this case are eliminated within wavelength range from 1300 nm to 2300 nm.

For direct comparison, we also plot the spectral positions of the undisturbed TM0 waveguide modes [black dashed line in Fig. 3(a)] and the MSP resonances of conventional SRR arrays without a wave guiding layer (i.e., directly placed on a semi-infinite epoxy substrate) with ay = 400 nm and ax varying from 900 nm to 1300 nm [black dotted line in Fig. 3(a)]. It is worth noting that the positions of TM0 waveguide modes are given by ${\lambda _{wg}} = {n_{eff}}{a_x}$, which is the case of m =1 and n = 0 in Eq. (4). The effective refractive index neff can be calculated by the following relations:

$$\tan ({t\kappa } )= \frac{{{\gamma _c} + {\gamma _s}}}{{\kappa \left( {1 - \frac{{{\gamma_c}{\gamma_s}}}{{{\kappa^2}}}} \right)}} \textrm{(for TE mode)}$$
$$\tan ({t\kappa } )= \frac{{\kappa \left( {\frac{{n_w^2}}{{n_s^2}}{\gamma_s} + \frac{{n_w^2}}{{n_c^2}}{\gamma_c}} \right)}}{{{\kappa ^2} - \frac{{n_w^4}}{{n_c^2n_s^2}}{\gamma _c}{\gamma _s}}} \textrm{(for TM mode)}$$
where t is the thickness of the waveguide layer, nc, ns, nw are refractive indices of the cover layer, substrate and waveguide layer, respectively, $\; \beta = \sqrt {k_0^2n_w^2 - {\kappa ^2}} $ is the propagation constant of the waveguide mode, κ is the propagation constant in waveguide layer, ${\gamma _c} = \sqrt {{\beta ^2} - k_0^2n_c^2} $ and ${\gamma _s} = \sqrt {{\beta ^2} - k_0^2n_s^2} $ describe transverse decay in cover layer and substrate, k0 = $\frac{{2\pi }}{\lambda }$ is the free-space wavevector and neff = $\frac{\beta }{{{k_0}}}$.

It could be seen from Fig. 3(a) that with increasing the periodicity ax the position of the undisturbed TM0 waveguide mode continuously red-shifts (black dashed line), while the MSP resonance of SRR array on the semi-infinite epoxy layer almost keeps a constant wavelength (black dotted line). Therefore, it is expected that TM0 waveguide mode could be tuned to overlap with and strongly couple to the MSP resonance at a certain periodicity ax, which is about ax = 1100 nm (corresponding to the crossing point of the black dashed and dotted lines). For a relatively small ax = 900 nm, it is found that the longer wavelength branch is closed to the MSP resonance (red solid circles and black dotted line) and the shorter wavelength branch is closed to the undisturbed TM0 waveguide mode (blue solid circles and black dashed line), indicating there is no coupling or only weak coupling between the MSP resonance and the TM0 waveguide mode. When the periodicity ax is increased to about 1100 nm, the positions of transmission dips present an obvious anti-crossing phenomenon (blue and red circles), which is a characteristic of the strong coupling between the TM0 waveguide mode and the MSP resonance. With further increasing ax, the position of the undisturbed TM0 waveguide mode will move away from the MSP resonance, which causes the disappearance of the strong coupling, and thus the longer wavelength branch is closed to the undisturbed TM0 waveguide mode (red solid circles and black dashed line) and the shorter wavelength branch is closed to the MSP resonance (blue solid circles and black dotted line).

Figure 3(b) summarizes the transmission dips extracted from the transmission spectra of the hybrid structure with a fixed ax = 400 nm and ay varying from 900 nm to 1400 nm. In this case, only the first order diffraction along the y-axis could contribute to the excitation of the TE0 waveguide mode. The spectral positions of the undisturbed TE0 waveguide modes (black dashed line) and the MSP resonances of conventional SRR arrays (black dotted line) are also plotted in Fig. 3(b). Around the crossing point of the black dashed line and dotted line, obvious anti-crossing phenomenon could be observed (red and blue solid circles) and thus indicating the occurrence of the strong coupling between the MSP resonance and TE0 waveguide mode. Away from this strong coupling regime, the transmission dip positions approximately follow the two black lines.

In order to study the influence of the strong mode coupling on SHG intensity of SRR array, the sources of SHG are applied to the surfaces of gold SRR. The enhancement of the mode coupling on the SHG intensity of the SRR array is shown in Fig. 4. Figure 4(a) shows the simulated SHG intensity of the SRR array with different ax under strong mode coupling, and the maximum SHG enhancement occurs at periodicity of ax = 1120 nm and ay = 400 nm, which is corresponding to the strong coupling regime in Fig. 3(a). The enhancement factors in Fig. 4(a) are normalized by the SHG intensity maximum of SRR array (without waveguide) at periodicity of ax = 1120 nm and ay = 400 nm. Similarly, we normalize the enhanced SHG intensity with different ay to the SRR array (without waveguide) at periodicity of ay = 1190 nm and ax = 400 nm in Fig. 4(b). Due to the strong coupling between the MSP resonances of gold SRRs and the waveguide modes, the SHG intensity is enhanced by 9 times and 15 times by tuning the periodicity in x and y directions, respectively.

 figure: Fig. 4.

Fig. 4. (a) Simulated SHG intensity of SRR array with different ax when fixing ay = 400 nm, and the simulated SHG intensity is normalized by the maximum SHG intensity of the conventional SRR array with ax = 1120 nm, ay =400 nm. (b) Simulated SHG intensity of SRR array with different ay when fixing ax = 400 nm, and the simulated SHG intensity is normalized by the maximum SHG intensity of the conventional SRR array with ay = 1190 nm, ax =400 nm.

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3. Summary

In this work, we proposed a hybrid structure consisting of a rectangular array of gold SRRs on top of a dielectric waveguide layer for SHG enhancement. By respectively varying the periodicity along the x-axis and y-axis of the rectangular array, we demonstrated that both the fundamental TE and TM waveguide modes could be strongly coupled into the MSP resonances of the SRRs. Associated with the strong coupling between the MSP resonance and the waveguide mode, the SHG intensity could be greatly enhanced by 9–15 times, compared with the conventional SRR arrays having the same periodicity. Our results offer a new way to enhance the nonlinear effect of metasurfaces, which could improve nonlinear conversion efficiency significantly and meanwhile reduce the density of micro-nano array and thus the processing cost.

Funding

National Key Research and Development Program of China (2017YFA0303700); National Natural Science Foundation of China (11774162, 11834007).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. Schematic diagram illustrating the geometry of the hybrid structure, the coordinate system, and the polarization configuration. The hybrid structure consists of a rectangular array (ax and ay: periods along the x- and y-axes) of gold SRRs on top of a slab waveguide layer with a thickness of t. The height (h), base length (l), the arm width (w) and the arm gap (g) of the SRRs are fixed to h = 30 nm, l = 200 nm, w= 80 nm, and g = 100 nm, respectively.

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Fig. 2.
Fig. 2. (a) Cut-off wavelengths for fundamental and first-order waveguide modes as a function of the waveguide thickness. Black vertical arrow line indicates a single-mode operation region for TE0 and TM0 waveguide modes supported in a t = 650 nm thick waveguide. (b) Normal incidence transmission spectrum of a rectangular array of SRRs with ax = 400 nm and ay = 900 nm on top of the waveguide layer. Left inset shows the Ex component of electric field distribution on the yz-plane at 1345 nm. Right inset shows the Hz component of magnetic field distribution on the xy-plane at 1750nm.

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Fig. 3.
Fig. 3. Transmission dips extracted from spectral positions with different ax (a) and ay (b). The red and blue solid circles represent the longer and the shorter wavelength transmission dips of the hybrid structure, respectively. The black dashed line represents the undisturbed waveguide modes, and the black dotted line represents the MSP positions of the SRR array based on the semi-infinite epoxy resin layer. In Fig.3a, the period of SRR array in y direction is fixed to ay = 400 nm; and the period in x direction is fixed to ax = 400 nm in Fig. 3(b).

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Fig. 4.
Fig. 4. (a) Simulated SHG intensity of SRR array with different ax when fixing ay = 400 nm, and the simulated SHG intensity is normalized by the maximum SHG intensity of the conventional SRR array with ax = 1120 nm, ay =400 nm. (b) Simulated SHG intensity of SRR array with different ay when fixing ax = 400 nm, and the simulated SHG intensity is normalized by the maximum SHG intensity of the conventional SRR array with ay = 1190 nm, ax =400 nm.

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Equations (6)

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λ T E _ c u t = 2 π t ε w g − ε s u b j π + tan − 1 ( ε s u b − ε a i r ε w g − ε s u b )
λ T M _ c u t = 2 π t ε w g − ε s u b j π + tan − 1 ( ε w g ε a i r ε s u b − ε a i r ε w g − ε s u b )
k → w g = k → / / + m G x x ^ + n G y y ^
n e f f λ w g = m a x + n a y
tan ⁡ ( t κ ) = γ c + γ s κ ( 1 − γ c γ s κ 2 ) (for TE mode)
tan ⁡ ( t κ ) = κ ( n w 2 n s 2 γ s + n w 2 n c 2 γ c ) κ 2 − n w 4 n c 2 n s 2 γ c γ s (for TM mode)
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