## Abstract

Nanoplasmonic waveguides based on lithium niobate (LN) are shown to provide the light-matter interaction required for next-generation developments in nonlinear frequency-conversion nanostructures. Here, we numerically investigate second harmonic generation of a 1550 nm, 100 fs pulse in metal-LN-metal (MLNM) nanoplasmonic and LN hybrid-plasmonic (LNHP) waveguides. In comparison to a photonic LN waveguide, a 2.1 µm-long LNHP waveguide exhibits a conversion efficiency improvement of 11 times, whereas a 20 µm-long MLNM nanoplasmonic waveguide is shown to have a conversion efficiency of 1.1 × 10^{−4}. The MLNM nanoplasmonic and LNHP waveguides have the potential to operate as sources of optical radiation for on-chip photonic systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The nanophotonics field is continuing to evolve with the development of novel light-based devices having unique functionalities. It is envisioned that this field will gradually merge with the more mature field of electronics for the enhanced performance of devices, such as photonic logic, high-speed switching, and signal processing. In particular, this is brought closer to realization when light confinement is implemented in nanoplasmonic structures. Nanoplasmonic devices have been exploited to guide, modulate, and generate electromagnetic radiation, while simultaneously performing as electronic interconnects.

While nonlinear electronic elements such as diodes, transistors, and amplifiers have made electronics prevalent, it is essential for nanoplasmonic devices to benefit from optical nonlinearity to allow for a higher level of device versatility. In the nonlinear optical regime, the extreme light confinement effect of nanoplasmonics has proven valuable by enhancing light-matter interaction. When considering the visible and near-infrared optical regimes of the electromagnetic spectrum, the nanoplasmonic confinement effects enhance nonlinear light generation in nanofabricated structures such as Si-gold nanoplasmonic waveguides [1,2], gold-shelled cylindrical waveguides [3], and gold nanoantennas [4]. The nanoscale size of the nonlinear plasmonic structures is essential for the miniaturization of radiation sources for on-chip nanooptoelectronic systems. Of the numerous nonlinear dielectrics that have been investigated, lithium niobate (LiNbO_{3}, denoted as LN) has prevailed as the dominant crystal for nonlinear frequency-conversion processes across a wide electromagnetic spectrum. Such acclaim is due to LN’s high nonlinear coefficient and well-established nanofabrication techniques.

Recently, there has been insightful theoretical [5–8] and experimental [9] work performed in combining LN with various plasmonic configurations for second harmonic generation (SHG) and optical parametric amplification. 2D and 3D metal-LN-metal (MLNM) waveguides [5–7], a hybrid air-nanogap plasmonic waveguide [8], and a Kretschmann configuration [9] were investigated to show their potential for enhanced nonlinear generation. However, for nanoplasmonic applications, the planar nature of the 2D waveguides and the Kretschmann configuration are inherently large-footprint structures; whereas the fabrication of a quasi-phase-matched periodically poled LN hybrid air-nanogap plasmonic waveguide is inconceivable on the nanoscale.

In modeling the LN plasmonic waveguides discussed in [5–8], a two-step, segregated process is implemented that numerically solves for the waveguide modes and incorporates the resulting distributions with the time-harmonic 1D coupled mode theoretical equations. Since only a single, or a few, second harmonic modes are being considered, the generation of certain second harmonic modes are neglected, thus overlooking important mode-beating effects that influence the nonlinear conversion efficiency. Additionally, some of the theoretical models ignore depletion of the pump electric field and neglect the implementation of several second-order nonlinear susceptibility elements.

In the above mentioned LN plasmonic waveguides [5–8], another key limitation in modelling the nonlinear frequency-conversion process is that the interaction is restricted to the single frequency (i.e. continuous-wave) regime. This is a consequence of the derived coupled mode theory equations using the constraint of time-harmonic fields and the fact that sum frequency mixing effects are neglected. However, since the nature of the frequency-conversion process is intensity-dependent, the high-peak powers delivered by ultrashort pulses are necessary for efficient light generation. Furthermore, the commonly-used coupled mode theory equations make the assumption that the radiation is described by the slowly-varying-amplitude approximation (SVAA). It has been shown that the SVAA cannot be ignored for the situation where nonlinear materials are excited by ultrashort pulses or highly-intense radiation [10]. Specifically, if this approximation is incorrectly implemented when the second-order differential terms in the coupled mode theoretical equations are relevant, then the SVAA approximation unintentionally neglects the generation of backwards-travelling waves [10]. For ultrashort pulses of appreciable intensities, the SVAA breaks down before the damage threshold of LN is reached [10]. Although the SVAA is valid for pump pulses having moderate intensities in LN (i.e. on the order of tens or hundreds of GW/cm^{2}), it breaks down for high intensity pump pulses (≳1 TW/cm^{2}) [10]. Clearly, a more comprehensive modelling technique must be developed to investigate the broadband (i.e. ultrashort laser pulse) excitation of LN plasmonic waveguides. Such a model must incorporate material and waveguide dispersions to describe the spatial and temporal separation of the ultrashort pulses, the full second-order nonlinear susceptibility tensor, pump laser pulse depletion effects, frequency-conversion mode beating, and mixing within the bandwidth of the ultrashort laser pulse.

Herein, LN plasmonic waveguides are investigated using a fully-integrated (i.e. one-step) comprehensive model based of the finite-difference time-domain (FDTD) technique, which solves the full-vector Maxwell’s equations without invoking the SVAA formalism. To date, this FDTD simulation procedure is the most rigorous model describing nonlinear interaction in LN plasmonic waveguides. We use this technique to investigate ultrashort pulse electric field generation in a MLNM nanoplasmonic and a LN hybrid-plasmonic (LNHP) nonlinear waveguide. By exciting the nonlinear nanoplasmonic waveguides with a 1550 nm, 100 fs pulse, spectral components are generated near the wavelength of 775 nm. In both the MLNM nanoplasmonic and LNHP waveguides, a highly directional electric field polarization and significant spatial overlap between the pump mode and the SHG mode allow for high nonlinear conversion. The LNHP waveguide exhibits a maximum conversion efficiency of >10^{−5} over a length of a few microns, while the MLNM nanoplasmonic waveguide conversion efficiency reaches values >10^{−4} at lengths approaching 20 µm. In comparison to a photonic LN waveguide, the LNHP waveguide exhibits improvements in the conversion efficiency by an order of magnitude. These nanoplasmonic waveguides are envisioned for use as sources of optical radiation that occupy a small footprint and which can be integrated into optoelectronic systems. The rigorous FDTD procedure used in this analysis can model sub-wavelength nonlinear generation effects and could therefore be used to investigate nonlinear frequency-conversion waveguides having ultra-small footprints (i.e. lengths on the order of a wavelength).

## 2. Simulation method

Nonlinear effects are incorporated into the FDTD simulations by using the full second-order nonlinear coefficient tensor, which is given as,

*d*are the second-order nonlinear coefficients. This tensor is used to determine the second-order nonlinear polarization components as,

_{ij}*ω*is the frequency being generated,

*ε*is the permittivity of free space,

_{0}*E*are the electric field components along the

_{x,y,z}*x*,

*y*, and

*z*crystal-directions, respectively, and {

*E*⁎

_{i}*E*}(

_{j}*ω*) represents the frequency-domain convolution operation of the

*i*and

^{th}*j*components of the electric field. Notably, the convolution operation arises because we are considering mixing between all of the frequency components present in a broadband pulse. Using Eqs. (2a)-(2c), the components of the displacement field can be written as,

^{th}*ε*,

_{r,xx}*ε*, and

_{r,yy}*ε*are the diagonal components of the relative permittivity tensor. Notably, dispersion of the

_{r,zz}*d*values is negligible across the spectral regions being investigated, such that the

_{ij}*d*values are independent of frequency (i.e.

_{ij}*d*(

_{ij}*ω*) =

*d*). These frequency-domain displacement field constitutive relations include both first-order and second-order polarization effects, and can be converted to the time-domain to obtain,

_{ij}*t*represents time and {

*ε*⁎

_{r,ii}*E*}(

_{i}*t*) is the time-domain convolution operation between the

*ii*relative permittivity tensor component and the

^{th}*i*component of the electric field. Equations (4a)-(4c) represent the time-domain constitutive relation used in the FDTD code to update the displacement field. Notably, while the above equations describe a general second-order nonlinear material, they can be simplified for the particular case of LN, which has the second-order nonlinear coefficient tensor,

^{th}*d*= 4 pm/V,

_{31}*d*= −2.4 pm/V, and

_{22}*d*= 20.6 pm/V at a pump wavelength of 1550 nm [11–16].

_{33}This nonlinear FDTD simulation procedure considers mixing between all the frequency components of a broadband, femtosecond pump pulse [see Eqs. (2a)-(2c) and (3a)-(3c)]. This is in contrast to previous theoretical investigations [5–8], where a major limitation arose from using the time-harmonic assumption. Such a key assumption restricts the aforementioned models to single-frequency (i.e. continuous-wave) investigations. Additionally, while other investigations often utilize an effective value for the second-order nonlinear coefficient, in the current investigation, we implement FDTD simulations that incorporate the full second-order nonlinear coefficient tensor in Eq. (1). Moreover, our simulations solve the full-vector Maxwell’s equations, such that no approximation is made on the amplitude of the electric and magnetic fields. As such, this provides a more accurate model than SHG investigations that solve the coupled-mode theoretical equations, since the SVAA is often utilized to simplify the computations.

## 3. Waveguide structures and material properties

Both the MLNM nanoplasmonic and LNHP frequency-conversion waveguides are supported by a SiO_{2} platform, as shown in Fig. 1(a) and 1(d), respectively. The MLNM nanoplasmonic waveguiding structure is composed of a thin LN layer of height *h*, width *w*, and length *l* that is sandwiched between two 100 nm-thick gold film layers. Alternatively, the LNHP waveguide is composed of stacked gold, LN, and Si layers. Here, the LN width, height, and length are denoted as *w*, *h*, and *l*, respectively, the gold thickness is set to 100 nm, and the Si layer thicknesses is set as 123 nm to allow for phase-matching. Since a substantial portion of the SHG and pump mode is situated in the Si, this layer is used to adjust the effective refractive indices of these modes. Notably, gold is chosen as the metal layer because it provides a stable chemical structure in air. This contrasts with silver, which is known to react with sulfur compounds in air and produce silver sulfide. In both of the waveguides, the *c*-axis of the LN is oriented along the *z*-axis and the crystal’s cross-section is cut along the <010> plane. The waveguides are excited by an electric field pulse having a central-wavelength of 1550 nm and a pulse duration of 100 fs. As such, the resulting pump mode generates photons near a wavelength of 775 nm via SHG as it propagates along the LN waveguides. Importantly, very detailed analyses have been performed on coupling into these plasmonic waveguides [17–20].

In the calculations, the ordinary and extraordinary refractive indices of the uniaxial LN crystal are obtained from [21], the linear optical properties of Si and SiO_{2} are both taken from [22], and the optical data for gold is attained from [23]. For the nonlinear optical properties, the simulations implement the complete second-order nonlinear coefficient tensor of LN [see Eq. (5)]. Furthermore, while nonlinear effects such as multiphoton absorption occur in the LN at intensities >100 GW/cm^{2} [24], the waveguide modes excited by the pump pulse all have a peak intensity of 10 GW/cm^{2}, thus these nonlinear effects do not influence the SHG process.

Figure 1(a) illustrates the cross-section of the *w* = 400 nm and *h* = 780 nm MLNM nanoplasmonic waveguide. At these dimensions, the MLNM nanoplasmonic waveguide supports the pump mode and the quasi-TM_{00} SHG mode displayed in Fig. 1(b) and 1(c), respectively. Notably, these modes exhibit significant spatial overlap and have the majority of their electric field polarization along the *z*-direction, such that the photons generated near 775 nm will couple to the quasi-TM_{00} SHG mode with a high efficiency. Similarly, the *w* = 400 nm and *h* = 100 nm LNHP waveguide [Fig. 1(d)] supports the pump mode displayed in Fig. 1(e) and the SHG mode presented in Fig. 1(f). Again, since these modes occupy the same spatial region and both possess a significant *z*-directed electric field polarization component, SHG is expected to occur at a high conversion efficiency.

## 4. Results and discussion

The phase-matching condition is investigated for both the MLNM nanoplasmonic and LNHP waveguides, where phase-matching is achieved by tuning the cross-sectional dimensions of the plasmonic waveguides. Figure 2(a) depicts the effective refractive index, *n _{eff}*, of the modes for both waveguides over the wavelength ranges of

*λ*= 1450 nm-1650 nm and

_{pump}*λ*= 725 nm-825 nm. By choosing

_{SHG}*h*= 780 nm for the MLNM nanoplasmonic and

*h*= 100 nm for the LNHP waveguides, the pump wavelength of

*λ*= 1550 nm is phase-matched to the SHG wavelength of

_{pump}*λ*= 775 nm. The coherence length,

_{SHG}*L*, is presented in Fig. 2(b), where the large peaks correspond to perfect phase-matching between

_{coh}*λ*= 1550 nm and

_{pump}*λ*= 775 nm. Importantly, a large coherence length is also obtained in the vicinity of

_{SHG}*λ*= 1550 nm. For example, at a pump wavelength of

_{pump}*λ*= 1560 nm, the MLNM nanoplasmonic and the LNHP waveguides have an

_{pump}*L*of 310 µm and 28 µm, respectively. Figure 2(c) show the propagation length of the pump and SHG wavelengths,

_{coh}*L*. The small

_{prop}*L*values (i.e. <40 µm) observed in both waveguides are primarily due to losses introduced by the gold layers at both the pump and SHG wavelengths. Clearly, the MLNM nanoplasmonic waveguide exhibits the largest

_{prop}*L*at the SHG wavelengths, since the associated quasi-TM

_{prop}_{00}mode is mainly confined to the LN core and only a small portion of the electric field interacts with the gold layers. In the MLNM nanoplasmonic waveguide, frequency-conversion ceases from the pump wavelengths having an

*L*<<

_{prop}*L*, even for spectral components away from the central pump wavelength of 1550 nm. On the other hand, wavelengths near 775 nm have an

_{coh}*L*<<

_{prop}*L*in the LNHP waveguide, such that a small

_{coh}*L*at the SHG wavelengths is the dominant mechanism limiting the length over which frequency-conversion occurs in these waveguides.

_{prop}SHG in the MLNM nanoplasmonic and LNHP waveguides can be analyzed from the time-averaged distributions of the electric field magnitude at the wavelength of 775 nm, which are shown in Fig. 3(a) and 3(b), respectively. For the MLNM nanoplasmonic waveguide [Fig. 3(a)], the electric field of the quasi-TM_{00} SHG mode increases in magnitude with increasing propagation distance along the *y*-direction. Since no electric field amplitude oscillations are observed, the MLNM nanoplasmonic waveguide does not support other phase-mismatched SHG modes. Furthermore, the quasi-TM_{00} nature of this SHG mode would allow for efficient coupling into photonic waveguide structures (e.g. LN core). For the LNHP waveguide [Fig. 3(b)], the SHG mode has the highest electric field magnitude at a distance of *y* = 4-6 µm from the input of the waveguide. As such, the optimal waveguide length of the LNHP waveguide is found to be between *y* = 4-6 µm. Beyond this propagation distance, the SHG field decreases as a result of high losses at the pump wavelengths and the SHG wavelengths. From Fig. 3(b), it is evident that part of the electric field propagates in the SiO_{2} substrate layer. This is due to the height of the Si layer being 123 nm, which is less than the wavelength of the second harmonic in the Si layer (i.e. *λ _{Si}*≈200 nm). An interesting observation from Fig. 3(b) is the presence of electric field amplitude oscillations, where each electric field maximum is separated by a distance of ~1.2 µm. This effect is due to a portion of the generated photons coupling to a phase-mismatched waveguide mode having an effective index of 2.74 at

*λ*= 775 nm.

_{SHG}Figure 3(c) shows the SHG time-domain electric field signals recorded near the regions where the field is strongest. Notably, the electric field is very high in both the MLNM nanoplasmonic and LNHP waveguides, exhibiting peak-to-peak values of 40 kV/cm and 16 kV/cm, respectively. For both the MLNM nanoplasmonic and LNHP waveguides, the second harmonic pulse durations are ~70 fs. Since SHG is dependent on the intensity of the pump pulse, the generated second harmonic signals have a shorter pulse duration than the 100 fs pump pulse. As seen from Fig. 3(d), the spectral density of the SHG electric field pulses show uniform spectral distributions centered around ~775 nm. This is expected, since *L _{coh}*>>

*L*at the pump wavelengths in the MLNM nanoplasmonic waveguide and

_{prop}*L*>>

_{coh}*L*at the SHG wavelengths in the LNHP waveguides [see Figs. 2(b) and 2(c)]. From these spectral density plots, the SHG full-width half maximum (FWHM) power bandwidths can be calculated as 12 nm (i.e. 6 THz). Therefore, in terms of frequency, the generated second harmonics have a larger bandwidth in comparison to the 4.4 THz FWHM power bandwidth associated with the 100 fs pump pulse. Again, this is due to the SHG dependence on the pump pulse intensity.

_{prop}The performance of the phase-matched frequency-conversion nanoplasmonic waveguides is studied by calculating the SHG conversion efficiency. Figure 4 demonstrates the enhanced frequency-conversion obtained from the MLNM nanoplasmonic and LNHP waveguides in comparison to phase-matched metal-LN (MLN) plasmonic and photonic waveguides. By choosing the photonic waveguide to have cross-sectional dimensions of 400 nm × 837 nm (see Fig. 4 inset), phase-matching is achieved between the TM_{00} mode at *λ _{pump}* = 1550 nm and the TM

_{02}mode at

*λ*= 775 nm. Similarly, the cross-sectional dimensions of the MLN plasmonic waveguide are chosen as 400 nm × 509 nm, with a 100 nm-thick top gold layer (see Fig. 4 inset). This allows phase-matching to be achieved between the plasmonic pump mode at

_{SHG}*λ*= 1550 nm and the quasi-TM

_{pump}_{00}SHG mode at

*λ*= 775 nm. The MLN plasmonic waveguide exhibits the lowest conversion efficiency (<9 × 10

_{SHG}^{−6}) at nearly all investigated lengths. This is due to the plasmonic pump mode being highly confined to the metal-LN interface, while the quasi-TM

_{00}SHG mode is primarily located in the LN central region, such that the spatial overlap of the electric fields is poor. Oscillations are evident in the conversion efficiency of the LN photonic waveguide, which arise from generation of the phase-mismatched TM

_{00}second harmonic mode. From the effective refractive index of the TM

_{00}pump mode (

*n*= 1.64) and the TM

_{eff}_{00}second harmonic mode (

*n*= 2.00), the coherence length of these modes is calculated to be

_{eff}*L*= 1.1 µm at

_{coh}*λ*= 1550 nm. The power converted to a second harmonic mode is back-converted to the fundamental frequency after the distance of 2

_{pump}*L*[25]. This effect is clearly observed in Fig. 4, where the spatial oscillations have a period of 2

_{coh}*L*= 2.2 µm. Notably, in this photonic waveguide, coupling between the TM

_{coh}_{00}pump mode and the phase-mismatched TM

_{00}second harmonic mode is unavoidable because of the significant spatial overlap they exhibit.

The MLNM nanoplasmonic waveguide and the photonic LN waveguide are pumped at the same peak intensity of 10 GW/cm^{2}, corresponding to pump energies of 0.83 pJ and 1 pJ, respectively. The *l* = 10 µm MLNM nanoplasmonic waveguide has a conversion efficiency of 6.5 × 10^{−5}, which is 2.5 times higher than the value attained from the photonic LN waveguide. Furthermore, the *l* = 20 µm MLNM nanoplasmonic waveguide reaches a high conversion efficiency of 1.1 × 10^{−4}, where this value is 1.15 times the value obtained from the photonic LN waveguide. Although the MLNM nanoplasmonic waveguide clearly exhibits the largest conversion efficiencies at waveguide lengths between *l* = 4.7-20 µm, this comes at the expense of having a large footprint (i.e. large *l*).

For lengths of *l* = 1.2-4.7 µm, the LNHP waveguide exhibits the highest conversion efficiencies, ranging from 0.5 to 2.3 × 10^{−5} and reaching its maximum value at *l* = 4.5 µm. In comparison to the photonic LN waveguide of *l* = 2.1 µm and 4.5 µm, the LNHP waveguide has a conversion efficiency that is 11 and 4 times higher, respectively. This dramatic enhancement is achieved despite the fact that the LNHP and photonic LN waveguides are being pumped at the different energies of 0.22 pJ and 1 pJ, respectively. Notably, this improvement in the conversion efficiency is achieved by confining the electric field to a smaller region in comparison to the LN photonic waveguide [26]. The *l*>4.5 µm LNHP waveguides are strongly influenced by losses at the pump and SHG wavelengths, such that the conversion efficiency decreases to ~0 at *l*≈20 µm. Clearly, the LNHP waveguides are well-suited for miniaturized radiation sources for on-chip nanooptoelectronic applications.

## 5. Summary

We have shown that nonlinear light-generation occurs at a higher conversion efficiency in LN nanoplasmonic waveguides in comparison to a photonic LN waveguide. Such a class of nanostructures enables the light-matter interaction necessary to reach the next level in the development of next-generation nonlinear nanoplasmonic devices. When considering a 2.1 µm-long LNHP waveguide, the SHG conversion efficiency is found to be 11 times higher than the value obtained from a LN photonic waveguide. For a 20 µm-long MLNM nanoplasmonic waveguide, a high conversion efficiency of 1.1 × 10^{−4} is achieved. These MLNM nanoplasmonic and LNHP waveguides have the potential to be used as optical radiation sources for on-chip photonic and optoelectronic systems.

## Funding

Natural Sciences and Engineering Research Council of Canada.

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