## Abstract

The paper introduces a wavelength converter composed of a metallic finite 2-dimensional particle grating on top of an optical waveguide. The particles sustain plasmonic resonances which will result in the near-field enhancement and therefore, high conversion efficiency. Due to near-field interaction of the grating field with the propagating modes of the waveguide, the generated third harmonic wave is phase-matched to a propagating mode of the waveguide, while the fundamental frequency component is not coupled into the output waveguide of the structure. The performance of this structure is numerically investigated using a full-wave transmission line method for the linear analysis and a three-dimensional finite-difference time-domain method for the nonlinear analysis.

© 2012 OSA

## 1. Introduction

Exploiting optical nonlinearity of metals has been a vigorous area of research and development in the last decades. In comparison with dielectrics, metals have intrinsically higher odd-order nonlinear susceptibility. This makes them suitable in realizations of optical up-converters with a variety of applications, such as tailoring the dynamics of plasmonic modes [1, 2], wavelength conversion using quasi-phase matched plasmonic photonic crystals [3], nonlinear optical excitation for dark field microscopy [4], and all-optical switching [5]. Moreover, enhancement of the optical near-field with localized plasmonic resonances of metallic nano-particles greatly increases the conversion efficiency of the nonlinear process. This effect has been utilized to achieve higher harmonic generation from a host medium using plasmonic resonances of bow-tie antennas [6]. More fundamental research has been carried out to utilize localized plasmonic resonances in metal-dielectric composite materials in the form of either Maxwell-Garnet [7] or layered photonic band gap structures [8].

Resonant grating waveguide structures which are composed of single or double diffraction gratings on top of a high refractive index slab waveguide have been extensively investigated in the literature [9–12]. The grating integrated on top of the waveguide enables the coupling of radiated beams into the propagating modes of waveguide by mapping the wave vector of waveguide modes into the light cone. Such structures have been exploited in extremely narrow line reflection filters [13, 14], enhancement of two-photon fluorescence excitation [15, 16], inverted Fabry-Pérot interferometers [17], optical biosensors [18, 19], and enhancement of quantum efficiency of quantum-well infrared detectors [20].

Considering time reversal, in the presence of the grating the guided modes can also excite radiation modes of the structure. Resonances in the reflection and transmission of such structures can be attributed to the interference of radiation due to diffraction of the guided modes and the directly reflected and transmitted radiation. It is noteworthy that most resonant grating waveguide structures studied so far are periodic in one dimension only, with translational invariance in the second direction. Only a few attempts have been made to investigate the coupling to photonic waveguides with gratings having 2-dimensional periodicity [21], as studied here.

We report on an optical wavelength converter, whose critical element is a plasmonic nonlinear grating coupler on the top surface of a slab waveguide. The nonlinear grating is comprised of gold nano-rod antennas. Their geometry is tuned such that their plasmonic resonance is at the fundamental frequency. Due to (nearly perfect) inversion symmetry of the gold nano-particles, the bulk second-harmonic generation is essentially forbidden. We therefore consider only the third-order nonlinear susceptibility of gold nano-particles. The period of the grating has been designed on the one hand to match the wave vector of the generated third-harmonic radiation to the phase-constant of the waveguide mode. The fundamental beam, on the other hand, is not coupled to the waveguide, i.e., it is reflected and transmitted only into the zeroth diffraction order as shown in Fig. 1
. Thus, there is no need to filter out the fundamental beam at the output of D_{WG} (Output detector), while both the fundamental and converted beams are present at the location of D_{T}. This detector is placed just below the substrate, enabling the detection of near-field intensities of the electromagnetic field. In [22] we had introduced a nonlinear dielectric grating, which couples second harmonic radiation into an underlying silicon-on-insulator waveguide. The nonlinear metallic grating introduced here uses plasmonic particle resonances for enhanced up-conversion of radiation. This is exclusively coupled into propagating modes of the underlying waveguide via the independently designed grating arrangement of the particles; the fundamental radiation is not coupled to the waveguide.

Since, at first glance, our structure design might seem similar to hybrid grating-waveguide approaches such as used in [23, 24], we briefly discuss the crucial distinctions. Whereas the latter uses 1-dimensional gratings, illuminated with p-polarization, we investigate 2-dimensional gratings, excited in s-polarization. In hybrid grating-waveguides two resonant systems are coupled at the fundamental frequency – often in a Fano-type fashion [23, 24] – to form new hybrid modes. In our design only the finite 3-dimensional metallic nano-rod antennas are individually resonant and may be thought of as the sole third harmonic generator. This third harmonic radiation is subsequently coupled into propagating modes of the waveguide, where they are sustained even at great distances from the converter-coupler area. Moreover, exploiting the 3-dimensional resonating system with nano-rods will result in a huge field-enhancement factor, about 2 orders of magnitude higher than the field enhancement achievable with 2-dimensional gratings. In toto, whereas the generation of third harmonic radiation, its coupling to, and propagation in the waveguide are intricately fused in hybrid structures, they appear conceptually and structurally separated in the device we propose here.

The numerical methods exploited so far to simulate nonlinear photonic structures fall into two broad categories: analytical modeling such as coupled mode theory [25], and numerical methods such as eigen-mode expansions [26], the time-domain beam-propagation method (TD-BPM) [27], the finite-difference frequency-domain (FDFD) and the finite-difference time-domain method (FDTD). Analytical modeling is not appropriate for structures of complex geometries. Eigen-mode expansion methods assume the slowly varying envelope approximation. A time differencing procedure is utilized to solve the two resulting coupled equations for the fundamental and the second-harmonic waves. TD-BPMs work with paraxial wave propagation. The frequency-domain methods like FDFD may also be used following the undepleted pump approximation. Among these methods, however, FDTD appears most versatile and suitable in modeling various nonlinear processes such as wave mixing and harmonic generation [22, 28, 29]. This method is used here to investigate the nonlinear performance of the structure.

## 2. Structure and linear analysis

Figure 2(a)
shows the structure of the proposed wavelength converter. This multi-layer structure is comprised of a finite array of gold nano-particles on a slab waveguide. Due to the in-plane inversion symmetry of the gold nano-particles used here, generation of second harmonic signal is only limited to the interface at which the inversion symmetry is broken, therefore, the effective second-order nonlinear susceptibility is of smaller magnitude than the third-order nonlinear susceptibility of gold. The grating is exposed to a beam at the fundamental frequency${\omega}_{0}$. The generated third harmonic signal (THS) is radiated into several diffraction orders, whereas for the fundamental frequency only the zeroth-order is available through a careful design of grating periodicity, i.e., L_{x}<<λ_{0} and L_{y}<<λ_{0}. In order to couple the THS to the propagating modes of the waveguide, the following phase matching condition must be satisfied:

_{x}is the periodicity of the structure in the $x$-direction, $m=\mathrm{...},-1,0,1,\mathrm{...}$ denotes different diffraction orders, and $\beta \left(3{\omega}_{0}\right)$is the propagation constant of the TE

_{y}or TM

_{y}modes of the waveguide at the third harmonic frequency. We assume that the fundamental beam is at a wavelength of ${\lambda}_{0}=800nm$, and that the HfO

_{2}layer used here has a thickness of 180nm. The analytical relation for the dispersion diagram of the bare slab waveguide (far away from the converter-coupler area) shows the values of ${\beta}_{{\rm T}{{\rm M}}_{y}}\left(3{\omega}_{0}\right)=4.797\times {10}^{7}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$ and ${\beta}_{{\rm T}{E}_{y}}\left(3{\omega}_{0}\right)=4.218\times {10}^{7}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$ for the TM

_{y}and TE

_{y}modes of the waveguide, respectively. The fundamental wavelength is λ

_{0}= 800nm, and the height of the HfO

_{2}slab waveguide is 180nm. Using Eq. (1), and considering${L}_{x}=140nm$, which is small enough in comparison with the fundamental wavelength, one can approximately compute the angles at which the phase-matching conditions are satisfied. These are $\theta ={7.54}^{\circ}$ and $\theta =-{6.58}^{\circ}$, for the TM

_{y}and TE

_{y}modes of the waveguide, respectively. The negative angle means that the second-order mode is in the opposite direction of propagation compared with the incident beam. This is because of the influence of the periodic grating which folds the band-diagram of the structures at the boundary of the Brillouin zone (X-point).

The Au nanoparticles used here as the elements of the nonlinear grating show plasmonic resonances at the fundamental wavelength of 800nm. Because of the quasi-1-dimensional Fabry-Pérot-like plasmonic resonance inside the nano-rods [30], a huge field enhancement is obtained with a considerable impact on the generated intensity of the THS. This can be described with Eq. (2),

In order to investigate the far-field response of the grating, the full-wave transmission line (TL) method [34] has been used to calculate the extinction spectrum ($-\mathrm{ln}\left(T\right)$) of an infinite periodic structure with the same particles as above. Figure 3
shows the calculated extinction spectrum versus the angle of incidence for the wavelengths $\lambda =600nm~950nm$. Clearly, the response to s-polarized excitation of a grating of gold nanoparticles is largely independent of the angle of incidence over the intervals $0\le \theta \le 45$ degrees and $600nm\le \lambda \le 950nm$. The absence of any strong grating response is due the period of the grating along the $x$-axis being much smaller than the wavelength of the incident wave, which leaves the plasmonic resonances uncoupled from the propagating modes of the slab waveguide. In other words, this structure does not support a waveguide-polariton quasi-particle like the one introduced in [2, 23, 24]. This fact is advantageous in the present case, since its enhancement factor is rather constant over a wide range of incident angles of the input beam. Figure 4
shows the extinction spectrum versus $\theta $ at $\lambda =267nm$, i.e., at the wavelength of the generated THS. Three sharp anomalies are visible in the extinction spectrum. The anomalies at $\theta ={6.6}^{\circ}$ and $\theta ={7.7}^{\circ}$ arein agreement with the results estimated above using Eq. (1). The anomaly at $\theta ={12.6}^{\circ}$can also be obtained using Eq. (1) with the propagation constant of the lower order TM_{y} mode. Figure 4(b) shows the magnitude of the E_{y} field component at λ = 267nm when the structure is exposed to a $y$-polarized plane wave at θ = 6.6°, θ = 7.7°, and θ = 12.6°, respectively. A sharp Fano resonance is visible at θ = 6.6° which is mainly due to the coupling of the two individual resonances at θ = 6.6° and θ = 7.7°.

## 3. Nonlinear analysis using FDTD method

In order to investigate the performance of the converter-coupler introduced in this paper, we apply a nonlinear, 3-dimensional FDTD method previously developed in our lab [22]. The simulation domain is discretized with uniform unit-cells of size 10 nm. We neglect second-order terms as discussed above. The relation between the polarization vector and the electric field is assumed to be:

_{2}and quartz is neglected in comparison to that of gold. The magnetic field and the electric displacement vector (D) are updated in every iteration. For updating the electric field, a Newton iteration procedure is utilized to compute the electric field from the previously updated electric displacement using Eq. (4) [29].

The simulation domain contains a total number of 36 nanoparticles on the HfO_{2} waveguide with a quartz substrate. The particles are arranged in two rows along the $y$-axis with${D}_{x}=40\text{\hspace{0.17em}}nm$, ${D}_{y}=120\text{\hspace{0.17em}}nm$, ${L}_{x}=140\text{\hspace{0.17em}}nm$, and ${L}_{y}=240\text{\hspace{0.17em}}nm$. In order to excite the plasmonic resonances of the gold nanoparticles, the structure is excited with a TM_{y} optical pulse of 50 fs duration and a carrier wavelength of 800nm.

As the spatial distribution of the optical pulse a Gaussian function with $W=1.5\lambda $ is used in which W is the beam-width of the optical pulse at the half-maximum intensity. The peak power of the excitation introduced here is P_{max} = 61kW, which is smaller than the one reported in [6], which is 100kW. The maximum intensity of the excitation (or magnitude of the Poynting vector) is ${I}_{\mathrm{max}}\approx 1.8\times {10}^{13}Wc{m}^{-2}$. Figure 5(a)
shows the average intensity of the electric field computed at test points located 10nm above the nanoparticles for the total and incident fields at normal incidence. Generation of higher harmonics up to an order of 11 is observed in this figure. The local field enhancement factor at the fundamental wavelength is approximately 60 dB just outside the top surface of the array structure. Nevertheless, the produced THS is not particularly strong, which is mainly due to the dissipative nature of gold at the third harmonic: ${\chi}^{\left(L\right)}\left(3{\omega}_{0}\right)=-0.5574+9.7525i$. That is, the produced THS is strongly absorbed inside the nanoparticles. This can also be computed from the diffraction efficiency spectrum. Normalized dissipated power is computed from ${P}_{d}=1-{\displaystyle \sum _{i,j}{r}_{ij}-}{\displaystyle \sum _{i,j}{t}_{ij}}$ in which ${r}_{ij}$ and ${t}_{ij}$ are the reflection and transmission coefficients. Such calculations show that the dissipated power is about 38% at $\lambda =267nm$.

Figure 5(b) and (c) show the spatial distribution for the magnitude of the Fourier-transform of the $z$-component of the electric field at the fundamental wavelength. Figure 5(b) depicts these values in the xy-plane 10nm above the structure, and Fig. 5(c) gives these values in the xz-plane passing through the first row of the nanoparticles 10nm away from the front side. From Fig. 5(c) we conclude that the maximum enhancement factor occurs just above the nanoparticles, and inside the nanoparticles the field penetrates with a skin-depth of about 10 nm. Figure 6(a) shows the average power computed at the position of the detectors depicted in Fig. 1. These power spectra are calculated as Fourier-transforms of the E and H fields, as:

_{y}and TE

_{y}modes of the waveguide. Due to the angular extent of the Gaussian excitation beam used here, TM

_{y}and TE

_{y}modes are excited simultaneously (see also Fig. 4). Curiously, though, they propagate away from the coupling region in opposite directions, as can be clearly seen from Fig. 6(b) and Fig. 6(c). A sharp dip is observed in the logarithmic power spectrum ${P}_{\text{\hspace{0.17em}}{D}_{WG}}$ at $\lambda =800nm$, which verifies the phase matching condition described by Eq. (1). The suppression is not perfect, though, presumably due to direct illumination of the detector in the waveguide by tails of the Gaussian excitation, which are refracted at the air/waveguide interface and diffracted at the finite grating. Thus a finite portion of the field strength of the fundamental frequency component is present at this location inside the waveguide. However, as these rays cannot excite the propagating fundamental frequency modes, these field components die off exponentially along $x$ in the waveguide. In an experimental study of such a converter coupler structure, the spectrum would be picked up at much greater distances than the detector simulated here. We expect a much improved suppression of the fundamental signal under these circumstances. The number of unit-cells used here along the $x$-direction is only 18. Obviously, a sharper and deeper resonance may be achieved if the number of the grating elements increases.

Figure 7
shows plots of the magnitude of the $x$-component of the Poynting vector at $\lambda =267nm$ at different cut through the structure as indicated by the red dashed lines. The Poynting flow was computed using${S}_{x}=\frac{1}{2}\mathrm{Re}\left\{E\left(3{\omega}_{0}\right)\times {H}^{\ast}\left(3{\omega}_{0}\right)\cdot \widehat{x}\right\}$. It clearly shows that the guided THS is mostly transformed into the TM_{y} mode of the waveguide. Figure 8
shows the guided power versus the angle of incidence for the input optical pulse. The optical pulse introduced here has a large broadening angle of $\Delta \theta ={8}^{\circ}$ such that different resonances of Fig. 4 are not resolvable. The guided power for the fundamental harmonic exhibits a deep minimum at $\theta ={8}^{\circ}$, which corresponds to the phase matching condition for larger conversion efficiency. However, the guided power of the THS shows a smooth and broadened resonance at the same angle. That is mainly due to the difference between the linear permittivity of the gold nanoparticles at the fundamental and third harmonics; i.e., ${\epsilon}_{r}\left({\omega}_{0}\right)=-26.2748+1.8493i$ and ${\epsilon}_{r}\left(3{\omega}_{0}\right)=0.44+9.75i$. The quality factor of any resonator made of such a medium at $\lambda =800nm$ is larger than that of a resonator at $\lambda =267nm$.

Equation (2) shows that the local field enhancement factor due to the localized plasmonic resonances has a great impact on the conversion efficiency of third harmonic generation. This fact is investigated here by computing the average-intensity spectrum for illuminating the same structure with an optical pulse with a center wavelength of ${\lambda}_{c}=1000nm$. The result is shown in Fig. 9 . It is evident that the local field enhancement has affected the intensity of the generated harmonics. Although the difference in the intensity of the electric field at the fundamental frequency is only 6dB, the intensity of the THS decreases by 22dB, in comparison with the resonant excitation of ${\lambda}_{c}=800nm$.

Another important factor to be considered is the propagation pattern of the generated third-harmonic beam inside the waveguide. Especially it is important to calculate the divergence angle of the third harmonic beam coupled to the waveguide. In order to do that, a finite array of 18 × 8 nano-rod antennas has been assumed. The other parameters, such as the dimensions of the nano-rod antennas, the height of the waveguide and the optical excitation are as before. Figure 10(a) and (b) show the field profile for the $z$-component of the electric field at a distance of 10 nm below the array and inside the waveguide, at the third-harmonic and fundamental frequencies, respectively.

Evidently, the grating of metallic nano-rod antennas exhibits characteristics of a lens, focusing the propagated third-harmonic radiation at a distance of approximately 4μm inside the waveguide. Moreover, it emerges from this figure that the size of the grating array has an important effect on the performance of this converter-coupler. From a practical point of view, the overall dimension of the array of nano-rod antennas should not exceed the spot size of theincident beam by much. Considering the time-reversed phenomenon, the guided third-harmonic radiation, diffracted by the grating in the absence of any incident radiation, will be coupled out of the waveguide into photonic modes above and below the structure. It is the interference of the THS propagated in the waveguide and that generated by the incident radiation, which gives rise to the anomalous extinction efficiencies observed in Fig. 4. If the domain of the array illuminated by the excitation beam would be much smaller than the whole array, a considerable amount of the propagating third-harmonic power will leak out of the waveguide. Figure 11 shows the guided power spectrum inside the waveguide, computed at a distance of 4 μm from the end of the array. The transmitted power is also shown in this figure. Evidently, the THS power propagating inside the waveguide is drastically higher than that at the fundamental frequency. The power scattered from the edges of the array, can hardly couple into the propagating modes of the waveguide, and is mostly radiated into the surrounding media. Considering the power spectrum of Fig. 11, a higher suppression of the fundamental beam at the output detector has been achieved than in the case of Fig. 6(a). On the one hand this is due to the larger grating in the second configuration. On the other hand, diffraction of the fundamental beam does not give rise to propagating modes in the waveguide. By considering even larger gratings and longer waveguides, still better suppression of the fundamental beam is possible.

In comparison with previously reported results, Fig. 11 shows rather good conversion efficiency of the third-harmonic generation process for the proposed configuration is about −35dB. For the 5th and 7th harmonics it is −45dB and −50dB, respectively. These values are higher than the conversion efficiency of −86dB reported in Ref [6]. for the 7th-Harmonic generation, though it should be also mentioned that the experiment carried out in Ref [6]. is with an excitation intensity of 10^{11} W cm^{−2}. Moreover, a hybrid photonic-plasmonic waveguide has been proposed in Ref [3]. in a phase-matched configuration, to enhance the second-harmonic generation process inside a periodically-poled LiNbO_{3}. Since the fundamental and produced second-harmonics are both propagating through a lossy silver waveguide, the efficiency very much depends on the length of the structure. Using a waveguide length of only 3μm and an incident intensity of 10^{7} W cm^{−2}, the reported conversion efficiency for the second-harmonic generation is about −33dB.

## 4. Conclusion

We have proposed and analyzed the case of generating third harmonic radiation which is exclusively coupled into the propagating modes of a slab waveguide. By controlling the characteristics of grating anomalies, the phase-matching condition for coupling of the up-converted incident wave is achieved. At the same time, the fundamental frequency component is reflected off the grating.

Our study suggests an elegant separation of two crucial optical device design aspects through the use of metallic nano-particles: First, they individually generate harmonic frequency radiation of an incident wave. To this end, plasmonic particle resonances give a convenient boost to the up-conversion efficiency. Second, they can be arranged in a phase-matched array geometry to act as a coupler to further structures. Beyond simple waveguides, these can include resonators, mixers, de-multiplexers, etc.

## Acknowledgments

N. Talebi and M. Shahabadi acknowledge the financial support of this project by Iran Telecommunication Research Center (ITRC).

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