## Abstract

We report efficient low dispersion light coupling into a silicon waveguide using an antenna consisting of two metallic nanoparticles. We find that strong multiple scattering between the nanoparticles dictates the coupling efficiency. We also explore directional coupling, by using different particles with a relative scattering phase, but find that optimum directionality corresponds to minimum efficiency. A dipole model highlights a subtle interplay between multiple scattering and directionality leading to a compromise allowing up to 30% transmission into a single direction. With a 500nm bandwidth near infrared telecoms bands, group delay dispersion is sufficiently low to faithfully couple pulses as short as 50fs.

© 2012 OSA

## 1. Introduction

The coupling of light into photonic waveguides is a well-known problem, but still topical due to the use of semiconductor materials for compact integrated photonic devices [1]. Semiconductor waveguides, used in photonics, are smaller than the vacuum wavelength and support high momentum modes that are poorly impedance matched to free space. In principle, prism and grating coupling can overcome these factors [2]; however, prisms are bulky with limited permittivity for coupling to semiconductor waveguides, and the most efficient gratings generally require complicated optimized designs [3]. Simple gratings are typically 20% efficient [4]. In addition to a trade-off between design complexity and efficiency, the greatest limitations of these techniques are low bandwidth, high dispersion and high angular sensitivity. Consequently, many researchers and commercial products still use direct end-fire excitation for its simplicity, despite efficiencies as low as just 1% [5–7].

Recent work has looked at reducing the size of grating couplers by using metallic nanoparticles [8–18]. Although the metal’s high permittivity contrast provides stronger scattering [9], the localized surface plasmon (LSP) resonances [10] of nanoparticles greatly enhance scattering cross-sections [11–13] despite being smaller than the vacuum wavelength [14]. A number of researchers have exploited this to realize compact couplers using just a few metal nanoparticles [15–18]. These works focused primarily on the directional aspect of the coupling without evaluating the potential coupling efficiency.

In this letter, we report an optical antenna that couples a focussed beam into a silicon waveguide mode. Our approach uses the strong light-matter interactions and compactness of plasmonics, while avoiding the high loss associated with plasmonics by radiating rapidly into the photonic mode. Remarkably, we find total coupling efficiencies (into both waveguide directions) of 20% for a single particle and 40% for an identical particle pair at *λ* ≈ 1.55μm, dropping by only 10% over a broad 400nm range. We find optimal coupling into a single direction of 27% by using two differently shaped particles.

## 2. Plamsonic-photonic coupling

We investigate two dimensional scattering of light focussed onto gold particles with variable length (*L*) and thickness (*T*) embedded in SiO_{2} and placed in the vicinity of a photonic Silicon waveguide of thickness (*W*) (Fig. 1(a)). The particles each support a LSP with dipole moment predominantly along the *x̂* direction. An external Gaussian beam, polarized along *x̂*, excites LSPs, which subsequently re-radiate into the waveguide mode. The coupling rate, *κ*, between LSPs and waveguide mode is largest when the mode confinement is strongest. There is thus a trade-off between the mode size and the *x̂* component of the electric field at the position of the particles. Hence, the coupling rate *κ* ∝ |*E _{x}*(

*x*

_{NP})|

^{2}/ ∫

*E*(

*x*) ×

*H*

^{*}(

*x*)

*x̂*d

*z*. A waveguide thickness of

*W*= 220nm confines the fundamental Transverse Magnetic (TM) mode for optimal coupling near a wavelength of 1.55μm. Meanwhile, the particles are placed as close to the silicon as possible where the mode intensity is greatest (Fig. 1(b)). A 5nm gap avoids direct hybridization of LSPs with the silicon. Finally, the LSP resonances are tuned, by varying the particle length,

*L*, so that the peak in scattering cross section matches the maximum radiative coupling rate of the particle-waveguide system. 20 × 250nm

^{2}gold particles embedded in SiO

_{2}produce a good response (Fig. 1(c)).

We first calculate the transmission into the waveguide by a single gold particle. We use a finite element solver (Comsol Multiphysics) in all numerical calculations. We find 4% transmission into the fundamental TM mode at one end of the waveguide from a Gaussian beam focussed to a 7μm diameter spot (Fig. 1(d)). When we add a second particle a distance *D* from the first, the effective scattering cross section nearly doubles leading to a transmission of just over 7% into one direction, as shown in Fig. 2(a). Here, the de-focussed beam allows us to vary the relative positions of particles with a nearly uniform input field; later, we show how a focused beam significantly increases the transmission.

## 3. Coupled dipole oscillator model

The transmission amplitude, *t*(*x*), into forward and backward propagating waveguide modes can be modelled by interference of the radiation from two forced coupled dipoles [19].

*a*are the dipole amplitudes of the LSPs; and

_{i}*k*is the propagation constant of the mode. The left and right transmission coefficients become,

*β*=

_{i}*κ*are the proportions of light coupled to the waveguide mode;

_{i}/γ_{i}*c*are the coupling coefficients between the input beam and the LSPs and are dependent on the scattering cross-sections of the particles;

_{i}*γ*are the total LSP extinction rates, including radiation, waveguide coupling and absorption;

_{i}*γ*̄

*=*

_{i}*γ*are the total extinction rates normalised to the frequency;

_{i}/ω*α*=

_{i}*β*cos

_{i}*ϕ*;

_{i}*ψ*=

*kD*; and

*s*= (1 –

*β*

_{1}

*β*

_{2}

*γ*̄

_{1}

*γ*̄

_{2}

*α*

_{1}

*α*

_{2}/4e

^{i(ϕ1+ϕ2–2ψ)}) describes high order multiple scattering. Note that we have defined the phase of each resonator as $\text{tan}{\varphi}_{i}=\left({\omega}_{i}^{2}-{\omega}^{2}\right)/{\gamma}_{i}\omega $ and therefore cos

*ϕ*is the amplitude response across the tuning range, |

_{i}*ϕ*| <

_{i}*π*/2.

Following the example of other authors [16, 17], we examine a simplified solution without multiple scattering by excluding the diagonal terms and setting *s* = 1, giving the waveguide mode transmission,

*ϕ*

_{2}–

*ϕ*

_{1}. For a single particle the transmission into one direction is

*T*

_{±}=

*T*

_{1}(Fig. 2(a)). To achieve directional coupling, we must control the relative phase of the two particles by adjusting their length,

*L*.

Figure 1(d) shows the transmission and its phase for a single particle as we detune the LSP at a fixed wavelength of 1.55μm. The transmission and phase approximately follow the expected dependence of the dipole model, *T _{i}*(

*ϕ*) ≈

_{i}*T*(0) cos

_{i}^{2}

*ϕ*. The relative phase between particles can clearly be tuned by more than

_{i}*π*/2. (Full 2

*π*phase control can be achieved with more complex 3D nanoparticle geometries [20].) The transmission curve’s shape, which resembles those in recently reported literature [16, 20], occurs due to resonance broadening for longer particles. From Eq. (3), the best directional coupling into the forward direction only occurs when both,

*n,m*∈ . However, the total transmission for optimal directionality (Δ =

*π*/2), given by is a minimum. For two particles, the total transmission is greatest when

*T*

_{1}=

*T*

_{2}and

*ϕ*

_{2}=

*ϕ*

_{1}= 0, which also corresponds to least directionality. Furthermore, since

*T*∝ cos

_{i}^{2}

*ϕ*the efficiency for an optimal directional coupler is less than that of an identical particle pair. Directional coupling and optimal efficiency are therefore mutually exclusive. Nevertheless, directional coupling can be effective as shown in Fig. 2(b), where Δ =

_{i}*π*/4. Here, a maximum transmission of 11% into one direction can be achieved.

## 4. Multiple scattering and directional coupling

The asymmetric transmission lineshapes in Figs. 2(a) and 2(b) cannot be explained by the simple interference of two dipoles. A good correspondence between the numerical simulation and Eq. (2) requires strong multiple scattering between the particles. To make the comparison, we have used the analytical transmission trend in Fig. 1(d) with the approximation that *β*_{1} ≈ *β*_{2}, *γ*̄_{1} ≈ *γ*̄_{2} and *c*_{1} ≈ *c*_{2}, giving only two unknown fitting parameters: the peak transmission of a single particle with zero phase, *T*_{0} = *T*(*ϕ*= 0) = 3.5% and the multiple scattering parameter, *β*^{2}*γ*̄ ≈ 0.4. The large value of the scattering parameter highlights the strong coupling between nanoparticles and waveguide mode. Furthermore, multiple scattering is sufficiently strong to observe higher order scattering, where *s*^{−1} ≈ 1.2. In the absence of multiple scattering, two identical particles should provide four times the single particle transmission, *T*_{0}, due to constructive interference, as shown by Eq. (3), for *T*_{1} = *T*_{2} = *T*_{0}; *ϕ*_{1} = *ϕ*_{2} = 0. However, with multiple scattering the total transmission for two identical particles is just 4*T*_{0}/(1 + *β*^{2}*γ*̄)^{2} ≈ 2*T*_{0}. Thus, multiple scattering suppresses the constructive interference of the dipoles and reduces the total transmission (Fig. 2(a)). Interestingly, the suppression of interference is less pronounced for Δ = *π*/4 (Fig. 2(b)).

To further understand the role of multiple scattering, we investigate the two particle coupler’s transmission for various phase differences. To this end, the length of one particle is varied, controlling *ϕ*_{2}, while the length of the other is kept constant at *L* = 250nm, thus fixing the value of *ϕ*_{1} ≈ 0. Figure 3(a) shows the peak transmission for each direction as a function of *ϕ*_{2}. While multiple scattering significantly reduces the coupling efficiency for identical particles, the effect is less pronounced for different particles, with a phase difference between them. Hence, when increasing Δ, the transmission also increases approaching the transmission we would expect for constructively interfering particles, near Δ = *π*/4. It is interesting to note that for Δ > *π*/4, the transmission surpasses even that of the simple constructively interfering dipoles. Meanwhile, the effect of directional coupling reduces transmission for Δ > *π*/4, leaving the maximum transmission near Δ ≈ *π*/4. Multiple scattering is a natural consequence of the extremely strong coupling between nanoparticle and waveguide. For weaker coupling (*β*^{2}*γ*̄ ≪ 1) we have observed the constructive interference of each nanoparticle and a total transmission of 4*T*_{0}. This suggests that, if *β* can be maintained at a high value while *β*^{2}*γ*̄ ≪ 1, the reduction of multiple scattering could yield even greater coupling efficiencies.

## 5. Efficient low dispersion coupling

Simply focussing the beam down to a 2μm spot significantly increases the coupling efficiency with the total transmission into one direction reaching 13%, 20% and 27% for the single particle, identical pair and *π*/4 phase-shifted pair, respectively (Fig. 3(b)). Furthermore, the coupling efficiency is sustained over a bandwidth of at least 500nm. Since scattered light from the plasmonic antenna also only incurs a *π* phase change across the wide transmission band, we can also expect low group delay dispersion (GDD) of light passing through the coupler, c.f. Fig. 4(a). To calculate pulse broadening we determined the total phase (*ϕ*) induced by both the coupler and mode propagation over 4μm of the waveguide. The corresponding pulse broadening map is shown in Fig. 4(b). Interestingly, we find anomalous dispersion of our particle coupler that compensates for the normal dispersion on propagating 15μm through the silicon waveguide. With the broad bandwidth and low dispersion, these antennas could faithfully couple the spectral components of pulses as short as 50fs.

## 6. Conclusion

In conclusion, we have presented a sub-wavelength optical coupler that is more compact and easier to design than conventional grating couplers yet provides comparable efficiency over a much broader spectrum with low dispersion. Our approach offers directional control through the scattered phase by simply varying each particle’s length. However, coupling efficiency and directionality are mutually exclusive. Meanwhile, multiple scattering between nanoparticles due to the high coupling efficiency suppresses the constructive interference expected of an identical particle pair. Since multiple scattering scales with *β*^{2}*γ*̄, we have proposed to reduce it by using resonators with higher Q factors, at the expense of dispersion. To this end, 3D nanoparticles coupled to TM or TE waveguide modes could mitigate multiple scattering. Such waveguide couplers are a viable approach for integrated semiconductor photonics technology due to their ability to achieve high coupling efficiency over a broad wavelength range with extremely low group delay dispersion.

## Acknowledgments

This work was sponsored by the UK Engineering and Physical Sciences Research Council (EPSRC). R.F.O. is supported by an EPSRC Fellowship ( EP/I004343/1) and Marie Curie IRG ( PIRG08-GA-2010-277080).

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