## Abstract

The integration of optical metamaterials within silicon integrated photonic circuitry bears significantly potential in the design of low-power, nanoscale footprint, all-optical functionalities. We propose a novel concept and provide detailed analysis of an on-chip ultrafast all-optical modulator based on an hyperbolic metamaterial integrated in a silicon waveguide. The anisotropic metamaterial based on gold nanorods is placed on top of the silicon waveguide to form a modulator with a 300x440x600 nm^{3} footprint. For the operating wavelength of 1.5 μm, the optimized geometry of the device has insertion loss of about 5 dB and a modulation depth of 35% with a sub-ps switching rate. The switching energy estimated from nonlinear transient dynamic numerical simulations is 3.7 pJ/bit when the transmission is controlled optically at a wavelength of 532 nm, resonant with the transverse plasmonic mode of the metamaterial. The switching mechanism is based on the control of the hybridization of eigenmodes in the metamaterial slab and the Si waveguide.

© 2014 Optical Society of America

## 1. Introduction

A photonic integrated circuit (PIC) is a general concept of device that provides multiple functions, based on optically or electronically controlled optical signals. PICs may simultaneously contain several functional components, such as optical filters, modulators, amplifiers, lasers as well as photodetectors. While the majority of electronic components are based on a silicon technology, basic photonic functionalities, such as light modulation or light generation, require usage of different material components. For example, electro-optical materials are widely used for integrated optical modulation together with silicon waveguides [1, 2]. However, electro-optical modulation still relies on electric connections leading to a limit in the total switching speed to up to a few tens of GHz [3–6]. All optical modulation overcomes this limit allowing for a higher modulation bandwidth as the modulating signal and recovery time are typically short and well above the GHz regime. It has been recently shown that by taking advantage of different ultrafast nonlinear processes like two-photon absorption, free carrier heating, or the Franz-Keldysh effect, it is possible to achieve modulation speeds exceeding 100 Gbit/s. Furthermore, it was also shown that large electric fields, characteristic of surface plasmon excitations in metallic nanostructures, further enhance the nonlinear response allowing for a more efficient all-optical modulation using highly integrated devices [7–11]. In particular, plasmonic metamaterials composed of metallic nanorods have shown a strong and ultrafast nonlinear response taking advantage of the non-local effects of the supported plasmonic modes [10].

In this context, we propose an integrated optical modulator design based on coupling a plasmonic metamaterial to a silicon waveguide. We show that the hyperbolic dispersion observed in this kind of metamaterials [12, 13] leads to a high sensitivity of the Si-waveguide mode to changes in the metamaterial’s permittivity due to strong coupling between the modes of the Si waveguide and the metamaterial slab. The proposed device has a footprint of a few hundreds of nanometers, at least one order of magnitude smaller than typical silicon electro-optical ring-resonator modulator [1], and a modulation speed in the upper GHz range, maintaining a modulation depth performance exceeding 35%. Furthermore, this metamaterial device is compatible with typical silicon waveguide making it a strong candidate for highly integrated all-optical modulation for silicon nanophotonics.

## 2. Theory

The basic geometry of the device is depicted in Fig. 1(a).The device is designed to operate at a wavelength of 1.5 μm. It consists of a standard 300 nm wide and 340 nm high Si waveguide supporting a plasmonic metamaterial slab. The latter is made of strongly interacting Au nanorods aligned with their long axes oriented normally to the waveguide’s top surface. The metamaterial behaves as an effective medium with optical properties sensitive only to the average size and position of rods within the assembly [12].

The anisotropic permittivity of the metamaterial can be represented by a diagonal tensor of effective permittivities having elements ε_{x} = ε_{y}≠ε_{z}, where the z-direction [Fig. 1(a)] is the optical axis along the long axis of the rods. The permittivity tensor, determined by the nanorod material and dimensions, including rod diameter and spacing, as well as the host medium, was obtained within the effective medium theory (EMT) as and ${\epsilon}_{x}=[p{\epsilon}_{m}{\epsilon}_{d}+(1-p){\epsilon}_{d}({\epsilon}_{d}+1/2({\epsilon}_{m}-{\epsilon}_{d}))]/[p{\epsilon}_{d}+(1-p)({\epsilon}_{d}+1/2({\epsilon}_{m}-{\epsilon}_{d})]$ where ε_{d} is the permittivity of the host medium, ε_{m} is the permittivity of the metal, and , where *r* is the nanorod radius and *d* is the average inter-rod distance in the assembly [12]. The wave equation in such an anisotropic material imposes a specific relationship between the magnitudes of the propagation wavevectors components which is formulated for ordinary and extraordinary waves, respectively, as [14]

*k*are the wavevector’s components in the metamaterial within the referential of Fig. 1(a) and k

_{x,y,z}_{0}= 2π/λ

_{0}is the wavevector of light in vacuum. For the nanorod metamaterial, these relations exhibit a particular behavior when ε

_{z}<0 and ε

_{x}>0. Then, for extraordinary waves (Eq. (2) maps an hyperboloid rather than the ellipsoid observed for ordinary waves (Eq. (1). Considering a waveguide geometry formed by the metamaterial slab, the boundary conditions imposed on the metamaterial restrict waves sustained by the waveguide to modes with specific propagation constant β. In general, for a planar anisotropic slab, these modes can be divided in extraordinary and ordinary modes, but the two-dimensional confinement of the waveguide mixes polarizations, leading to hybrid electric (HE) modes with both ordinary and extraordinary components. This hybrid nature of the fundamental mode supported by the metamaterial waveguide along with the anisotropic hyperbolic dispersion of the metamaterial form the basis for the switching mechanism reported in this paper.

## 3. Mode matching technique and eigenmode analysis

In the hyperbolic regime, the modulator geometry of Fig.
1(a) allows the strong coupling of the HE modes of the silicon waveguide to the HE modes
of the metamaterial. In order to determine the insertion properties of the modulator, we have
performed an eigenmode analysis and determined effective refractive indices of the modes
supported by both the Si waveguide and the modulator. Both transmission and modulation
performance of the waveguide/metamaterial device are then determined and optimized using the
mode matching approach [15]. Both polarization of the
HE_{11} mode are considered as an input. The HE_{11,x} mode with an electric
field mainly polarized along the *x*-axis has a spatial distribution of its
intensity in the Si waveguide shown in Fig. 1(a). The
transmission though the modulator’s interface [Fig.
2(a)] is given by the modified Fresnel formula, taking into account the spatial overlap
between the modes [15]

*i*and

*j*are the eigenmodes of the waveguide and the modulator, respectively, n

_{j}

^{mod}is the effective refractive index of the

*j*mode in the modulator, n

_{i}

^{wg}is the effective refractive index of the

*i*mode in the waveguide and the integral is performed over the modes cross-section. The total transmission through the waveguide/modulator interface with respect to a given eigenmode

*i*of the waveguide consists of the sum of all

*T*, corresponding to every eigenmode

_{ij}*j*supported by the modulator. Using this general formalism (Eq. (3), we can estimate the overall transmission of the lowest

*x*and

*z*polarized HE waveguided modes through the modulator’s interface at the operating wavelength of 1.5 μm as a function of metamaterial geometry. The transmission of the HE

_{11,x}mode is shown in the color map of Fig. 2(a) as a function of the parametric variables of the metamaterial geometry. A strong sensitivity of the transmission to rod length and spacing has been observed with an overall change in transmission ranging from 10% to 95% for different metamaterial geometrical parameters in the modulator. A similar analysis shows more modest transmission values for the HE

_{11,z}waveguided mode, only reaching about 40% transmission change, but nevertheless also very sensitive to the metamaterial geometrical parameters [Fig. 2].

It is instructive to understand the origin of the strong local minima in transmission observed in Fig. 2(a). Intuitively, the modulator’s structure can be regarded as a system of two coupled waveguides, one made of Si and the other made of metamaterial. At the chosen operation wavelength, the dispersion of the metamaterial is hyperbolic in all ranges of the geometrical parameters considered. Considering first the changes of the transmission of the modulator with the increase of nanorod height, before the first transmission minimum in reached [Fig. 2(a)], the real part of the refractive index is Re(*n _{zz}*)~0 and 1.9<Re(

*n*)<2.3. In these conditions, both Si and metamaterial waveguides are weakly coupled, and high transmission [Fig. 2(a)] is achieved with a waveguided mode propagating in the Si waveguide and evanescent in the metamaterial. For the HE

_{xx}_{11,x}mode, the mode profile is then essentially that of an ordinary wave, which propagation in the metamaterial is mainly governed by the values of

*n*.

_{xx}With further increase of the nanorod height, a small but important component of the electric field along the extraordinary axis of the metamaterial, present because of the lateral confinement of the mode in a rectangular Si waveguide, leads to the resonant coupling of the waveguided mode in the Si waveguide to an extraordinary mode in the metamaterial waveguide. It takes place when the wavevector of the extraordinary mode β_{ext} in the metamaterial waveguide, controlled solely by metamaterial height here, coincides with the wavevector of the mode (β_{ext}) supported by the Si waveguide [15]. This resonant coupling reduces the transmission through the modulator. A further increase in the height of the metamaterial layer leads to the loss of resonant coupling as the wavevectors β_{ext} and β_{wg} mismatch. The subsequent confinement of the waveguided mode to the Si section of the modulator restores high transmission.

The relatively large variations of the transmission through the interface between the Si waveguide and the modulator, to small variations in the modulator’s geometry imply that similar changes in transmission could be achieved dynamically on the basis of the metamaterial’s non-linear optical response [10]. We have simulated the nonlinear optical response of the metamaterial within the Random Phase Approximation (RPA) using a two-temperature model including the dependence of electron scattering to both electron and lattice temperatures [16, 17]. In brief, the pump beam increases the electrons temperature on a time scale on the order of a few tens of femtoseconds, leading to an imbalance between electrons and lattice temperatures. The system then relaxes back to the initial state depending on both electron temperature diffusion rate and energy coupling to the lattice via electron-phonon scattering. While in general this model includes both interband and intraband transitions, at the telecom wavelength of 1.5 μm the predominant contribution to the metal’s permittivity comes from intraband dynamics. Therefore the RPA formulation can be reduced to a Drude-like model where the permittivity of the metal is expressed as

_{p}= 2.168x10

^{15}rad/s is the plasma frequency, γ

_{intra}(ω,T

_{e},T

_{L}) is the electron scattering rate, including both electron-electron and electron-phonon scattering as a function of the temperatures of electrons (T

_{e}) and the lattice (T

_{L}), respectively, and is the high-frequency limit of the permittivity. The transmission through the input interface of the modulator was calculated within the RPA model at both room temperature and for an electron temperature in the Au nanorods of T

_{e}= 3000 K, corresponding, to a good approximation, to the nanorod metamaterial under pulsed optical laser excitation [10, 18]. The modulator’s working point was chosen so that this state leads to maximum transmission of the device (the ON state). The OFF state minimizes modulator transmission and corresponds to an electron temperature of T

_{e}= 300 K, or room temperature. Figure 2(b) shows the differential transmission between the two states, demonstrating a maximum change in the transmission exceeding 60% in the first-order transmission minimum.

## 4. 3D numerical simulation

To determine both the transmission and modulation performance of the proposed device, a 3D numerical modeling was performed on the basis of the optimal modulation geometry [Fig. 2(b)]. Calculations were first performed by modeling the nanorod metamaterial within the EMT to determine the modulator’s switching performance as a function of length. For the considered operating point, the length of the modulator was chosen to be 600 nm as a tradeoff between insertion losses and maximized modulation depth. The transmission of the modulator as the function of the wavelength is plotted in Fig. 3(a) in both ON and OFF states. Note that Fig. 3(a) corresponds to the full transmission of the device defined as the ratio between the time-averaged integrals of the Poynting vector calculated at the output over that at the input of the modulator. In agreement with the single interface simulations, the transmission of the device shows a distinct transmission minimum of about 37% at the operating wavelength of 1.5 µm. Figures 3(b)-3(c) shows the plot of the absolute value of the normalized electric field amplitude along the propagation direction for both OFF and ON states. The corresponding mode distribution is depicted in Figs. 1(b)-1(c), respectively. As predicted analytically, the variation of the mode’s spatial confinement in the metamaterial layer as a function of operating conditions is clearly visible. In the ON state, the fundamental mode stays mainly confined to the silicon waveguide decaying exponentially in the nanorod metamaterial. In contrast, the OFF state shows hybridization of the modes, as a significant coupling to extraordinary waves is observed within the metamaterial layer.

Confirmation of the expected modulator performance is obtained from the full 3D FEM simulations accounting for the internal structure of the metamaterial [Fig. 3(a)]. The agreement with the EMT calculations is good, recovering both the spectral behavior and the modulation performance quantitatively.

## 5. Nonlinear analysis

Nonlinear transient dynamic simulations were carried out using a 50 fs control pulse at a center wavelength of 532 nm, resonant with the metamaterial’s transverse resonance providing an extinction exceeding 90%. The optical pump [Fig. 1(a)] controls the transmission of the modulator via the change of the temperature-dependent permittivity of the Au nanorods [16]. Typically, the gradient of the electron temperature results in the equalization over the whole metamaterial before a sensible energy exchange between electrons and lattice occurs. For a uniform distribution of electron temperature across the metamaterial, electron energy is given by [17]

where C_{e}is the heat capacitance of electrons (67.96 J/m

^{3}K

^{2}) and

*V*is the total volume of the metal inside the metamaterial. For the considered geometry, an estimated energy of 3.7 pJ is required in order to achieve the 35% modulation [Fig. 3(b)]. Furthermore, the modulator switching time can be estimated in the framework of the two-temperature model. In the weak excitation regime assumed here, electron-phonon coupling is linear with a decay rate that can be estimated as –g/C

_{mod}_{e}= −2.943 × 10

^{14}K/s for gold, leading to a decay rate of (C

_{e}/g)(T(ON)-T(OFF)) = 9.1 ps between ON and OFF states. Since diffusion effects were not accounted for in this estimate, this time corresponds to an upper limit to the practical relaxation time of the device that could, therefore, exceed 100 GHz. Indeed previous studies for similar metamaterial geometries have shown relaxation times on the order of hundreds of femtoseconds leading to THz speed light modulation [10].

## 6. Conclusion

In conclusion, a new integrated metamaterial-based optical modulator for Si waveguides has been demonstrated. The optimized geometry of the modulator consists of a hyperbolic metamaterial slab deposited on top of a Si waveguide. Its dimensions of width 300 nm, height 440 nm, and length 600 nm result in insertion losses of 5.1 dB and a modulation depth of 6.33 dB/μm. The active control over the modulator’s transmission is achieved optically by controlling the level of hybridization between the fundamental mode supported by the Si waveguide and extraordinary modes with high density of states supported by the hyperbolic metamaterial slab. Strong coupling within the modulator corresponds to a mode delocalization in the metamaterial leading to increased insertion losses and a consequent minimum in the modulator’s transmission. The switching energy estimated from nonlinear dynamics simulations is 3.7 pJ/bit at rates approaching THz frequencies when modulation is controlled optically at a wavelength of 532 nm, resonant with the transverse plasmonic mode of the metamaterial.

## Acknowledgments

This work has been funded in part by ESPRC (UK) and the ERC iPLASMM project (321268). A.Z. acknowledges support from the Royal Society and the Wolfson Foundation. A.N. acknowledges King’s College London Postgraduate School for KORS studentship. G.W. is grateful for support from the People Programme (Marie Curie Actions) of the EC FP7 project 304179.

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