Giant Transverse Optical Forces in Nanoscale Slot Waveguides of Hyperbolic Metamaterials

Here we demonstrate that giant transverse optical forces can be generated in nanoscale slot waveguides of hyperbolic metamaterials, with more than two orders of magnitude stronger compared to the force created in conventional silicon slot waveguides, due to the nanoscale optical field enhancement and the extreme optical energy compression within the air slot region. Both numerical simulation and analytical treatment are carried out to study the dependence of the optical forces on the waveguide geometries and the metamaterial permittivity tensors, including the attractive optical forces for the symmetric modes and the repulsive optical forces for the anti-symmetric modes. The significantly enhanced transverse optical forces result from the strong optical mode coupling strength between two metamaterial waveguides, which can be explained with an explicit relation derived from the coupled mode theory. Moreover, the calculation on realistic metal-dielectric multilayer structures indicates that the predicted giant optical forces are achievable in experiments, which will open the door for various optomechanical applications in nanoscale, such as optical nanoelectromechanical systems, optical sensors and actuators.


Introduction
Optical forces arising from the gradient of light field have been extensively employed to realize exciting applications for light-matter interactions, such as optical amplification and cooling of mechanical modes [1], actuation of nanophotonic structures [2][3][4], optomechanical wavelength and energy conversion [5], and optical trapping and transport of nanoparticles and biomolecules [6,7]. It has been shown that optical forces can be remarkably enhanced with coupled high-Q optical resonators where the circulating optical power is considerably amplified due to the long photon lifetime [8,9]. Besides, such gradient optical forces can also be significantly enhanced through the compression of optical energy into deep subwavelength scale. Recently, strongly enhanced optical forces have been obtained in dielectric slot waveguides [10,11] and hybrid plasmonic waveguides [12]. Since the optical field enhancement is proportional to the index-contrast at the slot interfaces, a material with a higher refractive index is desirable to further boost the optical force in slot waveguide structures. Metamaterials can be carefully designed to exhibit ultrahigh refractive indices [13][14][15], which are not available in naturally occurring materials at optical frequencies. Especially, hyperbolic metamaterials constructed with metal-dielectric multilayers supports huge wave vectors and therefore also ultrahigh refractive indices, due to the extreme anisotropy of permittivity tensor [16][17][18][19].
In this paper, we will demonstrate that the transverse optical forces in slot waveguides of hyperbolic metamaterials can be over two orders of magnitude stronger than that in conventional dielectric slot waveguides [11]. The mechanism of such optical force enhancement will be investigated both numerically using Maxwell's stress tensor integration, and analytically using a 2D approximation of the 3D slot waveguide system. Moreover, the relation between the optical force and the waveguide mode coupling strength is derived based on the coupled mode theory analysis [20]. The comprehensive understanding of the enhanced transverse optical forces in metamaterial slot waveguides will be very useful for nanoscale optomechanical applications, such as optical tweezers [21,22], optomechanical device actuation [23] and sensitive mechanical sensors [24].

Deep mode confinement and giant optical force
Figure 1(a) shows the schematic of the hyperbolic metamaterial slot waveguides. Two identical waveguides with width L x = 40 nm and height L y = 30 nm are separated with a nanoscale air gap g along the y direction. In each waveguide, the metamaterial is constructed with alternative thin layers of silver (Ag) and germanium (Ge). The multilayer metamaterial can be regarded as a homogeneous effective medium and the principle components of the permittivity tensor can be determined from the effective medium theory (EMT) [25] x where f m is the volume filling ratio of silver, ε d and ε m are the permittivity of germanium and silver, respectively. ε d = 16, and ε m (ω) = ε ∞ -ω p 2 /(ω 2 + iωγ) from the Drude model, with a background dielectric constant ε ∞ = 5, plasma frequency ω p = 1.38 × 10 16 rad/s and collision frequency γ = 5.07 × 10 13 rad/s. Figure 1(b) shows the dependence of the permittivity tensor on the silver filling ratio f m at the telecom wavelength λ 0 = 1.55 μm. All the components of the permittivity tensor will grow in magnitude as the filling ratio increases. For instance, the permittivity tensors of hyperbolic metamaterial are ε y = 29.2 + 0.12i, ε x = ε z = −39.8 + 2.1i for f m = 0.4, and ε y = 76.3 + 1.4i, ε x = ε z = −81.7 + 3.6i for f m = 0.7, respectively. It should be noted that the wavelength λ 0 = 1.55 μm is merely used as an example throughout the paper, and this design can actually work in a broadband frequency range due to the non-resonant nature of hyperbolic metamaterials [26]. The strong mode coupling between the two closely spaced waveguides will generate mode splitting of the individual waveguide mode and result in two eigenmodes; one is the symmetric mode (denoted by M s ) and the other is the anti-symmetric mode (denoted by M a ). Both of the two modes are TM-like (in which H x , E y and E z components are dominant), so an incident light with E y polarization is necessary to efficiently excite these two modes. The effective indices n eff,z ≡ k z /k 0 and the propagation length L m ≡ 1/2Im(k z ) corresponding to the two eigenmodes are obtained by finite-element method (FEM) with the software package COMSOL (where k 0 is the wave vector in free space and k z is the wave vector along the propagation direction z). Figure 1(c) and (d) show that the dependences of n eff,z and L m on the gap sizes are distinct for the two eigenmodes. As the gap size g shrinks, n eff,z grows dramatically for mode M s but decreases slightly for mode M a . In fact, the magnitude of the effective index variation is equivalent to the mode coupling strength between two identical waveguides [12], and therefore the distinguished variations of effective indices for two eigenmodes imply a strong coupling strength for mode M s and a weak coupling strength for mode M a . Furthermore, the opposite effective index variations for two eigenmodes indicate an attractive force for mode M s and a repulsive force for mode M a . The propagation length L m decreases for mode M s and increases for mode M a as the gap size gets narrower, due to the tradeoff between the optical mode confinement and the propagation loss. The effective index and the propagation length for the unperturbed mode of the individual waveguide are shown in Fig. 1(c) and (d) for comparison (denoted by M 0 ).
The optical mode profiles of field components E y , H x and S z for slot waveguides with g = 10 nm are shown in Fig. 2. As can be seen from Fig. 2(a), distinct behaviors in electric field E y are obtained for modes M s and M a , where a strong (weak) electric field is localized in the gap region for M s (M a ) mode. It has been proposed that the optical field could be tightly confined and greatly enhanced in the nanoscale slot region due to the large discontinuity of normal electric fields (E y in our case) at the high-index-contrast interface [10]. Here we demonstrate that strong optical field confinement is achievable in the slot region for the symmetric modes, due to the constructively interfered electric field. While for the antisymmetric modes, weak optical field confinement is obtained in the slot region. Figure 2(b) shows that the magnetic fields are always tightly confined within the hyperbolic metamaterials for both eigenmodes due to the absence of magnetic response for the metamaterials at an optical frequency. Accordingly, a large amount of energy flow is guided in the slot region for the symmetric mode, while a negligible amount of energy is confined in the slot region for the anti-symmetric mode [see where S  is a surface enclosing one metamaterial waveguide, e y  is the unit vector along the y direction. Here the transverse optical force f opt is normalized to the total optical power P z confined in the coupled waveguides. The optical field intensity at the slot region directly indicates the magnitude of transverse optical forces between the two coupled waveguides, and therefore the optical force for the symmetric mode is expected to be much stronger than that for the anti-symmetric mode, due to the distinguished optical field distributions for the two modes (see Fig. 2). Figure 3 shows the effective indices n eff,z and the optical forces through the integration of Maxwell's stress tensor for both the symmetric mode and the anti-symmetric mode. n eff,z for the symmetric mode increases noticeably as the gap size shrinks [see Fig. 3(a)], implying a strong mode coupling strength between the two waveguides. Accordingly, the attractive optical force for the symmetric mode grows dramatically with the decreased gap sizes, resulting in optical forces up to 8 nNμm −1 mW −1 for L y = 30 nm and 4 nNμm −1 mW −1 for L y = 80 nm [see Fig. 3(b)], over two orders of magnitude larger than that in a dielectric slot waveguide [11]. On the contrary, n eff,z for the anti-symmetric modes show negligible variation with gap sizes [see Fig. 3(c)], so that the repulsive optical forces for the anti-symmetric modes just increase slightly when the gap size shrinks [see Fig. 3(d)]. As a result, optical forces for the anti-symmetric mode are much weaker than that of symmetric modes, in sharp contrast to the case in dielectric slot waveguides, where the optical forces for the symmetric mode and the anti-symmetric mode are comparable in magnitude [4,11,28]. It is the strong interaction between the two waveguides that leads to the distinct mode coupling strengths and the distinguished optical forces obtained from the two eigenmodes. Furthermore, stronger optical forces are achieved in slot waveguides with a smaller cross section, for both the symmetric modes and the anti-symmetric modes, due to larger effective indices and thus stronger mode coupling strength.

Mode coupling analysis
In order to provide a comprehensive understanding of the relation between the gradient optical forces and the waveguide mode coupling, here we give an analytical expression to solve the optical forces in the metamaterials slot waveguides. Since the optical mode profiles (see Fig. 2 where Φ(y) = cosh(γy)/cosh(γg/2) for the symmetric mode and Φ(y) = sinh(γy)/sinh(γg/2) for the anti-symmetric mode, ϕ is the phase shift at the middle of each waveguide due to the mode coupling. The wave vector inside metamaterial k y and the field decay rate in air γ are related to the propagation wave vector k z through the dispersion relations and f ± corresponds to the optical forces for the symmetric modes ( + ) and the anti-symmetric modes (-), respectively. In the coupled mode theory, sin 2 (θ) corresponds to the maximum power transfer efficiency from one waveguide to the other in a coupled waveguide system [20]. For a coupled system with two identical waveguides, sin(θ) = 1, and Eq. (9) is reduced to This formula reveals that the transverse optical forces are proportional to the variation rate of the mode coupling strength as the two waveguides approach each other adiabatically. To check the validity of this formula, the calculated optical forces using Eq. (9) are displayed in Fig. 5. It is shown that the CMT formula can give optical forces exactly the same as that rigorously calculated from the integration of Maxwell's stress tensor for both the symmetric modes and the anti-symmetric modes.

Broadband operation
In contrast to the metamaterials based on electric or magnetic resonances, the hyperbolic metamaterial based on metal-dielectric multilayer structure is intrinsically non-resonant and thus can support giant optical transverse force in a broad wavelength range. As shown in Fig.   6(a), giant optical forces are obtained for the symmetric modes with operating wavelength λ 0 from 1 μm to 2 μm, which verify the broadband nature of our design. The optical forces for the anti-symmetric modes are relatively weak and drop very fast as the operating wavelength increases [ Fig. 6(b)].

Realistic multilayer structures
Finally, a realistic hyperbolic metamaterial slot waveguide is constructed using alternative silver and germanium layers with a period of 10 nm and a silver filling ratio of f m = 0.4. The comparison between the multilayer structures and the ideal effective medium is shown in Fig.  7. With the height of L y = 80 nm, the slot waveguides of multilayer structures can reproduce the results of the effective indices n eff,z and the optical forces f opt calculated based on the waveguides of ideal effective media, for both the symmetric modes and the anti-symmetric modes. While with L y = 30 nm, both n eff,z and f opt become smaller than the results of the effective media calculation, which is due to that the large wave vector along the y direction becomes close to the Brillouin zone of periodic multilayer structures, so that the waveguide mode profiles begin to deviate from those predicted from the effective medium theory. Considering that a realistic hyperbolic metamaterial from metal-dielectric multilayer structure is always lossy due to the energy dissipation in metal, the optical force will gradually decrease along the propagation direction for a certain incident optical power. Specifically, the optical force can be expressed as f opt (z) = f opt (z = 0)·exp[-2Im(n eff,z )k 0 z], where the optical force is normalized to the incident optical power at z = 0 [12]. However, the optical force is still considerably strong even if the energy dissipation is taken into account.

Conclusions
In conclusion, we have demonstrated giant transverse optical forces up to 8 nNμm −1 mW −1 in nanoscale slot waveguides of hyperbolic metamaterials, due to the strong optical field confinement and extreme optical energy compression within the air slot region. The influences of the waveguide geometries and the metal filling ratios of the hyperbolic metamaterials on both the symmetric modes and the anti-symmetric modes are studied numerically using the Maxwell stress tensor integration method, together with an analytical approach using a 2D approximation of the 3D coupled waveguides. Furthermore, the relation between the transverse optical forces and the waveguide mode coupling strength is derived from the coupled mode theory, revealing the mechanism of optical force enhancement in slot waveguide system. Finally, it is shown that the predicted giant optical force is achievable in realistic metal-dielectric multilayer structures. The strongly enhanced optical forces in slot waveguides of hyperbolic metamaterial will open a new realm in many exciting nanoscale optomechanical applications.