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The properties of integrated-photonics directional couplers composed of near-field-coupled arrays of metal nanoparticles are analyzed theoretically. It is found that it is possible to generate very compact, submicron length, high field-confinement and functionality devices with very low switch energies. The analysis is carried out for a hypothetical lossless silver to demonstrate the potential of this type of circuits for applications in telecom and interconnects. Employing losses of real silver, standalone devices with the above properties are still feasible in optimized metal nanoparticle structures.
©2011 Optical Society of America
1. Introduction
Nanophotonics in general and plasmonics in particular have received much attention in recent years, fuelled by a general interest in nanotechnology but also by rapid advances in integrated photonics over the years, primarily brought about by using silicon/air or quartz interfaces, giving a larger refractive index contrast than previously employed [1,2]. One can show that the minimum lateral spatial field width for a planar silicon waveguide in air is ~300 nm, with a wavelength in the medium of ~500 nm, at a vacuum wavelength of 1550 nm. Thus, any attempts in nanophotonics integration should surpass these values, as measured in proportion to the relevant vacuum wavelength. To increase spatial integration, it is necessary to find a successor to the current silicon nanowire technology. Such successors seem to have to rely on materials with negative ε, notably metals [3,4] since these offer a possibility for increasing the integration density in photonic lightwave circuits in two ways: (i) By using the properties of materials with negative ε to concentrate light in ways other than employing total internal reflection and (ii) employing metal based metamaterials to generate artificially very large effective media refractive indices, larger than those of silicon waveguides. Both methods could allow denser lateral packing of waveguides as well as shorter resonators and filters. A main and seemingly detrimental problem for many applications has been the optical loss that is associated with these metal-based metamaterials.
Waveguides made from arrays of near- resonantly-operated and near-field-coupled metal nanoparticles in the shape of e.g. spheres have attracted considerable attention [3,5–10]. Nanoarray waveguides, based on passive nanoparticles, such as silver nanoparticles are, however, very lossy, see e.g. Ref. 7, and thus are of limited use for some albeit not all applications. By operating the nanoparticles appropriately, radiation losses can be made small and basically only (the very large) losses due to Joule heating remain. However, in anticipation of a possible future breakthrough in developing metamaterials with at least a factor of 10 lower losses [11,12], or in achieving loss compensation [13], or employing stand-alone submicron devices, we analyze in this paper the ability of two adjacent arrays of metal nanoparticles to achieve extremely compact directional couplers. Such couplers [14] form the basis of generic types of integrated photonics devices such as modulators and switches [15]. Specifically, here, the coupling length lc, the spectral characteristics of the coupling, and switching characteristics are investigated.
2. Models
We have considered arrays comprising 50-nm diameter metal particles spaced by d=75 nm center to center within those arrays, excited by transverse waves polarized perpendicularly to the plane of the device. We use a hybrid finite-element-method and multi-level fast-multipole-algorithm (FEM/MLFMA) to simulate the structure in the frequency-domain with 3D resolution of the particles [16–18]. Here, the FEM handles the interior electromagnetic fields of the particles, while finite-element boundary-integration (FE-BI) equations are used on the particle surfaces. The particle permittivity is described by the Drude model for a lossless metal , where the plasma frequency ħω p=6.18 eV is used to fit the experimental permittivity of silver [7,19] at the single particle resonance energy 3.5 eV. Using lossless Ag simplifies our analysis, facilitates good transmission, with only small radiative losses for waveguide k numbers to the right of the light line in the host, cf. Ref. 7. The single arrays were also modelled using the point-dipole approximation [7,13] to calculate dispersion properties, essential for e. g. filtering applications involving the directional couplers, e. g. utilizing differential dispersion [14,20]. The two models give very similar results for the transmission band of the nanoparticle arrays. Using the FEM/MLFMA we obtain good transmission for the wavelengths λ0=359-372 nm, in very good agreement with the part of the dispersion that is to the right of the light line in the point-dipole model, see Fig. 3 of Ref. 7, corresponding to the wavelength range λ0=355-368 nm. A small red shift of the transmission band as seen using the former model can be expected since it considers the finite particle size.
Fig. 3 (a) Optical switch based on a mismatched nanoarray waveguide directional coupler. The nanoparticles in the upper waveguide have the plasma frequency ħωp,upper=6.18 eV, while those of the lower waveguide are shifted as indicated in the figure. (b) Classical directional coupler behavior of the electric field amplitudes in the upper (|B(x)|) and lower (|C(x)|) waveguides, respectively, according to Eqs. (1,2), for increasing mismatch δ. The increment of the mismatch parameter δ between successive graphs has been chosen to resemble the simulation results of the upper panel. (c) E-field magnitudes 3 nm above the upper (|E upper(x)|) and lower (|E lower(x)|) nanoarray waveguides for the respective plasma frequencies ħωp,upper=6.18 eV and ħωp,lower=6.165 eV. The electric field amplitudes |B(x)| and |C(x)| of the classical directional coupler for the mismatch δ/κ=1.25 is indicated by dashed lines for comparison.
In addition, the point-dipole model is employed to assess achievable switch lengths using longitudinal waves on arrays of prolate spheroids in the latter part of Sec. 3.2.
3. Results
We analyze structures where two particle arrays, which may be identical or dissimilar, are placed in a proximity of one another.
3.1 Directional coupler structures
The case of the two identical arrays with a center-to-center spacing of c=90-130 nm, excited by transverse waves from a dipole source, is shown in Fig. 1(a) . A spatially periodic sinusoidal-like coupling of the surface plasmon polariton excitations on the arrays is shown numerically. Extremely short coupling lengths, e.g. 490 nm for c=90 nm, are demonstrated. This is about one order of magnitude shorter than in new directional couplers demonstrated based on Si nanowires [21] and photonic crystals [22], and about three orders of magnitude shorter than in conventional directional couplers based on usual materials, like silica or lithium niobate. The coupling length can also be calculated as l c=π/(k even-k odd), where k even and k odd are the propagation constants of the even and odd supermodes.
Fig. 1 (a) Top view of the lossless Ag nanoparticle arrays showing the E-field magnitude in a plane 3 nm above the particle surfaces. The particle positions are indicated by black dots. The periodic coupling as well as the influence of the increasing array separation c is evident. (b) The coupling length l c versus the center-to-center spacing c of the nanoparticle arrays.
In contrast with the case for dielectric waveguides, where the coupling length increases exponentially as the waveguide separation increases owing to the exponential decline of the evanescent fields, here the coupling length can be very well fitted by a power of the spacing c of the nanoparticle arrays, Fig. 1(b). This is to be expected, since the near fields of the nanoparticles do not follow an exponential decay with the distance. However the coupling between the arrays is complicated, involving not only the nearest neighbour particles of the two arrays, but rather each nanoparticle is substantially coupled to several particles in the opposing array. Further analysis of a wider range of array spacings c should be necessary to determine this dependence more conclusively.
Further, we investigated the effect of having staggered arrays offset by half a period, d/2, to each other on the coupling characteristics, in comparison to the aligned arrays in registry with each other. Longer coupling lengths were obtained for the staggered case, as could be expected from the larger particle-particle separations belonging to different arrays for the same center-to-center separation c between the arrays, Fig. 2 . In addition, the staggered configuration was found to have a stronger wavelength dependence of the coupling length, Fig. 2, though the reason for this is still unclear. The stronger wavelength dependence enables efficient filtering, e.g. corresponding to a 2.4-nm bandwidth (full width at half maximum of the cross-coupled intensity) for a 3-μm long coupler, indeed very good characteristic for an integrated optics filter.
3.2 Optical switching
The impact of phase mismatch between the two nanoarray waveguides is analyzed as follows. The phase mismatch can be brought about by artificially changing the plasma frequency or by changing the host index or particle shape for one of the arrays, in order to get a propagation constant mismatch δk between the two waveguides. The operation of such a device is illustrated in Fig. 3(a), where the signal appears in the cross-coupled port or in the driven bar waveguide depending on the mismatch, at a specific wavelength of excitation in the driving branch of the coupler. For the conventional lossless dielectric waveguide directional coupler, the coupling behaviour is then given by [14]
where C(x) is the cross-coupled field after the distance x along the coupler, and B(x) the field in the launch (bar) waveguide. Here, the coupling coefficient , and . According to Eqs. (1),2), with no mismatch there is a spatially sinusoidally varying coupling between the guides; increasing mismatch results in a higher spatial coupling frequency and lower coupling to the crossover waveguide, as seen in Fig. 3(b). The simulated results for a nanoparticle-array directional-coupler switch, where the plasma frequency of the particles in the lower waveguide has been changed artificially, and excited by transverse waves from a dipole source, is shown in Fig. 3(a). Again, the behaviour is very analogous to the conventional coupler, Fig. 3(b), and we see that for an approximately 1.4 μm long coupler, i.e. one coupling length in the uppermost phase-matched case, a change of the plasma frequency by 0.32% (ħωp,lower=6.18→6.16 eV) is enough for high-extinction-ratio switching, from a cross state to a bar state. Large shifts of the Fröhlich plasmon resonance energies of Au nanorods have been observed due to electron injection from chemical reductants, indicating a plasma frequency shift of about 5% [23], i.e. more than 15 times the shift required here. A detailed comparison of the electric field magnitudes just above the two nanoarrays with the classical result of Eqs. (1,2) is shown in Fig. 3(c). The analogous behavior is quite clear, apart obviously from the periodic variation due to the nanoparticles, and radiative losses giving a field attenuation of about 50% over the simulated length.Another way of effecting switching is to change the host refractive index for one of the arrays, thereby changing the single-particle (Fröhlich) resonance frequency. In order to calculate the required host index changes in an electrooptic host material, we need to use a dispersion and, as an example, Fig. 4 shows this relation for the longitudinally excited arrays of prolate spheroids. The dispersion is obtained within a point-dipole model including the effects of retardation [13]. Here too, the permittivity of the particles is described by the lossless Drude function, but a different effective plasma frequency ħω p=8.70 eV is used in order to fit the experimental permittivity of silver [19] at the pertinent Fröhlich resonance energy ħω 0=1.773 eV. By using spheroids the Fröhlich resonance frequency
where N=0.174 is the depolarization factor [3] (N=1/3 for spheres) and is the host permittivity, is red-shifted, thereby shifting the dispersion to more technologically viable wavelengths, also in terms of loss. Further, the bandwidth ΔE=0.230 eV between the extremes of the dispersion curve is smaller than for the spheres used above, increasing the wavevector shift δk that can be achieved in this longitudinal case. Now, by changing the host index , we find from Eq. (3) thatFig. 4 Dispersion relation in a longitudinally excited array of prolate spheroids of size 50×25×25 nm3. Shifting the host refractive index by δn H gives a shift δk of the waveguide k number at the operation point. The shift is exaggerated compared to the example in the text for clarity.
For a modulator, it is the change δk in the wave vector at a given frequency that is interesting, and δk times the device length L should in general be equal to π in order to switch from cross to bar state or vice versa. A rough estimate can be made by writing
Inserting n H=2.20 (as in LiNbO3), we get m using Eqs. (4,5). Thus, a 5 μm long device would require δn H=4×10−3 host index change, reachable with LiNbO3. Much larger refractive index shifts, e.g. δn H=0.1, are achievable using gallium lanthanum sulphide (GLS) and other chalcogenide glasses [24,25], indicating switching lengths of approximately 200 nm or only a few nanoparticles. The small dimensions of even the 5-μm-long device imply an extremely low power dissipation device, where the necessary stored RF electric energy for switching in the above example would be on the order of fJ. A detailed calculation requires a numerical analysis of the structure to analyze the impact of the non-homogenous RF field in the host, brought about by the metal spheres, on the propagation characteristics. This feature of the RF field should actually increase the efficiency, since the highest RF fields should in general overlap well with the optical fields. However, possible ways of utilizing these favourable characteristics are elusive and yet to be explored due to the very high light propagation losses, see Sec. 4 below.
4. Discussion and conclusions
In the treatment above, we have used hypothetical, lossless silver. The array structure we have analyzed has been treated rather extensively in the literature, see e. g. Refs [5–10,26,27]. One of the issues investigated for the metal nanoparticle arrays has been the optical loss. The highest attenuations of over 100 dB/μm, e.g [27], have been obtained for transverse polarization in an air host, the structure used in the first part of this paper. So admittedly the results obtained here cannot be directly applied using actual Ag with loss. The attenuation tends to be higher for transversely polarized waves due to low group velocity and the fact that operation close to the light line, which entails radiative losses, is necessary to accomplish positive group velocities. However, lower attenuations below 10 dB/μm have been calculated for transverse and longitudinal polarization for Ag nanoparticle arrays in host materials [8]. In particular for the longitudinal polarization low losses of 5-6 dB/μm have been obtained even considering the additional surface scattering of electrons in nanoparticles [6,26]. Such scattering may increase the nonradiative damping substantially in metallic nanostructures [28]. In the case of transverse polarized waves full compensation of the attenuation due to Ag material losses has been reported either using gain in the host medium [9] or in quantum dots (QDs) in arrays of composite metal-nanoshell/QD nanoparticles [13]. From this, and the results of this paper, we conclude that high performance, submicron length stand alone devices, based on longitudinal propagation or gain-assisted transverse propagation, with the functionality described above, are feasible with today’s materials. This includes directional coupler structures that are orders of magnitude more compact than conventional dielectric-based ones in lateral and longitudinal sizes, e.g. 500×200×100 nm3 and potentially with switch energies in the fJ range. This structure can be used as the basis for a number of devices such as more complex couplers, filters based on waveguides with differential dispersion [20], ring resonators (with low Q, on the order of 50) etc. One could conceivably use the inherent periodic structures of these devices as intermediate phase matching elements for e. g. two parallel waveguides and introduce superperiods for generating Bragg reflectors. Coupling in and out can be accomplished by tapers [29] or different antenna-like structures. For the directional couplers, the standard methods of using S shaped bends can be used to access the closely separated waveguides.
However, for the creation of densely integrated circuits, the losses are far too high, even if amplification is used [30]: amplification degrades power dissipation as well as signal-to-noise-ratio performance. Any evolution to high density plasmonics integrated circuits carrying “the photonic” Moore’s law [2] beyond silicon and III-V materials will have to rely on Joule losses decreased by one to two orders of magnitude in existing materials such as silver. This paper points to the great potential of such hypothetical novel materials.
Acknowledgement
P.H. and L.T. acknowledge support from the Swedish Research Council and the Foundation for Strategic Research, Sweden. A.B. has been partially supported by DARPA, USA ‖Grant No. HR0011-05-3-0002).
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