To enable multiple functions of plasmonic nanocircuits, it is of key importance to control the propagation properties and the modal distribution of the guided optical modes such that their impedance matches to that of nearby quantum systems and desired light-matter interaction can be achieved. Here, we present efficient mode converters for manipulating guided modes on a plasmonic two-wire transmission line. The mode conversion is achieved through varying the path length, wire cross section and the surrounding index of refraction. Instead of pure optical interference, strong near-field coupling of surface plasmons results in great momentum splitting and modal profile variation. We theoretically demonstrate control over nanoantenna radiation and discuss the possibility to enhance nanoscale light-matter interaction. The proposed converter may find applications in surface plasmon amplification, index sensing and enhanced nanoscale spectroscopy.
© 2012 OSA
Surface plasmon polaritons (SPPs) are promising for the realization of optical nanocircuits  and provide possible solutions to control light-matter interaction on the nanometer scale , paving the way to quantum optics with surface plasmons [3, 4]. Recently, prototype nanocircuits using gap nanoantennas [5, 6] in connection with plasmonic two-wire transmission lines (TWTLs) have been theoretically proposed [7–9] and experimentally studied [10–12], showing a realizable approach to the manipulation of electromagnetic field for enhanced spectroscopy [13, 14]
In order to guide the localized optical fields, various plasmonic nanowaveguides have been proposed and studied [15–20]. Two commonly used waveguides are metal nanowires [15, 18–20] and nanogrooves in metallic films [16, 17] representing the insulator-metal-insulator (IMI) and metal-insulator-metal (MIM) waveguides , respectively. Plasmonic TWTLs consisting of two parallel metallic nanowires with a nanosized dielectric gap may function both as IMI and MIM waveguides depending on the phase difference ΔΦ between the displacement currents on the wires. With in-phase displacement currents (ΔΦ = 0), the charge distribution is symmetric across the gap, hence the electric field hardly enters the gap and the power is mostly guided through the outer surface of the two wires. Such a spatially less confined mode has a transverse magnetic (TM) character and is analogous to the guided fundamental mode on an IMI waveguide. With out-of-phase currents (ΔΦ = ± π), the opposite charges result in a highly confined field in the dielectric nanogap [7, 22, 23] and the polarization of the electric field is well transverse to the propagation direction, i.e. transverse electric (TE).
In view of power and signal transmission in a functional nanocircuit, the TM mode on a TWTL is more competent since it has higher group velocity and longer propagation length thanks to the less-confined field distribution. For the compensation of inherent loss in plasmonic nanocircuits, such large and loose modal distribution also allows for collecting energy from more excited gain materials [24–29] On the other hand, for enhancing nanoscale light-matter interaction, guided TE mode on a TWTL is more promising since it provides extreme spatial confinement, large intensity enhancement and well-defined polarization. Considering the interaction between a polarized electric field and a dye molecule, the excitation efficiency is proportional to ,with being the dipole moment operator and and being the excited and ground state of the dye molecule, respectively  It is obvious that both the intensity and the polarization of the electric field play important roles in the light-matter interaction. To realize functional plasmonic nanocircuits, single quantum systems inside the nanogap are of great interest since they may serve as nanosized transistors for surface plasmons  In order to fill the requirements of various circuit functions and to manipulate the light-matter interaction, it is, therefore, important to have the ability to control the guided modes such that the impedance matches to that of nearby nanoobjects. Figure 1 shows the schematic of the proposed mode converter and illustrates the idea of manipulating the nanoscale light-matter interaction by mode conversion. The guided TM mode with in-phase currents is injected from the left side and is converted by the mode converter into a highly concentrated TE mode in the dielectric nanogap. While the guided TM mode exhibits very low field intensity in the gap and bypasses the quantum systems, the guided TE mode is well-polarized and highly concentrated inside the nanogap and therefore strongly interacts with quantum systems integrated in the gap.
Here, we present efficient mode converters for controlling the optical impedance and light-matter interaction in an integrated plasmonic nanocircuit. To achieve deterministic mode conversion, additional phase difference between the guided SPPs on individual wires is introduced by three means, namely by differentiating the path length, the waveguide cross section and the surrounding refractive index. Compared to conventional Mach-Zehnder interferometers, the proposed mode converters rely on strong near-field coupling of guided SPPs instead of interference of guided photonic modes. With the proposed mode converter, we demonstrate improved impedance matching and controlled emission from a nanoantenna in a complex nanocircuit. We also discuss possible applications in index sensing and enhancing nanoscale light-matter interaction, and provide realizable fabrication strategies. Our plasmonic mode converter is of great interest for the realization of practical multi-functional nanocircuits and enhanced spectroscopy using nanoscopic enhanced optical field.
2. Analysis on plasmonic waveguiding
Theoretical and numerical analyses on plasmonic single-wire waveguide and TWTL are provided in this section. Our numerical studies are performed using three-dimensional finite-difference time-domain (FDTD) method and eigen-mode solver based on finite-difference frequency-domain (FDFD) method  at a fixed frequency of 361.196 THz, corresponding to a wavelength of 830 nm in vacuum. We have chosen the operational wavelength in order to conform to our current equipped laser system for further experiments. However, we emphasize here that the design principle demonstrated in the work is general and not limited to 830 nm. The dielectric function of gold is fitted to experimental values  and the minimum mesh size is set to 1 nm3. Nanowires with rectangular cross section are surrounded either by vacuum or by a homogeneous dielectric material. In three dimensional FDTD simulations, all boundaries are set to contain 12 perfectly matched layers and are positioned at least half wavelength away from the structure to avoid absorption of the near field.
2.1. Waveguiding on a single nanowire
For a guided mode on a single metallic nanowire, the electric field phasor in the dielectric medium at a position r along the wire has a form of , where is the initial amplitude, is the complex wavevector, θ denotes the phase shift to a reference time-harmonic field, typically the excitation source, and is the imaginary unit. The complex wavevector can be expressed as a product of the complex effective index and the corresponding wave number in free space by
Changing the cross sectional geometry [33–35], permittivity of metal  or dielectric function of the surrounding medium [37–39] can lead to a strong modification of the wavevector of the guided SPPs. Such unique properties of plasmonic waveguides find no counter parts in low-frequency electronic circuitry where metals behave as perfect conductors.
Figure 2(a) shows the propagation constant and the propagation length as a function of the width of wire cross section. The propagation constant decreases and the propagation length increases as the width increases. For thicker wires (width > 60 nm), the propagation constant gradually saturates due to the finite penetration depth, which is about 30 nm for gold at 361.196 THz. A consequence is that the contribution from higher order modes becomes significant for thicker wires [19, 20]. To avoid the complexity, we use only very thin nanowires with rectangular cross section of 30 × 30 nm2 and consider only the contribution from the 0th-order TM mode on a single gold nanowire . With such a small cross section, the contribution from higher order modes is negligible due to either very short propagation length or very low initial amplitude [7, 19].
Figure 2(b) shows the dependence of the propagation constant and propagation length on the refractive index n of the surrounding medium. As can be seen, both quantities are sensitive to the variation of the surrounding refractive index. The propagation constant exhibits roughly a linear dependence on the refractive index with a slope around 830 nm (RIU, refractive index unit). The slope increases with decreasing the wire cross section and represents the sensitivity to the index change which may find applications in plasmonic sensing . We will later exploit such dependence of wavevector on the refractive index to control the emission of a nanoantenna.
2.2. Waveguiding on a plasmonic TWTL
A nanosized TWTL is essentially a pair of nanowires with their guided SPPs transversely coupled through near-field interaction, similar to mode hybridization between strongly coupled plasmonic nanoparticle dimers [23, 40, 41]. Such coupling results in two distinct guided modes with anti-symmetric and symmetric charge distribution, corresponding to the fundamental TE and TM modes of a plasmonic TWTL, respectively. Figure 3(a) illustrates the wavevector splitting due to the coupling between guided SPPs, as well as the corresponding modal profiles on a TWTL. The wavevector difference between guided TE and TM mode increases as the gap size decreases due to enhanced coupling strength. For guided TE mode, the opposite charges on the two wires result in a highly confined and enhanced field in the dielectric nanogap. The extremely concentrated field in the gap exhibits a well-defined polarization transverse to the propagation direction, which is of great potential for the enhancement of light-matter interaction at the nanoscale [6, 14, 42]. The guided TM mode, on the other hand, shows very low field intensity in the gap and, consequently, does not interact significantly with nanomatter in the gap.
While the coupling between nanoparticle dimmers are usually observed by the shift in the resonant frequency, the coupling between guided SPPs at a fixed operation frequency is revealed in the wavevector splitting. Such strong coupling is missing for guided photonic modes on dielectric or photonic crystal waveguides and cannot be exploited in conventional optical interferometry, where the signal modulation is a result of pure interference effect. From an experimental point of view, operating the circuit at a constant frequency can be beneficial since variation due to the dispersion of material dielectric function or frequency-dependent polarizability of the quantum systems can be avoided. Figure 3(b) plots the propagation constant and propagation length as a function of the gap size. The splitting of the propagation constant decreases with increasing gap size, indicating a weaker coupling strength for larger gap size. For gap size larger than 300 nm, the propagation constants for the two modes reach a common value of , which coincides the value for a solitary single wire with the same cross section (red circled data points in Fig. 2). The gap size of 300 nm thus marks the distance at which the SPPs on two wires no longer couple.
For deeper investigations over the coupling between guided SPPs on individual wires, we have recorded the near-field intensity for a TWTL with a Y-split, as shown in Fig. 4 . Either TE or TM mode is injected from the left side. The near-field intensity is recorded inside the gap for TE injection and 5 nm away from the outer surface of the wire for TM mode. With a + 45 and −45 degree bend of each of the two wires, the TWTL splits into two parallel single nanowires separated with a distance of 610 nm, which ensures a negligible coupling between the wires. The two open ends of the nanowires then reflect the guided SPPs and result in standing wave patterns that allow for analyzing the effective wavelength and wavevector of the guided modes. It can be clearly seen that, regardless of the injected modes, after the Y-split the wavevector restores to the value of guided SPPs on single nanowires, i.e. . Such simple structure may serve as a model system for the study of the coupling between guided SPPs. It is worth noting that our Y-split is not adiabatic and significant power loss is expected. To estimate the total loss caused by the Y-split, one needs to know the power reflection at the Y-split due to impedance mismatch and the power loss after the Y-split owing to the wire bending. Since different injected modes exhibit distinct characteristic impedance, they experience different degree of impedance mismatch at the Y-split. Analyzing the standing wave pattern on the TWTL for each mode should provide the information of power reflection . Such analysis requires, however, the TWTL to be very long due to the long propagation length of the TM mode and is out of the scope of this work. Since the injected mode on the TWTL, i.e. TE or TM mode, is fully converted back to the guided mode on a single wire, the loss can be readily obtained by analyzing the bending loss of a single wire waveguide. In our design, the loss introduced by the 45 degree bending is optimized to about 0.2 dB.
3. Plasmonic mode converters
3.1. Conversion mechanisms
Having analyzed the coupling between the guided SPPs on solitary wires and the guided modes on a TWTL, now we may start constructing our mode converters. In order to switch between modes on a TWTL, additional phase difference between the current elements on individual wires needs to be introduced. The phase difference can be expressed asFig. 5(a) illustrates such kind of mode converter. Alternatively, the required phase difference can be introduced by modifying the propagation constant of the guided SPPs on individual wires. The required condition for successful mode conversion isFig. 5(a), middle panel) or the refractive index of the surrounding (Fig. 5(a), bottom panel). Note that we increased the gap size in the converter region in order to reduce the wire coupling such that propagation constant and length for a solitary wire (Fig. 2) can be directly used.
Figure 5(b) plots the maps of the enhancement of electric field intensity on a plane cutting through the structure at the middle height. After passing through the mode converter, the loose field distribution of the injected TM mode is clearly switched to a concentrated TE mode in the nanogap. Here, we terminated the TWTL with an open end in order to build up a clear standing wave pattern for the analysis of the wavevector and impedance matching . By analyzing the near-field intensity undulation, where the distance between two maxima corresponds to half of the wavelength of the guided mode, the propagation constant of guided mode can be obtained. As shown in the top panel of Fig. 5(b), the field inside the gap clearly reveals a distance of 127 nm between two maxima, corresponding to a propagation constant of , which is very close to the value for purely TE mode (). Although we have chosen to inject TM guided mode, all converters work well with TE injection. It is worth mentioning that the three tuning principles illustrated here work well independently but may also be combined to obtain higher flexibility. In view of practical use, mode converters using the difference in path length or cross sectional geometry require careful design and the conversion efficiency is fixed once the fabrication process is completed. In contrast, converter using the difference in the refractive index is more powerful since the index can be tuned actively by using active materials such as photoconductive semiconductors [43–45], phase transition materials [46, 47] and liquid crystals [48–50].
3.2. Conversion efficiency
In order to evaluate the conversion efficiency, we monitored the power flux on the outer wire surfaces and inside the nanogap to characterize the variation of the modal profile. We defined a mode characteristicFig. 5 are 35.4%, 22.4% and 21.5%, for the length-difference (top), cross section-difference (middle) and index-difference converter (bottom), respectively. The current metric for conversion efficiency is defined under the unique spatial mode distribution. Future direction in obtaining more accurate measures on the conversion efficiency may be performed in the Fourier-domain to associate the power related to each mode.
4. Applications in a complex integrated nanocircuit
Due to the distinct modal profile and propagation constant, the input impedance of the two modes on TWTL can be very different. One explicit example is the difference in the capability of driving optical nanoantennas. TE modes are capable of driving the bright mode of a linear gap nanoantenna from the central feed gap and the near-field optical energy can then be re-emitted to the free space as propagating photons through the antenna [7, 41]. In contrast, the guided TM mode will be strongly reflected since its impedance matches only to the dark antenna mode, with which the localized energy cannot be converted into propagating photons . From this viewpoint, our converters offer possibility to control the radiation from a nanoantenna by manipulating the optical impedance of the guided power.
As an example for the application of our mode converter, we show controlled and improved power transmission in a complex integrated nanocircuit consisting of a single metal stripe in connection with a TWTL followed by an index mode converter and terminated by an optimized nanoantenna as a near-to-far-field coupler, as depicted in Fig. 6(a) . Single metal stripes and nanowires are commonly used waveguides for SPPs [18, 20, 21, 51]. However, the mismatch in the modal distribution makes it hard to excite the radiative mode of a gap nanoantenna using single metal stripe. In the following, we show that using our mode converter, “unbalanced” optical energy guided on a single metal stripe can be used to excite radiative antenna mode and the optical power transmitted from the near to the far field can be controlled. Our mode converter, therefore, serves as a plasmonic counter part of “balun”, which improves the impedance matching between “unbalanced” signals guided by the single metal stripe and “balanced” signals on the TWTL. We demonstrate mode conversion by an index mode converter because this converter can be actively controlled and is of potential for index sensing. The cross sectional geometry of the single metal stripe is the same as that of the TWTL, except that the gap is filled up by gold. The total length of the nanoantenna has been optimized to achieve best radiation efficiency.
Let us start with the transmission of the guided power from the single metal stripe to the TWTL. Considering the modal distribution of the guided SPPs on single metal stripe, it is obvious that the TM mode on the TWTL matches guided SPPs much better than the TE mode does. The junction between single metal stripe and TWTL indeed serves as a mode filter, which only allows transmission of TM mode on TWTL and SPPs on a single metal stripe. As a result, the optical power from the single metal stripe couples only to the TM mode on the TWTL, whereas the TE mode cannot be excited due to impedance mismatch. After successful transmission of power to the TWTL, the guided TM mode then passes through the mode converter. With same path length and cross section of each wire in the converter, the conversion efficiency is completely determined by the refractive index of the surrounding media. If the refractive index of the material inside the green box area is identical to that of the rest surrounding, i.e. n = 1.0, there would be no conversion of mode and the guided power will be reflected at the nanoantenna terminal since the TM mode cannot drive the radiative antenna mode. Figure 6(b) clearly shows that the guided TM optical power is reflected back to the nanocircuit and a pronounced standing wave pattern is generated around the outer surface of the TWTL. With refractive index inside the green box swept from 1.0 to 2.0, the guided TM mode is gradually converted into TE mode and the emitted power therefore increases and reaches a maximum at n = 1.5, corresponding to the highest overall conversion efficiency. Figure 6(c) shows the intensity map of the circuit with the refractive index in the green box set to 1.5. From the modal distribution recorded at different sections of the circuit, shown in Fig. 6(d), it is obvious that the TM mode is successfully converted into TE mode when n = 1.5 and the radiative mode of the nanoantenna is successfully excited. Following the impedance matching argument and according to the reciprocity theorem , it becomes obvious that the bonding and anti-bonding mode of a symmetric gap nanoantenna  can be used to efficiently excite the guided TE and TM modes on a plasmonic TWTL, respectively.
Since the radiative mode of the nanoantenna can only be driven by the TE mode, the emitted power by the antenna can be expressed as7]. It is clear that the emission power of the nanoantenna is a linear function of the overall conversion efficiency. Since , the emission power is controlled by . Figure 6(e) plots the mode conversion efficiency , power conversion efficiency and the overall conversion efficiency , along with the emission power from the optimized nanoantenna as a function of the refractive index of the material inside the green box. It can be seen that the power conversion efficiency is relatively constant with respect to the change of surrounding refractive index and the radiated power follows the trend of . Taking the slope s of the linear relationship between the wavevector and the index of refraction, as shown in Fig. 2(b), the mode conversion efficiency can be expressed as a function of the difference in refractive index of media surrounding the two wires in the converter area
5. Perspective on nanoscale light-matter interaction
Plasmonic nanostructures show general similarities to natural chemical molecules  and the optical impedance of nanoantennas and quantum emitters can be described under an unified framework . As we have demonstrated, using TWTL and our mode converter, we may control the emission of a nanoantenna by controlling the optical impedance of the guided power. Replacing the nanoantenna with a quantum emitter that has its own optical impedance, such as a single molecule, we may, in principle, manipulate the light-matter interaction in a similar way. In view of nanoscale light-matter interaction, the much reduced mode volume Vmode substantially enhances the coupling strength between the nanocavity and single quantum systems . In addition, the well-defined polarization of the electric field may further improve the interaction between the guided field and the quantum systems residing in the nanogap. From classical point of view, the energy of a molecule with dipole moment in a polarized electric field is just their dot product,54, 55]. Using recently proposed hybrid fabrication approach for atomically smooth plasmonic nanostructure  and the “stacked” geometry for extremely small gap , it is possible to integrate thin layer of various materials, including gain, liquid crystal, photoconductive semiconductor or graphene to enable multiple functions of nanocircuits.
We have presented realizable plasmonic mode converters to manipulate the guided modes on a plasmonic TWTL. The mode conversion is accomplished through the introduction of additional phase difference between guided SPPs on individual wires by differing the path length, the cross section, and by changing the surrounding index of refraction. We analyzed the conversion efficiency of the mode converter and demonstrated its capability in manipulating the optical impedance of guided mode and therefore the emission power of a nanoantenna in a complex nanocircuit. Being able to manipulate the guided mode is important for the control over guiding properties and optical impedance of the mode and provides possibility to achieve active manipulation of the interaction between optical field and nanoobjects. One may think about building up a nanocavity consisting of a piece of TWTL terminated by two gap nanoantennas, the capability to control the mode and near-to-far field conversion efficiency then means the ability to tune the reflectivity and thus the quality factor of the cavity. Our converter may find interesting applications in optical impedance control, nanosensors and nanoresonators. We anticipate a number of applications in optical nanocircuits and active control of the light-matter interaction at the nanoscale.
The work was supported by the National Science Council of Taiwan under grants NSC 99-2113-M-007-020-MY2 and NSC 100-2112-M-007-007-MY3. J.-S. H. thanks C.-S. Huang and Dr. T. Lankau for valuable discussions.
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