A nonlinear plasmonic resonator design is proposed for three-state all-optical switching at frequencies including near infrared and lower red parts of the spectrum. The tri-stable response required for three-state operation is obtained by enhancing nonlinearities of a Kerr medium through multiple (higher order) plasmons excited on resonator’s metallic surfaces. Indeed, simulations demonstrate that exploitation of multiple plasmons equips the proposed resonator with a multi-band tri-stable response, which cannot be obtained using existing nonlinear plasmonic devices that make use of single mode Lorentzian resonances. Multi-band three-state optical switching that can be realized using the proposed resonator has potential applications in optical communications and computing.
© 2014 Optical Society of America
Surface plasmons result from coherent resonant interactions of metals’ conduction-band free electrons . They can be excited very efficiently by incident electromagnetic fields at optical frequencies [2, 3] and they support highly localized and very strong fields in the vicinity of metal/dielectric interfaces allowing for drastic enhancement of weak nonlinear processes [4,5]. This concept of plasmonic-enhanced nonlinearity has recently been used in designing photonic devices with bi-stable responses. The nonlinear bi-stability, i.e., the reversible change in the response of a device, enables ultrafast changes in the output due to variations in the input [5, 6] suggesting possible applications in switching. Additionally, the presence of two-states in the output can be used in digital logic systems. Indeed, static random access memory applications, which make use of nonlinearities in output states of photonic devices, are under development [5,6]. Recent work on nonlinear photonic devices has focused on using bi-stability to switch on and off certain characteristics of optical nanoantennas, cloaks, and negative and near-zero refractive index materials [7–12] using self-modulation schemes. Along the same lines, a step-like transmission, similar to nonlinear bi-stability, has been obtained through an asymmetric Fano resonance generated by a Kerr layered/grating structure . Similarly, a one-dimensional grating consisting of alternating layers of dielectric and chalcogenide glass has been used to obtain bi-stability in the reflected field [14–16]. The field localization on a two-dimensional photonic crystal has also been exploited to generate bi-stability for a low input power level but at a narrow bandwidth .
Oftentimes, singly resonant photonic structures with high Q factors are used for enhancing weak nonlinear processes introduced by medium properties. This reduces the input power level required by the design to operate within the nonlinear region since it scales with 1/Q2 [14–16]. However, typically, high Q factors come at the cost of reduced bandwidth of operation. Lossless dielectric cavities [18, 19] or under-damped (lossy) singly resonant plasmonic resonators [7, 8] have been used for enhancing nonlinearities but with a limited bandwidth of operation.
The significant improvements that come with using three-state switches (instead of their two-state counterparts) for communications and computing in the field of digital electronics have been well-understood. In data communication, the process of line coding converts the raw digital data to digital signals . Several multi-level line coding schemes, which benefit from the use of three-state output, have been developed; these include the well-known 4B/3T and 8B/6T encoding schemes . An example of 8B/6T line coding scheme is described in Fig. 1(a): 8-bits (1 byte) of binary data are encoded to 6-bits of ternary digital signal. A signal described using ternary states instead of binary ones helps in better synchronization and increasing the data rate of the communication system . In Fig. 1(b) a three-state logic gate is described; this type of gates are used in digital computing systems whenever multiple devices are connected together through a bus . Here, the additional “enable” input is responsible for connecting or isolating the device output from the rest of the system, which cannot be achieved with gates that support standard two-state output. Three-state logic gates are essentially used in computing and communication applications to provide time-division multiplexing for several devices connected to a single channel. No useful communication can take place if all the devices simultaneously try to communicate their digital data over the bus waveguide. However, if the devices support three-state outputs, where the lowest state correponds to an open circuit (high impedance state) the devices could communicate to the bus waveguide (one device at a time), [Fig. 1(c)] . It should be clear from the discussion here that for the examples described in Figs. 1(a), 1(b) and 1(c) the device support with three-state output is essential. In digital electronics, three-state logic devices are realized with TTL or CMOS technology .
Similarly, all-optical communication and computing systems can benefit from advantages that come with three-state switches. The optical implementation of three-state switches making use of Kerr nonlinearity has previously been achieved using photonic bandgap structures [22–24] and coupled waveguide-cavity resonators . In this work, a plasmonic resonator that supports three states in its output at frequencies ranging from infrared to optical parts of the spectrum is designed for that purpose. The optical response required for three-state output is achieved by enhancing nonlinearities of a Kerr medium through multiple higher order plasmons excited on resonator’s metallic surfaces. Exploitation of multiple plasmons equips the proposed resonator with characteristics that cannot be obtained using existing nonlinear device designs that make use of single-mode Lorentzian resonances: (i) It supports nonlinear response at multiple bands of frequencies, which cumulatively increases the bandwidth of operation. (ii) It supports nonlinear tri-stable output characteristics that cannot be achieved using existing Kerr nonlinear devices with bi-stable output states [8, 14–16].
2. Design details and results
To achieve multi-band multi-stable characteristics in the design output three different criteria are set for a plasmonic nanostructures: (i) Ability to generate higher order modes. (ii) Support of high and uniform fields inside a smaller region where the Kerr-nonlinear material is located. (iii) Ability to operate at normal incidence at optical frequencies. The plasmonic resonator in Fig. 2 satisfies these criteria.
This design consists of a square gold frame and a rectangular gold rod. The geometry dimensions shown in Fig. 2 are R1 = 120 nm, R2 = 90 nm, L = 50 nm, W = 20 nm, g = 2 nm, and t = 10 nm. It should be noted here that selection of g = 2 nm ensures that a strong and (essentially) uniform electromagnetic field is generated inside the gap; this field is denoted by Eg. The gap between the frame and the rod is filled with a material with relative permittivity εg. The intrinsic nonlinearity of gold is assumed to be negligible and the tabulated values of its permittivity are used in the simulations . Additionally the photo darkening and thermo-optic effects are assumed negligible . The resonator is excited by a normally incident plane wave with electric field polarized parallel to the rod (Fig. 2). The input intensity of this field is denoted by I0. It should be noted here that multiple higher order surface plasmon resonances are induced as a result of the broken geometrical symmetry (with respect to the excitation) [28–32] obtained by shifting the rod along the horizontal axis of the frame. This shift allows odd-symmetric higher order modes to be excited on the structure.
2.2. Linear simulations
To demonstrate the presence of multiple (higher order) surface plasmons in the response of the resonator, enhancement factor |Eg|/|E0| is computed at frequencies between 175 THz and 500 THz for I0 = 21 W/cm2 and various values of εg. |Eg|/|E0| is plotted against frequency in Fig. 3. The resonance frequencies of the plasmons are clearly identified as three well-separated peaks in the figure. It is well-known that the bi-stability is a result of the fact that the spectral location of a single plasmonic resonance is sensitive to variations in εg . Similarly, as shown in Fig. 3, multiple plasmonic resonances are also sensitive to variations in εg. This suggests that when the gap is filled with a nonlinear medium, i.e., when εg changes nonlinearly with Eg (or I0), the multiple peaks in the |Eg|−εg curves will provide a mechanism for the generation of multi-band tri-stability. This is demonstrated next.
2.3. Nonlinear simulations
For the “nonlinear” simulations of the resonator, the gap is assumed to be filled with Ag-BaO, which is modeled as a third order nonlinear χ(3) Kerr material . Its relative permittivity is given by33].
The nonlinear simulation is carried out using the well-known graphical method . At a given frequency and I0, “linear” simulations are carried out to compute Eg for various values of εg. Then, |Eg| is plotted against εg. For linear simulations, since |Eg| scales linearly with , additional curves for different values of I0 are obtained by simply shifting the initial |Eg|−εg curve. The intersection points of these curves with the curve of the nonlinear relation in Eq. (1) are found. These intersection range defined by values of I0, |Eg|, and εg are the solutions to the nonlinear simulation. The I0 used in the graphical method is determined by the desired output characteristics of the device.
The graphical method can be explained better with an example (Fig. 4). The nonlinear simulation of the resonator is carried out at 410 THz. First, for I0 = 19.5 MW/cm2, Eg is computed for a set of values of εg between 1 and 18, and the |Eg|−εg (green) curve is plotted. Then, the |Eg|−εg (blue) curve for I0 = 3.74 GW/cm2 is obtained by simply scaling the curve for I0 = 19.5 MW/cm2. The curve for I0 = 19.5 MW/cm2 intersects the (black) curve of the nonlinear relation in Eq. (1) at three different points. The middle point corresponds to the solution for the unstable branch of the bi-stability. Similarly, the curve for I0 = 3.74 GW/cm2 has five intersection points (Fig. 4) that correspond to a combined bi- and tri-stability.
Next, the graphical method described above is used to find the response of the resonator for a band of frequencies between 175 THz and 500 THz for I0 = 0.22 GW/cm2. Figure 5 compares the extinction cross section (CS) spectrum of the proposed nonlinear resonator and a linear resonator with its gap loaded with a material with εg = 2.52 for I0 = 0.22 GW/cm2 and I0 = 21 W/cm2, respectively. It is clearly seen that the high input intensity self-modulates the response of the nonlinear resonator and four bi-stability regions are clearly identified in the extinction CS spectrum.
At the frequency 410 THz, the two steps of graphical method applied in the above example in Fig. 4 [scaling of the curves and finding their intersections with the curve of Eq. (1)] are repeated for a set of values of I0 between 2 MW/cm2 and 10 GW/cm2. Figure 6(a) plots εg against I0. Let Ti, i = 1, ..., 6, represent the thresholds of I0 for stable resonator operation. It is clearly shown in Fig. 6(a) that between T1 and T2, εg forms a connected double S chain representing the bi-stability in the response of the resonator. Between T3 and T6 both bi- and tri-stability are observed. Note that the tri-stability condition is achieved when three stable states are simultaneously generated within the same range of I0.
The nonlinear simulation described above is repeated at 498 THz. Figure 6(b) plots εg against I0. A small bi-stable region between T3 and T4 is added on top of a broader bi-stable region between T1 and T2, making this region exhibit tri-stability.
It should be noted here that the resonator’s geometry is designed to ensure that the specific features visible in its far-field originate from its near-field response . As a consequence, the effects of nonlinearity on Eg/εg as shown in Figs. 6(a) and 6(b) are expected to appear in the resonator’s far-field response. Indeed, bi-, multi-bi-, and tri-stability are achieved in the extinction CS spectrum of the resonator. This is clearly demonstrated in Figs. 6(c) and 6(d), where the extinction CS of the resonator is plotted against I0 at 410 THz and 498 THz, respectively. At 410 THz, the extinction CS forms an oval shape for the initial bi-stability between T1 and T2 and a connected double inverted S chain in the form of multi-bi- and tri-stable states between T3 and T6 [Fig. 6(c)]. Similarly, Fig. 6(d) shows a tri-stable extinction CS between T3 and T4 at 498 THz. In this case, a small bi-stable region between T3 and T4 is added on top of a broad bi-stable region between T1 and T2, making this region exhibit tri-stability. Figures 6(a) and 6(b) and 6(c) and 6(d) also show that multi-bi-stability and tri-stability states are excited at higher values of I0 between 1 GW/cm2 and 10 GW/cm2 in comparison with values of I0 required for bi-stable only operation.
Figure 5 clearly demonstrates the multi-band characteristics of the proposed resonator’s nonlinear response. Compared to conventional Lorentzian based resonators [8, 14–16], the proposed design, can be made bi-stable within all these frequency regions without any modifications on the geometry. Consequently, the bandwidth of operation for nonlinear functionalities including switching can be significantly increased.
In addition to optical bi-stability, the multi-stability supported by the proposed resonator in the forms of multi-bi-stability and tri-stability in Figs. 6(a)–(d) provides a mechanism for realizing multi-state output for optical switching and memory access applications  as described next. The stable branches within the tri-stability region (T2 < I0 < T3) are highlighted in Fig. 7(a). In this figure, branches between output intensity levels O1 − O2, O3 − O4, and O5 − O6 are assigned to output states 1, 2 and 3, respectively. The state flow diagram in Fig. 7(b) describes the states of all-optical switching and memory access applications that can be achieved using the proposed resonator. The switching behavior requires transition among the three states of flow diagram (blue and yellow arrows) [Fig. 7(b)]. The transition (or switching) to another state requires the corresponding input intensity I0 to be varied accordingly (shown on the arrows of diagram). For the memory access application, “set” and “hold” conditions should be satisfied. The memory set condition is same as the switch condition that requires a desired transition among the states (blue and yellow arrows). The closed feedback loops for all three states with input intensity condition T2 < I0 < T3 represent the memory hold condition (purple arrows).
The switching contrast in the extinction CS of the proposed resonator is provided in Table 1 for various switching (input) intensities identified with threshold levels in Figs. 6(c) and 6(d). These refer to corresponding switching between states depicted in the state flow diagram in Fig. 7(b).
For pratical applications the plasmonic resonator must be cascaded with an external input intensity controller to provide switching and memory functionalities through self-modulation. An ideal candidate for this controller is a Vertical Cavity Surface Emitting Laser (VCSEL), whose output intensity is linearly proportional to the injected current under stimulated emission condition [6, 8, 12]. The plasmonic resonator can be placed in the path of VCSEL’s output beam [Fig. 7(c)]. Finally, the schematic device model in Fig. 7(c) summarizes the proposed resonator’s use in optical communication and computing. The self-modulated input intensity I0 controls the three-state resonator output “1/2/3”, which is compatible with 8B/6T “−/0/+” ternary line encoding schemes described in Fig. 1(a). It should also be emphasized here that the three-state switching capability is useful for communicating the multiplexed digital data that results from logic operations in computing devices sharing a common channel [Figs. 1(b) and 1(c)].
The portion of curves within the unstable regions are primary source of error for digital switching characteristics. During transients of the system, the output may reside in one of these unstable branches leading to ambiguity in the output state. However, once close to steady state the system will rapidly switch to one of its stable solutions. Also at high input intensity required for Kerr nonlinear operation the Joule heating may induce unwanted thermo-optic effects in the resonator. This may alter the predicted theoretical multi-stable response from the experimental case.
For certain applications switching response time should be known. Transient analysis can be done using time-domain simulators for this purpose. Several open-source finite-difference time-domain (FDTD) simulators can be used but they are known to suffer from accuracy and stability problems especially for nonlinear and dispersive media for three dimensional models . It was previously demonstrated that Kerr nonlinear response time of a thin-film Ag-BaO was 210 fs . As the nanoscale localized surface plasmon resonances have low Q-factors, consequently it is expected that the switching times will be on the order of couple of picoseconds [5, 15]. It is estimated that with this time response the resulting switching operation will support a signal rate close to Tb/sec . It should be also noted here that the maximum data rate is not only determined by the signal rate. The three-state output supported by the design increases the data rate by increasing number of digital symbols per signal from two to three, see 8B/6T scheme in Fig. 1(a).
Finally, a comparison of the proposed design to existing multi-stable output resonators is done from a practical implementation perspective: [22–25] (i) The thickness of the proposed device t in the propagation direction is much smaller than those of band-gap structures and coupled resonators [22–25]. This small thickness permits nanoscale integration of the device into an optical system and faster switching time. (ii) Unlike band-gap structures [22–24] and coupled resonators  which work in a narrow band of frequencies, the proposed resonator has a broad band of operation. The mutli-stability is supported for visible and near infrared frequencies due to higher order plasmon modes generated on the resonator. (iii) Compared to dielectric resonators [22–25] the plasmonic resonator supports highly localized and strong near fields |Eg| inside the gap allowing drastic enhancement of weak nonlinear χ(3) process. Due to this enhancement the input intensity required for nonlinear operation is reduced.
A plasmonic resonator, which is capable of supporting multi-bi- and/or tri-stability in its response at the infrared and optical frequencies, is described. The capability to generate multilevel states can be envisioned as multi-state digital logic system that can operate beyond the traditional binary on-off switching. Such a device can be used in all-optical computing and communication applications.
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