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Silicon-on-insulator Mach-Zehnder interferometer structures that utilize a photonic crystal nanobeam waveguide in each of two connecting arms are proposed here as efficient 2 × 2 resonant, wavelength-selective electro-optical routing switches that are readily cascaded into on-chip N × N switching networks. A localized lateral PN junction of length ~2 μm within each of two identical nanobeams is proposed as a means of shifting the transmission resonance by 400 pm within the 1550 nm band. Using a bias swing ΔV = 2.7 V, the 474 attojoules-per-bit switching mechanism is free-carrier sweepout due to PN depletion layer widening. Simulations of the 2 × 2 outputs versus voltage are presented. Dual-nanobeam designs are given for N × N data-routing matrix switches, electrooptical logic unit cells, N × M wavelength selective switches, and vector matrix multipliers. Performance penalties are analyzed for possible fabrication induced errors such as non-ideal 3-dB couplers, differences in optical path lengths, and variations in photonic crystal cavity resonances.
© 2015 Optical Society of America
1. Introduction
Photonic crystal electro-optical nanobeams (PC EO NBs) have been exploited recently in 1 × 1 optical modulators, however, four-port 2 × 2 EO switching, an important on-chip network application, has not yet been investigated. To remedy that deficiency, we present new designs and theoretical simulations of 2 × 2 and N × N PC EO NB switches that are based upon a 2 × 2 Mach-Zender intereferometer (MZI) architecture in which identical NBs form the two connecting arms, and where the two TEo NB resonances are shifted in unison. After examining the NB transmission and reflection, we derive the 2 × 2 switching characteristic. Then, multi-stage switching networks are formulated and presented schematically for the cases of N × N path switching, EO logic unit cells, and the wavelength-multiplexed applications of vector matrix multiplication and N × M color routing.
2. Background discussion of nanobeam technology
The devices discussed here are resonant; they switch a narrow wavelength band and they switch only the TE polarization of light. Admittedly, these are limitations. However, the proposed devices offer ultralow < 500 aJ/bit switching energies not attainable with the broad-spectrum approaches. Silicon-on-insulator (SOI) nanobeams (NBs) have been investigated for applications in the 1550-nm data-communications band. The prior-art NB simulations of Hendrickson et al. [1] and the experimental work of Shakoor et al. [2,3] provide a solid foundation for future NB development. Those results are a starting point for our dual NB work. Shakoor used a rib-waveguide geometry with a localized lateral PIN diode in silicon of length Lm = 3.5 μm for electron-and-hole injection into the strip-shaped intrinsic Si core of the waveguide, whereas Hendrickson had a localized PN junction of Lm = 2 μm within the Si strip/rib in order to deplete electrons and holes from the resonator region by a sweepout voltage as shown in Fig. 1 of [1]. In the PIN NB of Shakoor [2], the Si channel waveguide core was not intentionally doped and was not lossy. That allowed a quality factor of 20,000 to be achieved. By comparison, the PN NB of [1] exhibited an inherent Q of 10,000 before the P and N dopings were put into the Si rib platform. When those acceptor and donor impurities were introduced [1], the intrinsic Q then became reduced to 8600 due to loading of the cavity by the P and N doped regions, although even at Q = 8600 the information bandwidth was a useful 230 GHz. The estimated PN switching rate was above 10 Gb/s.
Within the low-transmission forbidden spectral band of the NB, there are typically two widely spaced transmission resonances, plus some band-edge states. One transmission peak, the fundamental TE-mode resonance at λres is utilized here, and the NB quality factor determines the FWHM linewidth Δλres of the NB as Δλres = λres/Q. The free-carrier effect, whether PIN or PN, serves to shift the transmission spectral profile along the wavelength axis by an amount Δλsh, the “resonance shift.” Unlike 1 × 1 modulation where “small” Δλsh are acceptable, complete 2 × 2 switching requires a shift Δλsh of at least two linewidths Δλres. For example, in the PN device just cited, Δλres = 177 pm and we thus seek Δλsh ≥ 354 pm.
Figure 1 shows top views of the air-hole 1D photonic-crystal lattice pattern and the waveguided mode field pattern in the SOI PN NB where a quadratic in-out tapering of hole diameter along the waveguide axis was used. An advantage of this “zero point defect” cavity approach is that silicon is located at the cavity center as shown by the vertical dashed line, unlike the lattice arrangement employed in [2,3] where an air hole was placed at cavity center. Placing a portion of the PN junction at cavity center is a strategy that maximizes the Δλres that can be attained in practice. The “air center” lattice in [2,3] may explain the relatively small Δλres that they observed during carrier injection, and we speculate that they might have obtained Δλres ~100 pm with the Fig. 1 arrangement.
The EO NB has an AC switching energy and a DC holding energy, where the latter is the energy required to hold the device in either of its two states. Those states are selected by an applied voltage Va = V1 and Va = V2, respectively. Comparison of [1] and [2,3] shows that the PN technique is the lowest-energy approach. Examining the PN junction in more detail [1], there is a built in voltage Vbi of 0.935 V. That is why the width of the depletion region Wd is not zero at zero bias. By bucking out Vbi with V1 = + 0.935 V forward bias, Wd shrinks to 10 nm, but that bias is not the best choice. At Lm = 2 μm and V1 = Vbi, the junction capacitance becomes Cfor = 200 fF (Fig. 7 of [1]) as compared to Cfor = 0.55 fF at the bias of V1 = + 0.53V. It was found [1] that the wavelength λres is 1517.65 nm, 1518.10 nm, and 1518.50 nm at Va of + 0.93, + 0.53, and −2.16 V, respectively. Therefore, Δλsh of 850 pm could be attained for a voltage swing from + 0.93 to −2.16 V, however, that high-C strategy would violate the attojoules switching-energy goal, resulting in a consumption of several femtojoules. Instead, for the ultralow energy targeted here it is best to use a forward bias of V1 = + 0.53 V as the initial condition of the 2 × 2 switch, then, to reach the second state of the switch, the bias is changed to V2 = −2.16 V reverse where Crev = 0.17 fF. Here Wd widens to 110 nm, thereby producing Δλsh = 400 pm. Driven by one voltage source, the two NBs are connected electrically in parallel. The 2 × 2 energy per bit is E = 2Eb, where Eb is the NB energy 1/4(CforV12 + CrevV22) according to Eq. (6) of [1]. If we then substitute into the equation the above capacitance and voltage values, we find that E = 474 aJ/bit.
Switches employing SOI micro ring resonators (MRRs) can be seen as competitors to the dual nanobeam (DNB) devices. One or two electro-optical MRRs are often side-coupled to two SOI strip waveguides for the purposes of 2 × 2 EO switching. It is revealing to compare the relative advantages of the MRR and DNB approaches in multi-spectral (wavelength multiplexed) applications, especially in the cases of N × M on-chip integrated routing networks. We find in this paper that the DNBs are functionally equivalent to the MRRs with the distinction that MRRs offer a periodic sequence of resonances within a free spectral range governed by ring diameter, whereas the properly designed NB has in essence only one resonant mode. Another finding of the present paper is that the energy consumed in the MRR devices is generally higher than that in the DNBs. That happens in part because the circumference of the ring must be electrically driven, and that length constitutes a longer active region than that of the DNBs. The resonant mode volume of DNBs at 0.03 λ3 is generally smaller than that in MRRs.
The free carrier plasma effect in both the PIN and PN geometries serves to change the effective index of the waveguide in the cavity region, creating Δλres. The free carrier absorption induces a small but not negligible damping. For the PIN NB, the injection current at forward bias defines the energy consumption of the device. For example, in the PIN NB simulations of Ebrahimy et al. [4] with Q = 2400, starting at V1 = 0.7 V, an increase to V2 = 1.05 V produced Δλres = 3000 pm, but at a current of 904 μA. Thus, Δλres is traded off against energy in the PIN.
MOS gating of SOI NBs is feasible as an electro-optical alternative to “junction control.” However, the details of the structure have not yet been worked out. An HfO2 or SiO2 gate layer [5] could be placed on top of the NB or within the NB waveguide core as a horizontal oxide slot. Then, bias applied to the gate electrode will give free carrier accumulation at the oxide interfaces, producing Δλres. The gate electrode induces optical attenuation of the resonant transmission and that loading needs to be quantified. Beyond the free-carrier actuation discussed here, there is evidence of a viable low-energy alternative, the silicon-organic hybrid (SOH) NB approach. For example, a second-order nonlinear organic polymer may be embedded in a vertical nanoslot within the NB [6,7]. Alternatively, the NLO polymer could be fabricated within a horizontal NB slot [8]. With an available Pockels-effect coefficient of r = 150 pm/V [7] the induced index perturbation and the Δλres could be larger than that in the PN method. This is a topic for future study. Regarding the optical polarization, a fishbone-like NB structure has been shown [9] to operate on both TE and TM modes, and those corrugated NBs may be a pathway to polarization-independent 2 × 2 DNBs.
As a final note on attojoules photonics, we want to point out that the various NB routing switches are part of a larger picture of reconfigurable optical-interconnect datacom networks on a chip. In this chip scenario, every one of the electrically actuated integrated-photonic components is a NB device including the LEDs, laser diodes, modulators, and photodetectors. Attaining attojoules/bit for each chip-system component seems plausible. The NB construction could, for example, consist of an active Ge or GeSn or SiGeSn gain region (or detection region) with volume about λ3/30 grown selectively inside a tiny trench at the PIN or PN SOI central “diode space” thereby creating a NB heterodiode of P-Si/i-GeSn/N-Si to function in light emission or detection. This all-NB communication system, operating perhaps at ~2 μm wavelengths rather than at 1.55 μm, has been described [10,11]. The DNB “generic approach” in this paper applies immediately to the 2 μm band as well as to 1.55 μm and can employ GeOI [12] as well as SOI.
3. Proposed two-nanobeam 2 x 2 switches
The waveguided 2 × 2 MZI is proposed here as an ideal structure for DNB applications, and we suggest that by employing an identical NB in each connecting arm, a high-performance EO building-block switch is created. This is illustrated in the top view of Fig. 2 where SOI strip waveguides are shown everywhere except in the NB regions where there is a thin 50-nm Si rib platform underneath the Si strip (see Fig. 1 of [1]). The lateral P and N doped regions are quite localized within the rib as shown and the Fig. 1 lattice is assumed.
The conventional 3-dB directional couplers shown in Fig. 2 are excellent choices although a smaller footprint for the overall switch device is achievable by using a pair of compact 3-dB multimode-interference (MMI) couplers [13] as illustrated in the top view of Fig. 3. This is an available alternative. Returning to Fig. 2, we have labeled the four ports as input, drop, through, and add. We can analyze the optical power Pout that exits from drop and through and add when unit power is launched in the input port.
Fig. 3 A waveguided version of Fig. 1 that employs multimode couplers instead of 3-dB directional couplers.
To find the 2 × 2 switching characteristic of Pout versus Va, we first turn to the individual NB properties, where T refers to the fraction of incident light power that is transmitted by the NB and where R is the input power reflected by that NB. Next, we calculate the 2 × 2 responses by writing a product of transfer matrices for the input and output light waves (the scattering matrices are not shown here). The result of that modeling is expressed in terms of the above T and R, and we find that the add port has zero output, while Pout = T for the through port and Pout = R for the drop port, where T and R are functions of Va. We want the roles of T and R to become reversed with applied voltage swing because we seek high T at Va = V1 and subsequently high R at Va = V2. At a given voltage, the sum T + R is slightly less than unity because of loss.
To quantify the 2 × 2 model, we have performed Lumerical FDTD simulations on the PN NB having a lattice of 24 air holes with a = 350 nm and r = 88 to 125 nm [1] and with W × H = 500 nm × 220 nm, assuming that Lm = 2 μm and that Va = V1 = + 0.53V and later, that Va = V2 = −2.16 V. The PN region was 285 nm wide 5 × 1017 cm−3 P-Si and 215 nm wide 1 × 1018 cm−3 N-Si. We find that V2 - V1 = ΔV = 2.7V produces a Δλ res of 400 pm. A reflection monitor was put into our simulations to find R as well as T. The key metrics of 2 × 2 switch performance are the insertion loss IL and the optical crosstalk CT, where we consider the input-to-through cross state of the switch at Va = V1 and the input-to-drop bar state at Va = V2, specifically, IL(dB) cross = 10 log T(V1), CT(dB) cross = 10 log R(V1), IL(dB) bar = 10 log R(V2), and CT(dB) bar = 10 log T(V2). The simulation results are presented in Fig. 4 where, taking λin = λres(V1) = 1518.1 nm, we find IL cross = 2 dB, CT cross = −16 dB, IL bar = 0.5 dB, and CT bar = −16 dB with an information bandwidth B(V1) = c/Qλres(V1) = 230 GHz. When Va = V2 the information bandwidth is wider than that at V1 because the spectral passband at λres in Fig. 4 widens at V2.
The Fig. 2 device is an elemental 2 × 2 “building block” with which to construct an interconnected multi-stage cascade of 2 × 2s in a “higher order” switch. The spatial routing diagram of this 2 × 2 is presented in Fig. 5 for monochromatic light at λin = λres. As mentioned, the cross and bar states in Fig. 4 at λin = 1518.1 nm appear to be practical for the cascading of 2 × 2s that we shall now show in generic N × N block diagrams.
Fig. 5 Functional diagram of the wavelength-selective switch in Figs. 2 and 3 in its cross and its bar states.
4. N x N optical data routing
Arguably, the most important N × N is the path-independent-loss PILOSS switch [14,15] of Fig. 6, where we label the elemental four ports per Fig. 5. Another practical layout is the Benes matrix [16] of Fig. 7. Referring now to the Fig. 2 device that is working as the cross-bar of Fig. 5, we shall now show in Fig. 8 the specific embodiment of a DNB matrix switch which is the 4 x 4 version of the 8 x 8 device depicted in Fig. 6. In addition to the Fig. 8 switching network, we could easily configure the DNB 2 × 2 interconnections as per Fig. 7.
Fig. 6 Waveguide-connected architecture of the 8 × 8 strictly nonblocking path-independent loss network switch.
Fig. 8 Layout of DNB PILOSS 4 × 4 router with the electro-optical control leads for each PN dual-NB indicated as a wire pair fanned out to the side.
5. Wavelength-selective switching and optical computation
Now we propose a group of DNB applications for wavelength-division multiplexing and computing. We begin by noting that each NB reflects fully the incident light whose wavelength is more than 1 nm away from the λres at V1. That is why a group of “off resonance” wavelengths at the input port will be sent “without loss” to the drop port. To be specific, let us assume that we want to operate upon four incoming wavelengths, and that we shall devote one 2 × 2 device to each wavelength with the first switch designed for λres = λ1, the second for λres = λ2, etc. The four resulting switches are represented in the spectrum diagram of Fig. 9. Here, the intersections of the vertical color lines with the switching-response curves show the high-or-low V1-multi-spectral output powers for each switch device. To illustrate the general wavelength-multiplexed switching of a 2 × 2, we shall assume that the device is resonant at the wavelength of λ2 for V1 and we then show in Fig. 10 the resulting color routing for the six input wavelengths. The multiple color switching and color transferring in Fig. 10 becomes the basis of several DNB applications:
Fig. 9 An array of four 2 × 2 DNBs with their V1 resonances spaced 3 nm apart. The collective switching response to four color inputs is shown.
Fig. 10 Wavelength routing properties of the λ2-resonant 2 × 2 DNB (in its cross and bar states) for four wavelengths at the input port, plus two different wavelengths put in at the add port.
First is the N × N wavelength-selective switch (WSS) of Fig. 11 for N = 3 which provides selected combinations of colors at each of the three column outputs when the matrix input is a wavelength-multiplexed light stream that travels, for example, in an optical fiber. This is an unusual kind of WSS that is related to the optical vector matrix multiplier (OVMM) discussed below. In Fig. 11, the ability to transmit N colors to any output port is traded off against an optical power division (insertion loss) at the input. To achieve such multi-wavelength outputs, it is necessary to send all N wavelengths into each row of the matrix which is done by introducing a 1/N optical power divider after the input fiber as shown in Fig. 11. For electrical addressing of the WSS, all N2 switches are initially in the V2 bar state as per Fig. 10. Then, the matrix is programmed by actuating a set of V1 cross-state choices for the N switches in each column of the matrix. The desired routing of wavelength signals requires an optical summation within a waveguide path that travels vertically within a column. To attain an additive column path we use SOI channel waveguides as interconnects and begin with the light exiting the through port of the bottom-row switch. That port is connected to the add port of the switch above it. Then, the through port of that switch is connected to the add port of the switch above it and, finally, the through port of the top switch is connected to a matrix output. In this way, a summation of different-wavelength signals traveling in one waveguide is attained. Figure 11 presents an example of color- combined outputs.
Fig. 11 Wavelength-selective switching by an array of interconnected 2 × 2 DNBs that routes three input colors in any chosen combination to each of three ports.
The second application is the OVMM in which a vector B1…N represents an intensity of a color at λ1, an intensity at λ2,… and an intensity at λN. Then, there is a weighting matrix A which, when multiplied with B, gives the desired optical computation, the optical output vector C shown here for the three color case:
Now we can use exactly the architecture of Fig. 11 to enable the OVMM as illustrated schematically in Fig. 12. Here, at the output waveguides, we require three spectrally broadband photodiodes whose electrical output signal is proportional to the summation of the three (incoherent) optical signals incident upon it. The i-th vertical element Ci of the output C-vector is created by the optical addition (described above) taking place in the i-th column of the switch array.
Fig. 12 Vector matrix multiplication by an array of waveguide-connected 2 × 2 DNBs. The matrix weighting A of three electrically modulated color-coded inputs (electric vector input B) results in an output electric vector C as illustrated.
Each box in Fig. 12 is an elemental switch labeled with its particular λres at V1. We see that this wavelength sequence is different in each column of the matrix. To explain the operation of the OVMM, let us look at the representative vector-element C2 = A21Sa + A22Sb + A23Sc, where Sa, Sb, and Sc are different-wavelength optical signals circulating in the matrix. We find that C2 is the same for any color sequence of signals because the broad photodetector responds the same to all colors and because we deliberately make all signals S have the same amplitude. The operation of the Fig. 12 OVMM is somewhat different that in Fig. 11 because in Fig. 12 each elemental EO switch operates within the “grey zone” between the bar state and the cross state. Specifically Aij(V1) = 1 and Aij(V2) = 0. For a voltage V somewhere between V1 and V2, Aij(V) is the fraction of optical power transmitted by the switch element. For programming the OVMM, it would be helpful if Aij(V) were a linear function of V, but this A vs V dependence is actually “somewhat nonlinear.”
As a final matrix application, it is illuminating to compare the wavelength-multiplexed N × N crossbar shown schematically in Fig. 13 with the Fig. 11 matrix. Figure 13 differs from Fig. 11 because input power division is avoided and is replaced by a “lossless” passive WDDM, yielding a distinct wavelength input to each row of the crossbar. The crossbar gives a nonblocking 1-to-1 mapping of inputs to outputs as illustrated by the red, green, and blue example. For electrical addressing, all N2 elements (Fig. 2) are at first in the V2 bar state to send light along each row. Then, N elements (one per row) are actuated into the V1 cross state. The lightwave in a given row is routed to any of N column outputs using only one cross-state 2 × 2 switch; therefore, the insertion loss of the matrix is that of only one 2 × 2 as shown in Fig. 13.
6. Electro optical Logic
Xu and Soref [17] made an extensive investigation of EO logic and invented an EO logic unit cell presented in Fig. 5 of [17]. These authors showed that essentially all of the possible (complicated) logic functions could be obtained by chaining a sequence of different unit cells. The meaning of EO logic is that the logic signals are fed into the cell from the electrical domain; those signals are then processed in the optical domain, and the computational “answers” are read out in the electrical domain. The unit cell is made from two stages in cascade, each of which has a cross-bar 2 × 2 switch and a passive signal-pass 1 × 1 NB. The construction of that cell was illustrated as being constituted in MRRs [17,18]. By contrast, we find here that it is feasible to realize the unit cell completely in NBs using the 2 × 2 EO DNBs and the passive 1 × 1 NBs as illustrated in Fig. 14 for an all-nanobeam logic. For “dynamic use” in universal logic functions, it is necessary to be able to reconfigure each 2 × 2 and each 1 × 1 using reconfiguration voltages. To accomplish reconfiguration, tiny thermo-optic (TO) heaters could be added within each nanobeam. However, it is likely that the DC holding energy of the TO strips would violate the attojoules guideline of the present work and, therefore, we have decided not to employ TO reconfiguration in Fig. 14. A useful alternative is to replace dynamic reconfiguration by permanent configurations that offer a dedicated logic function from a fixed logic switch fabric. Let us consider the multi-spectral logic in Fig. 8 of [17] where a laser wavelength λi is sent into the i-th row of the logic switch fabric comprised of unit cells. Now we have design rules for configuring the cells. The configuration of each 2 × 2 and of each 1 × 1 in Fig. 14 is obtained by design of the 1D PhC lattice in each DNB or NB in order to yield a desired λres. To be specific, the 2 × 2 cross-bar configuration is attained by making λres(i) = λi, whereas the bar-cross configuration comes from λres(i) = λi – 0.4 nm, and the bar-bar configuration stems from λres(i) = λi + 0.4 nm. In the 1 × 1, the pass-signal configuration is λres(i) = λi while the block-signal comes from λres(i) = λi + 0.4 nm. Our purpose in presenting Fig. 14 is to suggest that this is an attojoules per cell embodiment of EO logic.
Fig. 14 SOI waveguide layout of the electro-optical logic unit cell for computing universal logic functions. The cell is constructed entirely of 2 × 2 DNBs and 1 × 1 passive NB pass or block filter.
Here is a final note on wavelength routing. The devices of Figs. 2 and 3 are voltage-reconfigurable add-drop multiplexers, ROADMs. In addition, those devices can serve in the role of voltage-controlled wavelength-multicasting devices such as the multicasters depicted in Fig. 2(b) of Su et al. [19] where the output power was divided between the two output ports. Here, Va triggers the partial add-drop and, as discussed above for the OVMM, Va is about halfway into the grey-scale range between the cross and bar voltages.
7. Optimization of 2 x 2 metrics
The main optimization issue is to reduce IL(cross) to 1 dB or less. The secondary issue is to reduce CT(cross) and CT(bar) to below −20 dB. The IL(bar) is already 0.5 dB. An effective way to reduce both CT values is to increase the Q of the NB at V1 without reducing the transmission T(V1). We believe this can be done by increasing the number of air holes in the 1D lattice [20,21] from 24 to 30 or 36. The reduction of IL(cross) requires simultaneous adjustment of many NB parameters. We would make the following modifications and “tunings” in the IL(cross) optimization: (1) change Lm into the 1.0 to 2.0 um range so that the depleted Si pattern overlaps less of the “anti-nodes” and more of the ”nodes” in the mode-pattern, (2) center the 40-nm depletion zone in the waveguide center when V1 = + 0.53 by adjusting the position of the vertical junction, (3) adjust the rib height, (4) adjust W and H, and (5) adjust P and N slightly.
8. Switching characteristics of non-ideal 2 x 2 devices
Dimensional errors invariably arise during the process of constructing the 2 × 2 switch. Those fabrication errors can be kept very small by manufacturing the switch in a modern 45-nm SOI factory. Nevertheless, deviations from the ideal structure will always be present. The purpose of this section is to quantify the penalties in 2 × 2 performance that result from fabrication errors. In particular, we have investigated performance reductions from three types of errors: (1) deviation of 3-dB directional couplers from a perfect 50/50 split, (2) an inequality ΔL in the optical path length of the connecting waveguide “arms” in the MZI structure, and (3) a difference δλres in the fundamental resonance-mode wavelength between the two nanobeams. To measure the performance penalties, we have examined the increase in IL and the increase in CT produced by each type of fabrication error, considering both the cross and bar states of the device.
The complete layout of the MZI structure was taken into account in analytical calculations of non-ideal 2 × 2 device structures. For each NB, the curves in Fig. 4 were fitted with a Lorentzian to give equations for the NB transmission and reflection, first at V1, and then at V2 (cross and bar). Next, transfer matrices were set up to describe the full MZI response, following the general outline in [22]. At each port of the switch there is either forward-traveling transmitted or backward-traveling reflected light. Separate matrix equations were set up for the transmitted and reflected components and for both the cross and bar states. These four matrix equations allowed us to solve for all eight light-components as a function of wavelength, coupler ratio, and optical path length. The non-ideal switching results are presented in Figs. 15-17. A general finding in Figs. 15-17 is that a reflected component of light, called “crosstalk” here, appears at the input port and at the add port when the switch parameters deviate from ideal (such reflections are zero for the ideal). For Figs. 15-17, the IL has been plotted along with the three CT results - the four-port behavior. In Fig. 15, the input coupler was fixed at 3-dB while the coupling ratio of the output coupler was varied from 2.7 to 3.3-dB. For Fig. 16, the optical path length difference in the MZI arms, ΔL, gives rise to an optical phase difference Δϕ between the two arms, where Δϕ = 2πnΔL/λres. Thus in Fig. 16, the IL and CT are plotted versus Δϕ in the range from zero to π radians. Finally, in Fig. 17 the abscissa of the two graphs is δλres expressed in picometers. In this situation, the resonant wavelength in one arm of the MZI is held constant while the resonant wavelength in the other arm is shifted through a range of ± 200pm.
Fig. 15 Insertion loss and crosstalk at three ports of the non-ideal 2 × 2 as a function of non-ideal coupling ratio in the second hybrid coupler. The first hybrid coupler is fixed at 3-dB.
Fig. 16 Insertion loss and crosstalk at three ports of the non-ideal 2 × 2 as a function of the optical phase difference between the two waveguided arms of the MZI device.
Fig. 17 Insertion loss and crosstalk at three ports of the non-ideal 2 × 2 as a function of the wavelength difference between the fundamental resonance-wavelength of each nanobeam.
We see in Figs. 15-17 the specific sensitivity of the switching characteristics to the errors in fabrication, and we interpret these results as follows. Figure 15 is a favorable result for cross and bar. There, the IL and the main CT are not sensitive to a change in the coupler splitting ratio, although the weaker CTs have some sensitivity to ratio. Although the IL and main CT in Fig. 16 are not sensitive to Δϕ in cross and bar, there is clearly a problem with the weaker CTs because they increase quite rapidly with Δϕ in both states. An inference drawn from Fig. 16 is that Δϕ should be held to only 5 or 10 degrees in order to achieve acceptably low levels of reflected light at the add port (cross state) and input port (bar state). Figure 17 is a favorable result because, in the bar state, a difference δλres of ± 50 pm between the NB resonances does not degrade the IL and CTs. However, there is an issue in the cross state where the IL does increase slowly with δλres as do all three CTs. A useful strategy is to require a δλres of ± 20 pm, or less, for a high performance cross state.
9. Conclusion
The switches proposed and analyzed in this paper are intended to fulfill an unsatisfied need in the emerging field of “attojoules photonics.” Attojoules photonics (AJP) describes an on-chip “system”, a network in which each electrically actuated waveguided photonic component consumes less than 1 fJ/bit. This AJP system, in our opinion, could be an everything-in-nanobeams circuit. We have presented theoretical simulations of a fast, compact, resonant 2 × 2 EO switch that uses identical NBs in the two connecting arms of an SOI Mach-Zehnder interferometer. A localized PN junction in each NB of length 2 μm along the waveguide axis is proposed as the lowest-energy switching approach. FDTD simulations project a switching energy of 474 aJ/bit for complete cross-bar switching at λin = λres = 1518.1 nm where the TEo mode resonance is at 0.53V forward bias, the loaded Q was 8600, and the data bandwidth was 230 GHz (a spectral bandwidth of 177 pm). Swinging to 2.16 V reverse bias (ΔV = 2.7V), widened the PN carrier depletion zone to 110 nm, thereby shifting the initial resonance by 400 pm. Our numerical simulations indicate 0.5 to 2.0 dB IL with −16 dB CT during cross-bar switching. This 2 x 2 is a wavelength-selective voltage-reconfigurable add-drop multiplexer. Using schematic drawings we show how the dual-NBs apply to Benes and PILOSS N × N switches, EO logic, VMMs, and WSSs.
Acknowledgments
RS is grateful for the support of the Air Force Office of Scientific Research (AFOSR) on grant FA9550-14-1-0196 and of the UK EPSRC on Project Migration. JH also acknowledges the support of the Air Force Office of Scientific Research (AFOSR) (PM: Gernot Pomrenke) under LRIR 15RYCOR159.
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