Realization of true all-optical AND logic gate based on nonlinear coupled air-hole type photonic crystal waveguides

In this manuscript we propose an easily scalable true all-optical AND logic gate for pulsed signal operation based on band-gap transmission within nonlinear realistic air-hole type coupled photonic crystal waveguides (C-PCW). We call it "true" all-optical AND logic gate, because all AND gate topologies operate with temporal solitons that maintain a stable pulse envelope during the optical signal processing along the di erent C-PCW modules yielding ultrafast full-optical digital signal processing. We directly use the registered (output) signal pulse as new input signal between multiple concatenated nonlinear C-PCW modules (i.e. AND gates) to setup a multiple-input true all-optical AND logic gate. Extensive full-wave computational electromagnetic analysis proves the correctness of our theoretical studies and the proposed operation principle of the multiple-input AND logic gate is vividly demonstrated for realistic C-PCWs. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (230.7370) Waveguides; (230.5298) Photonic crystals; (290.5825) Scattering theory; (260.2110) Electromagnetic optics. References and links 1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987). 2. K. Yasumoto ed., Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2005). 3. J.-M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, "High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide," Opt. Express 16, 4177–4191 (2008). 4. Y. Ishizaka, Y. Kawaguchi, K. Saitoh and M. 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Introduction
Photonic crystals (PhCs) have inspired a lot of interest due to their wide application for controlling light within small length scales [1,2].Due to their dispersion engineering features PhCs have enabled the implementation of linear functionalities into ultra-compact photonic devices.However, for the realization of true all-optical signal processing, the optical system needs to have nonlinear properties.Nonlinear PhCs are one of the most promising candidates to enable dense full-optical signal processing on the same chip leading to much lower production and operating costs even if they have to compete with ultra-compact state-of-the art optoelectronics counterparts such as e.g high-speed PhC electro-optic modulator schemes [3].When a linear defect is introduced into the PhC, a photonic crystal waveguide (PCW) is formed and the electromagnetic wave, whose frequency lies in the band-gap region can be guided along the defect channel.If two or three PCWs are placed in a close proximity, a coupled PCW (C-PCW) is formed and the optical power is e ciently transferred from one PCW to another [2].
All-optical logic gates, which can fulfill various logical functions operations and capable of performing complex light control in all-optical circuits, have received much attention for their potential application in ultrafast information processing and optical computing systems.Various PhC structures have already been proposed to realize all-optical logic gates [4][5][6][7][8][9][10].The most PhC-based logic gates rely either on resonator cavities with finite filling time or branching structures with a very confined interaction volume that is both highly sensitive to production variations and in need of pre-defines input signal phases.In this regard, the proposed formulation is advantageous due to the travelling-wave nature of the band-gap transmission, namely the resulting nonlinear traveling-wave interaction.Moreover, it is very important to mention that PhC with a triangular lattice made of air holes in a background nonlinear Kerr-type medium with a linear refractive index n 0 = 2.95, where h is the period of the PhC and w = p 3h is the width of each of the coupled W1 defect waveguides.The holes are parallel to the z-axis.Number of layers of the upper and lower PhCs is 8.The quantity (3) stands for the third-order nonlinear optical susceptibility of the nonlinear background medium.The length of the photonic crystals is 30h.most of the PhC logic gates are operated with continuous wave (CW) signals, whereas this work demonstrates a realistic pulse operation that is very apt for digital signal processing.Solitons maintain a stable pulse envelope along the di erent C-PCW modules yielding ultrafast full-optical digital signal processing that is characterized only by the peculiarities of the propagation of nonlinear traveling-waves within each gate.
In this manuscript we propose, actually for the first time, the realization of a true all-optical AND logic gate based on the phenomenon of band-gap transmission [11,12], that could o er a promising scheme for ultra-high throughput optical digital signal processing in the framework of e.g.ultra-fast parity bit checking [13].A realistic PCW structure, such as the planar air-hole type PhC with a nonlinear silicon background material is studied.Planar silicon PhCs are promising candidates for the successful realization of various nonlinear optical devices on a chip [14][15][16].The e ect of band-gap transmission takes place when the frequency of the injected signal is very close to the band edge of the PhC.In the linear case no transmission is supposed to occur, whereas in the nonlinear case above some threshold value of the signal amplitude, the propagation of solitons takes place.Such a bifurcation could be observed in both spatial and temporal domains [17].PhC guarantee a good confinement of the light in the slow light regime, which is a necessary condition for the enhancement of the nonlinear e ects [18].The idea of band-gap transmission was already successfully applied within earlier conceptual studies where we proposed an amplification e ect of the signal in both, subwavelength metallic waveguides [19] and coupled PhC waveguides (C-PCWs) [20,21].

Formulation of the problem
The realization of the true all-optical AND logic gate is analyzed based on the three adjacent C-PCWs each consisting of a single line defect (i.e. a W1 waveguide) within a planar PhC with a triangular lattice.The cross-sectional view of the structure is shown in Fig. 1.The triangular lattice of air holes is formed in a 2-D nonlinear background dielectric medium where its linear (e ective) refractive index is set to n 0 = 2.95 in conjunction with a Kerr-type nonlinearity.
Here we aim at a typical semiconductor material system (e.g.Si, GaAs, InP) with a relatively high refractive index yielding complete photonic bandgap.In particular in silicon photonics the emergent Kerr-type nonlinearities in bulk c-Si are weak and accompanied by strong two-photon absorption (TPA) associated to high free carrier induced loss [22].For practical application it would be better to use modified material systems with enhanced third order nonlinearities (Kerr-type) based on e.g.amorphous Si [23], hybrid organic-Si-on-insulator compounds [22,23], or as possible alternative Ag-polymer based low refractive index PhC slabs with prominent nonlinear optical properties [24].
A number of layers of the upper and lower PhCs is 8 with a lattice constant h and a radius of the air-holes of r = 0.32h.The width of each W1 waveguide forming the PCW is w = p 3h and the length of the PhC amounts to 30h.This 2-D model has already proven to be a good approximation of a realistic planar 3-D PCW structure [25,26], thus enabling a substantial reduction in computation time and what is more important, allows us to reliably demonstrate the realization of true all-optical logic gates.Under the adjusted parameters, the air hole PhC yields a photonic band-gap for the H-polarized field H z , E x , E y covering the frequency range of 0.230 < !h/2⇡c < 0.310, where c is velocity of light and ! is the angular frequency of the excitation.The dispersion diagrams of the three C-PCWs (Fig. 1) are studied based on our original fast rigorous and accurate analysis taking into account the Lattice Sums technique [27] combined with the transition matrix (T-matrix) approach and a recursive algorithm to calculate the generalized reflection and transmission matrices for the PhC [28,29].The propagation constant h/2⇡ of three C-PCWs (cf.Fig. 1) for both the symmetric ( S1 and S2 ) and antisymmetric A modes versus the normalized frequency is calculated and plotted in Fig. 2 by blue, green and red lines, respectively.The schematic distributions of the magnetic field for the symmetric and antisymmetric modes are presented within the corresponding insets.The dispersion diagrams are plotted only in the normalized frequency range 0.230 < !h/2⇡c < 0.270, since only this region is eligible for the realization of all-optical logic gates.Note that within this region there exist only two symmetric modes and one antisymmetric mode, which are formed by splitting the fundamental mode of the single W1 PCW due to coupling with the adjacent PCWs.
The underlying principle for the realization of the true all-optical logic AND gate -please note that a similar analysis can be applied to realization of OR all-optical logic gate -in the nonlinear C-PCW is as follows: first, we properly choose the operating frequency !located at the edge of the dispersion curve where no propagating modes are excited (cf.red circle in Fig. 2).At this frequency a continuous signal with amplitude 2A is launched into Port 2. The input signal excites the combination of both symmetric modes A ( S1 + S2 ).Note that in the linear regime (i.e.small amplitudes of the CW signal) no waves will propagate inside the bulk PhC.Increasing the amplitude of the signal, its frequency acquires a nonlinear shift.Particularly, the dispersion diagram is shifted to the lower frequencies and crosses the black horizontal line (see Fig. 2).As a result, a band-gap transmission of one of the symmetric modes (blue line) occurs and it starts to propagate in the nonlinear C-PCWs.Other two modes are not propagating.The threshold amplitude could be calculated from the relation [12,21]: where = 3! (3) /4, the quantity (3) stands for the third-order nonlinear optical susceptibility and includes an enhancement e ect due to the pulse slowdown in the coupled PCWs [18].Its enhancement is accounted for by multiplying (3) with the square of the slowdown factor -in our case slowdown factor is equal to 16.The coe cient  takes into account the weakening of the nonlinearity due to the existence of air holes in the planar background material [21].If we properly choose the amplitude of the CW signal to be below the threshold amplitude A th for band-gap transmission, no waves will propagate inside the nonlinear C-PCWs.Next, a Gaussian signal pulse (i.e. a carrier signal with Gaussian pulse envelope) with amplitude injected into the upper and/or the lower waveguide of the three C-PCWs can now trigger the band-gap transmission process.
Formally, this results in the excitation of a combination of modes according to p 2 that enhances the amplitude of the symmetric mode S1 reaching the condition that allows for band-gap transmission whereas the other modes remain evanescent.Hence, the input pulse will be carried by the only propagating symmetric mode S1 that has now formed a propagating soliton inside the C-PCW structure with the shape given as: Here F is a soliton amplitude, ⇤ is a soliton width, !denotes the frequency shift of the localized modes due to the nonlinearity of the background material and v S1 denotes the group velocity for symmetric mode S1 .It is worth noting that knowing the excitation frequency ! and the amplitude F of the soliton, we can define the propagation constant using the relation , whereas the other parameters, such as the soliton width ⇤, the group velocity v S1 and the frequency shift !are given as follows: Key to the realization of true all-optical logic gates is perfect "digitalization" in the process of band-gap transmission.This means that, when operating frequency of the CW signal launched in Port 2 is located su ciently close to the photonic band edge, the weak signals injected into Port 1 and/or Port 3 are becoming capable to overcome the amplitude threshold and a soliton can be generated with amplitude and shape that is completely defined by the CW signal injected into Port 2. In other words, the weak input pulse just triggers the creation of a stable soliton with predefined amplitude yielding a complete regeneration of the input signal pulse prior (and during) the proposed optical digital signal processing [21,30,31].Note that within the context of the aforementioned "digitalization" Port 2 can be viewed as an "enable pin" of the proposed logic gate where the CW signal will act as the "enable signal" common in digital electronics.

Numerical results and discussions
For a reliable proof of concept comprehensive full-wave computational electromagnetics analyses are conducted based on the finite-di erence time-domain (FDTD) method using the Berenger's PML to confine the simulation domain [14,32] at the fixed normalized frequency !h/2⇡c = 0.232 marked by the red dot in Fig. 2. It should be emphasized that the frequency is located at the edge of the dispersion curve of the symmetric mode (blue line).
is the electric field amplitude of the CW signal launched into Port 2 of C-PCW structure.Figure 3(d) depicts the distributions of the propagating soliton inside the nonlinear C-PCWs.The output signal is registered at a distance x = 30h at the end of the C-PCWs where the magnetic field H z is depicted in Fig. 3(e).Since the amplitude of the CW is below the threshold, no soliton is generated and the amplitude of the output signal is almost zero.Next, we additionally inject a Gaussian pulse into Port 1 at the same normalized frequency !h/2⇡c = 0.232 with an amplitude = 0.72 as it is shown in Fig. 4(a), whereas no signal is injected in Port 3. The full width at half maximum pulse duration is 800h/c corresponding to values around 1.0-1.2ps for typical losses have been investigated using full-wave computational electromagnetic simulations as well as corresponding reflection/transmission measurements in [33].It has been shown that a backscattering is virtually negligible in the bandgap region.At the band edges, there exists backscattering loss, which exponentially decreases towards the bandgap region.

Conclusion
In conclusion it is worth emphasizing that based on the phenomenon of band-gap transmission we have proposed a functional true AND all-optical logic gate (1) using a realistic model of a coupled Kerr-type nonlinear air-hole type C-PCWs, and (2) that is operating with realistic optical signal pulses.Our numerical analyses have shown that even for the most challenging case, such as cascaded systems yielding all-optical multiple-input logic gates, the shape of the output signal (namely its amplitudes and associated widths) is in a very close agreement with those for the original input signal.Each C-PCW module underlying the logic gate is yet capable to provide all-optical regeneration (2R: reamplification and re-shaping) for signal pulses above the aforementioned threshold for soliton propagation.It is important to mention that our analysis can be also applied to realize the true all-optical multiple-input OR gate in a similar a cascaded topology.However, some modifications are needed, because the time for soliton formation when injecting the Gaussian pulses either into Port 1 or Port 3 is slightly di erent compared to the case where the Gaussian pulses are simultaneously injected into Port 1 and Port 3. Hence, proper timing issues are now under intense investigation.

Fig. 1 .
Fig.1.Schematic view of the three symmetric nonlinear C-PCWs embedded into a planar PhC with a triangular lattice made of air holes in a background nonlinear Kerr-type medium with a linear refractive index n 0 = 2.95, where h is the period of the PhC and w = p 3h is the width of each of the coupled W1 defect waveguides.The holes are parallel to the z-axis.Number of layers of the upper and lower PhCs is 8.The quantity(3) stands for the third-order nonlinear optical susceptibility of the nonlinear background medium.The length of the photonic crystals is 30h.

Fig. 2 .
Fig. 2. Dispersion curves of the symmetric (blue and green lines) and the antisymmetric (red line) (super-) modes for the H-polarized field (H z , E x , E y ) of the three C-PCWs as shown in Fig. 1.The normalized operation frequency is !h/2⇡c = 0.232 marked by the red dot.The distributions of the transversal magnetic field H z for two symmetric (blue and green lines) and one antisymmetric (red line) modes are shown in the corresponding insets.

Fig. 3 .
Fig. 3. Schematic distributions (H z ) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW input signal with the amplitude = 0.78 is launched through Port 2 and no input signals = 0 are injected into Port 1 (a) and Port 3 (c).Magnetic field H z of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (e).

Figures 3 -
5 illustrate the subsequent realization of an operational all-optical AND gate.First, a CW signal, whose transversal electric field is in plane to the planar PhC, with the normalized operation frequency !h/2⇡c = 0.232 and an amplitude of = 0.78 is launched into Port 2 of C-PCW structure [cf.Fig.3(b)], whereas no signal = 0 is launched into Port 1 [Fig.3(a)]and Port 3 [Fig.3(c)].The peak power of the injected signal pulses is chosen as (3) E 2 0 = 0.08, where E 0