Realization of True Logic Gates using Coupled Photonic Crystal Waveguides based on Bandgap

Transmission Phenomenon

Vakhtang Jandieri1, Ramaz Khomeriki2, Daniel Erni1

1 General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, Germany, vakhtang.jandieri@uni-due.de, daniel.erni@uni-due.de 2 Department of Physics, Tbilisi State University, 3 Chavchavadze ave., 0128 Tbilisi, Georgia, khomeriki@hotmail.com

Abstract — In this manuscript we propose an idea to realize easily scalable true logic gates for pulsed signal operation based on a bandgap transmission phenomenon within Kerr-type nonlinear realistic air-hole coupled photonic crystal waveguides (C-PCW). We call it “true” logic gate, because all gate topologies operate with pulsed signals (solitons) that maintain a stable pulse envelope during the signal processing along the different C-PCW modules yielding ultrafast digital signal processing. In the proposed gate topology a realistic pulse operation is used, which is very apt for digital signal processing. In this work based on the proposed idea we demonstrate only the realization of AND logic gate. Extensive full-wave computational electromagnetic analysis proofs the correctness of the theoretical studies.

Index Terms — Periodic structure, propagation, scattering, photonic crystals, digital signal processing.

I. INTRODUCTION

Photonic crystals (PhCs) are periodically modulated dielectric or metallic structures in which any electromagnetic wave propagation is forbidden within a fairly large frequency range (i.e. the photonic band-gap) [1, 2]. In nonlinear media, sufficiently high light intensities modify the properties of materials while enabling true interaction of light waves. Planar silicon Kerr-type nonlinear PhCs are one of the most promising candidates to enable signal processing on the same chip leading to much lower production and operating costs. When a linear defect is introduced into the PhC, a photonic crystal waveguide (PCW) is formed. If two or three PCWs are placed in a close proximity, a coupled PCW (C-PCW) is formed and the optical power is efficiently transferred from one PCW to another [1]. PhCs-based logic gates, which can fulfill various logical functions operations and capable of performing complex signal controlling in the circuits, have received much attention for their potential application in ultrafast information processing [3, 4]. The most of the PhC logic gates are operated with continuous wave (CW) signals, whereas this work demonstrates a realistic pulse operation (solitons) that is very apt for digital signal processing. In this paper we propose, actually for the first time, the idea how to realize true logic gate based on the phenomenon of band-gap transmission [5, 6]. In this work we are focusing only on the realization of AND true logic gate, however, the idea can be applied to the realization of

OR logic gate also. A realistic PCW structure, such as the planar air-hole type PhC with a silicon Kerr-type background material is studied. The latter is usually used in experiments. The effect 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. Key to the realization of true logic gates is a perfect "digitalization" in the process of band-gap transmission. 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 digital signal processing. The idea of band-gap transmission we have already successfully applied within conceptual studies where we proposed an amplification effect of the signal in both, subwavelength metallic waveguides [7] and coupled PhC waveguides (C-PCWs) [8, 9].

II. NUMERICAL RESULTS AND DISCUSSIONS

The realization of the true 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 as inset in Fig. 1. The triangular lattice of air holes is formed in a 2-D nonlinear background dielectric medium where its linear (effective) refractive index is n0 =2.95 in conjunction with Kerr-type nonlinearity. The width of the two layers that confine the PCW is N=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 3w h= 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 [10, 11]. Under the adjusted parameters, the air hole PhC yields a photonic bandgap for the H-polarized field (Hz,

Ex, Ey) covering the frequency range of 0.230 < 2/h cω π

0.310< , where c is velocity of light and ω 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 based on the Lattice Sums technique [12, 13]. The propagation constant / 2hβ π of three

EuCAP 2018 1570409156

1

C-PCWs (cf. Fig. 1) for both the symmetric and antisymmetric

modes versus the normalized frequency 2/h cω π is calculated and plotted in Fig. 1 by blue, green and red lines, respectively.

The underlying principle for the realization of the true AND logic gate – please note that a similar analysis can be applied to realizarion of OR all-optical logic gate – in the nonlinear C-PCW is as follows: first, we properly choose the

operating frequency 2 0.232/h cω π = located at the edge of the dispersion curve, where no propagating modes are excited (cf. red circle in Fig. 1). The frequency is located at the edge of the dispersion curve of the symmetric mode (blue line). At this frequency a CW with an amplitude of δ =0.78 is launched into Port 2, whereas no signal δ =0 is launched into Port 1 and Port 3. The peak power of the injected signal pulses is chosen

as (3) 2

0 0.08Eχ = , where0E is the electric field amplitude of

the CW signal launched into Port 2 and(3)χ stands for the

third-order nonlinear optical susceptibility. Note that in the linear regime [i.e. small amplitudes of the CW signal] no waves will propagate inside the bulk PhC, as far as the excitation frequency is within the range of the photonic band-gap. Next, we additionally inject a Gaussian pulse into Port 1

at the same normalized frequency 2 0.232/h cω π = with an amplitude δ =0.72 as it is shown in Fig. 2 (a), whereas no signal is injected in Port 3 [Fig. 2 (c)]. An increased input power compared causes the dispersion to shift to lower frequencies because of the underlying Kerr-type nonlinearity,

however it cannot overcome the threshold value to reach an operation point that allows for a propagating soliton inside the C-PCW. Hence, the registered output signal at the end of theC-PCW is almost zero as displayed in Fig. 2 (d) (schematicdistributions of propagating soliton) and Fig. 2 (e) (amplitudeof its magnetic field). However, when we simultaneously injectan additional signal pulse into Port 3 (which is in phase to thesignal injected in Port 1), a soliton is generated and thecharacteristics of the output signal pulse at Port 2 (namely itsamplitude and width) are very close to those of the input signalpulse showing a relative error about 7% as shown in Fig. 3.

In conclusion it is very important to note that even for the most challenging case, such as cascaded systems yielding multiple-input logic gates, the shape of the output signals (namely its amplitudes and associated widths) are in a very close agreement with those for the original input signals. Hence, the realization of a true AND logic gate has been vividly demonstrated within a realistic pulse signal operation scenario apt for future digital circuits.

Fig. 1. Dispersion curves of the symmetric (blue lines) and the antisymmetric (red line) (super-) modes for the H-polarized field (Hz, Ex,

Ey) of the three C-PCWs, where 0.32r h= , n0 = 2.95 and 3w h= .

The normalized operation frequency is 2/h cω π 0.232= marked by

the red dot. The distributions of the transversal magnetic field Hz for two symmetric (blue and green lines) and one antisymmetric (red line) modes are shown in the corresponding insets. 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 is shown as inset.

Fig. 2. Schematic distributions (Hz) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW signal with the amplitude δ=0.78 is launched through Port 2 (b), and a Gaussian pulse with amplitude δ=0.72 is injected into Port 1(a), whereas no single is injected into Port 3 (c). Amplitude of the magnetic field Hz of the received signal at a distance z=30h inside the C-PCW associated to Port 2 (e).

Fig. 3. Schematic distributions (Hz) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW signal with the amplitude δ=0.78 is launched through Port 2 and Gaussian pulses with an amplitude δ =0.72 are injected into Port 1 and Port 3. Amplitude of the magnetic field Hz of the received signal at a distance z=30h inside the C-PCW associated to Port 2 (e).

2

REFERENCES

[1] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett., vol.58, pp.2059-2062, 1987.

[2] K. Yasumoto ed., Electromagnetic Theory and Applications for Photonic Crystals. Boca Raton, FL: CRC Press, 2005.

[3] P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,“ Journal of Optical Society of America B, vol. 26, pp.10-16, 2009.

[4] C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,“ Optics Express, vol. 19, no. 21, pp. 20681-20690, 2011.

[5] F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett., vol.89, pp.134102-134105, 2002.

[6] R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett., vol. 92, pp. 063905-063908, 2004.

[7] R. Khomeriki, J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A, vol.87, 053806-053809, 2013.

[8] V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters, vol. 27, pp. 639-641, 2015.

[9] V. Jandieri, R. Khomeriki, D. Erni and W. C. Chew, “Realization of

All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research, vol.158, pp.63-72, 2017.

[10] P. Strasser, R. Flückiger, R. Wüest, F. Robin and H. Jäckel, “InP-based compact photonic crystal directional coupler with large operation range,” Optics Express, vol. 15, no. 13, pp. 8472-8478, 2007.

[11] Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics, vol. 41, no. 1, pp. 76-84, 2005.

[12] K. Yasumoto, H. Toyama and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation, vol. 52, no. 10, pp. 2603-2611, 2004.

[13] V. Jandieri, K. Yasumoto and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves, vol. 30, pp. 1102-1112, 2009.

3

Realization of True Logic Gates UsingCoupled Nonlinear Photonic Crystal Waveguides Based

on Bandgap Transmission Phenomenon

Vakhtang Jandieri 1, Ramaz Khomeriki 2 and Daniel Erni 1

1 General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CENIDE – Center for

Nanointegration Duisburg-Essen, D-47048 Duisburg, Germany.2 Department of Physics, Tbilisi State University, 3 Chavchavadze,

0128 Tbilisi, Republic of Georgia.

London, UK, April 9-13, 2018

Introduction

• Photonic Crystals and Photonic Crystal Waveguides.

• Formulation of the Problem.

• Bandgap Transmission Phenomenon.

• Realization of „True“ All-Optical AND Logic Gates.

• Numerical Results and Discussions.

Unique feature to localize electromagnetic waves to specific arrays and to guide along certain directions at restricted frequencies.

1μm

1. K. Yasumoto, H. Toyama and T. Kushta, IEEE TAP, vol.52,pp.2603-2611, 2004.2. V. Jandieri, K. Yasumoto and J. Pistora, IEEE Transactionson Magnetics, vol.53, no.4, 1000306, 2017.3. V. Jandieri, P. Meng, K. Yasumoto and Y. Liu, JOSA A, vol.32,no.7, pp.1384-1389, 2015.4. P. Baccarelli, V. Jandieri, G. Valerio, and G. Schettini, EuCAP2017, Paris, France, 19-24/03/2017, pp. 3222-3223.5. V. Jandieri, P. Baccarelli, G. Valerio and G. Schettini, IEEETAP (to be submitted).

Photonic Crystals (Bandgap Structures)

0, 1, 2, ..,nxn nk iαβ = ± ±= +

02

nn

hπβ β= +

Modal field is represented as a

superposition of an infinite number of space harmonics

Space-harmonic phase constant

Modal attenuation (leakage) constant

PhotonicCrystalWaveguide

For the realization of true all-optical signalprocessing, the optical system needs to havenonlinear properties.In optically nonlinear media, the index of refractionis modified by the presence of a light signal and thismodification can be exploited to influence anotherlight signal, thereby performing an all-optical signalprocessing operation.Nonlinear Photonic Crystals (PCs) are one of themost promising candidates to enable dense full-optical signal processing on the same chip leading tomuch lower production and operating costs.

n0 = 3.4 (linear refractive index)

r = 0.2h (radius of the rods)

w = 2.0h (width of the waveguide)

Bandgap for TE modes(Ey, Hx, Hz). Pillar type PCs.

[ ]

0 0

2 3(30

)

,

( , )

y yx z

xzy y

E EH Hz t x t

HHn x z E E

x z tχ

µ µ

ε

∂ ∂∂ ∂= − =

∂ ∂ ∂ ∂∂∂ ∂

− = +∂ ∂ ∂

Initial are Maxwell’s equations:

(3)2

20

4,

3 n n cεχ =where 2n − Nonlinear refractive index.V. Jandieri and R. Khomeriki, “Linear amplification of

optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters, vol. 27, 639, 2015.

Formulation of the Problem (1)

In the first-order coupled-mode analysis for the weaklycoupled PC waveguides [1, 2], the guided field supportedby the waveguide system is approximated as follows:

( )0( , , ) ( , ) ( , ) . . yi z tE x z t F t z vt x z e c cβ ω−= − Φ +

[1] K. Yasumoto, V. Jandieri and Y. Liu, JOSA A, vol.30, no.1,pp. 96-101, 2013.

[2] V. Jandieri, K. Yasumoto and J. Pistora, JOSA A, vol.31, no.3,pp. 518-523, 2014.

0 ( ) :ω β Frequency of the fundamental symmetric or antisymmetric modes.

( , ) :x zΦ Transverse field distribution (periodic function with z).

( , ) :F t z vt− Evolution of the slowly varying envelope of electric field amplitude.

Formulation of the Problem (2)

[1] T. Taniuti and N. Yajima, “Perturbation method for a nonlinear wave modulation,” Journal of Mathematical Physics, vol.10, pp.1369-1372, 1969.

( )0 1 2 0 1 2( , , ) ( , , , , , , ) y

j jy

jE x z t E x z z z t t tε= ∑ � �

“Multiple scale expansion” [1] for slowly varying wave amplitude:

Multiple space and temporal scales: z0 = z, z1 = ε z, z2 = ε 2 z,… t0 = t, t1 = ε t, t2 = ε 2 t,…

( )0 0 0(1)0 1 2 0 1 2 1 2 1 2( , , , , , , ) ( , , , , ) i z t

yE x z z z t t t F x z z t t e β ω−=� � � �

2

0 1 2

2 2 22

2 2 20 0 1 1 0 2

2 2

z z z z

z z z z z z z

ε ε

ε ε

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂

�

�

Formulation of the Problem (3)

Formulation of the Problem (4)

22, ,

, , ,22 0S A S A

S A S A S A

FFi

Ft

Fω γξ

∂ ∂′′+ + =

∂ ∂

Maxwell equations are reduced to the nonlinear Schrodinger equation:

( )0, ,

,, ,

exp( )( , )

coshS A S A

S AS A S A

F i tF z t

z v t

δω=

− Λ

which has a soliton solution:

( )2, 0, , ,0

,

21 1,

4S A

S A S A S AS A

FF

ωδω γ

γ

′′Λ = =

4 4(3),3 4 ( , ) ( , ),S A x z dxdz x z dxdzκγ ω κχ

′Γ Γ

== Φ Φ∫ ∫where

n0 = 3.4 (linear refractive index)

r = 0.2h (radius of the rods)

w = 2.0h (width of the waveguide)

Bandgap for TE modes(Ey, Hx, Hz). Length of the device is 30h.

[1] K. Yasumoto, V. Jandieri and Y. Liu, JOSA A, vol.30, pp. 96-101, 2013.[2] V. Jandieri, K. Yasumoto and J. Pistora, JOSA A, vol.31, pp. 518-523, 2014.

To realize Bandgap Transmission an excitationfrequency of the injected signal should be properlychosen very close to band edge (red dot in thedispersion diagram). In the linear regime nopropagation (all the modes evanescent).

V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters, vol. 27, 639, 2015.

Bandgap Transmission (1)

Bandgap Transmission (2)

n0 = 3.4 (linear refractive index)

r = 0.2h (radius of the rods)

w = 2.0h (width of the waveguide)

Bandgap for TE modes(Ey, Hx, Hz). Length of the device is 30h.

( ) ( ) ( )20 4A AFω γβ ω β= −

( ) ( )( )

0

0

exp[ ],

cosh 2A A Ay A

A A

zF i tE x z

F z v t

β ω

γ ω

−= Φ

′′−

( ) 0 exp( ), Sy S S S zE i tx z F ω βΦ −= −

Frequency shift due to the Kerr effect.

PropagatingAntisymmetric.

EvanescentSymmetric.

FDTD simulation

Agreement of the numerical result with the nonlinear Schrödinger model (red line).

Numerical Results – Bandgap soliton (1)

FDTD simulationNumerical Results – Bandgap soliton (2)

The total output energy flux density of the signal registered at the output port does not depend on theinput power launched in Port 2 [1, 2].

A weak signal triggers the creation of the soliton not influencing much its shape and power.1. R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett., vol.92, 063905-063908, 2004.2. V. Jandieri, R. Khomeriki, D. Erni and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research, vol.158, pp.63-72, 2017.

Numerical Results – Bandgap soliton (3)

Numerical Results – Logic AND gate (1)

n0 = 2.95 (linear refractive index)

r = 0.32h (radius of the rods)

w = 1.73h (width of the waveguide)

Bandgap for TM modes(Hz, Ex, Ey). Air-Hole type PCs.Length of the device is 30h.

[1] K. Yasumoto, V. Jandieri and Y. Liu, JOSA A, 30, 96-101, 2013.[2] V. Jandieri, K. Yasumoto and J. Pistora, JOSA A, 31, 518-523, 2014.

Typical semiconductor material (Si, GaAs, InP) with relative highrefractive index is studied.

For practical application would be better to use modified materialsystems with enhanced third order nonlinearities based on amorphousSi, or hybrid organic-Si-on-insulator compounds [3]:

[3] J. Leuthold, C. Koos, W. Freude, ‘Nonlinear silicon photonics,’Nature Photonics, 4, 535, 2010.

Cascaded system to realizetrue all-optical logic AND gate

Carrier phases of the input Gaussian pulses are in line with the phase of CW (coherent carriers).Proposed scheme still operational when time difference between the Gaussian amount to 0.3 – 0.4 ps.Relative error between the output and input signals (its amplitude and width) is about 10 % -12 %.The peak level contrast between ‘1‘ and ‘0‘ output signal amounts to 16 dB.

Numerical Results – Logic AND gate (3)