## Abstract

We propose an all-optical half adder based on two different cross structures in two-dimensional photonic crystals. One cross structure contains nonlinear materials and functions as an “AND” logic gate. The other one only contains linear materials and acts as an “XOR” logic gate. The system is demonstrated numerically by the FDTD method to work as expected. The optimal operating speed without considering the response time of the nonlinear material, the least ON to OFF logic-level contrast ratio, and the minimum power for this half adder obtained were 0.91Tbps, 16dB and 436mW, respectively. The proposed structure has the potential to be used for constructing all-optical integrated digital computing circuits.

©2008 Optical Society of America

## 1. Introduction

Photonic Crystals (PhCs), a new class of artificial materials first predicted by Yablonovitch [1] and John [2], is a promising candidate as a platform to build future photonic integrated devices with dimensions in the order of the operating wavelength [3], due to their small sizes and unique capability to modify photon interaction with host materials.

In the last three decades of the 20th century, optical bistability and optical logic devices were studied extensively [4]. Yet it was blocked by difficulties in optical integrations on a small chip. The creations of PhCs bring new hopes to all-optical logic circuits integration on a small chip. First, optical bistability was found in PhC microcavity with nonlinear materials [5]. Then, optical switches using optical bistability in PhCs and basic optical logic gates using PhC optical switches were reported [6–10]. Besides, basic optical logic gates using PhC splitters were reported [11–13]. Also, we have seen some kind of optical integrated circuits on PhCs [14–15]. It is, however, only the very beginning in developing optical integrated logic circuits and systems on the platforms of PhCs. For example, until now, there is even no answer to what are the most typical and practical structures for the basic all-optical logic gate “ AND”, “OR”, and “NOT” built on the platforms of PhCs.

It is known that full adders are the basic parts of a central processing unit (CPU). Since full adders are generally built with half adders, it is important to investigate all-optical half adders (AOHA) for the realization of all-optical CPU. In this paper we show an AOHA on the platform of 2-D PhCs. Its operating speed is calculated to be greater than 0.9Tbps when neglecting the response time of the nonlinear material.

This paper is organized as follows. In Sec. 2, we first give the models of the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs, then a model of an all-optical half adder, and last a description of simulation method and parameters. In Sec. 3, the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs are first investigated and optimized, and then the all-optical half adder on the platform of two-dimensional PhCs is investigated. Finally in Sec. 4, we draw a brief conclusion.

## 2. Models

#### 2.1 Models of the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs

Figure 1 shows the models of all-optical “AND” (a) and “XOR” (b) logic gates on the platform of 2-D PhCs. In Fig. 1, the hollow circles indicate linear dielectric rods, and the black solid dots indicate Kerr-type nonlinear rods. In Fig. 1(a), a nonlinear diffraction rod is put right in the intersection center of the two waveguides. The radius of the three nonlinear rods is the same as that of the linear rods in the 2-D PhC. In Fig. 1(b), the center of the diffraction rod is at (1.21*a*, 1.21*a*) with (0, 0) being the center of the bottom-left rod at the waveguide-intersection region. The radius of the diffraction rod is 0.3*a*. These parameters are determined through optimization. The so called optimization in this paper means a process of minimizing the operating power, the response time, and the size of the devices, and maximizing the operational bandwidth and the ON (logic 1) to OFF (logic 0) logic-level contrast ratio in the systems by scanning operating parameters.

In the following simulations, each of the two structures in Fig. 1 is a 17*a*×17*a* 2-D square lattice PhC with a lattice constant *a*=520.8nm. The dielectric rods consisting of the PhCs are silicon (Si) cylinders, of which the relative permittivity *ε* and the radius are 11.56 and 0.18 *a*, respectively. The background material is air. The third-order nonlinear susceptibility of the Kerr-type nonlinear rods in Fig. 1(a) is chosen to be *χ*
^{(3)}=1.0.10^{-4}
*µ*m^{2}/V^{2}, which corresponds approximately to AlGaAs with a Kerr coefficient of *n*
_{2}=1.5×10^{-17}m^{2}/W [7,8,16].

For a uniform PhC without defects under the above operating parameters, we find out that there exists a large bandgap by a standard plane-wave expanding method: the light with wavelengths between 2.250*a* and 3.304*a* cannot pass through the uniform PhC and thus is completely reflected [17]. We also find that, a photonic-crystal waveguide formed by removing a row of rods in the uniform PhC can guide light with wavelength *λ*
_{0}=2.9762*a*=1.550µm, which is the operating wavelength we desired for applications in optical communications.

The basic idea for constructing the “AND” gate in Fig. 1(a) is as follows. The two black rods bring a uniformity break to the left waveguide in Fig. 1(a), so that reflection will be produced when there is a wave propagating in it. Nonlinear material is introduced to promote the ON to OFF logical-level contrast ratio. Finally, with proper choice of the distance between the uniformity-breaking point and the diffraction rod at the crosspoint of the two waveguides, the structure may function as an “AND” gate through the interference of the reflected waves from the uniformity-breaking part of the waveguide and the waves diffracted by the diffraction rod at the waveguide crosspoint.

The principle of the “XOR” gate in Fig. 1(b) is based on the wave-splitting property of the diffraction rod deviated from the waveguide-cross center. This is clearly seen from the field distributions obtained through simulations in section 3 in the following.

#### 2.2 A Model of an all-optical half adder on the platform of two-dimensional PhCs

It is known that a half adder adds up two one-bit binary numbers (*A* and *B*). The outputs of it are the sum *S*=*A*⊕*B* and the carry *C*=*A*·*B*. Then referring to the all-optical “XOR” and “AND” gates in Fig. 1, we may construct an AOHA, as shown in Fig. 2. The up-right rod and bottom-left rod of the four rods at the waveguide crosspoints in Fig. 2 play the role of splitters [13].

It is obvious that this structure is symmetrical for data *A* and data *B*, since the two signals entering into both *A* and *B* have to go over the same distance to reach the output port *S* and *C*.

In simulations of the AOHA, all parameters, including the nonlinear third-order susceptibility, the dielectric constant, the wavelength of the excitation source, the radius, and the position of the nonlinear rods and the diffraction rods are the same as that in the models shown in Fig. 1.

#### 2.3 Simulation method

In order to demonstrate the functions and to investigate the properties of the models presented above, we used the FDTD method for simulations. Since the FDTD method is well known and widely used in computational electromagnetism, we omit the description of the FDTD method in this paper. Here we just write out the related expressions concerning nonlinear materials in the models.

The Kerr effect of the nonlinear material is modeled by introducing an intensity-dependent increment of the refractive index:

where Δ*n* is the increment of the refractive index due to the nonlinearity of the material induced by the electric fields of the waves in the models, *I* is the local intensity of light and is proportional to |**E**|^{2}, _{sat}*I* is the saturation intensity of the nonlinearity, and _{2}
*n* is proportional to the third-order nonlinear susceptibility *χ*
^{(3)}.

Equation (1) can be rewritten in the following form:

where *n*̄_{2}=*βn*
_{2} and $u(x,y)=\sqrt{\frac{I}{\beta}}$ are the dimensionless nonlinear refractive index and the dimensionless field amplitude, respectively, and *α*=*β*/*I _{sat}* is a dimensionless coefficient. The constant

*β*is determined from the initial condition to be

where *P _{in}* is the total power per unit length launched into the crystal and the integral is taken along the phase front of the input field. The input field

*u*(

*x*,0) is normalized such that the integral length in Eq. (3) is equal to 1µm. In our computations, for simplicity, we assume that the fields are much less than that required for saturation, i.e.,

The increment in the refractive index is thus simplified from Eq. (2) to be

Denoting the linear refractive index by *n _{L}*, the overall dielectric constant of the Kerr materials can be written out to be

In order to include optical nonlinearity into the FDTD algorithm, a nonlinear polarization term is added to the linear polarization term in Maxwell’s equations [18]. Thus, the electric field is related to the displacement vector by

where

and

are the linear and nonlinear polarization term, respectively. Here *χ*
^{(1)}(*t*) and *χ*
^{(3)}(*t*) are respectively the first and third-order susceptibility, and **r**=(*x*,*y*) is the position vector [19].

We point out that, in simulations the input ports of the structures in Fig. 1 and 2 are excited by continuous wave (CW) sources. The electric-field polarization of the wave is chosen to be parallel to the *y*-axis, which is the axis of the dielectric rods. The wave propagates in the (*x*,*z*) plane.

We also point out that, in our simulations the mesh sizes in the *x*- and *z*- directions are set to be *a*/16 and the time step is set to be 8.33.10^{-2} fs, which meets the requirements of Courant stable condition. And the calculated area is surrounded by a perfect matched layer (PML) boundary.

Another point to be mentioned is that we use a 2-D rather than a 3-D geometry in our numerical simulations. According to Ref. [20], 2-D simulations can be used to estimate the power needed to operate a true 3-D device, reducing the computation time considerably, while still capturing the essential physics of the problem.

## 3. Numerical results and discussion

Since one needs “AND” and “XOR” gates to build the half adder and the performance of the half adder depends on the two gates used, we first study the two basic gates separately, and then investigate the half adder.

#### 3.1 The “AND” logic gate

To demonstrate the function of the structure in Fig. 1(a), a numerical experiment is performed. We first apply a CW signal with a power *P*
_{0}(*P*
_{0}=2.3×10^{-11}
*a*/*n*
_{2}=806 mW) at port *A* or *B* separately, and measure the output powers at steady state. Then we apply CW signals with the same power simultaneously both at *A* and *B*, and measure the output power at steady state.

We found that the output power is 0.014*P*
_{0} (logic 0) for separate excitation at *A*, 0.005*P*
_{0} (logic 0) for separate excitation at *B*, and 1.46*P*
_{0} (logic 1) for simultaneous excitations at both input ports. This demonstrates that the structure in Fig. 1(a) does operate as an optical “AND” logic gate. The smallest ON to OFF logic-level contrast ratio for the “AND” logic gate is calculated to be 20.2 dB.

For getting better insight into the physics of the structure in Fig. 1(a), the field distributions at steady state operation are illustrated in Fig. 3, from which we see clearly that the structure functions as an “AND” gate.

### 3.1.1 The optimal logic 0 to 1 turn-over threshold power

In this section, we repeat the experiment shown in Fig. 3(c) by varying the radius *r _{c}* of the nonlinear rod at the waveguide crosspoint. The present numerical experiment shows a very interesting phenomenon that the logic 0 to 1 turn-over threshold power varies with

*r*. Denoting the input power at a single port by

_{c}*P*and the steady state output power by

^{s}_{in}*P*, respectively, we obtain the result shown in Fig. 4. When

_{out}*r*is smaller than 0.34

_{c}*a*, the minimum power to observe the logic 0 to 1 turn-over is larger than 218mW and the transmittance is low. On the other hand, when

*r*is larger than 0.34

_{c}*a*, the resulting transmission decreases sharply. Thus, we get an optimal

*r*=0.34

_{c}*a*at which the power to realize the logic 0 to 1 turn-over is 218mW with the transmission for separate input power being approximately 95%.

We point out that the relative refractive index change of the nonlinear rods at the waveguide sidewalls to realize the OFF to ON transition is Δ*n*/*n*
_{0}=0.064 (at the rod boundary) ~0.15 (at the rod center), where *n*
_{0}=2.168 is the refractive index at weak field intensity. It seems that the relative index change is quite large and unrealistic for conventional Kerr materials. In the simulation, however, the third-order susceptibility of the nonlinear rods is taken to be approximately that of AlGaAs, so the simulation result is practical. The large change of refraction index is due to the high field intensity in the resonator and the relatively large input power (218mW). Yet, the relative index change of the nonlinear rod at the waveguide crosspoint is only Δ*n*/*n*
_{0}=0.06 (at the rod boundary) ~0.003 (at the rod center). We hope: with the development of new nonlinear materials with high third-order susceptibility, the input power will be greatly reduced.

The above optimal power level is many orders of magnitude lower than that required by other small all-optical ultra-fast switches. This is explained as follows. First, the transverse area of the modes in the PhC is only 0.67*λ*
^{2}; consequently, much less power is needed than in other systems with larger transverse mode area. Second, the nonlinear rods in the system introduced cavities which enhance the nonlinearity and reduce the threshold power similar to that described in Ref. [5].

### 3.1.2 The Contrast Ratio and the Bandwidth

The ON to OFF logic-level contrast ratio is shown in Fig. 5 for different input powers. From Fig. 5 we see that for a fixed input power, the operating bandwidth, which is defined as the region in which the contrast ratio is no less than a given value, is limited. We also find that the operating bandwidth is different with the input power. For an input power of 274mW, the bandwidth for the contrast ratio larger than 15dB is 6nm.

Furthermore, from Fig. 5 we can see that the higher the input power is, the longer the operating wavelength gets. This can be explained as follows. Each waveguide-uniformity-breaking region can be regarded as a resonator filled with nonlinear capacitive rods. Only waves with their wavelengths near the resonance wavelength of the resonator can tunnel through the resonator and reach the output port. Noting that the resonance wavelength is proportional to the square root of capacitance in a resonant circuit, the capacitance of a nonlinear rod is proportional to its refractive index, and the refractive index increases with the power, as can be seen from Eqs. (5) and (6), we conclude that the resonant wavelength will increase with the input power, in agreement with that shown in Fig. 5. So, the operating wavelength can be tuned by the input power.

#### 3.2 The “XOR” logic gate

Now we perform simulations with the structure shown in Fig. 1(b) in the same way as that in 3.1.

When we apply a single CW signal with power *P*
_{0} at port *A* (*B*), the output power from port *Y* at steady state is found to be 0.49 *P*
_{0} (0.48 _{0}
*P*), corresponding to logic 1, which is approximately equal to the output power from the idle port. When we excite the two ports *A* and *B* simultaneously each with power *P*
_{0}, the steady state output power at port *Y* is calculated to be 0.0004 *P*
_{0}, corresponding to logic 0. This demonstrates that the structure shown in Fig. 1(b) works as an optical “XOR” logic gate. The smallest ON to OFF logic-level contrast ratio for the “XOR” logic gate is found to be 30.8dB.

For getting better insight into the physics of the structure in Fig. 1(b), the field distributions at steady state operation are illustrated in Fig. 6, from which we see clearly that the structure functions as an “XOR” gate. As mentioned in Ref. [13], from Fig. 6 we see that there exists a phase difference of *π* between the wave at *Y* port in Fig. 6(a) and that in Fig. 6(b). This explains why the output logic at *Y* port becomes 0 when port *A* and *B* are excited separately and simultaneously by logic-1 signals.

To show the effect of the XOR, we have calculated the spectrum of the ON to OFF logic-level contrast ratio, as shown in Fig. 7, from which we may see that the bandwidth for the contrast ratio over 25dB is larger than 32nm. This indicates that XOR logic gate has a large bandwidth in the fiber-optic-communication wavelength band.

#### 3.3 Half Adder

Now we move to study the structure indicated in Fig. 2. Necessary parameters for simulations are given in Sec. 2.

Applying different signals at port *A* and *B*, and calculating the output power from port *S* and *C*, we obtain Table 1. For convenience of logic function verification, we transformed Table 1 into Table 2, which demonstrates clearly that the structure, indicated in Fig. 2, functions as a half adder, i.e., *S*=*A*⊕*B*, *C*=*A*·*B*.

In the same way as that in 3.1.2 we may get the ON to OFF logic-level contrast ratio of the half adder to be 16dB for an operating bandwidth of 3nm.

We have also calculated the operating speed, an important parameter for the half adder, through the time-domain response of the structure. The operating speed is defined as the inverse of the response time, which is the sum of the rising time *t _{r}* and the recovery time or falling time

*t*, as indicated in Fig. 8. We find that the structure can have an optimal operating speed of 0.91Tbit/s. This speed, influenced by the quality factor of the resonator introduced by three nonlinear rods, is obtained without considering the response time for action between the wave and the Kerr material as discussed in Ref. [21]. To promote the operating speed, the quality factor of the resonator introduced by the nonlinear rods should be as small as possible [22]. Also, ultra-fast wave-response-time materials should be used.

_{f}The operating speed may be affected by the operation power, as shown in Fig. 9, from which we can see that greater input power leads to higher operating speed. This may be explained as follows. As mentioned in section 3.2, the refractive index of nonlinear rods increases with the input power. Since increasing refractive index of the nonlinear rods means increasing uniformity break in the waveguides and also increasing reflection of waves by the nonlinear rods, thus the quality factor of the resonators introduced by the nonlinear rods decreases, so that the operating speed of the structure increases.

## 4. Conclusions

On the basis of the optical “XOR” and “AND” logic gates built with 2-D PhCs, we constructed an all-optical half adder. Numerical simulations demonstrated successfully by the FDTD method that the structure presented does function as an all-optical half adder. The ON to OFF logic-level contrast ratio for this half adder could reach at least 16dB and the optimal operating speed is found to be as high as 0.91Tbits/s when omitting the material-wave-response time of Kerr effect. This structure is useful in designing all-optical signal processing circuits and optical computer systems.

## Acknowledgments

We thank the supports from the Chinese Natural Science Foundation (Grant No. 60877034), the Guangdong Natural Science Foundation (Key Project No. 8251806001000004), and the Shenzhen Science Bureau.

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