**4.2 A PC Y-splitter**

136 Photonic Crystals – Innovative Systems, Lasers and Waveguides

Now, if we assume that the input signal launched only from the left (s2−=0) and that external

*ds is s s*

111 1 1 1 1 . *<sup>c</sup>*

Nonlinear Kerr type defects

(29)

Contour map of HY 1.0


(b) (c)

111 0 11 1 11 1 , <sup>2</sup>

 

*dt Q P* (30)

losses can be neglected (τ<sup>2</sup> →∞), (29) can be expressed in terms of just s1+ and s1− as:

Fig. 10. (a) A simple symmetric side-coupled cavity waveguide. The dark circles are assumed to be air holes in a GaAs substrate and the grey circles are assumed to contain the nonlinear elements. (b) Steady state response of the limiter for the linear case i.e. low input

1 1

*c ds s* 2

where Q is the quality factor of the cavity and P0 is characteristic power of the system (See

The PC used in this section to create a waveguide is a triangular lattice of holes in a GaAs substrate. The radius of the holes is chosen 0.3a. The triangular lattice of air holes can be more easily fabricated than the rod lattice version which is used for TM modes. To create a

In order to create a cavity mode in the bandgap region of a PC, a defect has to be added to the structure of the waveguide. The defect can later be doped with a material which exhibits strong Kerr effect (Fushman et al., 2008, Nakamura et al., 2004) so as to be able to create the desirable hysteresis effect. Doping quantum dots in to photonic crystals has long been known to create large Kerr type nonlinearity coefficients. Here, the defect is assumed to be doped using the method presented in (Nakamura et al., 2004) resulting in a refractive index equal to 2.6 and a nonlinear Kerr effect equal to n2 = 2.7×10-9 m2/W. The defect radius is rd = 0.25a. First the structure shown in Fig. 10a is chosen as the limiter. The grey circles are assumed to be the nonlinear elements. FDTD method is used for time domain analysis of

*i sss*

intensity (c) Transmission for high input intensity.

It can be shown that for Kerr type nonlinearity:

(Bravo-Abad et. al, 2007) for its definition.).

waveguide a row of holes is removed from the PC.

1

*dt*

(a)

As it was mentioned before, combination of a Y-junction and a limiter is proposed as the AND gate. Using methods presented in the literature (Wilson et al., 2003, Frandsen et. al, 2004, Yang et al, 2010, Danaie et. al, 2008) we have optimized the Y-splitter. A simple Yjunction and the optimized junction are shown in Fig. 12a and Fig. 12b respectively. By introducing modifications to the structure of the bends and branches the transmission spectrum can be improved (Wilson et al., 2003, Frandsen et. al, 2004, Yang et al., 2010). There have been some detailed topologies reported in literature which enhance the bandwidth (Borel et al., 2005, Têtu et al., 2005). (Yang et al., 2010) used a triangular lattice of dielectric rods to design a Y-Branch. They showed that the Y-branch can be treated as a cavity that couples with the input and output waveguides.

In (Yang et al., 2010) to enhance the transmittance, additional rods are added to the junction area and the corner rods are displaced. The movement of the corner rods increases the volume of their cavity and make the cavity mode resonant with the waveguide modes. It is shown in (Yang et al., 2010) that using the coupled mode theory the reflection coefficient can be expressed as (31).

PC Y-Junction PC Limiter

<sup>0</sup> 0.75 1.5 <sup>3</sup> 4.5 <sup>6</sup> <sup>0</sup>

Time (ps)

If only one signal is sent into the junction the power will be bellow the limiter threshold and the signal will not be able to reach the output port. However when the two signals are present, the output power of the splitter will be high enough to trigger the switching mechanism of the limiter. Therefore the signal will be directed to the output port. In order to have a better insight on the performance of the device; the time domain response is also depicted. First it is assumed that only one of the input signals is ON. Then both signals are turned ON for a while and one of them is turned OFF afterwards. The normalised output power is measured for each case. It can be seen that the transition time is less than 1ps.

Many researchers have worked on methods to efficiently couple an optical fiber to a photonic crystal waveguide. The issue of designing a coupling scheme becomes more important when photonic crystal waveguides with low group velocities are used. However, in our case since the group velocity of the W1 waveguides in inputs and outputs is not much low, the coupling problem to the W1 waveguides does not seem to be acute. In order to design the input and output couplers for the AND gate, the same efficient coupling structure implemented in (Ayre et al., 2005) can be used. The mentioned paper suggests a wideband Y-splitter in a GaAs substrate for TE modes. A 5μm ridge waveguide is coupled to the photonic crystal waveguide with period = 430 nm and 36.2% air filling factor (corresponding to a radius over period (r/a) ratio of 0.30) via a 9μm injector tapering to

Due to nearly exact same prameters, for the structure proposed as the AND gate and the Ysplitter in (Ayre et al., 2005), the fabrication procedure described in (Ayre et al., 2005) can

OFF ON

2.25 3.75 5.25

Output

0.4

A

B

**4.4 Implementation issues** 

provide a bandwidth of the order of 300 nm.

OFF ON

Fig. 13. The structure for the proposed AND gate.

0.8

1.2

Normalized Output Power

1.6

Input (A)

Input (B)

$$R = \left| \frac{-j(\alpha - \alpha\_0) + \frac{1}{\tau\_1} - \frac{1}{\tau\_2} - \frac{1}{\tau\_3}}{j(\alpha - \alpha\_0) + \frac{1}{\tau\_1} + \frac{1}{\tau\_2} + \frac{1}{\tau\_3}} \right|^2,\tag{31}$$

where, ω0 is the resonance frequency, τi is the time constant regarding the amplitude decay of the resonance into the ith port. It is seen that if the (32) is satisfied, the reflection will be zero for ω = ω0.

$$\frac{1}{\tau\_1} = \frac{1}{\tau\_2} + \frac{1}{\tau\_3}.\tag{32}$$

Fig. 12. (a) A simple photonic crystal Y-splitter consisting of a Y- junction (solid line) and two 60-degree bends (dashed line) (b) a Y-junction optimized to decrease the input reflection. The grey hole have radii equal to 0.4a, 0.31a, 0.15a, 0.11a respectively from left to right.

We already know that for our junction due to symmetry τ2 =τ3; therefore the mentioned condition can be simplified as: τ2 =2τ1. It means that the decay rate of resonance to the output ports should be half the decay rate of resonance to the input port. In a simple Yjunction all the three time constants are equal. The tapered structure of the holes inserted in our proposed design, improves the coupling strength between the resonator and the input waveguide. It results in the reduction of τ1. If the tapering is designed so that τ1 is reduced to half τ2, then the zero reflection condition can be satisfied. Numerical optimization was used to design such a tapering.

#### **4.3 Design of a PC AND gate**

In order to design a PC AND gate the structure depicted in Fig. 13 is proposed. It is comprised of the proposed Y-junction in Fig. 12b and the sharp limiter in Fig. 11a. Each signal is assumed to have a power equal to 10W. The junction section adds the signals together.

Fig. 13. The structure for the proposed AND gate.

If only one signal is sent into the junction the power will be bellow the limiter threshold and the signal will not be able to reach the output port. However when the two signals are present, the output power of the splitter will be high enough to trigger the switching mechanism of the limiter. Therefore the signal will be directed to the output port. In order to have a better insight on the performance of the device; the time domain response is also depicted. First it is assumed that only one of the input signals is ON. Then both signals are turned ON for a while and one of them is turned OFF afterwards. The normalised output power is measured for each case. It can be seen that the transition time is less than 1ps.

#### **4.4 Implementation issues**

138 Photonic Crystals – Innovative Systems, Lasers and Waveguides

<sup>111</sup> ( )

, <sup>111</sup> ( )

0

 

where, ω0 is the resonance frequency, τi is the time constant regarding the amplitude decay of the resonance into the ith port. It is seen that if the (32) is satisfied, the reflection will be

> 123 111 .

(a) (b)

Fig. 12. (a) A simple photonic crystal Y-splitter consisting of a Y- junction (solid line) and two 60-degree bends (dashed line) (b) a Y-junction optimized to decrease the input

reflection. The grey hole have radii equal to 0.4a, 0.31a, 0.15a, 0.11a respectively from left to

We already know that for our junction due to symmetry τ2 =τ3; therefore the mentioned condition can be simplified as: τ2 =2τ1. It means that the decay rate of resonance to the output ports should be half the decay rate of resonance to the input port. In a simple Yjunction all the three time constants are equal. The tapered structure of the holes inserted in our proposed design, improves the coupling strength between the resonator and the input waveguide. It results in the reduction of τ1. If the tapering is designed so that τ1 is reduced to half τ2, then the zero reflection condition can be satisfied. Numerical optimization was used

In order to design a PC AND gate the structure depicted in Fig. 13 is proposed. It is comprised of the proposed Y-junction in Fig. 12b and the sharp limiter in Fig. 11a. Each signal is assumed to have a power equal to 10W. The junction section adds the signals

*j*

*j*

*R*

zero for ω = ω0.

right.

together.

to design such a tapering.

**4.3 Design of a PC AND gate** 

0

2

(31)

(32)

123

123

Many researchers have worked on methods to efficiently couple an optical fiber to a photonic crystal waveguide. The issue of designing a coupling scheme becomes more important when photonic crystal waveguides with low group velocities are used. However, in our case since the group velocity of the W1 waveguides in inputs and outputs is not much low, the coupling problem to the W1 waveguides does not seem to be acute. In order to design the input and output couplers for the AND gate, the same efficient coupling structure implemented in (Ayre et al., 2005) can be used. The mentioned paper suggests a wideband Y-splitter in a GaAs substrate for TE modes. A 5μm ridge waveguide is coupled to the photonic crystal waveguide with period = 430 nm and 36.2% air filling factor (corresponding to a radius over period (r/a) ratio of 0.30) via a 9μm injector tapering to provide a bandwidth of the order of 300 nm.

Due to nearly exact same prameters, for the structure proposed as the AND gate and the Ysplitter in (Ayre et al., 2005), the fabrication procedure described in (Ayre et al., 2005) can

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also be employed for design and implementation of the gate. The AlGaAs/GaAs heterostructure can be grown by metal-organic chemical vapor-phase deposition (MOCVD); where a 300 nm silicon oxide is deposited over the wafer using plasma enhanced chemical vapor-phase deposition (PECVD), followed by a 200-nm-thick layer of PMMA as electronbeam resist. The pattern is then transferred into the oxide hard mask via reactive ion etching (RIE) using a fluorine process. The deep etching of the PC holes can be achieved using chemically assisted ion beam etching (CAIBE) in a Chlorine-Argon process (Ayre et al., 2005). The holes would thus be approximately 605nm deep. The input wavelength is assumed 1550nm which can be obtained using an Erbium doped fiber laser which results in the device size to be approximately equal to 5.95μm×12.90μm.
