**3. Case study 1: A PC all-optical switch**

126 Photonic Crystals – Innovative Systems, Lasers and Waveguides

nonlinear elements on the linear system. Transfer matrix method (TMM), perturbation theory, coupled mode theory (CMT) and FDTD can be used for this purpose. In order to model the Kerr effect using FDTD, several methods have been proposed. Here FDTD and perturbation

From the Maxwell equations, the wave equation for the nonlinear medium can be written

0 0 <sup>2</sup> ( , , , , ) 0. *<sup>r</sup> <sup>E</sup> E xyz E <sup>t</sup>*

2 2

*t t*

0 0 3 3 ( )2 . 8 8 *nn E n n E n n*

(11)

000 0 2 2 . *NL E P E n*

Based on the type of nonlinearity different approaches have to be made to solve the mentioned equation. The FDTD model for Kerr-type materials assumes an instantaneous

> (3) (3) <sup>2</sup> <sup>2</sup> 2 2 <sup>2</sup> 0 0 0

> > (3) <sup>2</sup> <sup>2</sup>

*n E* 

Perturbation theory is also a useful tool in engineering for analyzing systems with small nonlinearities. Using Maxwell equations, the following eigenvalue problem can be derived for linear time invariant PC systems (A waveguide is assumed in our case.). The Dirac

4

. <sup>3</sup>

2 0 0 00 *E* () . *r E c*

It is shown in (Bravo-Abad et al., 2007) that a change in the dielectric constant can result

0 0 0 0 0

But since for a waveguide *v k <sup>g</sup>* / then it can be shown that (Bravo-Abad et al., 2007)

<sup>2</sup> <sup>3</sup> 00 0 0 0 () () . 2 2 () () *E E dr r E r E E dr r E r*

2

(9)

(10)

(12)

(13)

2 3

(14)

theory are briefly reviewed. Also In section 4 CMT is used to analyze a PC limiter.

2

It is shown in (Joseph & A Taflov, 1997) that (9) can be rewritten as:

2 2

nonlinear response. The nonlinearity is modelled in the relation *D E* where:

Therefore the relation ship between E and D can be iteratively determined using:

0

specifies the Bloch eigenmode for the electric field.

in a variation in the original eigenvalue <sup>0</sup> as:

the following approximation is valid:

*<sup>D</sup> <sup>E</sup>*

as:

notation *E*<sup>0</sup>

The idea of a one dimensional PC all-optical switch was first proposed in (Scholz et al., 1998). Coupled cavity waveguides (CCW) with Kerr nonlinearity, were there after suggested for all-optical switching. The main drawback of CCW switches is that when the switch is in the OFF state, all the data signal is reflected back to the input port. Since the backscattered signal can affect other optical devices on an optical chip, it makes them unsuitable for all optical integrated circuit applications. A combination of directional couplers and nonlinear optical elements can be used to solve the mentioned problem (Yamamoto et al., 2006, Cuesta-Soto et al., 2004, Rahmati & N. Granpayeh, 2009).

In a directional coupler based switch, according to the ON or OFF state of the switch, most of the data signal power is guided to either of the two output ports. Usually only a very small amount leaks to the input port. It gives the designer the ability of using the switch in sequential optical circuits.

Here a PC directional coupler is designed first.The PC lattice used for this design is a two dimensional array of GaAs rods which is known to have Kerr type optical nonlinearity.

The band diagram of the PC, Which is obtained using Plane Wave Expansion (PWE) method, is shown in Fig. 1. The shaded region is the optical bandgap. No optical signal within the normalized frequency range of 0.28a/λ to 0.45a/λ; where a is the lattice constant and λ is free space wavelength, can propagate through the PC lattice. The value of a is chosen 635nm in our simulations. The radii of rods are equal to 0.2a and their refractive index is chosen equal to 3.4, which is equal to the refractive index of GaAs at the 1550nm wavelength.

Introduction of defects into PC lattice is the first step in designing PC devices. These are usually classified in two different categories of point defects and line defects, which can create resonators and waveguides respectively. As an Instance, in the mention PC, removing a column of rods (Fig. 2a) creates a waveguide mode in the bandgap region between the

. *LC ke ko*

0.3

(a) (b)

Coupling Section

Input Section Y-Splitter Y-Splitter Output Section

Fig. 3. The waveguide modes for (a) the W1 waveguide shown in Fig. 2(a) and (b) the directional coupler shown in Fig. 2(b). The super-cell for obtaining the directional coupler is

Dummy Input Port Output Port (2)

Switch Length (L)

(b) (c)

Coupling Length (LC Coupling Length ( L ) C )

Fig. 4. (a) A conventional PC directional coupler (b) f= 410a/λ (c) f= 0.396a/λ.

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.25

Wavenumber (2/a)

(a)

0.3

Input Port

depicted in the dashed box.

0.35

Frequency (a/)

0.4

0.45

0.5

*Odd Mode*

0.35

Frequency (a/)

0.4

0.45

0.5

(16)

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.25

*Even Mode* Decoupling

Wavenumber (2/a)

Point

Output Port (1)

normalized frequencies 0.340a/λ and 0.447a/λ (See Fig. 3a). A simple directional coupler can be obtained by removing two rows of rods adjacent to a central row (See Fig. 2b.) (Zimmermann et al., 2004, Nagpal, 2004). As discussed in (Nagpal, 2004), when two PC waveguides are placed close to each other, light propagating in one of the waveguides can be coupled to the neighboring waveguide.

Fig. 1. The band diagram for a 2D hexagonal array of GaAs rods, where the ratio of the rods radius to the lattice constant is 0.2.

Fig. 2. Different PC structures for TM modes: The dark circles are assumed to be dielectric and the background is air (a) a PC W1 waveguide (b) A PC directional coupler.

PWE method can also be used to analyze the structure in Fig. 2b. In this case, using a supercell, the two adjacent waveguides can be treated as a single symmetric line defect in the PC structure. The super-cell which is used for this purpose and the band diagram obtained in this case are shown is Fig. 3b. Here, the radii of the central rods, rC, is assumed 0.2a. An odd and an even defect mode are produced in this case.

It is shown (Zimmermann et al., 2004, Nagpal, 2004)] that the light that is travelling in one of these waveguides, can be periodically coupled to the other one after passing a certain distance referred to as the coupling length (LC). The coupling length is related to the propagation constants of the odd (ko) and even (ke) modes as follows:

normalized frequencies 0.340a/λ and 0.447a/λ (See Fig. 3a). A simple directional coupler can be obtained by removing two rows of rods adjacent to a central row (See Fig. 2b.) (Zimmermann et al., 2004, Nagpal, 2004). As discussed in (Nagpal, 2004), when two PC waveguides are placed close to each other, light propagating in one of the waveguides can

TM Band Structure

M K

2rC

Fig. 1. The band diagram for a 2D hexagonal array of GaAs rods, where the ratio of the rods

(a) (b)

and the background is air (a) a PC W1 waveguide (b) A PC directional coupler.

propagation constants of the odd (ko) and even (ke) modes as follows:

Fig. 2. Different PC structures for TM modes: The dark circles are assumed to be dielectric

PWE method can also be used to analyze the structure in Fig. 2b. In this case, using a supercell, the two adjacent waveguides can be treated as a single symmetric line defect in the PC structure. The super-cell which is used for this purpose and the band diagram obtained in this case are shown is Fig. 3b. Here, the radii of the central rods, rC, is assumed 0.2a. An odd

It is shown (Zimmermann et al., 2004, Nagpal, 2004)] that the light that is travelling in one of these waveguides, can be periodically coupled to the other one after passing a certain distance referred to as the coupling length (LC). The coupling length is related to the

be coupled to the neighboring waveguide.

0.0

and an even defect mode are produced in this case.

0.1

0.2

0.3

Frequency

radius to the lattice constant is 0.2.

 (a/

λ)

0.4

0.5

0.6

Fig. 3. The waveguide modes for (a) the W1 waveguide shown in Fig. 2(a) and (b) the directional coupler shown in Fig. 2(b). The super-cell for obtaining the directional coupler is depicted in the dashed box.

Fig. 4. (a) A conventional PC directional coupler (b) f= 410a/λ (c) f= 0.396a/λ.

Also:

Neglecting the LC term with regard to L2

Using (16), equation (19) can be simplified to:

relation for a PC directional coupler for TE modes.

changed using an optical pump signal as follows:

more accurate study reveals second order polynomial relation.

(fs) and δk in Fig. 6.

<sup>2</sup> <sup>1</sup> . *C C c LL L*

1 1 . ( ) *<sup>C</sup> ke ko <sup>L</sup> ke ko ke ke ko ko ke ko*

2 2 () () . ( ) *C C*

*kk k*

*e o ke ko ke ko L L <sup>L</sup>*

<sup>1</sup> *<sup>L</sup>* . *k* 

According to (20), the switch length is inversely proportional to δk; which is itself a function of δn. The relationship between δk and δn is dependant upon the structure geometry and is usually determined numerically using PWE method. In order to minimize the switch length and operational power, it is very important that for a slight variation in n, a large δk be obtained. In (Yamamoto et al., 2006) some methods are suggested to improve the δk and δn

The refractive index can be modified using electro-optic, thermo-optic or Kerr optical effect. From the mentioned optical phenomena, only Kerr effect can be used for all-optical applications. In materials that possess the Kerr effect, the refractive index can be linearly

0 2 *n n nI* ,

where, n0 is the refractive index in the linear regime, I is the optical field intensity and n2 is the nonlinear Kerr coefficient. Since δk is a function of δn, it can be seen from (20) that there is a trade-off between switch size and power, i.e. in order to reduce the operating power of the switch, longer device size is needed, or vice versa. The relationship between δk and δn should be calculated numerically (Danaie & Kaatuzian, 2011). Afterwards according to (20) the required switch length can be estimated for different signal frequencies. PWE method can be used for this purpose. We have depicted the relationship between signal frequency

In order to be able to optimize the switch length, we first try to fit an analytical expression on the curves shown in Fig. 6. Each of the five curves in Fig. 6 can be considered to be the bottom black curve multiplied by a γ factor. If γ is calculated for the four upper curves, it is seen that these factors are neatly placed on a line. On the other hand, the first glance at the curves shown in Fig. 6 suggests an exponential relation between δk and frequency; while a

C/δLC in (17) leads to:

2 2

(18)

*<sup>L</sup>* (17)

2

(20)

(21)

(19)

Directional couplers can be used as wavelength selective devices, in their linear regime, or switches when optical nonlinearity is introduced to their structure. A typical symmetrical directional coupler switch is shown in Fig. 4a. It consists of two input ports, two output ports and a central coupler. Assuming that a signal has entered from one of the input ports, it passes through the bends and enters the coupling region which has a length equal to L (several times larger than LC). The signal is then periodically transferred between the two waveguides and regarding the ratio between L and LC, it will be directed to either one of the output ports (or even both for a poor design).

Fig. 5. (a) The band diagram for the directional coupler shown in Fig. 2(b) when rC is reduced to 0.15a, (b) The spatial profile for the control mode at f = 0.295 (a/λ).

Fig. 4b and Fig.4c demonstrate the selective behavior of the device for two different input wavelengths. Since from (16) the coupling length is inversely related to the difference of ke and ko; then according to Fig. 3b, it is obvious that for each input signal frequency a different coupling length is perceived. This phenomenon can be used to separate different wavelengths (Fig. 4b and Fig. 4c).

Reduction of rC create a mode that can be used as the control signal. Fig. 5a shows the band diagram for the directional coupler in Fig. 2b when rC is 0.15 (a/λ). As seen in Fig. 5b the mentioned mode is highly confined. Since the rods are nonlinear, if the control signal power is large enough the refractive index of the middle row can be changed which will affect the system behaviour in regard to the linear case.

In a conventional directional coupler, if the refractive index of the central region between the two waveguides is somehow changed such as: n→n+δn, then the wavenumber of the odd and even modes change accordingly to ke + δke and ko + δko. From (16), it is obvious that the new coupling length for the modified structure will be slightly different. Assuming that the difference in the coupling length be δLC, and that m = L/LC, then if (m+1)δLC equals a coupling length, the light will be transferred to the other output port. It means that a change in the refractive index of the central row can provide a switching mechanism. From the previous equations, it can be concluded that (m+1)δLC = LC is the switching condition. Since m = L/LC then:

$$L = L\_{\mathbb{C}}^2 \times \frac{1}{\delta L\_c} - L\_{\mathbb{C}}.\tag{17}$$

Also:

130 Photonic Crystals – Innovative Systems, Lasers and Waveguides

Directional couplers can be used as wavelength selective devices, in their linear regime, or switches when optical nonlinearity is introduced to their structure. A typical symmetrical directional coupler switch is shown in Fig. 4a. It consists of two input ports, two output ports and a central coupler. Assuming that a signal has entered from one of the input ports, it passes through the bends and enters the coupling region which has a length equal to L (several times larger than LC). The signal is then periodically transferred between the two waveguides and regarding the ratio between L and LC, it will be directed to either one of the

(a) (b)

f=0.295 (a/λ)

Fig. 5. (a) The band diagram for the directional coupler shown in Fig. 2(b) when rC is reduced to 0.15a, (b) The spatial profile for the control mode at f = 0.295 (a/λ).

Fig. 4b and Fig.4c demonstrate the selective behavior of the device for two different input wavelengths. Since from (16) the coupling length is inversely related to the difference of ke and ko; then according to Fig. 3b, it is obvious that for each input signal frequency a different coupling length is perceived. This phenomenon can be used to separate different

Reduction of rC create a mode that can be used as the control signal. Fig. 5a shows the band diagram for the directional coupler in Fig. 2b when rC is 0.15 (a/λ). As seen in Fig. 5b the mentioned mode is highly confined. Since the rods are nonlinear, if the control signal power is large enough the refractive index of the middle row can be changed which will affect the

In a conventional directional coupler, if the refractive index of the central region between the two waveguides is somehow changed such as: n→n+δn, then the wavenumber of the odd and even modes change accordingly to ke + δke and ko + δko. From (16), it is obvious that the new coupling length for the modified structure will be slightly different. Assuming that the difference in the coupling length be δLC, and that m = L/LC, then if (m+1)δLC equals a coupling length, the light will be transferred to the other output port. It means that a change in the refractive index of the central row can provide a switching mechanism. From the previous equations, it can be concluded that (m+1)δLC = LC is the switching condition.

0 X (μm)

4

2

Normalized Power



output ports (or even both for a poor design).

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.25

wavelengths (Fig. 4b and Fig. 4c).

Since m = L/LC then:

system behaviour in regard to the linear case.

Wavenumber (2 /a)

0.3

0.35

Frequency (a/ )

0.4

0.45

0.5

$$
\delta L\_{\mathbb{C}} = \pi \left( \frac{1}{k\varepsilon - ko} - \frac{1}{k\varepsilon + \delta k e - ko - \delta k o} \right) \equiv \pi \frac{\delta k e - \delta k o}{\left(k\varepsilon - ko\right)^2}.\tag{18}
$$

Neglecting the LC term with regard to L2 C/δLC in (17) leads to:

$$L \equiv L\_{\mathbb{C}}^2 \times \frac{\left(ke - ko\right)^2}{\pi (\delta k\_e - \delta k\_o)} = L\_{\mathbb{C}}^2 \times \frac{\left(ke - ko\right)^2}{\pi \delta k}. \tag{19}$$

Using (16), equation (19) can be simplified to:

$$L \equiv \pi \times \frac{1}{8k} . \tag{20}$$

According to (20), the switch length is inversely proportional to δk; which is itself a function of δn. The relationship between δk and δn is dependant upon the structure geometry and is usually determined numerically using PWE method. In order to minimize the switch length and operational power, it is very important that for a slight variation in n, a large δk be obtained. In (Yamamoto et al., 2006) some methods are suggested to improve the δk and δn relation for a PC directional coupler for TE modes.

The refractive index can be modified using electro-optic, thermo-optic or Kerr optical effect. From the mentioned optical phenomena, only Kerr effect can be used for all-optical applications. In materials that possess the Kerr effect, the refractive index can be linearly changed using an optical pump signal as follows:

$$m = n\_0 + n\_2 I\_{\prime} \tag{21}$$

where, n0 is the refractive index in the linear regime, I is the optical field intensity and n2 is the nonlinear Kerr coefficient. Since δk is a function of δn, it can be seen from (20) that there is a trade-off between switch size and power, i.e. in order to reduce the operating power of the switch, longer device size is needed, or vice versa. The relationship between δk and δn should be calculated numerically (Danaie & Kaatuzian, 2011). Afterwards according to (20) the required switch length can be estimated for different signal frequencies. PWE method can be used for this purpose. We have depicted the relationship between signal frequency (fs) and δk in Fig. 6.

In order to be able to optimize the switch length, we first try to fit an analytical expression on the curves shown in Fig. 6. Each of the five curves in Fig. 6 can be considered to be the bottom black curve multiplied by a γ factor. If γ is calculated for the four upper curves, it is seen that these factors are neatly placed on a line. On the other hand, the first glance at the curves shown in Fig. 6 suggests an exponential relation between δk and frequency; while a more accurate study reveals second order polynomial relation.

When an optical pulse enters a medium with a low group velocity, it becomes squeezed in the time domain. In order for the pulse energy to remain constant its amplitude should increase (Soljacic´ & Joannopoulos, 2004). The term n2I in (25) can be therefore rewritten as α.n2.Iin(c0/vg); where Iin is the input power intensity of the control signal; vg is the group velocity, c0 is the free space light speed and α is a correction factor which is related to the control signal mode profile. It determines what ratio of the input power is concentrated on the nonlinear rods. The α factor should be calculated numerically for each frequency. Since vg and α are functions of the control frequency (fc), therefore equation (25) is then rewritten

.1550nm . 2 . (. . ) *s*

*an I b c f d f e*

0 2

*<sup>c</sup> an I f b c f d f e v f* 

*in c s s*

2 . ( )( ) (. . ) ( )

The relation between the group velocity and fc can be obtained using PWE. For fc = 0.295 (a/λ) α can be obtained from integration of control mode profile from x = -95nm to +95nm (the central rods section). Using a simple integration it can be seen that 0.35 percent of the

As a case study, we have designed a switch in this section. The control signal frequency is chosen fc = 0.295 (a/λ), so as to minimum the reflection in the control signal path. The signal

<sup>a</sup> , 2 17.5 0.17 .(1.25 1.08 0.24) *s s*

*n ff*

then if we decide the switch length L to be equal to 25a, then according to (27) the required δn will be equal to 0.204. We use 2D FDTD method to analyze the time domain behavior of the directional-coupler-based switch. The rods are assumed to have Kerr type nonlinearity

<sup>2</sup>

*s*

*g c*

.1550nm .

*s s*

(25)

(26)

Output Port (1)

Output Port (2)

(27)

 <sup>2</sup> 2

*<sup>f</sup> <sup>L</sup>*

2

Fig. 7. The all-optical switch designed using a directional coupler.

control power is located on the rods (Fig. 5b).

frequency is chosen fs = 0.41 (a/λ). Since:

*L*

*<sup>f</sup> <sup>L</sup>*

as:

Dummy Port

Input Port (1)

Control Port

Fig. 6. The relationship between signal frequency and δk of the directional coupler depicted in Fig. 2(b) with rC = 0.15a, obtained using PWE method for different values of central row's refractive index change.

Based on the previous observations the relationship between δk and δn can be assumed as:

$$\text{Risk}(\delta n, f\_s) = h(\delta n) \cdot \text{g}(f\_s)\_{\prime} \tag{22}$$

where fs is the signal frequency and *h* and *g* are first and second order polynomials dertermined as:

$$h\left(\delta n\right) = a.\delta n + b.$$

$$g\left(f\right) = c.f\_s^2 + d.f\_s + e.\tag{23}$$

For our structure (rC = 0.15a), the factors *a-f* are determined as follows using curve fitting techniques:

$$
\delta k \left( \delta n\_\prime \, f \right) = \left( 17.5 \delta n + 0.17 \right) \left( 1.25 f\_s^2 - 1.08 f\_s + 0.24 \right) .
$$

The previous equation can be used for optimization purpose; in which the *f* is considered per (a/λ) and the δk is per (2π/a). Since δn = n2I, then the relationship between the switch length (per lattice constant) and optical power intensity (on the rods) becomes as follows:

$$L \cong \pi \times \frac{1}{\left(\frac{2\pi}{\mathbf{a}}\right) \left(a.\delta n + b\right) \left(c.f\_s^2 + d.f\_s + e\right)} = \frac{\mathbf{a}}{2\left(a.n\_2 I + b\right) \left(c.f\_s^2 + d.f\_s + e\right)}.\tag{24}$$

In the above equation a is the latice constant. If the designer wishes to choose the input signal for the 1550nm wavelength, then the the lattice constant must be chosen equal to fs.1550nm; therefore the actual size of the switch becomes equal to:

δn/n0 =0.015 δn/n0 =0.029 δn/n0 =0.044 δn/n0 =0.058 δn/n0 =0.074

0.39 0.395 0.4 0.405 0.41 0.415

Frequency (a/λ)

Fig. 6. The relationship between signal frequency and δk of the directional coupler depicted in Fig. 2(b) with rC = 0.15a, obtained using PWE method for different values of central row's

Based on the previous observations the relationship between δk and δn can be assumed as:

where fs is the signal frequency and *h* and *g* are first and second order polynomials

*h n an b* . .

For our structure (rC = 0.15a), the factors *a-f* are determined as follows using curve fitting

<sup>2</sup> , 17.5 0.17 . 1.25 1.08 0.24 . *s s k nf n f f*

The previous equation can be used for optimization purpose; in which the *f* is considered per (a/λ) and the δk is per (2π/a). Since δn = n2I, then the relationship between the switch length (per lattice constant) and optical power intensity (on the rods) becomes as follows:

> <sup>2</sup> <sup>2</sup> <sup>2</sup> 1 a . <sup>2</sup> 2 . (. . ) ( ). (. . ) <sup>a</sup> *s s s s*

In the above equation a is the latice constant. If the designer wishes to choose the input signal for the 1550nm wavelength, then the the lattice constant must be chosen equal to

fs.1550nm; therefore the actual size of the switch becomes equal to:

*an I b c f d f e a n b cf df e* (24)

*k n* , ., *fs s h n g f* (22)

<sup>2</sup> . . *s s g f <sup>c</sup> <sup>f</sup> <sup>d</sup> <sup>f</sup> <sup>e</sup>* . (23)

0

refractive index change.

dertermined as:

techniques:

*L*

0.005

0.01

0.015

k (2 /a) 0.02

0.025

0.03

$$L \equiv \frac{f\_s.1550\,\text{nm}}{2\left(a.n\_2I + b\right)\left(c.f\_s^2 + d.f\_s + e\right)}.\tag{25}$$

When an optical pulse enters a medium with a low group velocity, it becomes squeezed in the time domain. In order for the pulse energy to remain constant its amplitude should increase (Soljacic´ & Joannopoulos, 2004). The term n2I in (25) can be therefore rewritten as α.n2.Iin(c0/vg); where Iin is the input power intensity of the control signal; vg is the group velocity, c0 is the free space light speed and α is a correction factor which is related to the control signal mode profile. It determines what ratio of the input power is concentrated on the nonlinear rods. The α factor should be calculated numerically for each frequency. Since vg and α are functions of the control frequency (fc), therefore equation (25) is then rewritten as:

$$L \equiv \frac{f\_s \cdot 1550 \,\text{nm}}{2 \left( a.n\_2 I\_m (\frac{c\_0}{v\_s (f\_c)}) \alpha(f\_c) + b \right) (c.f\_s^2 + d.f\_s + e)}. \tag{26}$$

The relation between the group velocity and fc can be obtained using PWE. For fc = 0.295 (a/λ) α can be obtained from integration of control mode profile from x = -95nm to +95nm (the central rods section). Using a simple integration it can be seen that 0.35 percent of the control power is located on the rods (Fig. 5b).

Fig. 7. The all-optical switch designed using a directional coupler.

As a case study, we have designed a switch in this section. The control signal frequency is chosen fc = 0.295 (a/λ), so as to minimum the reflection in the control signal path. The signal frequency is chosen fs = 0.41 (a/λ). Since:

$$L \cong \frac{\mathbf{a}}{2(17.58n + 0.17).(1.25f\_s^2 - 1.08f\_s + 0.24)},\tag{27}$$

then if we decide the switch length L to be equal to 25a, then according to (27) the required δn will be equal to 0.204. We use 2D FDTD method to analyze the time domain behavior of the directional-coupler-based switch. The rods are assumed to have Kerr type nonlinearity

assumed to be GaAs as before. In this section, first a sharp PC limiter is designed and then

One of the best methods to be able to observe a strong switching mechanism in a PC is to create a nonlinear PC cavity that is coupled to a PC waveguide. Optical nonlinearity can shift the position of the cavity, which can result in creating a hysteresis loop in the path of signal (i.e. an optical limiter can be created.). In order to do so, the PC structure, waveguide

Fig. 9. (a) Two different possible coupling schemes between a cavity and a waveguide (b) Schematic description of CMT model parameters for the side coupled cavity waveguide.

It was briefly mentioned that FDTD and perturbation methods can be used to analyze PCs which have optical nonlinearity. In this section another popular method is used to explain the behaviour of a nonlinear PC component. CMT is a general method that can be used for analyzing the behaviour of cavities that are weakly coupled to a waveguide (Saleh, 1991). Generally a cavity can be either placed at the end of a waveguide (directly coupled) or be

In the linear case, if we assume that the EM fields inside the cavity are proportional to the parameter A (Fig. 9b), this system can be modelled using CMT by the following set of

1 2

,

1 1

*dA iA A A s s*

 

where ωc is the resonant frequency of the cavity. τ1 is the decay constant of the cavity into the waveguide modes propagating to the right and to the left; while the magnitude τ2 relates to the decay rate due to cavity losses. The parameters s1+ and s1− (s2+ and s2−) denote the complex amplitudes of the fields propagating to the left and right at the input (output) of the waveguide. The parameter κ governs the input coupling between the resonant cavity and the propagating modes inside the waveguide. From (Bravo-Abad et. al, 2007) this

1 2

S1- S1+ S2- S2+

(28)

τ2

A

τ1

by combining it with a Y-junction, a PC gate is obtained.

topology and cavity specifications each play an important role.

resonator

(a) (b)

side-coupled to one (Yanik et al., 2003) (See Fig. 9a).

1 2 2 1

 

*dt*

*c*

*ss A ss A*

 

equations (Bravo-Abad et. al, 2007):

magnitude is given by κ = (1/τ1)(1/2).

resonator

**4.1 PC limiters** 

waveguide

waveguide

and the nonlinear Kerr coefficient n2 is assumed 1.5×10-17m2/W. Each unit cell is meshed 16×16. The final switch structure is shown in Fig. 7.

Fig. 8. FDTD time domain simulation results for the directional coupler shown in Fig. 7. The solid line represents the optical power directed to the first output port and the dashed line shows the optical power which goes to the second output port.

In order to be able to observe the time-domain characteristics of the switch, the control and signal inputs are turned on simultaneously. After 2ps the control signal is turned off and after 3.5ps the data signal is turned off as well. Fig. 8 show the power density versus time for the first and second output ports for the structure shown in Fig. 7. Since the group velocity of the data signal is higher than the control signal, it reaches the output section faster. Thus for a brief period of time (t = 0.4ps), it goes to the second output port. At t = 0.5ps the control signal reaches the output section (turns the rods nonlinear) and forces the data signal to travel to the first output port. When the control signal is turned off (t = 2ps), the rods are tuned linear again and the data signal travels back to the first output port.

#### **4. Case study 2: Design of an optical gate using CCWs**

Recently more research attention has been directed towards photonic crystal logic gates (Andalib & Granpayeh, 2009, Bai eta al., 2009). In (Zhu et al., 2006) a structures for Alloptical AND gate is proposed. The main problem of the structures is that the AND gate's inputs have different input wavelengths. In an ideal two-input AND gate, the wavelength of the inputs should be the same or the mentioned AND gate cannot be used in all-optical large-scale circuits. Here a photonic crystal AND gate with identical input characteristics is proposed. Most PC optical devices reported in the literature use TM photonic crystal structures, such as a square lattice of dielectric rods to create a switching mechanism; while here a triangular lattice of holes in a dielectric substrate is used instead (which can be used for TE modes). It can provide a considerably large bandgap and can easily be fabricated using integrated circuit manufacturing technology. The radius of the holes is chosen 0.3a. Choosing larger holes will reduce the guiding strength in vertical direction. The dielectric is assumed to be GaAs as before. In this section, first a sharp PC limiter is designed and then by combining it with a Y-junction, a PC gate is obtained.

#### **4.1 PC limiters**

134 Photonic Crystals – Innovative Systems, Lasers and Waveguides

and the nonlinear Kerr coefficient n2 is assumed 1.5×10-17m2/W. Each unit cell is meshed

Output port (2) Output port (1)

ON

ON

shows the optical power which goes to the second output port.

**4. Case study 2: Design of an optical gate using CCWs** 

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> <sup>0</sup>

Time (pSec)

Time (ps)

Fig. 8. FDTD time domain simulation results for the directional coupler shown in Fig. 7. The solid line represents the optical power directed to the first output port and the dashed line

In order to be able to observe the time-domain characteristics of the switch, the control and signal inputs are turned on simultaneously. After 2ps the control signal is turned off and after 3.5ps the data signal is turned off as well. Fig. 8 show the power density versus time for the first and second output ports for the structure shown in Fig. 7. Since the group velocity of the data signal is higher than the control signal, it reaches the output section faster. Thus for a brief period of time (t = 0.4ps), it goes to the second output port. At t = 0.5ps the control signal reaches the output section (turns the rods nonlinear) and forces the data signal to travel to the first output port. When the control signal is turned off (t = 2ps), the rods are tuned linear again and the data signal travels back to the first

Recently more research attention has been directed towards photonic crystal logic gates (Andalib & Granpayeh, 2009, Bai eta al., 2009). In (Zhu et al., 2006) a structures for Alloptical AND gate is proposed. The main problem of the structures is that the AND gate's inputs have different input wavelengths. In an ideal two-input AND gate, the wavelength of the inputs should be the same or the mentioned AND gate cannot be used in all-optical large-scale circuits. Here a photonic crystal AND gate with identical input characteristics is proposed. Most PC optical devices reported in the literature use TM photonic crystal structures, such as a square lattice of dielectric rods to create a switching mechanism; while here a triangular lattice of holes in a dielectric substrate is used instead (which can be used for TE modes). It can provide a considerably large bandgap and can easily be fabricated using integrated circuit manufacturing technology. The radius of the holes is chosen 0.3a. Choosing larger holes will reduce the guiding strength in vertical direction. The dielectric is

OFF

OFF

16×16. The final switch structure is shown in Fig. 7.

0.2

0.4

Normalized Power

Control

Signal

output port.

0.6

0.8

1

One of the best methods to be able to observe a strong switching mechanism in a PC is to create a nonlinear PC cavity that is coupled to a PC waveguide. Optical nonlinearity can shift the position of the cavity, which can result in creating a hysteresis loop in the path of signal (i.e. an optical limiter can be created.). In order to do so, the PC structure, waveguide topology and cavity specifications each play an important role.

Fig. 9. (a) Two different possible coupling schemes between a cavity and a waveguide (b) Schematic description of CMT model parameters for the side coupled cavity waveguide.

It was briefly mentioned that FDTD and perturbation methods can be used to analyze PCs which have optical nonlinearity. In this section another popular method is used to explain the behaviour of a nonlinear PC component. CMT is a general method that can be used for analyzing the behaviour of cavities that are weakly coupled to a waveguide (Saleh, 1991). Generally a cavity can be either placed at the end of a waveguide (directly coupled) or be side-coupled to one (Yanik et al., 2003) (See Fig. 9a).

In the linear case, if we assume that the EM fields inside the cavity are proportional to the parameter A (Fig. 9b), this system can be modelled using CMT by the following set of equations (Bravo-Abad et. al, 2007):

$$\begin{aligned} \frac{dA}{dt} &= i \mathbf{o}\_c A - \frac{1}{\tau\_1} A - \frac{1}{\tau\_2} A + \kappa \mathbf{s}\_{1+} + \kappa \mathbf{s}\_{2-} \\ \mathbf{s}\_{1-} &= \mathbf{s}\_{2-} - \kappa A \\ \mathbf{s}\_{2+} &= \mathbf{s}\_{1+} - \kappa A\_{\prime} \end{aligned} \tag{28}$$

where ωc is the resonant frequency of the cavity. τ1 is the decay constant of the cavity into the waveguide modes propagating to the right and to the left; while the magnitude τ2 relates to the decay rate due to cavity losses. The parameters s1+ and s1− (s2+ and s2−) denote the complex amplitudes of the fields propagating to the left and right at the input (output) of the waveguide. The parameter κ governs the input coupling between the resonant cavity and the propagating modes inside the waveguide. From (Bravo-Abad et. al, 2007) this magnitude is given by κ = (1/τ1)(1/2).

such a device. First we assume that the PC device is comprised of linear element (n2 =0); in such a case the input frequency is tuned at the cavity's notch. No signal can pass the waveguide in this case. Then we assume that the defect has Kerr effect which causes a shift in cavity's central frequency and allows the signal to pass. The switching mechanism of this

The mentioned structure cannot provide a sharp limiting operation. Based on our observations, if a number of cavities are used in cascade a sharper curve can be obtained. It should be noted that the distance between the cavities should be chosen long enough so that they cannot have loading effect on each other. The proposed structure is shown in Fig. 11a. In order to be able to observe the switching behaviour more accurately, the transmission ratio of the switch versus the input power has been depicted in Fig. 11b. We need that by doubling the input power a switching mechanism be observed in our design; therefore the

0.2

0.4

0.6

Transmission

Fig. 11. (a) The limiter topology used to design an optical gate, (b) The transmission curve

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

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

0.8

1

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>0</sup>

Input Power (W)

limiter is shown in Fig. 10b and Fig. 10c.

triple cascade combination can be used for our design.

(a) (b)

couples with the input and output waveguides.

for the limiter in fig. 11(a).

**4.2 A PC Y-splitter** 

be expressed as (31).

Now, if we assume that the input signal launched only from the left (s2−=0) and that external losses can be neglected (τ<sup>2</sup> →∞), (29) can be expressed in terms of just s1+ and s1− as:

$$\frac{ds\_{1-}}{dt} = \mathrm{i}o\_c s\_{1-} - \frac{1}{\tau\_1} s\_{1-} - \frac{1}{\tau\_1} s\_{1+}.\tag{29}$$

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 intensity (c) Transmission for high input intensity.

It can be shown that for Kerr type nonlinearity:

$$\frac{ds\_{1-}}{dt} = i\alpha\_c \left(1 - \frac{1}{2Q} \frac{\left|s\_{1-}\right|^2}{P\_0}\right) s\_{1-} - \frac{1}{\tau\_1} s\_{1-} - \frac{1}{\tau\_1} s\_{1+} \tag{30}$$

where Q is the quality factor of the cavity and P0 is characteristic power of the system (See (Bravo-Abad et. al, 2007) for its definition.).

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 waveguide a row of holes is removed from the PC.

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 such a device. First we assume that the PC device is comprised of linear element (n2 =0); in such a case the input frequency is tuned at the cavity's notch. No signal can pass the waveguide in this case. Then we assume that the defect has Kerr effect which causes a shift in cavity's central frequency and allows the signal to pass. The switching mechanism of this limiter is shown in Fig. 10b and Fig. 10c.

The mentioned structure cannot provide a sharp limiting operation. Based on our observations, if a number of cavities are used in cascade a sharper curve can be obtained. It should be noted that the distance between the cavities should be chosen long enough so that they cannot have loading effect on each other. The proposed structure is shown in Fig. 11a. In order to be able to observe the switching behaviour more accurately, the transmission ratio of the switch versus the input power has been depicted in Fig. 11b. We need that by doubling the input power a switching mechanism be observed in our design; therefore the triple cascade combination can be used for our design.

Fig. 11. (a) The limiter topology used to design an optical gate, (b) The transmission curve for the limiter in fig. 11(a).
