NNN n (HL) HD(HL) HD(HL) H Glass 0

Where, *H* and *L* denotes for high index (TiO2) and low index (SiO2) materials as *nH*=2.33 and *nL*=1.45, respectively for construction the basic resonators. Also, *N* is number of the repetition for the basic structure. It can be shown that by increasing *N,* light confinement in defect layer will be increased. Fig.1. Shows the basic structure of 1D-NCCW.

Fig. 1. Schematic illustration of 1D-NCCW with two defects.

As an example, defect layer, *D* is consist of CdSe which is an intensity dependent refractive index material.

The input pulse comes from left hand and during passing through whole of the 1D-NCCW, experiences delay time. This phenomenon is as a reason of reduction the speed of light propagation and confinement in the defect layer. It can be seen the enhancement of the electric field component of the incident light in the defect layer [21].

Nonlinear refraction is commonly defined either in terms of the optical field intensity *I* as [23]:

$$\mathbf{n}\_{\rm d} = \mathbf{n}\_{\rm d0} + \boldsymbol{\mathcal{Y}} \tag{1}$$

Dynamic All Optical Slow Light Tunability

when input I 0 .

by Using Nonlinear One Dimensional Coupled Cavity Waveguides 35

twin mini transmission frequencies band (resonance frequencies). It can be seen by increasing the input optical intensity, (form 0 to 6.4mW/cm2); the resonance modes shift

toward right side in the frequency domain (blue shift in wavelength domain).

Fig. 2. Illustration of twin resonance bands in transmission spectrum of 1D-NCCW

Fig. 3. Right side frequency shifting of the transmitted twin resonance modes when the

Under TB approximation, it can be shown that the group velocity at the resonance bands

input intensity changes from 0 to 6.4mW/cm2 .

can be written as [25]:

Or in terms of average of the square of the optical electric field <E2> as:

$$\mathbf{n}\_{\rm d} = \mathbf{n}\_{\rm d0} + \mathbf{n}\_2 \left< \mathbf{E}^2 \right>\tag{2}$$

Where *nd0* is the ordinary linear refractive index and γ is the nonlinear refractive index coefficient and *n2* is the nonlinear refractive index. The conversion between *n2* and γ can be written as [23]:

$$\text{tr}\_2[\text{cm}^3/\text{erg}] = (\text{cn}\_0/40\pi)\,\gamma[\text{m}^2/\text{W}] = 238.7 \,\text{n}\_0 \,\gamma[\text{cm}^2/\text{W}] \tag{3}$$

For CdSe γ=-147cm2/W and *nd0*=2.56. By choosing 0 1.55 m as the practical optical communication wavelength, and choosing quarter-wavelength optical thickness for high index and low index materials ( <sup>0</sup> nd nd HH LL <sup>4</sup> , where 0 1.55 m ), TiO2 and SiO2 respectively ( for constructing the basic resonators), it can be derived the central frequency for the basic resonator as <sup>15</sup> 1.21 10 Hz .

According to TB approximation, in the presence of defect layer between each resonators with initial half-wavelength optical thickness ( <sup>0</sup> n dd0 d <sup>2</sup> ), the central frequency of each resonator, will be split to two eigen frequency due to coupling of the individual cavity modes [15]. In the case which *Iin=0*, the transmission properties of the 1D-NCCW has been investigated by transfer matrix method (TMM) which is widely used for calculating the optical properties of alternative stack layers. TMM is based on solving the Maxwell's equations in each individual layer and considering the continuity conditions for the electric and magnetic components of the incident electromagnetic field yields to obtain the characteristics matrix of each individual layer as following:

$$\mathbf{M}\_{\mathbf{q}} = \begin{pmatrix} \cos \delta\_{\mathbf{q}} & \mathbf{i} \sin \delta\_{\mathbf{q}} \left/ \mathbf{n}\_{\mathbf{q}} \right. \\ \mathbf{i} \mathbf{n}\_{\mathbf{q}} \sin \delta\_{\mathbf{q}} & \cos \delta\_{\mathbf{q}} \end{pmatrix} \tag{4}$$

where n and q <sup>q</sup> denotes the refractive index and phase thickness ( q qq 0 2 nd / ) of the qth layer respectively and 0 is the wavelength of the incoming light. By multiplication of the characteristics matrix of each layer, transfer matrix of the multilayered structure can be obtained. Therefore, by using the TMM method the electric and magnetic components of the input and output signal through the whole of structure can be obtained. Hence, optical properties such as transmission, phase characteristics and dispersion and its higher order such as group velocity and third order dispersion of the structure can be derived. Fig.2 shows the transmission spectrum of the structure in the case which input I 0 .

For investigating the optical properties of the 1D-NCCW in the presence of input optical intensity signal, Eq.1 is used for determination of refractive index of the nonlinear defect layer. As an approximation method the transmission spectral characteristics of the 1D-NCCW can be obtained in the presence of nonlinear phenomena in the defect lyers. Painou *etal* and Johnson *etal*, have confirmed the convergence and correctness of this approximate approach [24]. Fig.3 shows the effect of increasing the input intensity on the position of the

Where *nd0* is the ordinary linear refractive index and γ is the nonlinear refractive index coefficient and *n2* is the nonlinear refractive index. The conversion between *n2* and γ can be

3 22 n [cm /er 2 0 g <sup>0</sup> ] (cn / 40 ) [m / W] 238.7 n [cm / W]

communication wavelength, and choosing quarter-wavelength optical thickness for high

respectively ( for constructing the basic resonators), it can be derived the central frequency

According to TB approximation, in the presence of defect layer between each resonators

resonator, will be split to two eigen frequency due to coupling of the individual cavity modes [15]. In the case which *Iin=0*, the transmission properties of the 1D-NCCW has been investigated by transfer matrix method (TMM) which is widely used for calculating the optical properties of alternative stack layers. TMM is based on solving the Maxwell's equations in each individual layer and considering the continuity conditions for the electric and magnetic components of the incident electromagnetic field yields to obtain the

q qq

 

> 

0 is the wavelength of the incoming light. By multiplication of

qq q cos isin / n <sup>M</sup> in sin cos 

denotes the refractive index and phase thickness ( q qq 0

 

the characteristics matrix of each layer, transfer matrix of the multilayered structure can be obtained. Therefore, by using the TMM method the electric and magnetic components of the input and output signal through the whole of structure can be obtained. Hence, optical properties such as transmission, phase characteristics and dispersion and its higher order such as group velocity and third order dispersion of the structure can be derived. Fig.2

For investigating the optical properties of the 1D-NCCW in the presence of input optical intensity signal, Eq.1 is used for determination of refractive index of the nonlinear defect layer. As an approximation method the transmission spectral characteristics of the 1D-NCCW can be obtained in the presence of nonlinear phenomena in the defect lyers. Painou *etal* and Johnson *etal*, have confirmed the convergence and correctness of this approximate approach [24]. Fig.3 shows the effect of increasing the input intensity on the position of the

, where 0

 1.55 m

 

<sup>2</sup> n n nE d d0 2 (2)

 1.55 m

), the central frequency of each

(4)

 2 nd / 

 ) of the

(3)

as the practical optical

), TiO2 and SiO2

Or in terms of average of the square of the optical electric field <E2> as:

For CdSe γ=-147cm2/W and *nd0*=2.56. By choosing 0

index and low index materials ( <sup>0</sup> nd nd HH LL <sup>4</sup>

with initial half-wavelength optical thickness ( <sup>0</sup> n dd0 d <sup>2</sup>

characteristics matrix of each individual layer as following:

q

shows the transmission spectrum of the structure in the case which input I 0 .

1.21 10 Hz .

for the basic resonator as <sup>15</sup>

written as [23]:

where n and q <sup>q</sup>

qth layer respectively and

twin mini transmission frequencies band (resonance frequencies). It can be seen by increasing the input optical intensity, (form 0 to 6.4mW/cm2); the resonance modes shift toward right side in the frequency domain (blue shift in wavelength domain).

Fig. 2. Illustration of twin resonance bands in transmission spectrum of 1D-NCCW when input I 0 .

Fig. 3. Right side frequency shifting of the transmitted twin resonance modes when the input intensity changes from 0 to 6.4mW/cm2 .

Under TB approximation, it can be shown that the group velocity at the resonance bands can be written as [25]:

Dynamic All Optical Slow Light Tunability

by Using Nonlinear One Dimensional Coupled Cavity Waveguides 37

group

Where *L* is the length of the 1D-NCCW and equal to 1.42×10-5 m.

Fig. 5. Slow down factor increasing as a function of input intensity.

Fig. 6. Group delay versus of input intensity.

As a consequence of Eqs.5,7, the input intensity dependent of group delay of the resonance modes during propagation through the 1D-NCCW is obvious. Fig.6 shows group delay for the first and second resonance modes when the input intensity changes from 0 to 6,4 mW/cm2. As an example, for the second mode, it can be seen by adjusting the input intensity between 0 to 6.4mW/cm2, the group delay can be tuned from 1.1ns to 0.05ns.

group

(7)

V L

$$\mathbf{V}\_{\text{g}} = -\kappa \mathbf{R} \mathcal{Q} \sin(\mathbf{k} \mathbf{R}) \tag{5}$$

where *ĸ* indicates coupling factor that is the value related to the overlapping of the electric field between the localized modes. <sup>6</sup> R 4.8 10 m , is the separation between each of defect mediums and *k* is the wave vector of the light traveling in 1D-NCCW.

The coupling factor can be written as [25]:

$$\kappa = \frac{\left(\Delta o o\_{\text{res}}\right)\_{\text{FWHM}}}{2\mathcal{Q}}\tag{6}$$

which indicates the inverse proportionality of the coupling factor to the quality factor ( Q ) of cavity modes ( <sup>1</sup> 2Q ). Fig.4 indicates the variation of the *Q* for the resonance modes in the presence of input power.

Fig. 4. variation the *Q* factor of the twin resonance modes versus optical input power.

As a result of increasing the input optical intensity, *Q* factor of each resonance modes of changes and hence, according to Eqs.5,6 coupling coefficient and group velocity at resonance modes alternate. Fig.5 shows the slow down factor, *S=Vg /C*, as a function of the input optical intensity, where *C* is the speed of light in vacuum.

It can be seen when the input optical intensity changes from 0 to 6.4mW/cm2, the slow down factor can be tuned from 0.43 10-4 to 5.6 10-4 for the first resonance mode and 0.43 10-4 to 9.2 10-4 for the second mode. Group delay of resonance modes propagation through the 1D-NCCW, can be derived as following:

V R sin(kR) <sup>g</sup> 

where *ĸ* indicates coupling factor that is the value related to the overlapping of the electric field between the localized modes. <sup>6</sup> R 4.8 10 m , is the separation between each of defect

> ( ) res FWHM 2

which indicates the inverse proportionality of the coupling factor to the quality factor ( Q )

Fig. 4. variation the *Q* factor of the twin resonance modes versus optical input power.

input optical intensity, where *C* is the speed of light in vacuum.

through the 1D-NCCW, can be derived as following:

As a result of increasing the input optical intensity, *Q* factor of each resonance modes of changes and hence, according to Eqs.5,6 coupling coefficient and group velocity at resonance modes alternate. Fig.5 shows the slow down factor, *S=Vg /C*, as a function of the

It can be seen when the input optical intensity changes from 0 to 6.4mW/cm2, the slow down factor can be tuned from 0.43 10-4 to 5.6 10-4 for the first resonance mode and 0.43 10-4 to 9.2 10-4 for the second mode. Group delay of resonance modes propagation

). Fig.4 indicates the variation of the *Q* for the resonance modes in

mediums and *k* is the wave vector of the light traveling in 1D-NCCW.

The coupling factor can be written as [25]:

2Q

of cavity modes ( <sup>1</sup>

the presence of input power.

(5)

(6)

$$
\tau\_{\text{group}} = \frac{\mathbf{V}\_{\text{group}}}{\mathbf{L}} \tag{7}
$$

Where *L* is the length of the 1D-NCCW and equal to 1.42×10-5 m.

Fig. 5. Slow down factor increasing as a function of input intensity.

As a consequence of Eqs.5,7, the input intensity dependent of group delay of the resonance modes during propagation through the 1D-NCCW is obvious. Fig.6 shows group delay for the first and second resonance modes when the input intensity changes from 0 to 6,4 mW/cm2. As an example, for the second mode, it can be seen by adjusting the input intensity between 0 to 6.4mW/cm2, the group delay can be tuned from 1.1ns to 0.05ns.

Fig. 6. Group delay versus of input intensity.

Dynamic All Optical Slow Light Tunability

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As mentioned before, photonic structures suffer from fundamental trade off between transmission bandwidth (FWHM of the transmitted resonance modes) and the optical delay, as delay bandwidth product (DBP).

The DBP variation in 1D-NCCW when the optical input intensity changes, is shown in Fig.7 As an example, for the second mode, it can be seen the DBP can be tuned from 17.4 to 15.6 by increasing the input optical intensity up to 6.4mW/cm2.

Fig. 7. DBP at the resonance modes versus input intensity.
