**3.1. Photonic crystal device for multiple wavelengths filtering**

By inserting photonic quantum–wells (PQWs) into an ideal photonic crystal, a series of the discrete defect states may be created and they provide the function of multiple channeled filtering. Many researches have been reported on how to generate the defect states. However, the frequencies of the defect states cannot be changed with freedom. In practice, the favorable design of optical multiple channeled filters need to pass arbitrarily preassigned frequencies. In this section, the issue of designing the specific PQWs which have the preassigned filtering channels is discussed. The aperiodic PQWs (APQWs) are sandwiched by two finite-length ideal photonic crystals, which consist of two alternately stacked layers *A* and *B* with different dielectric constants of *ε<sup>A</sup>* and *εB*, respectively. Their thicknesses are denoted by *dA* and *dB*, respectively, and *a* = *dA* + *dB* is the lattice constant of the one-dimensional (1D) photonic crystal. The APQWs are composed of two different alternately stacked basic constituent layers with the dielectric constants of *ε<sup>C</sup>* and *εD*. However, the thickness of each individual layer may not be equal and the individual layer thickness is determined by the merits of the desirable filters.

The transmission spectrum of designed APQW structures is calculated by using the transfer-matrix method. The transfer-matrix in each individual layer can be obtained by solving the Maxwell equations with a combination of boundary conditions. For a normally incident EM plane wave with the TE polarization, the transfer-matrix for the *j*-th layer is given by

$$
\hat{M}\_{\dot{\jmath}} = \hat{G}\_{\dot{\jmath}+1}^{-1} \hat{G}\_{\dot{\jmath}} \hat{P}\_{\dot{\jmath}\prime} \tag{28}
$$

for the *j*th layer in sample. *G<sup>j</sup>* reads

obtain the transmission probability.

*<sup>O</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>* <sup>∑</sup>*<sup>s</sup>*

decided by the SA algorithm.

with

where *<sup>ω</sup>*(*s*)


*ω*0(= *ωa*) < *ω<sup>o</sup>*

*G<sup>j</sup>* =

 *tn rn* = ∏ *j M j t*<sup>1</sup> *r*1 

*T* =

*<sup>α</sup>* <sup>|</sup>, *<sup>ω</sup>*(*o*)

*ε<sup>n</sup> ε*1 *tn t*1 2

For the wave which is not normal incident or TM mode, the similar approach can be used to

To design the APQW for producing specified defect states located at the preset frequencies *<sup>ω</sup>*(0) *<sup>α</sup>* within a given range of [*ω<sup>a</sup> <sup>ω</sup>b*], a perfect PC should be selected to serve as the prototype photonic crystal, into which the APQW is implanted. It is required that the chosen prototype PC should have an appropriate PBG located at this frequency range and with a certain width, not narrower than the range of [*ω<sup>a</sup> ωb*]. After determining this prototype PC, the APQW structure is determined by using the SA algorithm. The objective function is defined as

*<sup>α</sup>* , *<sup>ω</sup>*(*s*)

<sup>1</sup> <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> <sup>&</sup>lt; .... <sup>&</sup>lt; *ωα*−<sup>1</sup> <sup>&</sup>lt; *<sup>ω</sup><sup>o</sup>*

generated in every routed configuration of the APQW during the SA routine. *ωα* denote a series of the frequencies to partition the whole region of [*ωa*, *ωb*] into several subregions. In each subregion, only one of the preassigned confined states appears. For instance, *<sup>ω</sup>*(*o*)

located in the subregion of [*ωα*−1, *ωα*]. In the SA routine, more than one defect states may occur in one subregion, thus, *s* may be larger than 1, therefore, the sum over *s* should be take into account. The optimal design of the APQWs corresponds to a search for the minimum of *O*. The sandwiched part in the sample is divided into *n* unit blocks with the thickness *δd*; the dielectric constant of each individual block is chosen as one of the binary of *ε<sup>C</sup>* and *εD*,

An APQW is design for achieving two channeled filtering in a given range of [0.352 0.506](2*πc*/*a*), *c* is the speed of light in the vacuum. The dielectric constants are selected as *ε<sup>A</sup>* = *ε<sup>C</sup>* = 13.0 and *ε<sup>B</sup>* = *ε<sup>D</sup>* = 1.0. The thicknesses of the constituent layers *A* and *B* in the prototype PC are set to be *dA* = *dB* = 0.5*a*, thus, the second PBG of the prototype PC just is located at [0.352 0.506](2*πc*/*a*). In the following calculations, five *AB* layers, (*AB*)5,

*<sup>α</sup>* denotes the frequencies of the defect states appearing in [*ωα*−1, *ωα*], which are

for instance, the transmission probability is given by

 1 1 <sup>√</sup>*ε<sup>j</sup>* <sup>−</sup>√*ε<sup>j</sup>*

where *dj* denotes the thickness of the *<sup>j</sup>*-th layer, *kj* = <sup>2</sup>*π*√*εj*/*λ*, *<sup>ε</sup><sup>j</sup>* is the dielectric constant of the *j*-th layer; *λ* the wavelength of the incident light wave in vacuum. Thus, the total transfer matrix can be obtained by multiplying all individual transfer matrixes in sequence. The transmission and reflection coefficients of EM waves of the sample can be calculated from

, (31)

Optimization Design of Nonlinear Optical Frequency Conversion Devices Using Simulated Annealing Algorithm

105

, (32)

. (33)

*<sup>α</sup>* ∈ [*ωα*−1, *ωα*], *<sup>α</sup>* = 1, 2, 3.... (34)

*<sup>α</sup>* < *ωα*(= *ωb*), (35)

*<sup>α</sup>* is

where *j* ∈ {1, 2, ..., *N*} for *N* number of layers. *G<sup>j</sup>* is the transfer matrix. The propagating matrix *P<sup>j</sup>* reads

$$
\hat{P}\_1 = \begin{pmatrix} 1 \ 0 \\ 0 \ 1 \end{pmatrix}, \quad j = 1 \tag{29}
$$

for air at the most left-hand side of the sample and

$$
\hat{P}\_{\hat{l}} = \begin{pmatrix}
\exp(ik\_{\hat{l}}d\_{\hat{l}}) & 0 \\
0 & \exp(-ik\_{\hat{l}}d\_{\hat{j}})
\end{pmatrix}, \quad \hat{j} = 2, 3, 4... \tag{30}
$$

for the *j*th layer in sample. *G<sup>j</sup>* reads

12 Will-be-set-by-IN-TECH

amplification coefficient as a function of wavelength for signal light. There exist three strong expected peaks and some small dense oscillation as background, satellite peaks are quite low. The amplifier coefficients are 1.0240, 1.0240, and 1.0237 for signal wavelengths 1.7*μm*, 1.8*μm*, and 1.9*μm*, respectively. The average value < *G*(*λα*) > is 1.0239 and the maximal relative deviation is <sup>Δ</sup>*<sup>G</sup>* = [*max*{*G*(*λα*)} − *min*{*G*(*λα*)}]/ <sup>&</sup>lt; *<sup>G</sup>*(*λα*) <sup>&</sup>gt;<sup>=</sup> 2.9 <sup>×</sup> <sup>10</sup>−4. These data show that the constructed aperiodic optical superlattice can meet the predefined requirement well.

Photonic crystals have attracted extensive attentions in the past decades. Two remarkable characters of the photonic crystals are photonic band gaps (PBGs) and defect states [6]. Based

By inserting photonic quantum–wells (PQWs) into an ideal photonic crystal, a series of the discrete defect states may be created and they provide the function of multiple channeled filtering. Many researches have been reported on how to generate the defect states. However, the frequencies of the defect states cannot be changed with freedom. In practice, the favorable design of optical multiple channeled filters need to pass arbitrarily preassigned frequencies. In this section, the issue of designing the specific PQWs which have the preassigned filtering channels is discussed. The aperiodic PQWs (APQWs) are sandwiched by two finite-length ideal photonic crystals, which consist of two alternately stacked layers *A* and *B* with different dielectric constants of *ε<sup>A</sup>* and *εB*, respectively. Their thicknesses are denoted by *dA* and *dB*, respectively, and *a* = *dA* + *dB* is the lattice constant of the one-dimensional (1D) photonic crystal. The APQWs are composed of two different alternately stacked basic constituent layers with the dielectric constants of *ε<sup>C</sup>* and *εD*. However, the thickness of each individual layer may not be equal and the individual layer thickness is determined by the merits of the desirable

The transmission spectrum of designed APQW structures is calculated by using the transfer-matrix method. The transfer-matrix in each individual layer can be obtained by solving the Maxwell equations with a combination of boundary conditions. For a normally incident EM plane wave with the TE polarization, the transfer-matrix for the *j*-th layer is given

*<sup>M</sup> <sup>j</sup>* <sup>=</sup> *<sup>G</sup>*<sup>−</sup><sup>1</sup>

 1 0 0 1 

0 exp(−*ikjdj*)

*P* <sup>1</sup> =

exp(*ikjdj*) 0

for air at the most left-hand side of the sample and

*P<sup>j</sup>* =

where *j* ∈ {1, 2, ..., *N*} for *N* number of layers. *G<sup>j</sup>* is the transfer matrix. The propagating

*<sup>j</sup>*+1*GjP<sup>j</sup>*, (28)

, *j* = 1 (29)

, *j* = 2, 3, 4... (30)

**3. Design of photonic crystal devices**

filters.

by

matrix *P<sup>j</sup>* reads

on these two characters, many devices can be designed.

**3.1. Photonic crystal device for multiple wavelengths filtering**

$$
\hat{G}\_{\hat{\jmath}} = \begin{pmatrix} 1 & 1 \\ \sqrt{\mathbb{E}\_{\hat{\jmath}}^{\hat{\jmath}}} - \sqrt{\mathbb{E}\_{\hat{\jmath}}^{\hat{\jmath}}} \end{pmatrix}, \tag{31}
$$

where *dj* denotes the thickness of the *<sup>j</sup>*-th layer, *kj* = <sup>2</sup>*π*√*εj*/*λ*, *<sup>ε</sup><sup>j</sup>* is the dielectric constant of the *j*-th layer; *λ* the wavelength of the incident light wave in vacuum. Thus, the total transfer matrix can be obtained by multiplying all individual transfer matrixes in sequence. The transmission and reflection coefficients of EM waves of the sample can be calculated from

$$
\begin{pmatrix} t\_n \\ r\_n \end{pmatrix} = \prod\_j \hat{M}\_j \begin{pmatrix} t\_1 \\ r\_1 \end{pmatrix} \tag{32}
$$

for instance, the transmission probability is given by

$$T = \sqrt{\frac{\varepsilon\_n}{\varepsilon\_1}} \left| \frac{t\_n}{t\_1} \right|^2. \tag{33}$$

For the wave which is not normal incident or TM mode, the similar approach can be used to obtain the transmission probability.

To design the APQW for producing specified defect states located at the preset frequencies *<sup>ω</sup>*(0) *<sup>α</sup>* within a given range of [*ω<sup>a</sup> <sup>ω</sup>b*], a perfect PC should be selected to serve as the prototype photonic crystal, into which the APQW is implanted. It is required that the chosen prototype PC should have an appropriate PBG located at this frequency range and with a certain width, not narrower than the range of [*ω<sup>a</sup> ωb*]. After determining this prototype PC, the APQW structure is determined by using the SA algorithm. The objective function is defined as

$$O = \sum\_{\mathfrak{a}} \sum\_{\mathfrak{s}} |\omega\_{\mathfrak{a}}^{(o)} - \omega\_{\mathfrak{a}}^{(s)}|, \ \omega\_{\mathfrak{a}}^{(o)}, \omega\_{\mathfrak{a}}^{(s)} \in [\omega\_{\mathfrak{a}-1\prime} \quad \omega\_{\mathfrak{a}}], \ \mathfrak{a} = 1, 2, 3... \tag{34}$$

with

$$
\omega\_0 (= \omega\_a) < \omega\_1^0 < \omega\_1 < \dots < \omega\_{a-1} < \omega\_a^0 < \omega\_a (= \omega\_b) \tag{35}
$$

where *<sup>ω</sup>*(*s*) *<sup>α</sup>* denotes the frequencies of the defect states appearing in [*ωα*−1, *ωα*], which are generated in every routed configuration of the APQW during the SA routine. *ωα* denote a series of the frequencies to partition the whole region of [*ωa*, *ωb*] into several subregions. In each subregion, only one of the preassigned confined states appears. For instance, *<sup>ω</sup>*(*o*) *<sup>α</sup>* is located in the subregion of [*ωα*−1, *ωα*]. In the SA routine, more than one defect states may occur in one subregion, thus, *s* may be larger than 1, therefore, the sum over *s* should be take into account. The optimal design of the APQWs corresponds to a search for the minimum of *O*. The sandwiched part in the sample is divided into *n* unit blocks with the thickness *δd*; the dielectric constant of each individual block is chosen as one of the binary of *ε<sup>C</sup>* and *εD*, decided by the SA algorithm.

An APQW is design for achieving two channeled filtering in a given range of [0.352 0.506](2*πc*/*a*), *c* is the speed of light in the vacuum. The dielectric constants are selected as *ε<sup>A</sup>* = *ε<sup>C</sup>* = 13.0 and *ε<sup>B</sup>* = *ε<sup>D</sup>* = 1.0. The thicknesses of the constituent layers *A* and *B* in the prototype PC are set to be *dA* = *dB* = 0.5*a*, thus, the second PBG of the prototype PC just is located at [0.352 0.506](2*πc*/*a*). In the following calculations, five *AB* layers, (*AB*)5, on either side of the APQWs are employed. The sandwiched part is divided into *n* = 100 blocks and the thickness of the basic block *δd* = 0.02*a* is selected. Two filtering frequencies are preassigned as *<sup>ω</sup>*(*o*) <sup>1</sup> <sup>=</sup> 0.420(2*πc*/*a*) and *<sup>ω</sup>*(*o*) <sup>2</sup> = 0.480(2*πc*/*a*). The transmission spectrum of the designed sample is displayed in Fig. 7. The frequency increment in the scan is taken as *δω* <sup>=</sup> 1.0 <sup>×</sup> <sup>10</sup>−5(2*πc*/*a*) to ensure that any unwanted extra stray frequency peak does not occur. Two dashed vertical lines remark the positions of the second PBG of the prototype (*AB*)<sup>5</sup> photonic crystal. It is evident that there exist only two expected defect states in the desired frequency range. The frequencies of the defect states accord exactly with the preset values. Their transmittances are 0.90 and 0.86 for *ω*<sup>1</sup> = 0.420(2*πc*/*a*) and *ω*<sup>2</sup> = 0.480(2*πc*/*a*), respectively.

For simplicity, a one-dimensional layer structure is used as the sample. The incident wave is normally impinged upon the surface of the sample. For the *l* − *th* layer, the electric field

> (1)2 *l*

 *E*(1)

*<sup>l</sup>* (*z*) = −*k*

 *d*<sup>2</sup> *dz*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*

(2)2 *l*

 *E*(2)

*<sup>l</sup> e ik*(1)

*<sup>l</sup>* <sup>=</sup> 1 and *<sup>B</sup>*(1)

*<sup>l</sup> <sup>e</sup>*−*ik*(1)

 *d*<sup>2</sup> *dz*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*

(2) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(2)

*E*(1)

*<sup>l</sup>* (*z*) = *<sup>A</sup>*(1)

can be obtained. *N* is the total number of layers in the sample.

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *B*(1)

*<sup>l</sup> k*10, *k*

The solution of Equation 36 has the form

*<sup>l</sup>* and *<sup>B</sup>*(1)

and the initial conditions ( *A*(1)

−2*k*<sup>2</sup> <sup>20</sup>*χ<sup>l</sup> k* (2)2 *l*

Using the initial conditions *A*(2)

defined respectively as follows

*<sup>l</sup> <sup>e</sup>ik*(2)

*A*(1) *<sup>l</sup> <sup>B</sup>*(1)

respectively. *C*<sup>21</sup> and *C*<sup>22</sup> can be obtained as

<sup>2</sup> ) of the FW (SHG), neglecting the pump power depletion, satisfies the following

*<sup>l</sup>* ) is the refractive index of the *l* − *th* layer for the wavelength of FW (SHG), *c* is the velocity of the light in vacuum and *<sup>χ</sup><sup>l</sup>* is the nonlinear optical coefficient of the *<sup>l</sup>* <sup>−</sup> *th* layer.

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *B*(1)

respectively. Utilizing the continuous condition at each interface, the transfer matrix method,

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *C*21*ei*2*<sup>k</sup>*

*<sup>l</sup>* represent the amplitudes of the forward and backward SHGs at interface,

<sup>20</sup>*χ<sup>l</sup> <sup>A</sup>*(1)<sup>2</sup> *l*

20*χlB*(1)<sup>2</sup> *l*

, *<sup>η</sup>*(2)

(1)2 *l*

(1)2 *l*

*back* <sup>=</sup> <sup>|</sup>*B*(2)

1 | 2


,

Similarly, the SHG electric field in the *l* − *th* layer of photonic crystal can be expressed as

*<sup>C</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>C</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>N</sup>* <sup>|</sup>*A*(2) *N* | 2

*n*(1) <sup>1</sup> <sup>|</sup>*A*(1) <sup>1</sup> |<sup>2</sup>

<sup>1</sup> <sup>=</sup> 0, *<sup>B</sup>*(2)

*f orth* <sup>=</sup> *<sup>n</sup>*(2)

*<sup>η</sup>*(2)

*k* (2)2 *<sup>l</sup>* − 4*k*

*k* (2)2 *<sup>l</sup>* − 4*k*

can be derived. The conversion efficiencies of the forward and backward SHG waves are

As an example, the photonic crystal structure that can implement multiple wavelengths SHG is designed. The PQWs gives localized states at the frequencies of the FW and SHG,

(2)2 <sup>20</sup> *<sup>χ</sup><sup>l</sup>*

> *<sup>l</sup> e* −*ik*(1)

*<sup>l</sup>* represent the amplitudes of forward and backward FW at interface,

(1)

*<sup>l</sup>* , (39)

(*z*)*E*(1)<sup>2</sup>

*<sup>l</sup> k*20, *k*<sup>10</sup> = *ω*/*c*, *k*<sup>20</sup> = 2*ω*/*c*, *ω* is the frequency of the FW,

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>)

*<sup>N</sup>* = 0), the electric field of the FW in each layer

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *C*22*e*−*i*2*<sup>k</sup>*

*<sup>N</sup>* = 0, the electric field of the SHG at each interface

*<sup>l</sup>* (*z*) = 0, (36)

Optimization Design of Nonlinear Optical Frequency Conversion Devices Using Simulated Annealing Algorithm

*<sup>l</sup>* (*z*), (37)

107

, (38)

(1) *<sup>l</sup>* (*z*−*zl*−<sup>1</sup>)

. (40)

. (41)

*E*(1) *<sup>l</sup>* (*E*(1)

equations.

where *k*

where *A*(1)

*E*(2)

*<sup>l</sup>* and *<sup>B</sup>*(2)

*A*(2)

*<sup>l</sup>* (*z*) = *<sup>A</sup>*(2)

*n*(1) *<sup>l</sup>* (*n*(2)

(1) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(1)

**Figure 7.** Transmission spectrum of the designed APQW sample for two channeled filtering at the preset frequencies of *ω*(*o*) <sup>1</sup> <sup>=</sup> 0.420 and *<sup>ω</sup>*(*o*) <sup>2</sup> = 0.480(2*πc*/*a*).

### **3.2. Photonic crystal device for multiple wavelengths' second harmonic generation**

When a defect is introduced into a perfect photonic crystal, the defect mode will appear in the band gap. Furthermore, the light with the frequency corresponding to the defect mode will be located around the defect layer; the intensity of the located wave has been improved 3-4 orders comparing with the intensity of the incident wave. Therefore, if a nonlinear material is used as the defect medium, the nonlinear effect can be greatly enhanced. It was found that when the frequency of the fundamental wave (FW) was tuned to the defect state, the SHG can be greatly enhanced. Due to the strong localization, low group velocity, and spatial phase locking in the PC, the giant enhancement of SHG for each FW has been achieved.

For simplicity, a one-dimensional layer structure is used as the sample. The incident wave is normally impinged upon the surface of the sample. For the *l* − *th* layer, the electric field *E*(1) *<sup>l</sup>* (*E*(1) <sup>2</sup> ) of the FW (SHG), neglecting the pump power depletion, satisfies the following equations.

14 Will-be-set-by-IN-TECH

on either side of the APQWs are employed. The sandwiched part is divided into *n* = 100 blocks and the thickness of the basic block *δd* = 0.02*a* is selected. Two filtering frequencies are

of the designed sample is displayed in Fig. 7. The frequency increment in the scan is taken as *δω* <sup>=</sup> 1.0 <sup>×</sup> <sup>10</sup>−5(2*πc*/*a*) to ensure that any unwanted extra stray frequency peak does not occur. Two dashed vertical lines remark the positions of the second PBG of the prototype (*AB*)<sup>5</sup> photonic crystal. It is evident that there exist only two expected defect states in the desired frequency range. The frequencies of the defect states accord exactly with the preset values. Their transmittances are 0.90 and 0.86 for *ω*<sup>1</sup> = 0.420(2*πc*/*a*) and *ω*<sup>2</sup> = 0.480(2*πc*/*a*),

**Figure 7.** Transmission spectrum of the designed APQW sample for two channeled filtering at the preset

When a defect is introduced into a perfect photonic crystal, the defect mode will appear in the band gap. Furthermore, the light with the frequency corresponding to the defect mode will be located around the defect layer; the intensity of the located wave has been improved 3-4 orders comparing with the intensity of the incident wave. Therefore, if a nonlinear material is used as the defect medium, the nonlinear effect can be greatly enhanced. It was found that when the frequency of the fundamental wave (FW) was tuned to the defect state, the SHG can be greatly enhanced. Due to the strong localization, low group velocity, and spatial phase

<sup>2</sup> = 0.480(2*πc*/*a*).

**3.2. Photonic crystal device for multiple wavelengths' second harmonic**

locking in the PC, the giant enhancement of SHG for each FW has been achieved.

<sup>2</sup> = 0.480(2*πc*/*a*). The transmission spectrum

<sup>1</sup> <sup>=</sup> 0.420(2*πc*/*a*) and *<sup>ω</sup>*(*o*)

preassigned as *<sup>ω</sup>*(*o*)

respectively.

frequencies of *ω*(*o*)

**generation**

<sup>1</sup> <sup>=</sup> 0.420 and *<sup>ω</sup>*(*o*)

$$\left[\frac{d^2}{dz^2} + k\_l^{(1)2}\right] E\_l^{(1)}(z) = 0,\tag{36}$$

$$
\left[\frac{d^2}{dz^2} + k\_l^{(2)2}\right] E\_l^{(2)}(z) = -k\_{20}^{(2)2} \chi^l(z) E\_l^{(1)2}(z),\tag{37}
$$

where *k* (1) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(1) *<sup>l</sup> k*10, *k* (2) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(2) *<sup>l</sup> k*20, *k*<sup>10</sup> = *ω*/*c*, *k*<sup>20</sup> = 2*ω*/*c*, *ω* is the frequency of the FW, *n*(1) *<sup>l</sup>* (*n*(2) *<sup>l</sup>* ) is the refractive index of the *l* − *th* layer for the wavelength of FW (SHG), *c* is the velocity of the light in vacuum and *<sup>χ</sup><sup>l</sup>* is the nonlinear optical coefficient of the *<sup>l</sup>* <sup>−</sup> *th* layer. The solution of Equation 36 has the form

$$E\_l^{(1)}(z) = A\_l^{(1)} e^{i k\_l^{(1)}(z - z\_{l-1})} + B\_l^{(1)} e^{-i k\_l^{(1)}(z - z\_{l-1})},\tag{38}$$

where *A*(1) *<sup>l</sup>* and *<sup>B</sup>*(1) *<sup>l</sup>* represent the amplitudes of forward and backward FW at interface, respectively. Utilizing the continuous condition at each interface, the transfer matrix method, and the initial conditions ( *A*(1) *<sup>l</sup>* <sup>=</sup> 1 and *<sup>B</sup>*(1) *<sup>N</sup>* = 0), the electric field of the FW in each layer can be obtained. *N* is the total number of layers in the sample.

Similarly, the SHG electric field in the *l* − *th* layer of photonic crystal can be expressed as

$$\begin{array}{c} E\_l^{(2)}(z) = A\_l^{(2)} e^{i k\_l^{(2)}(z - z\_{l-1})} + B\_l^{(1)} e^{-i k\_l^{(1)}(z - z\_{l-1})} + \mathbb{C}\_{21} e^{i 2 k\_l^{(1)}(z - z\_{l-1})} + \mathbb{C}\_{22} e^{-i 2 k\_l^{(1)}(z - z\_{l-1})} \\ \ - \frac{2 k\_{20}^{(2)} \chi}{k\_l^{(2)2}} A\_l^{(1)} B\_l^{(1)} \end{array} \tag{39}$$

*A*(2) *<sup>l</sup>* and *<sup>B</sup>*(2) *<sup>l</sup>* represent the amplitudes of the forward and backward SHGs at interface, respectively. *C*<sup>21</sup> and *C*<sup>22</sup> can be obtained as

$$\mathbf{C}\_{21} = \frac{-k\_{20}^2 \chi\_l A\_l^{(1)2}}{k\_l^{(2)2} - 4k\_l^{(1)2}},$$

$$\mathbf{C}\_{22} = \frac{-k\_{20}^2 \chi\_l B\_l^{(1)2}}{k\_l^{(2)2} - 4k\_l^{(1)2}}.\tag{40}$$

Using the initial conditions *A*(2) <sup>1</sup> <sup>=</sup> 0, *<sup>B</sup>*(2) *<sup>N</sup>* = 0, the electric field of the SHG at each interface can be derived. The conversion efficiencies of the forward and backward SHG waves are defined respectively as follows

$$
\eta\_{orth}^{(2)} = \frac{n\_N^{(2)} |A\_N^{(2)}|^2}{n\_1^{(1)} |A\_1^{(1)}|^2}, \\
\eta\_{back}^{(2)} = \frac{|B\_1^{(2)}|^2}{|A\_1^{(1)}|^2}. \tag{41}
$$

As an example, the photonic crystal structure that can implement multiple wavelengths SHG is designed. The PQWs gives localized states at the frequencies of the FW and SHG,

**Figure 8.** Transmission spectrum of the designed photonic quantum well structure. The dash lines indicate *λ*1,1 = 1.064*μm*, *λ*1,2 = 1.136*μm*, *λ*1,3 = 1.188*μm*, *λ*2,1 = 0.532*μm*, *λ*2,2 = 0.568*μm*, and *λ*2,3 = 0.594*μm*.

**Figure 9.** Wavelength dependence of the conversion efficiency of the photonic quantum well structure

*<sup>α</sup>*,*<sup>s</sup>* and *η<sup>k</sup>*

states and corresponding conversion efficiencies generated from every transit structure during the SA process. *s* is the number of the FW to be designed and it was selected as *s* = 3 in this

Figure 8 presents the transmission spectrum of the designed photonic quantum well structure. The band structure of the PC (*AB*)<sup>10</sup> is almost unchanged. The wavelengths designed appear at *λ*1,1 = 1.064*μm*, *λ*1,2 = 1.136*μm*, *λ*1,3 = 1.188*μm*, *λ*2,1 = 0.532*μm*, *λ*2,2 = 0.568*μm*, and

The properties of the SHG for this structure is also investigated. The largest nonlinear coefficient *d*<sup>33</sup> of *LiNbO*<sup>3</sup> is used for achieving high conversion efficiency. The intensity of the incident FW is selected as 0.021*GW*/*m*<sup>2</sup> for each wavelength. The wavelength dependence of the SHG is shown in Fig. 9. Only three expected wavelengths appear. The conversion efficiencies of the forward SHGs are *η*<sup>1</sup> = 0.0943, *η*<sup>2</sup> = 0.0912, and *η*<sup>3</sup> = 0.0926, respectively. The conversion efficiencies have been enhanced nearly 103 time comparing with the periodically poled lithium niobate structure with identical length. The conversion efficiencies of the forward SHGs are nearly identical, which correspond to the designed aim well. Comparing with the structure with only the FW located at the defect state, the

*λ*2,3 = 0.594*μm* respectively. They agree with the required wavelengths quite well.

1,2 <sup>&</sup>lt; *<sup>λ</sup>*1,2 <sup>&</sup>lt; ... <sup>&</sup>lt; *<sup>λ</sup><sup>o</sup>*

2,2 < *<sup>λ</sup>*2,2 < ... < *<sup>λ</sup><sup>o</sup>*

1,*<sup>s</sup>* < *λ*1,*s*(= *λb*);

Optimization Design of Nonlinear Optical Frequency Conversion Devices Using Simulated Annealing Algorithm

109

2,*<sup>s</sup>* < *λ*2,*s*(= *λd*).

1,*<sup>s</sup>* denote the wavelengths of the defected

1,1 <sup>&</sup>lt; *<sup>λ</sup>*1,1 <sup>&</sup>lt; *<sup>λ</sup><sup>o</sup>*

2,1 <sup>&</sup>lt; *<sup>λ</sup>*2,1 <sup>&</sup>lt; *<sup>λ</sup><sup>o</sup>*

for the forward SHG.

*λ*1,0(= *λa*) < *λ<sup>o</sup>*

*λ*2,0(= *λc*) < *λ<sup>o</sup>*

work. The PQW structure is designed by the SA.

conversion efficiencies have also been mightily enhanced.

*<sup>β</sup>*<sup>1</sup> and *<sup>β</sup>*<sup>2</sup> are two adjustable constants. *<sup>λ</sup>*(*k*)

with

respectively. The preset wavelengths of FWs are *λ<sup>o</sup>* 1,1 <sup>=</sup> 1.064*μm*, *<sup>λ</sup><sup>o</sup>* 1,2 = 1.136*μm*, and *λo* 1,3 <sup>=</sup> 1.188*μm*, and the wavelengths of the corresponding second harmonics are *<sup>λ</sup><sup>o</sup>* 2,1 = 0.532*μm*, *λ<sup>o</sup>* 2,2 = 0.568*μm*, and *<sup>λ</sup><sup>o</sup>* 2,3 = 0.594*μm*, respectively. The conversion efficiencies for three FWs are required to be nearly equal. The periodic structure (*AB*)<sup>5</sup> is selected as the prototype photonic crystal. *A* and *B* are *LiNbO*<sup>3</sup> and air, respectively. The widths of the A and B layers are selected as *dA* = 0.1814*μm* and *dB* = 0.1330*μm*, respectively. This prototype photonic crystal has two band gaps [0.916*μm* 1.314*μm*] and [0.482*μm* 0.612*μm*]. The FWs and SHGs are located in the two band gaps, respectively. Then a aperiodic structure is designed and insert into the prototype photonic crystals. The thickness of *C* and *D* unit cell is *dC* = *dD* = 0.04*μm*, the number of unit cells is 300. However, each cell is selected from *C* or *D* which is determined by the SA algorithm with an objective function as

$$O = \sum\_{a} \sum\_{s} \sum\_{k} \left[ |\lambda\_{a,s}^{o} - \lambda\_{a,s}^{k}| + \beta\_1 |\eta^{0} - \eta\_{1,s}^{(k)}| \right] + \beta\_2 |\max(\{\eta\_{1,s}^{(k)}\}) - \min(\{\eta\_{1,s}^{(k)}\})| \tag{42}$$

where

$$\begin{array}{l} \lambda\_{1,s}^{o}\lambda\_{1s}^{(k)} \in [\lambda\_{1,s-1}\lambda\_{1,s}] \in [\lambda\_{a\prime}\lambda\_{b}],\\ \lambda\_{2,s\prime}^{o}\lambda\_{2,s}^{(k)} \in [\lambda\_{2,s-1\prime}\lambda\_{2,s}] \in [\lambda\_{c\prime}\lambda\_{d}],\\ \alpha = 1,2 \qquad s = 1,2,3,...,\end{array}$$

**Figure 9.** Wavelength dependence of the conversion efficiency of the photonic quantum well structure for the forward SHG.

with

16 Will-be-set-by-IN-TECH

**Figure 8.** Transmission spectrum of the designed photonic quantum well structure. The dash lines indicate *λ*1,1 = 1.064*μm*, *λ*1,2 = 1.136*μm*, *λ*1,3 = 1.188*μm*, *λ*2,1 = 0.532*μm*, *λ*2,2 = 0.568*μm*, and

1,3 <sup>=</sup> 1.188*μm*, and the wavelengths of the corresponding second harmonics are *<sup>λ</sup><sup>o</sup>*

*D* which is determined by the SA algorithm with an objective function as

*λo* 1,*s*, *<sup>λ</sup>*(*k*)

*λo* 2,*s*, *<sup>λ</sup>*(*k*)

*<sup>α</sup>*,*s*<sup>|</sup> <sup>+</sup> *<sup>β</sup>*1|*η*<sup>0</sup> <sup>−</sup> *<sup>η</sup>*(*k*)

for three FWs are required to be nearly equal. The periodic structure (*AB*)<sup>5</sup> is selected as the prototype photonic crystal. *A* and *B* are *LiNbO*<sup>3</sup> and air, respectively. The widths of the A and B layers are selected as *dA* = 0.1814*μm* and *dB* = 0.1330*μm*, respectively. This prototype photonic crystal has two band gaps [0.916*μm* 1.314*μm*] and [0.482*μm* 0.612*μm*]. The FWs and SHGs are located in the two band gaps, respectively. Then a aperiodic structure is designed and insert into the prototype photonic crystals. The thickness of *C* and *D* unit cell is *dC* = *dD* = 0.04*μm*, the number of unit cells is 300. However, each cell is selected from *C* or

> 1,*s* |

1,*<sup>s</sup>* ∈ [*λ*1,*s*−1, *<sup>λ</sup>*1,*s*] ∈ [*λa*, *<sup>λ</sup>b*],

2,*<sup>s</sup>* ∈ [*λ*2,*s*−1, *<sup>λ</sup>*2,*s*] ∈ [*λc*, *<sup>λ</sup>d*],

*α* = 1, 2 *s* = 1, 2, 3, ....,

1,1 <sup>=</sup> 1.064*μm*, *<sup>λ</sup><sup>o</sup>*

2,3 = 0.594*μm*, respectively. The conversion efficiencies

<sup>+</sup> *<sup>β</sup>*2|*max*({*η*(*k*)

1,*<sup>s</sup>* }) <sup>−</sup> *min*({*η*(*k*)

1,2 = 1.136*μm*, and

1,*<sup>s</sup>* })| (42)

2,1 =

respectively. The preset wavelengths of FWs are *λ<sup>o</sup>*

2,2 = 0.568*μm*, and *<sup>λ</sup><sup>o</sup>*

*<sup>O</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>* <sup>∑</sup>*<sup>s</sup>* <sup>∑</sup>

*k* |*λo <sup>α</sup>*,*<sup>s</sup>* <sup>−</sup> *<sup>λ</sup><sup>k</sup>*

*λ*2,3 = 0.594*μm*.

0.532*μm*, *λ<sup>o</sup>*

*λo*

where

$$\lambda\_{1,0}(=\lambda\_a) < \lambda\_{1,1}^{\vartheta} < \lambda\_{1,1} < \lambda\_{1,2}^{\vartheta} < \lambda\_{1,2} < \ldots < \lambda\_{1,s}^{\vartheta} < \lambda\_{1,s}(=\lambda\_b);$$

$$\lambda\_{2,0}(=\lambda\_c) < \lambda\_{2,1}^{\vartheta} < \lambda\_{2,1} < \lambda\_{2,2}^{\vartheta} < \lambda\_{2,2} < \ldots < \lambda\_{2,s}^{\vartheta} < \lambda\_{2,s}(=\lambda\_d).$$

*<sup>β</sup>*<sup>1</sup> and *<sup>β</sup>*<sup>2</sup> are two adjustable constants. *<sup>λ</sup>*(*k*) *<sup>α</sup>*,*<sup>s</sup>* and *η<sup>k</sup>* 1,*<sup>s</sup>* denote the wavelengths of the defected states and corresponding conversion efficiencies generated from every transit structure during the SA process. *s* is the number of the FW to be designed and it was selected as *s* = 3 in this work. The PQW structure is designed by the SA.

Figure 8 presents the transmission spectrum of the designed photonic quantum well structure. The band structure of the PC (*AB*)<sup>10</sup> is almost unchanged. The wavelengths designed appear at *λ*1,1 = 1.064*μm*, *λ*1,2 = 1.136*μm*, *λ*1,3 = 1.188*μm*, *λ*2,1 = 0.532*μm*, *λ*2,2 = 0.568*μm*, and *λ*2,3 = 0.594*μm* respectively. They agree with the required wavelengths quite well.

The properties of the SHG for this structure is also investigated. The largest nonlinear coefficient *d*<sup>33</sup> of *LiNbO*<sup>3</sup> is used for achieving high conversion efficiency. The intensity of the incident FW is selected as 0.021*GW*/*m*<sup>2</sup> for each wavelength. The wavelength dependence of the SHG is shown in Fig. 9. Only three expected wavelengths appear. The conversion efficiencies of the forward SHGs are *η*<sup>1</sup> = 0.0943, *η*<sup>2</sup> = 0.0912, and *η*<sup>3</sup> = 0.0926, respectively. The conversion efficiencies have been enhanced nearly 103 time comparing with the periodically poled lithium niobate structure with identical length. The conversion efficiencies of the forward SHGs are nearly identical, which correspond to the designed aim well. Comparing with the structure with only the FW located at the defect state, the conversion efficiencies have also been mightily enhanced.

### **3.3. Photonic crystal device for multiple wavelengths' coupled third harmonic generation**

For the coupled third harmonic generation, the electric field *E*(1) *<sup>l</sup>* (*E*(2) *<sup>l</sup>* , *<sup>E</sup>*(3) *<sup>l</sup>* ) of the FW (SHG, CTHG) for the *l* − *th* layer must satisfy the following equations:

$$
\left[\frac{d^2}{dz^2} + k\_l^{(1)2}\right] E\_l^{(1)}(z) = 0,\tag{43}
$$

three band gaps as [*λ<sup>a</sup>* = 1.283*μm*, *λ<sup>b</sup>* = 1.835*μm*], [*λ<sup>c</sup>* = 0.667*μm*, *λ<sup>d</sup>* = 0.848*μm*], and [*λ<sup>e</sup>* = 0.477*μm*, *λ<sup>f</sup>* = 0.533*μm*]. The double wavelengths CTHG is considered here, the

with that for multiple wavelengths SHG. The transmission spectrum of the designed structure is shown in Fig. 10. The peak wavelengths appear at *λ*1,1 = 1.458*μm*, *λ*1,2 = 1.578*μm*, *λ*2,1 = 0.729*μm*, *λ*2,2 = 0.789*μm*, *λ*3,1 = 0.486*μm*, and *λ*3,2 = 0.526*μm*, respectively. They agree well with the required wavelengths. Figure 11 presents the dependence of conversion

**Figure 10.** Transmission spectrum of the designed photonic quantum well structure for double wavelengths CTHG. The dashed lines represent *λ*1,1 = 1.458*μm*, *λ*1,2 = 1.578*μm*, *λ*2,1 = 0.729*μm*,

*λ*2,2 = 0.789*μm*, *λ*3,1 = 0.486*μm*, and *λ*3,2 = 0.526*μm*, respectively.

**Figure 11.** Conversion efficiency of the forward CTHG.

1,2 <sup>=</sup> 1.578*μm*, *<sup>λ</sup><sup>O</sup>*

3.2 = 0.526*μm*, respectively. The object function in this case is similar

2,1 <sup>=</sup> 0.729*μm*, *<sup>λ</sup><sup>O</sup>*

Optimization Design of Nonlinear Optical Frequency Conversion Devices Using Simulated Annealing Algorithm

2,2 = 0.789*μm*,

111

1,1 <sup>=</sup> 1.458*μm*, *<sup>λ</sup><sup>O</sup>*

preset wavelengths are *λ<sup>O</sup>*

3,1 <sup>=</sup> 0.486*μm*, and *<sup>λ</sup><sup>O</sup>*

*λO*

$$
\left[\frac{d^2}{dz^2} + k\_l^{(2)2}\right] E\_l^{(2)}(z) = -k\_{20}^{(2)2} \chi\_l(z) E\_l^{(1)2}(z),\tag{44}
$$

$$
\left[\frac{d^2}{dz^2} + k\_l^{(3)2}\right] E\_l^{(3)}(z) = -2k\_{30}^{(2)2} \chi\_l(z) E\_l^{(1)}(z) E\_l^{(2)}(z),\tag{45}
$$

where *k* (1) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(1) *<sup>L</sup> k*10, *k* (2) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(2) *<sup>L</sup> k*20, *k* (3) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(3) *<sup>L</sup> k*30, *k*<sup>10</sup> = *ω*/*c*, *k*<sup>20</sup> = 2*ω*/*c*, and *k*<sup>30</sup> = 3*ω*/*c* are wave vectors of the FW, SHG, and CTHG, respectively. *<sup>ω</sup>* is the frequency of the FW, *<sup>n</sup>*(1) *l* (*n*(2) *<sup>l</sup>* , *<sup>n</sup>*(3) *<sup>l</sup>* ) is the refractive index of the *l* − *th* layer for the wavelength of FW (SHG, CTHG), *<sup>c</sup>* is the velocity of the light in vacuum and *<sup>χ</sup><sup>l</sup>* is the nonlinear optical coefficient of the *<sup>l</sup>* <sup>−</sup> *th* layer. Similarly, the expression of CTHG electric field in the *l* − *th* layer is

$$\begin{array}{l} E\_{l}^{(3)}(z) = A\_{l}^{(3)} e^{i k\_{l}^{(3)}(z - z\_{l-1})} + B\_{l}^{(3)} e^{-i k\_{l}^{(3)}(z - z\_{l-1})} + \mathsf{C}\_{31} e^{i [k\_{l}^{(1)} + k\_{l}^{(2)}](z - z\_{l-1})} + \mathsf{C}\_{32} e^{-i [k\_{l}^{(1)} + k\_{l}^{(2)}](z - z\_{l-1})} \\ \quad + \mathsf{D}\_{31} e^{i [k\_{l}^{(1)} - k\_{l}^{(2)}](z - z\_{l-1})} + \mathsf{D}\_{32} e^{-i [k\_{l}^{(1)} - k\_{l}^{(2)}](z - z\_{l-1})} + \mathsf{E}\_{31} e^{i 3 k\_{l}^{(1)}(z - z\_{l-1})} \\ \quad + \mathsf{E}\_{32} e^{-i 3 k\_{l}^{(1)}(z - z\_{l-1})} + \mathsf{F}\_{31} e^{i \bar{k}\_{l}^{(1)}(z - z\_{l-1})} + \mathsf{F}\_{32} e^{-i \bar{k}\_{l}^{(1)}(z - z\_{l-1})}, \end{array} \tag{46}$$

and all parameters in this equation can be expressed as

*<sup>C</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> <sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> <sup>A</sup>*(2) *l k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* +*k* (2) *<sup>l</sup>* )<sup>2</sup> , *<sup>C</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> 30*χlB*(1) *<sup>l</sup> <sup>B</sup>*(2) *l k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* +*k* (2) *<sup>l</sup>* )<sup>2</sup> , *<sup>D</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> <sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> <sup>B</sup>*(2) *l k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* −*k* (2) *<sup>l</sup>* )<sup>2</sup> , *<sup>D</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> <sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(2) *<sup>l</sup> <sup>B</sup>*(1) *l k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* −*k* (2) *<sup>l</sup>* )<sup>2</sup> , *<sup>E</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> <sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> C*2*<sup>l</sup> k* (3)2 *<sup>l</sup>* −9*k* (1)2 *l* , *<sup>E</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> 30*χlB*(1) *<sup>l</sup> C*<sup>22</sup> *k* (3)2 *<sup>l</sup>* −9*k* (1)2 *l* , *<sup>F</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> 30*χlB*(1) *<sup>l</sup> C*2*<sup>l</sup> k* (3)2 *<sup>l</sup>* −*k* (1)2 *l* <sup>+</sup> <sup>2</sup>*k*<sup>2</sup> 30*k*<sup>2</sup> 20*χ*(2) *<sup>l</sup> <sup>χ</sup><sup>l</sup> <sup>A</sup>*(1)<sup>2</sup> *<sup>l</sup> <sup>B</sup>*(1) *l k* (2)2 *<sup>l</sup>* (*k* (3)2 *<sup>l</sup>* −*k* (1)2 *<sup>l</sup>* ) , *<sup>F</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup> <sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> C*<sup>22</sup> *k* (3)2 *<sup>l</sup>* −*k* (1)2 *l* <sup>+</sup> <sup>2</sup>*k*<sup>2</sup> 30*k*<sup>2</sup> 20*χ*(2) *<sup>l</sup> <sup>χ</sup>lB*(1)<sup>2</sup> *<sup>l</sup> <sup>A</sup>*(1) *l k* (2)2 *<sup>l</sup>* (*k* (3)2 *<sup>l</sup>* −*k* (1)2 *<sup>l</sup>* ) . (47)

By using the initial conditions *A*(2) <sup>1</sup> <sup>=</sup> 0, *<sup>B</sup>*(2) *<sup>N</sup>* <sup>=</sup> 0, *<sup>A</sup>*(3) <sup>1</sup> <sup>=</sup> 0, and *<sup>B</sup>*(3) *<sup>N</sup>* = 0, the electric fields of CTHG at each interface can be obtained. Therefore, the conversion efficiencies of the forward and backward waves are defined respectively as

$$\eta\_{orth}^{(3)} = \frac{n\_N^{(3)} |A\_N^{(3)}|^2}{n\_1^{(3)} |A\_1^{(1)}|^2}, \quad \eta\_{back}^{(3)} = \frac{|B\_1^{(3)}|^2}{|A\_1^{(1)}|^2}. \tag{48}$$

Similarly, a prototype photonic crystal (*AB*)<sup>10</sup> is designed. The thicknesses of *A* and *B* layers are *dA* = 0.2557*μm* and *dB* = 0.1875*μm*, respectively. Thus this photonic crystal has three band gaps as [*λ<sup>a</sup>* = 1.283*μm*, *λ<sup>b</sup>* = 1.835*μm*], [*λ<sup>c</sup>* = 0.667*μm*, *λ<sup>d</sup>* = 0.848*μm*], and [*λ<sup>e</sup>* = 0.477*μm*, *λ<sup>f</sup>* = 0.533*μm*]. The double wavelengths CTHG is considered here, the preset wavelengths are *λ<sup>O</sup>* 1,1 <sup>=</sup> 1.458*μm*, *<sup>λ</sup><sup>O</sup>* 1,2 <sup>=</sup> 1.578*μm*, *<sup>λ</sup><sup>O</sup>* 2,1 <sup>=</sup> 0.729*μm*, *<sup>λ</sup><sup>O</sup>* 2,2 = 0.789*μm*, *λO* 3,1 <sup>=</sup> 0.486*μm*, and *<sup>λ</sup><sup>O</sup>* 3.2 = 0.526*μm*, respectively. The object function in this case is similar with that for multiple wavelengths SHG. The transmission spectrum of the designed structure is shown in Fig. 10. The peak wavelengths appear at *λ*1,1 = 1.458*μm*, *λ*1,2 = 1.578*μm*, *λ*2,1 = 0.729*μm*, *λ*2,2 = 0.789*μm*, *λ*3,1 = 0.486*μm*, and *λ*3,2 = 0.526*μm*, respectively. They agree well with the required wavelengths. Figure 11 presents the dependence of conversion

18 Will-be-set-by-IN-TECH

*<sup>l</sup>* (*E*(2)

*<sup>l</sup>* (*z*) = 0, (43)

*<sup>l</sup>* (*z*)*E*(2)

*<sup>L</sup> k*30, *k*<sup>10</sup> = *ω*/*c*, *k*<sup>20</sup> = 2*ω*/*c*, and *k*<sup>30</sup> = 3*ω*/*c*

*<sup>l</sup>* ](*z*−*zl*−<sup>1</sup>) + *C*32*e*−*i*[*<sup>k</sup>*

30*χlB*(1) *<sup>l</sup> <sup>B</sup>*(2) *l*

<sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(2) *<sup>l</sup> <sup>B</sup>*(1) *l*

30*χlB*(1) *<sup>l</sup> C*<sup>22</sup>

(1) *<sup>l</sup>* (*z*−*zl*−<sup>1</sup>)

*<sup>l</sup>* , *<sup>E</sup>*(3)

*<sup>l</sup>* (*z*), (44)

*<sup>l</sup>* (*z*), (45)

(1) *<sup>l</sup>* +*k* (2) *<sup>l</sup>* ](*z*−*zl*−<sup>1</sup>)

*<sup>N</sup>* = 0, the electric fields of

. (48)

*l*

(46)

(47)

*<sup>l</sup>* ) of the FW (SHG,

**3.3. Photonic crystal device for multiple wavelengths' coupled third harmonic**

*<sup>l</sup>* (*z*) = −*k*

*<sup>l</sup>* (*z*) = −2*k*

are wave vectors of the FW, SHG, and CTHG, respectively. *<sup>ω</sup>* is the frequency of the FW, *<sup>n</sup>*(1)

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *C*31*ei*[*<sup>k</sup>*

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) <sup>+</sup> *<sup>F</sup>*32*e*−*ik*(1)

*<sup>l</sup>* )<sup>2</sup> , *<sup>C</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>l</sup>* )<sup>2</sup> , *<sup>D</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

, *<sup>E</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>N</sup>* <sup>=</sup> 0, *<sup>A</sup>*(3)

, *<sup>η</sup>*(3)

CTHG at each interface can be obtained. Therefore, the conversion efficiencies of the forward

Similarly, a prototype photonic crystal (*AB*)<sup>10</sup> is designed. The thicknesses of *A* and *B* layers are *dA* = 0.2557*μm* and *dB* = 0.1875*μm*, respectively. Thus this photonic crystal has

(1) *<sup>l</sup>* −*k* (2)

(3) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(3) (2)2

(2)2 <sup>30</sup> *<sup>χ</sup>l*(*z*)*E*(1)

*<sup>l</sup>* ) is the refractive index of the *l* − *th* layer for the wavelength of FW (SHG, CTHG), *<sup>c</sup>* is the velocity of the light in vacuum and *<sup>χ</sup><sup>l</sup>* is the nonlinear optical coefficient of the *<sup>l</sup>* <sup>−</sup> *th*

<sup>20</sup> *<sup>χ</sup>l*(*z*)*E*(1)<sup>2</sup>

(1) *<sup>l</sup>* +*k* (2)

*<sup>l</sup>* ](*z*−*zl*−<sup>1</sup>) + *E*31*ei*3*<sup>k</sup>*

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>),

*k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* +*k* (2) *<sup>l</sup>* )<sup>2</sup> ,

*k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* −*k* (2) *<sup>l</sup>* )<sup>2</sup> ,

*k* (3)2 *<sup>l</sup>* −9*k* (1)2 *l* ,

<sup>1</sup> <sup>=</sup> 0, and *<sup>B</sup>*(3)

1 | 2


*back* <sup>=</sup> <sup>|</sup>*B*(3)

For the coupled third harmonic generation, the electric field *E*(1)

CTHG) for the *l* − *th* layer must satisfy the following equations:

(1)2 *l*

(2)2 *l*

(3)2 *l*

 *E*(1)

 *E*(2)

 *E*(3)

*<sup>L</sup> k*20, *k*

layer. Similarly, the expression of CTHG electric field in the *l* − *th* layer is

*<sup>l</sup> <sup>e</sup>*−*ik*(3)

<sup>+</sup> <sup>2</sup>*k*<sup>2</sup> 30*k*<sup>2</sup> 20*χ*(2) *<sup>l</sup> <sup>χ</sup><sup>l</sup> <sup>A</sup>*(1)<sup>2</sup> *<sup>l</sup> <sup>B</sup>*(1) *l*

<sup>+</sup> <sup>2</sup>*k*<sup>2</sup> 30*k*<sup>2</sup> 20*χ*(2) *<sup>l</sup> <sup>χ</sup>lB*(1)<sup>2</sup> *<sup>l</sup> <sup>A</sup>*(1) *l*

*k* (2)2 *<sup>l</sup>* (*k* (3)2 *<sup>l</sup>* −*k* (1)2 *<sup>l</sup>* ) ,

*k* (2)2 *<sup>l</sup>* (*k* (3)2 *<sup>l</sup>* −*k* (1)2 *<sup>l</sup>* ) .

<sup>1</sup> <sup>=</sup> 0, *<sup>B</sup>*(2)

*n*(3) <sup>1</sup> <sup>|</sup>*A*(1)|<sup>2</sup> 1

*<sup>N</sup>* <sup>|</sup>*A*(3) *N* | 2

*<sup>l</sup>* ](*z*−*zl*−<sup>1</sup>) + *D*32*e*−*i*[*<sup>k</sup>*

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) <sup>+</sup> *<sup>F</sup>*31*eik*(1)

and all parameters in this equation can be expressed as

<sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> <sup>A</sup>*(2) *l*

<sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> <sup>B</sup>*(2) *l*

<sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> C*2*<sup>l</sup>*

30*χlB*(1) *<sup>l</sup> C*2*<sup>l</sup>*

<sup>30</sup>*χ<sup>l</sup> <sup>A</sup>*(1) *<sup>l</sup> C*<sup>22</sup>

*<sup>η</sup>*(3)

*f orth* <sup>=</sup> *<sup>n</sup>*(3)

 *d*<sup>2</sup> *dz*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*

 *d*<sup>2</sup> *dz*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*

 *d*<sup>2</sup> *dz*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*

> (2) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(2)

*<sup>l</sup>* (*z*−*zl*−<sup>1</sup>) + *B*(3)

*<sup>L</sup> k*10, *k*

*<sup>l</sup> <sup>e</sup>ik*(3)

(1) *<sup>l</sup>* −*k* (2)

(1)

*<sup>C</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>D</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>E</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>F</sup>*<sup>31</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

*<sup>F</sup>*<sup>32</sup> <sup>=</sup> <sup>−</sup>*k*<sup>2</sup>

By using the initial conditions *A*(2)

*k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* +*k* (2)

*k* (3)2 *<sup>l</sup>* −(*k* (1) *<sup>l</sup>* −*k* (2)

*k* (3)2 *<sup>l</sup>* −9*k* (1)2 *l*

*k* (3)2 *<sup>l</sup>* −*k* (1)2 *l*

*k* (3)2 *<sup>l</sup>* −*k* (1)2 *l*

and backward waves are defined respectively as

+*D*31*ei*[*<sup>k</sup>*

+*E*32*e*−*i*3*<sup>k</sup>*

**generation**

where *k*

(*n*(2) *<sup>l</sup>* , *<sup>n</sup>*(3)

*E*(3)

(1) *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>*(1)

*<sup>l</sup>* (*z*) = *<sup>A</sup>*(3)

**Figure 10.** Transmission spectrum of the designed photonic quantum well structure for double wavelengths CTHG. The dashed lines represent *λ*1,1 = 1.458*μm*, *λ*1,2 = 1.578*μm*, *λ*2,1 = 0.729*μm*, *λ*2,2 = 0.789*μm*, *λ*3,1 = 0.486*μm*, and *λ*3,2 = 0.526*μm*, respectively.

**Figure 11.** Conversion efficiency of the forward CTHG.

20 Will-be-set-by-IN-TECH 112 Simulated Annealing – Single and Multiple Objective Problems **Chapter 0**

efficiency of the CTHG on the wavelength of FW of the designed structure. Only for the two preset wavelengths, the conversion efficiency is quite high. The conversion efficiencies are *η*1,1 = 0.0917 for *λ*1,1 = 1.458*μm* and *η*1,2 = 0.0938 for *λ*1,2 = 1.578*μm*, the conversion efficiencies are nearly identical which corresponds to the required aim well.

**Simulated Annealing and Multiuser Scheduling**

**Chapter 6**

Adaptive modulation and coding (AMC) is an effective way for improving the spectral efficiency in wireless communication systems. By increasing the size of the modulation scheme constellation, the spectral efficiency can be improved, generally at the cost of a degraded error rate. A similar trade-off is possible by using a higher rate channel code. By an appropriate combination of the modulation order and channel code rate, we can design a set of modulation and coding schemes (MCSs), from which an MCS is selected in an adaptive fashion in each transmission-time interval (TTI) in order to maximize system throughput under different channel conditions. The use of AMC yields a rich variety of scheduling strategies [25]; [12]. In practice, a commonly encountered constraint is that the probability of erroneous decoding of a Transmission Block should not exceed some threshold value [11]. Multiple orthogonal channelization codes (multicodes) can be used to transmit data to a single user, thereby increasing the per-user bit rate and the granularity of adaptation [11, 16]. In Wideband Code-Division Multiple Access (WCDMA), the channelization codes are often referred to as Orthogonal Variable Spreading Factor (OVSF) codes. The number of OVSF codes per base station (BS) is quite limited due to the orthogonality constraint [11] and thus OVSF codes and transmit power are scarce resources. Fig. 1 shows the number of OVSF codes as a function of the spreading factor for WCDMA. Note that a lower value of spreading factor corresponds to a higher bit rate and vice versa. According to Fig. 1, if a spreading factor of 2 is needed, the system can allocate at most two such OVSF codes. On the other hand, if a spreading factor of 4 is required, a total of 4 such codes can be allocated. In High Speed Downlink Packet Access (HSDPA), a fixed spreading factor of 16 has been specified, thereby

The allocation of the number of OVSF codes (or multicodes) and the MCS level for each user depends on the strength of the received signal at the user, which, in turn, depends on 1) the quality of the wireless channel, and 2) the level of the transmit power to the respective user. <sup>1</sup> In principle, 16 OVSF codes can be used. However, one code is allocated for other purposes such as signalling. Thus,

> ©2012 Kwan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Kwan et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**in Mobile Communication Networks**

Raymond Kwan, M. E. Aydin and Cyril Leung

http://dx.doi.org/10.5772/45869

limiting the number of OVSF codes to 161.

a maximum of 15 codes can be allocated for data traffic [11].

cited.

**1. Introduction**

Additional information is available at the end of the chapter
