**1. Introduction**

Laser has become to be a fundamental light source in modern communication, scientific research, and industrial applications. More and more laser frequencies are required for various applications. However, common laser crystals can provide only some fixed frequencies which cannot satisfy various requirement. Nonlinear optical process presents an alternative approach for generating rich laser frequencies. The traditional nonlinear optical processes usually require the so called phase-matching condition [1, 2], which requires the nonlinear optical crystals with birefringence. The phase-matching condition raises a restriction of the choice of natural birefringence materials in the applications of frequency conversion. Quasi-phase-matching method uses periodic modulation of the nonlinear property of a crystal to compensate the mismatch between the wave vectors of the interaction light beams [3]. This method allows utilization of the large component of the nonlinear susceptibility tensor, which is usually inaccessible with the common phase matching. Periodic optical superlattice provides a reciprocal vector to compensate the phase mismatch between the interacting light beams. Thus, only one nonlinear process can be performed in the periodic optical superlattice. This idea can be naturedly expended to the aperiodic optical supperlattice which can provide a series of reciprocal vectors. The reciprocal vectors can be preset for special nonlinear process. The key problem is how to design different aperiodic optical supperlattice for matching the specified nonlinear optical process with high conversion efficiency.

In this chapter, the simulated annealing (SA) method is used to successfully design nonlinear optical frequency conversion devices for achieving different nonlinear optical processes, for example, multiple second harmonics generation and coupled third harmonic generation in the aperiodic optical superlattice, multiple wavelengths parametric amplification, multiple wavelengths second harmonics generation and coupled third harmonic generation in the defective nonlinear photonic crystals. The simulation results demonstrate that the SA method is an effective algorithm for nonlinear optical frequency conversion devices design. The designed devices can archive the preset goal well.

©2012 Zhang, 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 cited. ©2012 Zhang, 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.

## **2. Design of aperiodic optical supperlattice**

Unlike the periodic optical supperlattice, the aperiodic optical superlattice can provide more spatial Fourier components of structure, therefore, some coupled parametric processes may be realized in this kind of devices. Firstly, the optimal design problem of the aperiodic optical superlattice is described in the real-space representation. Then several model designs are carried out to demonstrate the effectiveness of the present design method. The *LiTaO*<sup>3</sup> is selected as basic crystal for polarizing. The direction of polarization vectors in successive domain are opposite, thus are the signs of the nonlinear optical coefficients. However, the width of each individual domain is no longer equal and should be determined by the specified nonlinear optical processes.

the parameter *ξ*

(*s*)

*ξ* (*s*)

*eff*(*λ*) = <sup>1</sup>

the polarization direction of each block.

thickness of the unit block is Δ*x* = *l*

function in the SA algorithm is chosen as

and with a period of *a* = 2Δ*x* = *l*

a search for the maximum of *ξ*

*l* (*s*) = *sinc*

where *sinc*(*x*) = *sin*(*πx*)/(*πx*). It can be seen that *ξ*

for *q* = 0, 1, 2, ....(*N* − 1). Equation 6 can be evaluated as

*N*Δ*x* <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> ˜*d*(*xq*)

> Δ*x l* (*s*) *<sup>c</sup>* (*λ*)

to the unit block and depends on the Δ*x* and the coherence length *l*

(*s*)

has its maximum value 2/[*π*(2*m* + 1)] when Δ*x* = [(2*m* + 1)/2]*l*

 <sup>1</sup> *<sup>N</sup>* <sup>∑</sup>*N*−<sup>1</sup>

*eff*(*λ*) depends on the polarization direction of each domain. Assume that the thickness of the each domain of the aperiodic optical superlattice is Δ*x*, thus the number of blocks in the sample is *N* = *L*/Δ*x*. The position of each domain is *xq* = *q*Δ*x*

> *xq*+Δ*<sup>x</sup> xq <sup>e</sup>i*(2*πx*/*<sup>l</sup>*

by the interference effect among the blocks and depends on the configuration of domains and

The optimization design of the aperiodic optical superlattice for the SHG can be ascribed as

has its maximum value of 1 in the case of the periodic structure with a period of *a* = 2Δ*x* =

*<sup>c</sup>* (*λ*), *d*(*xq*) takes opposite sign between two consecutive blocks. In this case, the phase lagging factor perfectly matches the reversal of the domain orientation between two adjacent blocks and the ideal constructive interference emerges. However, in the case of the aperiodic structure, it is quite difficult to find a solution and can be solved with the SA algorithm.

In order to demonstrate the effectiveness of the SA algorithm for design of aperiodic optical superlattices, a simple example of a single wavelength SHG is considered here. The parameters are chosen as follows: the wavelegth of the incident beam is *λ* = 1.064*μm*, the

2142. Thus the total length of sample is *L* = 8298.8*μm*. The refractive indexes of material at corresponding wavelengths are evaluated by the Sellmeier equation [4]. The objective

where *ξ*<sup>0</sup> is a preset value in guiding the SA procedure. Fig. 1 sketches a flowchart of the SA algorithm for constructing aperiodic optical superlattice. As shown in Fig. 1, the initial temperature and dropping rate are selected at the beginning of program, a random sign modulation ˜*d*(*xi*) is substituted into Equation 8 to calculate the initial object function *<sup>E</sup>*0. A

difference Δ*E* = *E*<sup>1</sup> − *E*<sup>0</sup> is obtained. A random number *p*, 0 ≤ *p* ≤ 1, is generated by the computer. If *p* satisfies *p* ≤ exp(−Δ*E*/*T*), this change will be accepted and *E*<sup>0</sup> will be replaced by *E*1, else the sign of this unit block will be retrieved and *E*<sup>0</sup> does not change. After test all of blocks, this procedure will be repeated with a new temperature *T* = *T* × Δ*T* until no sign

Finally, a perfect periodical structure with a pari of antiparallel domains for each unit block

(*s*)

(*s*)

*<sup>E</sup>* <sup>=</sup> <sup>|</sup>*ξ*<sup>0</sup> <sup>−</sup> *<sup>ξ</sup>*

new object function *E*<sup>1</sup> is calculated with changing the sign of a unit block in *d*

changes in the ˜*d*(*x*). Thus, the stable modulation left is the optimal one.

(*s*)

(*s*) *<sup>c</sup>* (*λ*))*dx* 

*<sup>q</sup>*=<sup>0</sup> ˜*d*(*xq*)*ei*[2*π*(*q*+0.5)Δ*x*/*<sup>l</sup>*

*eff*(*λ*) with respect to ˜*d*(*q*Δ*x*). The first factor in Equation 7

(*s*)

(*s*) *<sup>c</sup>* (*λ*)] 

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

*eff*(*λ*) has two contributions: one belongs

(*s*)

(*s*)

*<sup>c</sup>* (*λ*)/2 = 3.8713*μm*, and the number of blocks is

*<sup>c</sup>* as expected can be obtained. The reduced effective

*eff*(*λ*)|, (8)

, (7)

95

*<sup>c</sup>* (*λ*); the other is caused

*<sup>c</sup>* (*λ*). The second factor

(*xi*) and the

#### **2.1. Design method**

We use the second harmonic generation (SHG) process for a single wavelength as an example. Assume that a laser beam with frequency *ω*<sup>1</sup> = *ω* is perpendicularly incident on the surface of an aperiodic optical superlattice. In order to use the largest nonlinear coefficient *d*33, let the propagation and polarization directions of the input light are along the *x* and *z* axes, respectively. Two optical fields are involved in the SHG process. one is the fundamental wave with *ω*<sup>1</sup> = *ω* and other is the second harmonic wave with *ω*<sup>2</sup> = 2*ω*. Under the slowing-wave and small-signal approximations, the conversion efficiency *ηSHG* from the fundamental wave to second harmonic wave can be written as:

$$\eta\_{SHG} = \frac{I\_{2\omega}}{I\_{\omega}} = \frac{8\pi^2 |d\_{33}|^2 I\_{\omega} L^2}{c\varepsilon\_0 \lambda^2 n\_{2\omega} n\_{\omega}^2} \left| \frac{1}{L} \int\_0^L e^{i(k\_{2\omega} - 2k\_{\omega})x} d(x) dx \right|^2,\tag{1}$$

where *k<sup>ω</sup>* = *nωω*/*c* (*k*2*<sup>ω</sup>* = *n*2*ω*2*ω*/*c*) is the wave number of the fundamental (second harmonic) wave, *c* is the speed of light in vacuum, *n<sup>ω</sup>* (*n*2*ω*) is the refractive index of crystal at the fundamental (second harmonic) wavelength, *ε*<sup>0</sup> is the permittivity of vacuum, and *L* is the total length of the sample. *I<sup>ω</sup>* (*I*2*ω*) is the intensity of the fundamental (second harmonic) wave beam, and ˜*d*(*x*) only takes binary values of 1 or -1 which depends on the polarization direction. The coherence length *l* (*s*) *<sup>c</sup>* (*λ*) for the SHG is defined as:

$$l\_c^{(s)}(\lambda) = \frac{2\pi}{\Delta k} = \frac{2\pi}{k\_{2\omega} - 2k\_{\omega}} = \frac{\lambda}{2(n\_{2\omega} - n\_{\omega})}.\tag{2}$$

Thus Equation 1 can be rewritten as

$$\eta\_{SHG} = \mathbb{C}\_{\text{s}} \mathbb{C}^2(\lambda) \mathfrak{F}\_{eff}^{(s)2}(\text{s}),\tag{3}$$

with

$$\mathcal{C}\_{\mathcal{S}} = \frac{8\pi^2 |d\_{33}|^2 I\_{\omega} L^2}{c\varepsilon\_0} \,\mathrm{}^{\prime} \tag{4}$$

$$\mathcal{C}(\lambda) = \frac{1}{\lambda \sqrt{n\_{2\omega}} n\_{\omega}},\tag{5}$$

and

$$\xi\_{eff}^{(s)}(\lambda) = \left| \frac{1}{L} \int\_0^L e^{i(2\pi \ge /l\_\epsilon^{(s)}(\lambda))} \tilde{d}(\mathbf{x}) d\mathbf{x} \right|,\tag{6}$$

the parameter *ξ* (*s*) *eff*(*λ*) depends on the polarization direction of each domain.

2 Will-be-set-by-IN-TECH

Unlike the periodic optical supperlattice, the aperiodic optical superlattice can provide more spatial Fourier components of structure, therefore, some coupled parametric processes may be realized in this kind of devices. Firstly, the optimal design problem of the aperiodic optical superlattice is described in the real-space representation. Then several model designs are carried out to demonstrate the effectiveness of the present design method. The *LiTaO*<sup>3</sup> is selected as basic crystal for polarizing. The direction of polarization vectors in successive domain are opposite, thus are the signs of the nonlinear optical coefficients. However, the width of each individual domain is no longer equal and should be determined by the specified

We use the second harmonic generation (SHG) process for a single wavelength as an example. Assume that a laser beam with frequency *ω*<sup>1</sup> = *ω* is perpendicularly incident on the surface of an aperiodic optical superlattice. In order to use the largest nonlinear coefficient *d*33, let the propagation and polarization directions of the input light are along the *x* and *z* axes, respectively. Two optical fields are involved in the SHG process. one is the fundamental wave with *ω*<sup>1</sup> = *ω* and other is the second harmonic wave with *ω*<sup>2</sup> = 2*ω*. Under the slowing-wave and small-signal approximations, the conversion efficiency *ηSHG* from the fundamental wave

<sup>2</sup> *I<sup>ω</sup> L*<sup>2</sup>

*ω*

where *k<sup>ω</sup>* = *nωω*/*c* (*k*2*<sup>ω</sup>* = *n*2*ω*2*ω*/*c*) is the wave number of the fundamental (second harmonic) wave, *c* is the speed of light in vacuum, *n<sup>ω</sup>* (*n*2*ω*) is the refractive index of crystal at the fundamental (second harmonic) wavelength, *ε*<sup>0</sup> is the permittivity of vacuum, and *L* is the total length of the sample. *I<sup>ω</sup>* (*I*2*ω*) is the intensity of the fundamental (second harmonic) wave beam, and ˜*d*(*x*) only takes binary values of 1 or -1 which depends on the polarization

 1 *L L* 0 *e*

*<sup>c</sup>* (*λ*) for the SHG is defined as:

<sup>=</sup> *<sup>λ</sup>*

(*s*)2

<sup>2</sup> *I<sup>ω</sup> L*<sup>2</sup>

*<sup>c</sup>* (*λ*)) ˜*d*(*x*)*dx*

 

*cε*<sup>0</sup>

<sup>√</sup>*n*2*ωn<sup>ω</sup>*

2(*n*2*<sup>ω</sup>* − *nω*)

*k*2*<sup>ω</sup>* − 2*k<sup>ω</sup>*

*<sup>i</sup>*(*k*2*ω*−2*k<sup>ω</sup>* )*<sup>x</sup>* ˜*d*(*x*)*dx*

 

*eff* (*s*), (3)

, (4)

, (5)

, (6)

2

, (1)

. (2)

**2. Design of aperiodic optical supperlattice**

nonlinear optical processes.

to second harmonic wave can be written as:

direction. The coherence length *l*

Thus Equation 1 can be rewritten as

with

and

*<sup>η</sup>SHG* <sup>=</sup> *<sup>I</sup>*2*<sup>ω</sup>*

*l* (*s*)

> *ξ* (*s*) *eff*(*λ*) =

*Iω*

<sup>=</sup> <sup>8</sup>*π*2|*d*33<sup>|</sup>

(*s*)

*<sup>c</sup>* (*λ*) = <sup>2</sup>*<sup>π</sup>*

*c�*0*λ*2*n*2*ωn*<sup>2</sup>

<sup>Δ</sup>*<sup>k</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

*ηSHG* = *CsC*2(*λ*)*ξ*

*Cs* <sup>=</sup> <sup>8</sup>*π*2|*d*33<sup>|</sup>

*<sup>C</sup>*(*λ*) = <sup>1</sup> *λ*

 1 *L L* 0 *e i*(2*πx*/*l* (*s*)

**2.1. Design method**

Assume that the thickness of the each domain of the aperiodic optical superlattice is Δ*x*, thus the number of blocks in the sample is *N* = *L*/Δ*x*. The position of each domain is *xq* = *q*Δ*x* for *q* = 0, 1, 2, ....(*N* − 1). Equation 6 can be evaluated as

$$\begin{array}{lcl} \boldsymbol{\xi}\_{eff}^{(s)}(\lambda) &= \frac{1}{N\Delta x} \left| \sum\_{q=0}^{N-1} \tilde{d}(\mathbf{x}\_{q}) \int\_{\mathbf{x}\_{q}}^{\mathbf{x}\_{q} + \Delta \mathbf{x}} e^{i(2\pi \mathbf{x} / l\_{\epsilon}^{(s)}(\lambda))} d\mathbf{x} \right| \\ &= \left| \text{sinc} \left[ \frac{\Delta \mathbf{x}}{l\_{\epsilon}^{(s)}(\lambda)} \right] \right| \left| \left\{ \frac{1}{N} \sum\_{q=0}^{N-1} \tilde{d}(\mathbf{x}\_{q}) e^{i[2\pi(q+0.5)\Delta \mathbf{x}/l\_{\epsilon}^{(s)}(\lambda)]} \right\} \right| . \end{array} \tag{7}$$

where *sinc*(*x*) = *sin*(*πx*)/(*πx*). It can be seen that *ξ* (*s*) *eff*(*λ*) has two contributions: one belongs

to the unit block and depends on the Δ*x* and the coherence length *l* (*s*) *<sup>c</sup>* (*λ*); the other is caused by the interference effect among the blocks and depends on the configuration of domains and the polarization direction of each block.

The optimization design of the aperiodic optical superlattice for the SHG can be ascribed as a search for the maximum of *ξ* (*s*) *eff*(*λ*) with respect to ˜*d*(*q*Δ*x*). The first factor in Equation 7 has its maximum value 2/[*π*(2*m* + 1)] when Δ*x* = [(2*m* + 1)/2]*l* (*s*) *<sup>c</sup>* (*λ*). The second factor has its maximum value of 1 in the case of the periodic structure with a period of *a* = 2Δ*x* = *l* (*s*) *<sup>c</sup>* (*λ*), *d*(*xq*) takes opposite sign between two consecutive blocks. In this case, the phase lagging factor perfectly matches the reversal of the domain orientation between two adjacent blocks and the ideal constructive interference emerges. However, in the case of the aperiodic structure, it is quite difficult to find a solution and can be solved with the SA algorithm.

In order to demonstrate the effectiveness of the SA algorithm for design of aperiodic optical superlattices, a simple example of a single wavelength SHG is considered here. The parameters are chosen as follows: the wavelegth of the incident beam is *λ* = 1.064*μm*, the thickness of the unit block is Δ*x* = *l* (*s*) *<sup>c</sup>* (*λ*)/2 = 3.8713*μm*, and the number of blocks is 2142. Thus the total length of sample is *L* = 8298.8*μm*. The refractive indexes of material at corresponding wavelengths are evaluated by the Sellmeier equation [4]. The objective function in the SA algorithm is chosen as

$$E = |\mathfrak{f}^0 - \mathfrak{f}\_{eff}^{(s)}(\lambda)| \,\,\, \tag{8}$$

where *ξ*<sup>0</sup> is a preset value in guiding the SA procedure. Fig. 1 sketches a flowchart of the SA algorithm for constructing aperiodic optical superlattice. As shown in Fig. 1, the initial temperature and dropping rate are selected at the beginning of program, a random sign modulation ˜*d*(*xi*) is substituted into Equation 8 to calculate the initial object function *<sup>E</sup>*0. A new object function *E*<sup>1</sup> is calculated with changing the sign of a unit block in *d* (*xi*) and the difference Δ*E* = *E*<sup>1</sup> − *E*<sup>0</sup> is obtained. A random number *p*, 0 ≤ *p* ≤ 1, is generated by the computer. If *p* satisfies *p* ≤ exp(−Δ*E*/*T*), this change will be accepted and *E*<sup>0</sup> will be replaced by *E*1, else the sign of this unit block will be retrieved and *E*<sup>0</sup> does not change. After test all of blocks, this procedure will be repeated with a new temperature *T* = *T* × Δ*T* until no sign changes in the ˜*d*(*x*). Thus, the stable modulation left is the optimal one.

Finally, a perfect periodical structure with a pari of antiparallel domains for each unit block and with a period of *a* = 2Δ*x* = *l* (*s*) *<sup>c</sup>* as expected can be obtained. The reduced effective

4 Will-be-set-by-IN-TECH 96 Simulated Annealing – Single and Multiple Objective Problems Optimization Design of Nonlinear Optical Frequency Conversion Devices Using Simulated Annealing Algorithm <sup>5</sup>

nonlinear coefficient *ξ* (*s*) *eff* reaches its theoretical maximum value of 2/*π* = 0.6366. This means that the SA algorithm is appropriate for dealing with the above mentioned inverse source problem.

of 0.3 − 3. Five wavelengths are 0.972*μm*, 1.082*μm*, 1.283*μm*, 1.364*μm*, and 1.568*μm*. Other parameters are selected as: the total length of the sample *L* = 8295*μm*, the number of blocks

The obtained results are shown in Fig. 2. The wavelength is scanned with a interval of 0.05*nm* which is much small than that in the design procedure. There exist six strong peaks with almost identical peak value. Five of them are located at the expected wavelengths. One strong peak with an unexpected wavelength *λ* = 0.981*μm* appears very close to the expected wavelength *λ* = 0.972*μm*. There are also some stray peaks appearing in the lower wavelength

> (*s*) *eff*(*λα*)

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

97

regions and small dense oscillation structures as a background. The average value of *ξ*

**Figure 2.** Calculated results for the constructed aperiodic optical superlattice that implements multiple

In order to further reveal the characteristic of the SHG in the constructed aperiodic optical

the imping surface of incident light is shown in Fig. 3. It can be obviously seen that all curves exhibit nearly linearly increasing behavior with a nearly identical slope, which hints that the arrangement of domains is relatively favorable to the SHG process. The individual

Third harmonic generation (THG) has a wide application as a mean to extend coherent light sources to the short wavelengths. THG can be directly created using a third-order

*eff*(*λα*) with the optical propagating distance *x* from

(*s*)

contribution is accumulated with each in the constructive interference state.

**2.3. Coupled third harmonic generation for multiple wavelengths**

wavelengths SHG with an identical nonlinear optical coefficient.

superlattic, the plot of the variation of *ξ*

for preset five peaks is 0.1927 and the nonuniformity is 3.18 <sup>×</sup> <sup>10</sup>−4.

*N* = 2765, and the wavelength sampling interval is 1*nm*.

**Figure 1.** Flowchart of the SA algorithm for designing aperiodic optical superlattice.
