**2.2. Multiple wavelength SHG**

A particular design of the aperiodic optical supperlattice is carried out. It is expected that it can implement multiple wavelength SHG with an identical effective nonlinear coefficient *ξ* (*s*) *eff*(*λα*) = *<sup>ξ</sup>*<sup>0</sup> with *<sup>ξ</sup>*<sup>0</sup> is a constant. The thickness of each layer <sup>Δ</sup>*<sup>x</sup>* <sup>=</sup> 3.0*μ<sup>m</sup>* for matching the state of the art of microfabrication in practice. The objective function in the SA algorithm is set as

$$E = \sum\_{a} [|\xi^{0} - \xi\_{eff}^{(s)}(\lambda\_{a})|] + \beta [\max\{\mathfrak{f}\_{eff}^{(s)}(\lambda\_{a})\} - \min\{\mathfrak{f}\_{eff}^{(s)}(\lambda\_{a})\}] \,\tag{9}$$

where the function max{...} (min{...}) manifests to take their maximum (minimum) value among all the quantities including into {...}. *β* is an adjustable parameter taking a value 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 *N* = 2765, and the wavelength sampling interval is 1*nm*.

4 Will-be-set-by-IN-TECH

that the SA algorithm is appropriate for dealing with the above mentioned inverse source

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

*eff*(*λα*)|] + *β*[max{*ξ*

A particular design of the aperiodic optical supperlattice is carried out. It is expected that it can implement multiple wavelength SHG with an identical effective nonlinear coefficient

*eff*(*λα*) = *<sup>ξ</sup>*<sup>0</sup> with *<sup>ξ</sup>*<sup>0</sup> is a constant. The thickness of each layer <sup>Δ</sup>*<sup>x</sup>* <sup>=</sup> 3.0*μ<sup>m</sup>* for matching the state of the art of microfabrication in practice. The objective function in the SA algorithm is

where the function max{...} (min{...}) manifests to take their maximum (minimum) value among all the quantities including into {...}. *β* is an adjustable parameter taking a value

(*s*)

*eff*(*λα*)} − min{*ξ*

(*s*)

*eff*(*λα*)}], (9)

**2.2. Multiple wavelength SHG**

*<sup>E</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*

[|*ξ*<sup>0</sup> <sup>−</sup> *<sup>ξ</sup>*

(*s*)

*ξ* (*s*)

set as

*eff* reaches its theoretical maximum value of 2/*π* = 0.6366. This means

nonlinear coefficient *ξ*

problem.

(*s*)

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 regions and small dense oscillation structures as a background. The average value of *ξ* (*s*) *eff*(*λα*) for preset five peaks is 0.1927 and the nonuniformity is 3.18 <sup>×</sup> <sup>10</sup>−4.

**Figure 2.** Calculated results for the constructed aperiodic optical superlattice that implements multiple wavelengths SHG with an identical nonlinear optical coefficient.

In order to further reveal the characteristic of the SHG in the constructed aperiodic optical superlattic, the plot of the variation of *ξ* (*s*) *eff*(*λα*) with the optical propagating distance *x* from 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 contribution is accumulated with each in the constructive interference state.

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

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

with

and

*ξ* (*ct*) *eff* (*λ*) =

with

The variable *ξ*

*ξ* (*ct*) *eff* (*λ*) =

(*ct*)

Equation 15 will be rewritten as

 2 (*N*Δ*x*)<sup>2</sup> <sup>∑</sup>*N*−<sup>1</sup>

= 2 1 Δ*x* <sup>Δ</sup>*<sup>x</sup>*

= 2 *sinc*

× 1 *<sup>N</sup>* <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d*

× 1 *<sup>N</sup>*<sup>2</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d*

Δ*u* = <sup>2</sup>

coherence length *l*

correction.

= <sup>1</sup> *N*<sup>2</sup> *l* (*s*) *<sup>c</sup>* (*λ*) *iπ*Δ*x*

*<sup>L</sup>*<sup>2</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d* (*xq*)

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

*Ct* <sup>=</sup> <sup>144</sup>*π*4|*d*33<sup>|</sup>

(*λ*) = <sup>1</sup> *λ*2*n*3/2 *<sup>ω</sup> n*2*<sup>ω</sup>*

*C*�

*xq*+Δ*<sup>x</sup>*

*π*Δ*x l* (*s*) *<sup>c</sup>* (*λ*)

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

*xq*<sup>+</sup><sup>1</sup>

*iπ*Δ*x l* (*s*) *c* (*λ*)

> (*s*) *<sup>c</sup>* (*λ*)+1/*l*

one strong sharp peak at the preset wavelength of *λ* = 1.570*μm* with *ξ*

found that the designed structure performs the preset goal well.

*<sup>e</sup>*

*<sup>q</sup>*=<sup>0</sup> *<sup>e</sup>i*[2*πxq* (1/*<sup>l</sup>*

From Equation 16, it can been found that *ξ*

(*t*) *<sup>c</sup>* (*λ*)] <sup>1</sup>

(*xq*)*ei*[2*πxq*/*<sup>l</sup>*

*xq dxei*[2*πx*/*<sup>l</sup>*

Δ*x* <sup>Δ</sup>*<sup>x</sup>*

(*t*) *<sup>c</sup>* (*λ*)]

(*t*) *<sup>c</sup>* (*λ*)] *x*

(*ct*)

one factor (double *sinc* functions) belongs to unit block and strongly depends on Δ*x* and the

reflects the interference effect among the blocks in sample, depending on the arrangement of domains and the phase lagging from one block to other block. Δ*u* contributes to a small

A model design of the AOS that achieves the coupled THG is carried out. The parameters used in the design are : Δ*x* = 3*μm*, *L* = 8067*μm*, *N* = 2689, and *λ* = 1.570*μm*. Fig. 4 displays the calculated results. The wavelength is scanned with an interval of 0.1*nm*. There exists only

Δ*x l* (*s*)

*<sup>c</sup>* (*λ*) <sup>+</sup> <sup>Δ</sup>*<sup>x</sup> l* (*t*) *<sup>c</sup>* (*λ*)

(*t*) *<sup>c</sup>* (*λ*)] 1 *<sup>N</sup>* <sup>∑</sup>*q*−<sup>1</sup> *<sup>p</sup>*=<sup>0</sup> *d*

*xq dxei*[2*πxq*/*<sup>l</sup>*

+ *<sup>i</sup>π*Δ*<sup>x</sup> l* (*t*) *<sup>c</sup>* (*λ*) *sinc*

> (*t*) *<sup>c</sup>* (*λ*)]

 *L* 0 *e i*[2*πx*/*l* (*t*)

 2 *L*2

*<sup>q</sup>*=<sup>0</sup> *d* (*xq*)

*π*Δ*x l* (*t*) *<sup>c</sup>* (*λ*)

<sup>0</sup> *dxei*[2*πx*/*<sup>l</sup>*

 *sinc* <sup>4</sup> *I*<sup>2</sup> *<sup>ω</sup> <sup>L</sup>*<sup>4</sup>

<sup>√</sup>*n*3*<sup>ω</sup>*

0 *e i*[2*πζ*/*l* (*s*)

> *<sup>p</sup>*=<sup>0</sup> *d* (*xp*)

(*xp*)*ei*[2*πxp*/*<sup>l</sup>*

*xq <sup>d</sup>ζei*[2*πζ*/*<sup>l</sup>*

(*t*,*s*) *<sup>c</sup>* (*λ*); the other factor which is included inside the second curly braces

<sup>∑</sup>*q*−<sup>1</sup> *<sup>p</sup>*=<sup>0</sup> *d*

(*s*) *<sup>c</sup>* (*λ*)]

, (13)

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

, (14)

 

. (15)

99

(*s*) *<sup>c</sup>* (*λ*)] 

*<sup>c</sup>* (*λ*)] <sup>+</sup> <sup>Δ</sup>*<sup>u</sup>*

*eff* = 0.1811. It can be

 , (16)

(17)

 

(*s*)

Δ*x l* (*t*) *<sup>c</sup>* (*λ*)

*<sup>c</sup>* (*λ*)] ˜*d*(*ζ*)*dζ*

*xp*+Δ*<sup>x</sup>*

(*s*)

(*xp*)*ei*[2*π*(*p*+0.5)Δ*x*/*<sup>l</sup>*

*iπ*Δ*x l* (*t*) *<sup>c</sup>* (*λ*) *sinc*

*eff* (*λ*) has the contributions from two factors:

(*ct*)

(*s*) *<sup>c</sup>* (*λ*)]

 − *e* *xp <sup>d</sup>ζei*[2*πζ*/*<sup>l</sup>*

*<sup>c</sup>* (*λ*)] <sup>+</sup> <sup>Δ</sup>*<sup>u</sup>*

*c*2*�*<sup>2</sup> 0

*<sup>c</sup>* (*λ*)] ˜*d*(*x*)*dx <sup>x</sup>*

(*t*) *<sup>c</sup>* (*λ*)] <sup>∑</sup>*<sup>q</sup>*

<sup>0</sup> *<sup>d</sup>ζei*[2*πζ*/*<sup>l</sup>*

Assuming that the thickness of unit block is Δ*x*, the number of the blocks in sample is *N* = *L*/Δ*x*. The position of blocks is coordinated with *xq* = *q*Δ*x*, for *q* = 0, 1, 2, 3...(*N* − 1).

*eff* (*λ*) is the reduced coupled effective nonlinear coefficient for the CTHG.

**Figure 3.** Variation of *ξ* (*s*) *eff*(*λα*) with the optical propagating distance *x*.

nonlinear process, however, this method is of little practical importance because of intrinsic weak third-order optical nonlinearity. An efficient THG can be achieved by cascading two second-order nonlinear process. Two nonlinear optical crystal are involved: the first one is for SGH and the second one for sum frequency generation. THG can also be generated form the coupled parametric processess with high efficiency. The coupled THG (CTHG) is raised from the coupling effect of two nonlinear optical precesses: one is the SHG and the other is a sum frequency process. This two processes couple with each other in a single crystal. This coupling leads to a continuous energy transfer from the fundamental to the second, and then to the third harmonic fields. Thus, a direct third harmonic wave can be generated with high efficiency.

The THG process can be analyzed by solving the coupled nonlinear equations that describe interaction of these three fields: *Eω*, *E*2*ω*, and *E*3*<sup>ω</sup>* in the aperiodic optical superlattice. Under the small signal approximation, the third harmonic wave conversion efficiency is expressed by

$$\eta\_{\rm THG} = \frac{I\_{3\omega}}{I\_{\omega}} = \frac{144\pi^4 |d\_{33}|^4 I\_{\omega}^2 L^4}{c^2 \varepsilon\_0^2 \lambda^4 n\_{3\omega} n\_{2\omega}^2 n\_{\omega}^3} \left| \frac{2}{L^2} \int\_0^L e^{i(k\_{3\omega} - k\_{3\omega} - k\_{\omega})\cdot x} \tilde{d}(\mathbf{x}) \int\_0^x e^{i(k\_{3\omega} - 2k\_{\omega} \mathbb{I}\_z)} \tilde{d}(\mathbf{\zeta}) d\mathbf{\zeta} dx \right|^2 . \tag{10}$$

The coherence length *l* (*t*) *<sup>c</sup>* (*λ*) for the CTHG is defined as

$$d\_{\mathbb{C}}^{(t)}(\lambda) = \frac{2\pi}{k\_{3\omega} - k\_{2\omega} - k\_{\omega}} = \frac{\lambda}{(3n\_{3\omega} - 2n\_{2\omega} - n\_{\omega})}.\tag{11}$$

Thus, Equation 10 can be rewritten as

$$\eta\_{THG} = \mathbb{C}\_t \mathsf{C}^{\prime 2}(\lambda) (\xi\_{eff}^{(ct)}(\lambda))^2,\tag{12}$$

with

6 Will-be-set-by-IN-TECH

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

nonlinear process, however, this method is of little practical importance because of intrinsic weak third-order optical nonlinearity. An efficient THG can be achieved by cascading two second-order nonlinear process. Two nonlinear optical crystal are involved: the first one is for SGH and the second one for sum frequency generation. THG can also be generated form the coupled parametric processess with high efficiency. The coupled THG (CTHG) is raised from the coupling effect of two nonlinear optical precesses: one is the SHG and the other is a sum frequency process. This two processes couple with each other in a single crystal. This coupling leads to a continuous energy transfer from the fundamental to the second, and then to the third harmonic fields. Thus, a direct third harmonic wave can be generated with high

The THG process can be analyzed by solving the coupled nonlinear equations that describe interaction of these three fields: *Eω*, *E*2*ω*, and *E*3*<sup>ω</sup>* in the aperiodic optical superlattice. Under the small signal approximation, the third harmonic wave conversion efficiency is expressed

*<sup>i</sup>*(*k*3*ω*−*k*2*ω*−*k<sup>ω</sup>* )*<sup>x</sup>* ˜*d*(*x*)

<sup>=</sup> *<sup>λ</sup>*

(*ct*)

<sup>2</sup>(*λ*)(*ξ*

 *x* 0 *e*

(3*n*3*<sup>ω</sup>* − 2*n*2*<sup>ω</sup>* − *nω*)

*<sup>i</sup>*(*k*2*ω*−2*k<sup>ω</sup> <sup>ζ</sup>* ˜*d*(*ζ*)*dζdx*

*eff* (*λ*))2, (12)

 

. (11)

2

. (10)

**Figure 3.** Variation of *ξ*

efficiency.

*<sup>η</sup>THG* <sup>=</sup> *<sup>I</sup>*3*<sup>ω</sup>*

*Iω*

The coherence length *l*

<sup>=</sup> <sup>144</sup>*π*4|*d*33<sup>|</sup>

0*λ*4*n*3*ωn*<sup>2</sup>

(*t*)

*l* (*t*)

Thus, Equation 10 can be rewritten as

*c*2*�*<sup>2</sup>

<sup>4</sup> *I*<sup>2</sup> *<sup>ω</sup> <sup>L</sup>*<sup>4</sup>

> 2*ωn*<sup>3</sup> *ω*

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

 2 *L*2  *L* 0 *e*

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

*k*3*<sup>ω</sup>* − *k*2*<sup>ω</sup>* − *k<sup>ω</sup>*

*ηTHG* = *CtC*�

by

(*s*)

$$C\_l = \frac{144\pi^4 |d\_{33}|^4 I\_\omega^2 L^4}{c^2 \epsilon\_0^2},$$

$$\mathbb{C}'(\lambda) = \frac{1}{\lambda^2 n\_{\omega}^{3/2} n\_{2\omega} \sqrt{n\_{3\omega}}},\tag{14}$$

and

$$\left| \mathfrak{T}\_{eff}^{(\text{ct})}(\lambda) = \left| \frac{2}{L^2} \int\_0^L e^{i[2\pi x/l\_\epsilon^{(\text{t)})}(\lambda)]} \tilde{d}(x) dx \int\_0^X e^{i[2\pi \tilde{\xi}/l\_\epsilon^{(\text{s)})}(\lambda)]} \tilde{d}(\zeta) d\zeta \right|. \tag{15}$$

The variable *ξ* (*ct*) *eff* (*λ*) is the reduced coupled effective nonlinear coefficient for the CTHG.

Assuming that the thickness of unit block is Δ*x*, the number of the blocks in sample is *N* = *L*/Δ*x*. The position of blocks is coordinated with *xq* = *q*Δ*x*, for *q* = 0, 1, 2, 3...(*N* − 1). Equation 15 will be rewritten as

*ξ* (*ct*) *eff* (*λ*) = 2 (*N*Δ*x*)<sup>2</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d* (*xq*) *xq*+Δ*<sup>x</sup> xq dxei*[2*πx*/*<sup>l</sup>* (*t*) *<sup>c</sup>* (*λ*)] <sup>∑</sup>*<sup>q</sup> <sup>p</sup>*=<sup>0</sup> *d* (*xp*) *xp*+Δ*<sup>x</sup> xp <sup>d</sup>ζei*[2*πζ*/*<sup>l</sup>* (*s*) *<sup>c</sup>* (*λ*)] = 2 1 Δ*x* <sup>Δ</sup>*<sup>x</sup>* <sup>0</sup> *dxei*[2*πx*/*<sup>l</sup>* (*t*) *<sup>c</sup>* (*λ*)] <sup>1</sup> Δ*x* <sup>Δ</sup>*<sup>x</sup>* <sup>0</sup> *<sup>d</sup>ζei*[2*πζ*/*<sup>l</sup>* (*s*) *<sup>c</sup>* (*λ*)] × 1 *<sup>N</sup>* <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d* (*xq*)*ei*[2*πxq*/*<sup>l</sup>* (*t*) *<sup>c</sup>* (*λ*)] 1 *<sup>N</sup>* <sup>∑</sup>*q*−<sup>1</sup> *<sup>p</sup>*=<sup>0</sup> *d* (*xp*)*ei*[2*πxp*/*<sup>l</sup>* (*s*) *<sup>c</sup>* (*λ*)] <sup>+</sup> <sup>Δ</sup>*<sup>u</sup>* = 2 *sinc π*Δ*x l* (*t*) *<sup>c</sup>* (*λ*) *sinc π*Δ*x l* (*s*) *<sup>c</sup>* (*λ*) × 1 *<sup>N</sup>*<sup>2</sup> <sup>∑</sup>*N*−<sup>1</sup> *<sup>q</sup>*=<sup>0</sup> *d* (*xq*)*ei*[2*π*(*q*+0.5)Δ*x*/*<sup>l</sup>* (*t*) *<sup>c</sup>* (*λ*)] <sup>∑</sup>*q*−<sup>1</sup> *<sup>p</sup>*=<sup>0</sup> *d* (*xp*)*ei*[2*π*(*p*+0.5)Δ*x*/*<sup>l</sup>* (*s*) *<sup>c</sup>* (*λ*)] <sup>+</sup> <sup>Δ</sup>*<sup>u</sup>* , (16)

with

$$\begin{array}{lcl} \Delta \mathfrak{u} = \frac{2}{l^{2}} \sum\_{q=0}^{N-1} \widetilde{d}(\mathbf{x}\_{q}) \int\_{\mathbf{x}\_{q}^{i}}^{\mathbf{x}\_{q+1}} dx e^{i[2\pi \mathbf{x}\_{q}/l\_{\varepsilon}^{(l)}(\lambda)]} \int\_{\mathbf{x}\_{q}}^{\mathbf{x}} d\xi e^{i[2\pi l\_{\varepsilon}^{\*}/l\_{\varepsilon}^{(s)}(\lambda)]} \\ = \frac{1}{N^{2}} \left( \frac{l\_{\varepsilon}^{(s)}(\lambda)}{l^{\pi} \Delta \mathbf{x}} \right) \left\{ e^{\frac{l^{(s)}}{l^{(s)}}(\lambda)} \frac{+i^{\text{tr}\Delta\mathbf{x}}}{l^{(l)}(\lambda)} \text{sinc} \left[ \frac{\Delta\mathbf{x}}{l\_{\varepsilon}^{(s)}(\lambda)} + \frac{\Delta\mathbf{x}}{l\_{\varepsilon}^{(l)}(\lambda)} \right] - e^{\frac{l^{\text{tr}}\Delta\mathbf{x}}{l}} \text{sinc} \left[ \frac{\Delta\mathbf{x}}{l\_{\varepsilon}^{(l)}(\lambda)} \right] \right\} \\ \times \sum\_{q=0}^{N-1} e^{\frac{l^{\text{tr}}\Delta\mathbf{x}}{l\_{\varepsilon}}(1/l\_{\varepsilon}^{(s)}(\lambda) + 1/l\_{\varepsilon}^{(l)}(\lambda))} \end{array} \tag{17}$$

From Equation 16, it can been found that *ξ* (*ct*) *eff* (*λ*) has the contributions from two factors: one factor (double *sinc* functions) belongs to unit block and strongly depends on Δ*x* and the coherence length *l* (*t*,*s*) *<sup>c</sup>* (*λ*); the other factor which is included inside the second curly braces reflects the interference effect among the blocks in sample, depending on the arrangement of domains and the phase lagging from one block to other block. Δ*u* contributes to a small correction.

A model design of the AOS that achieves the coupled THG is carried out. The parameters used in the design are : Δ*x* = 3*μm*, *L* = 8067*μm*, *N* = 2689, and *λ* = 1.570*μm*. Fig. 4 displays the calculated results. The wavelength is scanned with an interval of 0.1*nm*. There exists only one strong sharp peak at the preset wavelength of *λ* = 1.570*μm* with *ξ* (*ct*) *eff* = 0.1811. It can be found that the designed structure performs the preset goal well.

**Figure 4.** Calculated result for the constructed aperiodic optical superlattice that implements the CTHG.

The same method can be used to construct aperiodic optical superlattic that implements multiple wavelengths CTHG with identical effective nonlinear coefficient. The relevant parameters are: *λα* = [1.40, 1.60, 1.80] *μm*, Δ*x* = 3*μm*, *L* = 8067*μm*, and *N* = 2689. The dependence of *ξ* (*ct*) *eff* (*λ*) on the wavelength is depicted in Fig. 5. The behavior of *ξ* (*ct*) *eff* (*λ*) exhibits fairly good uniformity. The coupled effective nonlinear coefficient is almost identical for three different wavelengths and the average value is 0.04792.

#### **2.4. Multiple wavelengths parametric amplification**

Parametric generation provides unique possibility of generating widely tunable radiation from a single pump light source, so it has attracted extensive interest since parametric amplification was theoretically predicted in 1960's [5]. To derive the mathematical expressions for parametric amplification in a aperiodic optical superlattice, the related formulas of optical parametric process in a homogeneous nonlinear medium should be briefly described here. Consider three optical plane waves with the frequencies *ω*1, *ω*2, and *ω*<sup>3</sup> (*ω*<sup>3</sup> = *ω*<sup>2</sup> + *ω*1), the equations governing the propagation of electromagnetic waves are written as

$$\begin{array}{ll}\frac{dE\_1}{d\mathbf{x}} = -\frac{\sigma\_1}{2}\sqrt{\frac{\mu}{\epsilon\_1}}E\_1 - \frac{i\omega\_1}{2}\sqrt{\frac{\mu}{\epsilon\_1}}d\_{33}E\_3E\_2^\*e^{-i\Delta k\mathbf{x}}\\ \frac{dE\_2^\*}{d\mathbf{x}} = -\frac{\sigma\_2}{2}\sqrt{\frac{\mu}{\epsilon\_2}}E\_2^\* + \frac{i\omega\_2}{2}\sqrt{\frac{\mu}{\epsilon\_2}}d\_{33}E\_1E\_3^\*e^{i\Delta k\mathbf{x}}\\ \frac{dE\_3}{d\mathbf{x}} = -\frac{\sigma\_3}{2}\sqrt{\frac{\mu}{\epsilon\_3}}E\_3 - \frac{i\omega\_3}{2}\sqrt{\frac{\mu}{\epsilon\_3}}d\_{33}E\_1E\_2e^{i\Delta k\mathbf{x}}\end{array} \tag{18}$$

**Figure 5.** Calculated result for the constructed aperiodic optical superlattice that implements the

� *ni ωi*

with *ni* is the refraction index of crystal at *ωi*(*i* = 1, 2, 3) and assuming the loss coefficients are

<sup>2</sup> *A*<sup>∗</sup>

where *�*<sup>0</sup> is permittivity of vacuum. In this derivation, we have used the approximation

a constant. As the meaningful quantity is the relative phase between *A*<sup>1</sup> and *A*3, therefore, we can set *A*3(0) to be real. Therefore, the relationship between the field variables *A*1(*x*), *A*2(*x*)

> *M*<sup>11</sup> *M*<sup>12</sup> *M*<sup>21</sup> *M*<sup>22</sup>

⎞ ⎠ ⎛ ⎝

<sup>2</sup> *<sup>e</sup>*−*i*(Δ*k*)*<sup>x</sup>*

*d*33*E*3(0),

<sup>2</sup> and *<sup>ω</sup>*2|*A*2(*x*)<sup>|</sup>

⎞

<sup>2</sup> throughout the interaction region, thus *A*3(*x*) can be regarded as

*A*1(*x*0) *A*∗ <sup>2</sup> (*x*0)

<sup>2</sup> *<sup>A</sup>*1*ei*(Δ*k*)*x*,

*Ei*, *i* = 1, 2, 3, (19)

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

(20)

101

<sup>2</sup> both remain small

⎠ , (21)

coupled THG for three wavelength with an identical nonlinear coefficient.

*Ai* =

negligible small, we can rewrite the first two equations in Equation 18 as

*dA*<sup>1</sup> *dx* <sup>=</sup> <sup>−</sup>*ig*

*dA*∗ 2 *dx* = +*ig*

*g* = � ( *μ �*0 ) *ω*1*ω*2 *n*1*n*<sup>2</sup>

and *A*1(*x*0), *A*2(*x*0) in homogeneous medium can be expressed as follows:

� = ⎛ ⎝

of small signal. In addition, we assume that *ω*1|*A*1(*x*)|

�*A*1(*x*) *A*∗ <sup>2</sup> (*x*)

Introducing a new field variable

compared with *ω*3|*A*3(0)|

where

where Δ*k* = *k*<sup>3</sup> − *k*<sup>1</sup> − *k*2, and *σ<sup>i</sup>* (*�i*) is the loss coefficient (permittivity of crystal) for *ω<sup>i</sup>* (*i* = 1, 2, 3); *μ* is the magnetic permeability in vacuum. It is noted that these equations are coupled to each other via the nonlinear coefficient *d*33.

**Figure 5.** Calculated result for the constructed aperiodic optical superlattice that implements the coupled THG for three wavelength with an identical nonlinear coefficient.

Introducing a new field variable

$$A\_{\dot{i}} = \sqrt{\frac{n\_{\dot{i}}}{\omega\_{\dot{i}}}} E\_{\dot{i}\prime} \quad \text{i} = 1, 2, 3,\tag{19}$$

with *ni* is the refraction index of crystal at *ωi*(*i* = 1, 2, 3) and assuming the loss coefficients are negligible small, we can rewrite the first two equations in Equation 18 as

$$\begin{split} \frac{dA\_1}{d\mathbf{x}} &= -\frac{i\mathbf{g}}{2} A\_2^\* e^{-i(\Delta k)\mathbf{x}}\\ \frac{dA\_2^\*}{d\mathbf{x}} &= +\frac{i\mathbf{g}}{2} A\_1 e^{i(\Delta k)\mathbf{x}} \end{split} \tag{20}$$

where

8 Will-be-set-by-IN-TECH

**Figure 4.** Calculated result for the constructed aperiodic optical superlattice that implements the CTHG.

The same method can be used to construct aperiodic optical superlattic that implements multiple wavelengths CTHG with identical effective nonlinear coefficient. The relevant parameters are: *λα* = [1.40, 1.60, 1.80] *μm*, Δ*x* = 3*μm*, *L* = 8067*μm*, and *N* = 2689. The

exhibits fairly good uniformity. The coupled effective nonlinear coefficient is almost identical

Parametric generation provides unique possibility of generating widely tunable radiation from a single pump light source, so it has attracted extensive interest since parametric amplification was theoretically predicted in 1960's [5]. To derive the mathematical expressions for parametric amplification in a aperiodic optical superlattice, the related formulas of optical parametric process in a homogeneous nonlinear medium should be briefly described here. Consider three optical plane waves with the frequencies *ω*1, *ω*2, and *ω*<sup>3</sup> (*ω*<sup>3</sup> = *ω*<sup>2</sup> + *ω*1), the

equations governing the propagation of electromagnetic waves are written as

*�*<sup>1</sup> *<sup>E</sup>*<sup>1</sup> <sup>−</sup> *<sup>i</sup>ω*<sup>1</sup> 2 *μ*

*�*<sup>3</sup> *<sup>E</sup>*<sup>3</sup> <sup>−</sup> *<sup>i</sup>ω*<sup>3</sup> 2 *μ*

where Δ*k* = *k*<sup>3</sup> − *k*<sup>1</sup> − *k*2, and *σ<sup>i</sup>* (*�i*) is the loss coefficient (permittivity of crystal) for *ω<sup>i</sup>* (*i* = 1, 2, 3); *μ* is the magnetic permeability in vacuum. It is noted that these equations are coupled

for three different wavelengths and the average value is 0.04792.

**2.4. Multiple wavelengths parametric amplification**

*dE*<sup>1</sup> *dx* <sup>=</sup> <sup>−</sup>*σ*<sup>1</sup> 2 *μ*

*dE*∗ 2 *dx* <sup>=</sup> <sup>−</sup>*σ*<sup>2</sup> 2 *μ �*<sup>2</sup> *E*<sup>∗</sup> <sup>2</sup> <sup>+</sup> *<sup>i</sup>ω*<sup>2</sup> 2 *μ*

*dE*<sup>3</sup> *dx* <sup>=</sup> <sup>−</sup>*σ*<sup>3</sup> 2 *μ*

to each other via the nonlinear coefficient *d*33.

*eff* (*λ*) on the wavelength is depicted in Fig. 5. The behavior of *ξ*

*�*<sup>1</sup> *d*33*E*3*E*<sup>∗</sup>

*�*<sup>2</sup> *d*33*E*1*E*<sup>∗</sup>

*�*<sup>3</sup> *<sup>d</sup>*33*E*1*E*2*ei*Δ*kx*,

<sup>2</sup> *<sup>e</sup>*−*i*Δ*kx*

<sup>3</sup> *<sup>e</sup>i*Δ*kx*

(*ct*) *eff* (*λ*)

(18)

dependence of *ξ*

(*ct*)

$$g = \sqrt{(\frac{\mu}{\epsilon\_0})\frac{\omega\_1 \omega\_2}{n\_1 n\_2}} d\_{33} E\_3(0),$$

where *�*<sup>0</sup> is permittivity of vacuum. In this derivation, we have used the approximation of small signal. In addition, we assume that *ω*1|*A*1(*x*)| <sup>2</sup> and *<sup>ω</sup>*2|*A*2(*x*)<sup>|</sup> <sup>2</sup> both remain small compared with *ω*3|*A*3(0)| <sup>2</sup> throughout the interaction region, thus *A*3(*x*) can be regarded as a constant. As the meaningful quantity is the relative phase between *A*<sup>1</sup> and *A*3, therefore, we can set *A*3(0) to be real. Therefore, the relationship between the field variables *A*1(*x*), *A*2(*x*) and *A*1(*x*0), *A*2(*x*0) in homogeneous medium can be expressed as follows:

$$
\begin{pmatrix} A\_1(\mathbf{x}) \\ A\_2^\*(\mathbf{x}) \end{pmatrix} = \begin{pmatrix} M\_{11} \ M\_{12} \\ M\_{21} \ M\_{22} \end{pmatrix} \begin{pmatrix} A\_1(\mathbf{x}\_0) \\ A\_2^\*(\mathbf{x}\_0) \end{pmatrix} \tag{21}
$$

where

$$\begin{aligned} M\_{11} &= e^{-i(\Delta k/2)(\mathbf{x} - \mathbf{x\_0})} \left[ \cosh(b(\mathbf{x} - \mathbf{x\_0})) + \frac{i(\Delta k)}{2b} \sinh(b(\mathbf{x} - \mathbf{x\_0})) \right], \\ M\_{12} &= e^{-i(\Delta k/2)(\mathbf{x} - \mathbf{x\_0})} e^{-i\Delta k \mathbf{x\_0}} \left[ -i\frac{g}{2b} \sinh(b(\mathbf{x} - \mathbf{x\_0})) \right], \\ M\_{21} &= M\_{12}^\*, \\ M\_{22} &= M\_{11'}^\* \end{aligned}$$

and

$$b = 0.5\sqrt{g^2 - (\Delta k)^2}.\tag{22}$$

The initial conditions are known as *A*1(0) and *A*<sup>∗</sup>

amplification coefficient of the signal wave.

belongs to solving an inverse source problem.

1.025

1.005

*<sup>E</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*

1.01

1.015

G (Arb. Unit)

SA algorithm is chosen as

1.02

optical superlattice once, it is amplified by *G* = *Mtot*

*<sup>g</sup>*(*xq*) = *<sup>μ</sup>*

idler wave *A*2(0) is considered as zero, so after signal wave passes through the aperiodic

In the case of aperiodic optical superlattice, ˜*d*(*xq*) varies with *xq*, so *g* is a function of *xq* as

 *ω*1*ω*<sup>2</sup> *n*1*n*<sup>2</sup>

Owing to ˜*d*2(*xq*) = 1, the sign modulation of ˜*d*(*xq*) does not bring any effect on *b*. However, *g*(*xq*) appearing in the transfer matrix is feeling to this sign modulation alone. Consequently, it is expected that the modulated structure may bring some benefits due to considerable effect of the sign modulation of ˜*d*(*xq*) on parametric amplification process. In the case of multiple wavelengths parametric amplification, the situation becomes much more complicated, and it

1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 <sup>2</sup> <sup>1</sup>

λ (μ m)


The optimization design of aperiodic optical superlattice served as parametric amplifier for multiple wavelengths with identical amplification coefficient *G*(*λα*) is carried out. Three signal wavelengths *λα* are selected as 1.70*μm*, 1.80*μm*, and 1.90*μm*. The pump intensity is *Ip* = 1010*W*/*m*2. Considering the practical technique for poling and dispersion of crystal, thickness of unit block Δ*x* = 8*μm* is selected in the simulation. The objective function for the

where *β* is a adjustable parameter taking a value of 0.3 ∼ 3. In multiple wavelengths parameter amplification, there is a trade-off between the amplification coefficients and uniformity, adjusting the value of *β* can balance them. Figure 6 exhibits the calculated

**Figure 6.** Calculated result for the constructed aperiodic optical superlattice that implements the parametric amplification for three wavelengths with an identical amplification coefficient.


*�*0

<sup>2</sup> (0) at *x*<sup>0</sup> = 0. In the general case, the initial

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

<sup>11</sup> times, therefore, *G* is the so–called

103

(*xq*)*E*3(0). (26)

Equations 21 and 22 tell that only when *b* is a real number, the signal and idler lights can be amplified. It requires that *g* must be greater than Δ*k*. If the wavelengths of the pump and signal lights are selected as 1.064*μm* and 1.78*μm*, respectively, Δ*k* has the value of 0.20*μm*−1. Even considering the largest nonlinear coefficient *d*<sup>33</sup> = 21.6*pm*/*V* of *LiNbO*3, it also requires the intensity of pump light being larger than 1.35 <sup>×</sup> <sup>10</sup>17*W*/*m*2, which is impossible in practical applications.

For the aperiodic optical superlattice with block thickness of Δ*x*, the coordinate of each blocks can be denoted by *xq* = *q*Δ*x*, for *q* = 0, 1, 2, 3....*N*, *N* is the total number of blocks in sample. Since *d*<sup>33</sup> for each unit block remains constant, therefore, Equations 21 and 22 are still valid within each block. By using the transfer matrix method, the total transfer matrix can be established by cascading individual matrix associated with each block in sequence. For instance, the transfer matrix for the (*q* + 1)th block from its left interface at *xq* to its right interface at *xq*+<sup>1</sup> can be expressed as

$$
\begin{pmatrix} A\_1(\mathbf{x}\_{q+1}) \\ A\_2^\*(\mathbf{x}\_{q+1}) \end{pmatrix} = \begin{pmatrix} M\_{11} \ M\_{12} \\ M\_{21} \ M\_{22} \end{pmatrix} \begin{pmatrix} A\_1(\mathbf{x}\_q) \\ A\_2^\*(\mathbf{x}\_q) \end{pmatrix} = M(\mathbf{x}\_q \rightarrow \mathbf{x}\_{q+1}) \begin{pmatrix} A\_1(\mathbf{x}\_q) \\ A\_2^\*(\mathbf{x}\_q) \end{pmatrix} \tag{23}
$$

where

$$\begin{aligned} M\_{11} &= e^{-i(\Delta k/2)\Delta \mathbf{x}\_q} \Big[ \cosh(b(\mathbf{x}\_q)\Delta \mathbf{x}\_q) \Big) + \frac{i(\Delta k)}{2b(\mathbf{x}\_q)} \sinh(b(\mathbf{x}\_q)\Delta \mathbf{x}\_q) \Big], \\ M\_{12} &= e^{-i(\Delta k/2)\Delta \mathbf{x}\_q} e^{-i\Delta \mathbf{k} \mathbf{x}\_q} \Big[ -i \frac{\mathcal{g}(\mathbf{x}\_q)}{2b(\mathbf{x}\_q)} \sinh(b(\mathbf{x}\_q)\Delta \mathbf{x}\_q) \Big], \\ M\_{21} &= M\_{12}^\*, \\ M\_{22} &= M\_{11}^\*. \end{aligned}$$

and

$$
\Delta \mathbf{x}\_{\mathfrak{q}} = \mathbf{x}\_{\mathfrak{q}+1} - \mathbf{x}\_{\mathfrak{q}}.\tag{24}
$$

Finally, the total transfer matrix reads

$$
\begin{pmatrix} A\_1(\mathbf{x}\_N) \\ A\_2^\*(\mathbf{x}\_N) \end{pmatrix} = \begin{pmatrix} M\_{11}^{tot} \ M\_{12}^{tot} \\ M\_{21}^{tot} \ M\_{22}^{tot} \end{pmatrix} \begin{pmatrix} A\_1(\mathbf{x}\_0) \\ A\_2^\*(\mathbf{x}\_0) \end{pmatrix} = M^{tot}(\mathbf{x}\_0 \rightarrow \mathbf{x}\_N) \begin{pmatrix} A\_1(\mathbf{x}\_0) \\ A\_2^\*(\mathbf{x}\_0) \end{pmatrix},
$$

where

$$M^{tot}(\mathbf{x}\_0 \to \mathbf{x}\_N) = \Gamma \mathbf{I}\_{q=0}^{N-1} M(\mathbf{x}\_q \to \mathbf{x}\_{q+1}).\tag{25}$$

The initial conditions are known as *A*1(0) and *A*<sup>∗</sup> <sup>2</sup> (0) at *x*<sup>0</sup> = 0. In the general case, the initial idler wave *A*2(0) is considered as zero, so after signal wave passes through the aperiodic optical superlattice once, it is amplified by *G* = *Mtot* <sup>11</sup> times, therefore, *G* is the so–called amplification coefficient of the signal wave.

In the case of aperiodic optical superlattice, ˜*d*(*xq*) varies with *xq*, so *g* is a function of *xq* as

10 Will-be-set-by-IN-TECH

*cosh*(*b*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0)) + *<sup>i</sup>*(Δ*k*)

<sup>2</sup>*<sup>b</sup> sinh*(*b*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0))

 ,

*<sup>g</sup>*<sup>2</sup> − (Δ*k*)2. (22)

*sinh*(*b*(*x* − *x*0))

 ,

where

and

where

and

where

*M*<sup>11</sup> = *e*

*M*<sup>12</sup> = *e*

*M*<sup>21</sup> = *M*<sup>∗</sup>

*M*<sup>22</sup> = *M*∗

impossible in practical applications.

interface at *xq*+<sup>1</sup> can be expressed as

*A*1(*xq*+1) *A*∗ <sup>2</sup> (*xq*+1)

*M*<sup>11</sup> = *e*

*M*<sup>12</sup> = *e*

*M*<sup>21</sup> = *M*<sup>∗</sup>

*M*<sup>22</sup> = *M*∗

Finally, the total transfer matrix reads

 =

*A*1(*xN*) *A*∗ <sup>2</sup> (*xN*)  =

−*i*(Δ*k*/2)Δ*xq*

<sup>−</sup>*i*(Δ*k*/2)Δ*xq e*

*Mtot* <sup>11</sup> *<sup>M</sup>tot* 12

*Mtot* <sup>21</sup> *<sup>M</sup>tot* 22

12,

11,

*M*<sup>11</sup> *M*<sup>12</sup> *M*<sup>21</sup> *M*<sup>22</sup>

−*i*Δ*kxq* − *i*

−*i*(Δ*k*/2)(*x*−*x*0)

−*i*(Δ*k*/2)(*x*−*x*0)

12,

11,

*e* −*i*Δ*kx*<sup>0</sup> <sup>−</sup> *<sup>i</sup> <sup>g</sup>* 2*b*

*b* = 0.5

Equations 21 and 22 tell that only when *b* is a real number, the signal and idler lights can be amplified. It requires that *g* must be greater than Δ*k*. If the wavelengths of the pump and signal lights are selected as 1.064*μm* and 1.78*μm*, respectively, Δ*k* has the value of 0.20*μm*−1. Even considering the largest nonlinear coefficient *d*<sup>33</sup> = 21.6*pm*/*V* of *LiNbO*3, it also requires the intensity of pump light being larger than 1.35 <sup>×</sup> <sup>10</sup>17*W*/*m*2, which is

For the aperiodic optical superlattice with block thickness of Δ*x*, the coordinate of each blocks can be denoted by *xq* = *q*Δ*x*, for *q* = 0, 1, 2, 3....*N*, *N* is the total number of blocks in sample. Since *d*<sup>33</sup> for each unit block remains constant, therefore, Equations 21 and 22 are still valid within each block. By using the transfer matrix method, the total transfer matrix can be established by cascading individual matrix associated with each block in sequence. For instance, the transfer matrix for the (*q* + 1)th block from its left interface at *xq* to its right

> *A*1(*xq*) *A*∗ <sup>2</sup> (*xq*)

 *A*1(*x*0) *A*∗ <sup>2</sup> (*x*0)

*<sup>M</sup>tot*(*x*<sup>0</sup> <sup>→</sup> *xN*) = <sup>Π</sup>*N*−<sup>1</sup>

*cosh*(*b*(*xq*)Δ*xq*)) + *<sup>i</sup>*(Δ*k*)

*g*(*xq*) 2*b*(*xq*)

= *M*(*xq* → *xq*+1)

2*b*(*xq*)

*sinh*(*b*(*xq*)Δ*xq*)

<sup>=</sup> *<sup>M</sup>tot*(*x*<sup>0</sup> <sup>→</sup> *xN*)

*A*1(*xq*) *A*∗ <sup>2</sup> (*xq*)

> ,

*A*1(*x*0) *A*∗ <sup>2</sup> (*x*0)

*<sup>q</sup>*=<sup>0</sup> *M*(*xq* → *xq*+1). (25)

 ,

*sinh*(*b*(*xq*)Δ*xq*)

 ,

Δ*xq* = *xq*+<sup>1</sup> − *xq*. (24)

(23)

$$g(\mathbf{x}\_{\emptyset}) = \sqrt{\left(\frac{\mu}{\varepsilon\_0}\right) \frac{\omega\_1 \omega\_2}{n\_1 n\_2}} |d\_{33}| \tilde{d}(\mathbf{x}\_{\emptyset}) E\_3(0). \tag{26}$$

Owing to ˜*d*2(*xq*) = 1, the sign modulation of ˜*d*(*xq*) does not bring any effect on *b*. However, *g*(*xq*) appearing in the transfer matrix is feeling to this sign modulation alone. Consequently, it is expected that the modulated structure may bring some benefits due to considerable effect of the sign modulation of ˜*d*(*xq*) on parametric amplification process. In the case of multiple wavelengths parametric amplification, the situation becomes much more complicated, and it belongs to solving an inverse source problem.

**Figure 6.** Calculated result for the constructed aperiodic optical superlattice that implements the parametric amplification for three wavelengths with an identical amplification coefficient.

The optimization design of aperiodic optical superlattice served as parametric amplifier for multiple wavelengths with identical amplification coefficient *G*(*λα*) is carried out. Three signal wavelengths *λα* are selected as 1.70*μm*, 1.80*μm*, and 1.90*μm*. The pump intensity is *Ip* = 1010*W*/*m*2. Considering the practical technique for poling and dispersion of crystal, thickness of unit block Δ*x* = 8*μm* is selected in the simulation. The objective function for the SA algorithm is chosen as

$$E = \sum\_{\mathfrak{a}} |G\_0 - G(\lambda\_{\mathfrak{a}})| + \beta \left[ \max \{ G(\lambda\_{\mathfrak{a}}) \} - \min \{ G(\lambda\_{\mathfrak{a}}) \} \right],\tag{27}$$

where *β* is a adjustable parameter taking a value of 0.3 ∼ 3. In multiple wavelengths parameter amplification, there is a trade-off between the amplification coefficients and uniformity, adjusting the value of *β* can balance them. Figure 6 exhibits the calculated

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.
