## Abstract

One-dimensional defective photonic crystals containing nonlinear material for coupled third harmonic generation (CTHG) have been designed. The general solution of the CTHG in such structure has been derived. The wavelengths of the fundamental wave (FW), second harmonic generation (SHG), and CTHG have been designed to lie at the defect states in different photonic band gaps by employing the simulated annealing (SA) method. Due to the strong location, low group velocity, and spatial phase locking, the conversion efficiencies of the SHG and CTHG have been greatly enhanced. Designed structure for multiple frequencies CTHG is also demonstrated.

©2007 Optical Society of America

## 1. Introduction

Over past decades, photonic crystals (PCs) have opened up a new subject as a class of optoelectronic material. The PCs are periodic arrays of dielectric composites for controlling and manipulating the propagation of light [1]. They have attracted considerable interests due to many potential applications. The nonlinear frequency conversion has been researched for many years [2]. In the traditional nonlinear optical processes, the phase-matching condition, which has a great restriction to the choice of birefringence material, was concernment for obtaining high conversion efficiency [3]. The quasi-phase matching has also been utilized to obtain high conversion efficiency [4].

By introducing a defect into a perfect PC, the confined mode can be generated in the band gap. The field distribution of the confined mode can be greatly enhanced around the defective region, thus the nonlinear phenomena can be effectively improved. Therefore, the PCs can also realize the nonlinear optical effects that have played an important role in the development of optoelectronic devices. In previous investigations, most works have been devoted to research the second harmonic generation (SHG) when the wavelengths of the FW and SHG are tuned to the photonic band edges [5, 6]. Recently, the giant enhancement of the SHG has realized by utilizing dual-localized modes at the FW and SHG in a defective PC [7–9]. By combining the strong localization, low group velocity, and spatial phase locking, the nonlinear interaction has been greatly enhanced. Inspired by these works, a single defective PC with three localized modes is constructed by using the simulated annealing (SA) method for obtaining giant enhancement of the SHG and coupled third harmonic generation (CTHG). The third harmonic wave is generated by summing the FW and SHG for effectively utilizing the second nonlinear coefficient which is much larger than the third one. The conversion efficiency of the SHG and CTHG are designed to be equal for some special applications. The defective PC for multiple wavelengths CTHG is also designed and the simulation results show that the constructed sample can perform the designed function well.

## 2. Basic theory

The one-dimensional layer structure is considered here, the incident wave is normally impinged upon the surface of the sample. Neglecting pump power depletion, the electric field *E*
^{(1)}
* _{l}*(

*z*) (

*E*

^{(2)}

*(*

_{l}*z*),

*E*

^{(3)}

*(*

_{l}*z*)) of the FW (SHG, CTHG) for the

*l*-th layer must satisfy the following equations:

where *k _{l}*

^{(1)}=

*n*

_{l}^{(1)}

*k*

_{10},

*k*

_{l}^{(2)}=

*n*

_{l}^{(2)}

*k*

_{20},

*k*

_{l}^{(3)}=

*n*

_{l}^{(3)}

*k*

_{30},

*k*

_{10}=

*ω*/

*c*,

*k*

_{20}= 2

*ω*/

*c*, and

*k*

_{30}= 3

*ω*/

*c*are wave vectors of the FW, SHG, and CTHG, respectively.

*ω*is the frequency of the FW,

*n*

_{l}^{(1)}(

*n*

_{l}^{(2)},

*n*

_{l}^{(3)}) is the refractive index of the

*l*-th layer for the FW (SHG, CTHG),

*c*is the velocity of light in vacuum,

*χ*is the second nonlinear coefficient of the

_{l}*l*-th layer.

The solution of Eq. (1) has the form as

where *A*
^{(1)}
* _{l}* and

*B*

^{(1)}

*represent the amplitudes of the forward and backward FW at interfaces. By utilizing the continuous condition of the electric fields at each interfaces, the transfer matrix method, and the initial condition*

_{l}*A*

^{(1)}

_{1}= 1 and

*B*

^{(1)}

*= 0 , the electric field of the FW in the*

_{N}*l*-th layer can be obtained.

*N*is the total number of layers in the sample. Similarly, the SHG electric field in the

*l*-th layer can be expressed as

*A*
^{(2)}
* _{l}* and

*B*

^{(2)}

*represent the amplitudes of the forward and backward SHG at interfaces, respectively. Substituting Eq. (4) and Eq. (5) into Eq. (2), it can be obtained that*

_{l}Furthermore, the expression of CTHG electric field in the *l* -th layer is

$$\phantom{\rule{3.2em}{0ex}}+{D}_{l,31}{e}^{i\left({k}_{l}^{\left(1\right)}-{k}_{l}^{\left(2\right)}\right)x}+{D}_{l,32}{e}^{-i\left({k}_{l}^{\left(1\right)}-{k}_{l}^{\left(2\right)}\right)x}+{E}_{l,31}{e}^{i3{k}_{l}^{\left(1\right)x}}+{E}_{l,32}{e}^{-i3{k}_{l}^{\left(1\right)}x}$$

$$\phantom{\rule{3.2em}{0ex}}+{F}_{l,31}{e}^{i{k}_{l}^{\left(1\right)}x}+{F}_{l,32}{e}^{-i{k}_{l}^{\left(1\right)}x},$$

*A*
^{(3)}
* _{l}* and

*B*

^{(3)}

*represent the amplitudes of the forward and backward THG at interface, respectively. Substituting Eqs. (4), (5) and (7) into Eq. (3), all parameters in Eq. (7) can be drawn as*

_{l}$${D}_{l,31}=\frac{-{k}_{30}^{2}{\chi}_{l}{A}_{l}^{\left(1\right)}{B}_{l}^{\left(2\right)}}{{k}_{l}^{\left(3\right)2}-{\left({k}_{l}^{\left(1\right)}-{k}_{l}^{\left(2\right)}\right)}^{2}},\phantom{\rule{1.2em}{0ex}}{D}_{l,32}=\frac{-{k}_{30}^{2}{\chi}_{l}{A}_{l}^{\left(2\right)}{B}_{l}^{\left(1\right)}}{{k}_{l}^{\left(3\right)2}-{\left({k}_{l}^{\left(1\right)}-{k}_{l}^{\left(2\right)}\right)}^{2}},$$

$${E}_{l,31}=\frac{-{k}_{30}^{2}{\chi}_{l}{A}_{l}^{\left(1\right)}{C}_{l,21}}{{k}_{l}^{\left(3\right)2}-{9k}_{l}^{{\left(1\right)}^{2}}},\phantom{\rule{3.2em}{0ex}}{E}_{l,32}=\frac{-{k}_{30}^{2}{\chi}_{l}{B}_{l}^{\left(1\right)}{C}_{l,22}}{{k}_{l}^{\left(3\right)2}-{9k}_{l}^{{\left(1\right)}^{2}}},$$

$${F}_{l,31}=-\frac{{k}_{30}^{2}{\chi}_{l}{B}_{l}^{\left(1\right)}{C}_{l,21}}{{k}_{l}^{\left(3\right)2}-{k}_{l}^{{\left(1\right)}^{2}}}+\frac{2{k}_{30}^{2}{k}_{20}^{2}{\chi}_{l}^{\left(2\right)}{\chi}_{l}{A}_{l}^{\left(1\right)2}{B}_{l}^{\left(1\right)}}{{k}_{l}^{\left(2\right)2}\left({k}_{l}^{\left(3\right)2}-{k}_{l}^{{\left(1\right)}^{2}}\right)},$$

$${F}_{l,32}=-\frac{{k}_{30}^{2}{\chi}_{l}{A}_{l}^{\left(1\right)}{C}_{l,22}}{{k}_{l}^{\left(3\right)2}-{k}_{l}^{\left(1\right)2}}+\frac{2{k}_{30}^{2}{k}_{20}^{2}{\chi}_{l}^{\left(2\right)}{\chi}_{l}{B}_{l}^{\left(1\right)2}{A}_{l}^{\left(1\right)}}{{k}_{l}^{\left(2\right)2}\left({k}_{l}^{\left(3\right)2}-{k}_{l}^{\left(1\right)2}\right)}.$$

By using the initial conditions *A*
^{(2)}
_{1}= 0, *B*
^{(2)}
* _{N}*= 0,

*A*

^{(3)}

_{1}= 0, and

*B*

^{(3)}

*= 0, the electric fields of CTHG at each interface can be derived. The conversion efficiencies of the forward and backward waves are defined respectively as follows*

_{N}## 3. Simulation and discussion

The sample investigated is a photonic quantum well structure. It is a structure of alternatively stacked layers of nonlinear material and air sandwiched by two truncated PCs. At first, we design the perfect structure (AB)_{10}, where A is LiNbO_{3} which is a dispersive material. B is air, whose refractive index is *n _{B}* =1.0 for all wavelengths we are interested. The refractive indexes of LiNbO

_{3}for different frequencies are from Ref. 10. The thicknesses of A and B layers are

*d*= 0.2557

_{A}*μm*and

*d*=0.1875

_{B}*μm*, respectively. Thus this prefect PC has three band gaps [

*λ*=1283

_{a}*μm*

*λ*= 1.835

_{b}*μm*], [

*λ*= 0.667

_{c}*μm*

*λ*=0.848

_{d}*μm*], and [

*λ*=0.477

_{e}*μm*

*λ*= 0.533

_{f}*μm*] which can cover the wavelengths for harmonic generation. Then we construct the photonic quantum well (PQW) structure, which is composed of alternatively layers of two materials C (LiNbO

_{3}) and D (Air), sandwiched by two perfect structures (AB)

_{5}. The total thickness of the PQW is 12.0

*μm*. Since the resonator-like structure created by the photonic crystal with a defect PQW leads to Fabry-Perot resonances and enhanced fields, the intensity of FW wave in the LiNbO

_{3}will be greatly enhanced, thus the nonlinear transform efficiency will be improved.

As the first aim, we design the defective PC for single wavelength CTHG. The preset wavelengths are *λ ^{o}*

_{1,1}=1.500

*μm*,

*λ*

^{o}_{2,1}= 0.750

*μm*, and

*λ*

^{o}_{3,1}=0.500

*μm*for the FW, SHG, and CTHG, respectively. They are located in three different band gaps of the perfect PC (AB)

_{10}, respectively. The thickness of the unit layer is selected as

*d*=

_{C}*d*= 0.04

_{D}*μm*. However, the real thickness of each layer C or D is integral times of

*d*or

_{C}*d*. They are determined by use of the SA algorithm for satisfy the specified requirement [8]. The object function for the SA algorithm is:

_{D}$${\lambda}_{1,s}^{o},{\lambda}_{1,s}^{\left(k\right)}\in [{\lambda}_{1,s-1},{\lambda}_{1,s}]\in [{\lambda}_{a},{\lambda}_{b}],\phantom{\rule{.2em}{0ex}}{\lambda}_{2,s}^{o},{\lambda}_{2,s}^{\left(k\right)}\in [{\lambda}_{2,s-1},{\lambda}_{2,s}]\in [{\lambda}_{c},{\lambda}_{d}],$$

$${\lambda}_{3,s}^{o},{\lambda}_{3,s}^{\left(k\right)}\in [{\lambda}_{3,s-1},{\lambda}_{3,s}]\in [{\lambda}_{e},{\lambda}_{f}],\phantom{\rule{2.2em}{0ex}}\alpha =\mathrm{1,2,3};\phantom{\rule{.2em}{0ex}}s=\mathrm{1,2,3}\cdots $$

with

$${\lambda}_{\mathrm{2,0}}\left(={\lambda}_{c}\right)<{\lambda}_{\mathrm{2,1}}^{o}<{\lambda}_{\mathrm{2,1}}<{\lambda}_{\mathrm{2,2}}^{o}<{\lambda}_{\mathrm{2,2}}<\cdots <{\lambda}_{2,s}^{o}<{\lambda}_{2,s}\left(={\lambda}_{d}\right);$$

$${\lambda}_{\mathrm{3,0}}\left(={\lambda}_{e}\right)<{\lambda}_{\mathrm{3,1}}^{o}<{\lambda}_{\mathrm{3,1}}<{\lambda}_{\mathrm{3,2}}^{o}<{\lambda}_{\mathrm{3,2}}<\cdots <{\lambda}_{3,s}^{o}<{\lambda}_{3,s}\left(={\lambda}_{f}\right),$$

where *β*
_{1} and *β*
_{2} are the adjustable constants, *α* = 1,2,3 express the FW, SHG, and CTHG, respectively. *S* denotes the *S*-th preset FW. *λ*
_{α,s}
^{(k)} is the wavelength of the defect states originated from every temporal structure during the SA process. *η*
_{1,s}
^{(k)} is the corresponding conversion efficiency. The SA searches for the minimum of the object function by adjust the thickness of layers C and D. Finally, the photonic quantum well structure can be completely determined.

Figure 1 presents the designed well structure constituted with C and D between the band structures. Figure 2 presents the transmission spectrum of the designed PQW structure. The wavelengths designed appear at *λ*
_{1,1} =1.500*μm*, *λ*
_{2,1}=0.750*μm* and *λ*
_{3,1} =0.500*μm*, respectively. They agree with the required wavelengths quite well. However, there are two extra defect states in the band of [ *λ _{a}*=1.283

*μm*

*λ*=1.835

_{b}*μm*] with wavelengths

*λ*

_{1,2}=1.333

*μm*and

*λ*

_{1,3}= 1.685

*μm*. We turn to explore the properties of the SHG and CTHG of this structure. The largest nonlinear coefficient d

_{33}=47.0pm/V of the LiNbO

_{3}is used for achieving higher conversion efficiency. The intensity of the incident FW is 21MW/m

^{2}. The wavelength dependence of the SHG and CTHG is shown in Fig. 3. The conversion efficiencies are

*η*

_{forth}^{(2)}=0.0913 for the forward SHG and

*η*

_{forth}^{(3)}= 0.0906 for the forward CTHG, they are nearly identical. For the strong localization of the FW, SHG, and CTHG simultaneously, the conversion efficiency has been obviously enhanced. For two extra defect states in the band [

*λ*= 1.283

_{a}*μm λ*= 1.835

_{b}*μm*], since only the FW is located at the defect state, the conversion efficiencies of the forward SHG (CTHG) for these wavelengths are quite small.

Secondly, we design the defective PC for double wavelengths CTHG. The preset wavelengths are *λ ^{o}*

_{1,1}=1.458

*μm*,

*λ*

^{o}_{1,2}=1.578

*μm*for the FWs,

*λ*

^{o}_{2,1}=0.729

*μm*,

*λ*

^{o}_{2,2}= 0.789

*μm*for the SHGs, and

*λ*

^{o}_{3,1}= 0.486

*μm*,

*λ*

^{o}_{3,2}= 0.526

*μm*for the CTHGs. They are also respectively located in three band gaps of the perfect PC (AB)

_{10}. Figure 4 presents the transmission spectrum of the designed photonic quantum well structure. The wavelengths designed 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 5 describes the wavelength dependence conversion efficiency of the designed structure. Only for two predesigned wavelengths, the conversion efficiency is quite high. The values 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 required aim well.

## 4. Conclusion

In summary, we have employed the SA algorithm to construct PQWs for achieving high conversion efficiency of the CTHG. Due to the strong location of the FW, SHG and CTHG, the conversion efficiency of the CTHG has been greatly enhanced. The simulation results shows that the constructed sample can perform the designed functions well. Due to the feature of high efficiency and low power density, such defective PC can be expected as a compact and high efficiency nonlinear frequency conversion photonic device.

## Acknowledgments

This work was supported by the Chinese National Key Basic Research Special Fund under grant 2006CB302901 and the National Natural Science Foundation of China under grant 10604042 and 10674038.

## References and links

**1. **E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

**2. **F. Omenetto, A. Efimov, A. Taylor, J. Knight, W. Wadsworth, and P. Russell, “Polarization dependent harmonic generation in microstructured fibers,” Opt. Express **11**, 61–67 (2003). [CrossRef] [PubMed]

**3. **Y. R. Shen, *The Principles of Nonlinear Optics* (Wiley, 1984).

**4. **S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. **6**, 301–304 (1972). [CrossRef]

**5. **M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A. **56**, 3166 –3174 (1997). [CrossRef]

**6. **M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E. **64**, 046606 (2001). [CrossRef]

**7. **F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode,” Phys. Rev. B. **70**, 245109 (2004). [CrossRef]

**8. **L. M. Zhao and B. Y. Gu, “Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material,” Appl. Phys. Lett. **88**, 122904 (2006). [CrossRef]

**9. **L. M. Zhao and B. Y. Gu, “Enhanced second-harmonic generation for multiple wavelengths by defect modes in one-dimensional photonic crystals,” Opt. Lett. **31**, 1510–1512 (2006). [CrossRef] [PubMed]

**10. **D. S. Smith and H. D. Riccius, “Refractive indices of lithium niobate,” Opt. Commun. **17**, 332–335 (1976). [CrossRef]