## Abstract

Optical second harmonic generation (OSHG) in a two-dimension photonic crystal consisting of centro-symmetric dielectric is investigated. The calculation model and analyzing method for OSHG are discussed. Based on Finite Difference–Time Domain algorithm, the electromagnetic field distribution in the structure and the intensity of second harmonic generation along the waveguide are analyzed. The results show that the acute spatial variation of electro-magnetic field results in the radiation of OSHG, and the intensity of OSHG is proportional to the square of the waveguide length. When the beam intensity of the pumping wave with a wavelength of 10.6 µm is 1.3MW/mm^{2}, the power conversion efficiency is 0.268% for a silicon photonic crystal with a length of 40 µm.

©2004 Optical Society of America

## 1. Introduction

Second harmonic generation (SHG) is an important nonlinear phenomenon by which laser wavelength conversion can be realized and the coverage range of laser spectrum can be expanded [1]. However, nonlinear materials with high second order susceptibility are rare, and one kind of material can only be used for one special wavelength. Up to now even in visible band the entire coverage of laser frequency has not been realized. So it is significant to develop some new available nonlinear materials for different wave bands. The mechanisms of SHG were developed in the 1960s [2], which include electric dipole (ED) polarization, magnetic dipole (MD) polarization, electric quadrupole (EQ) polarization, and magnetic quadrupole (MQ) polarization etc. Because ED polarization is the largest in conventional nonlinear bulk material, ED approximation is adopted to analyze SHG usually. However, as the second-order susceptibilities *χ*
^{(2)} are zeros in materials with inversion symmetry, EQ becomes one of the major origin of nonlinearity. Since the susceptibilities of EQ polarization is 3~4 magnitudes less than *χ*
^{(2)}, it is hard to observe SHG in this kind of materials. However, the EQ polarization can be enlarged if we increase the spatial gradient of the electric field within the dielectric with special measures. In fact, through applying dc electric field, R. W. Terhune has observed the SHG from the crystal calcite possessing a center of inversion in 1960s [3].

Photonic crystal (PC) is a periodic dielectric structure with obvious photonic band gap (PBG) [4]. The electromagnetic wave in PCs can be modulated acutely [5]. The density of modes (DOM) is increased and decreased violently near the edge of PBG [6], and a large electric field gradient can be obtained in PCs even without applying dc electric field. Therefore it is possible to obtain the notable SHG in materials with inversion symmetry so long as a suitable configuration is designed. In 1998, the phase matched SHG was found in 3-D PCs consisting of dye-coated polystyrene spheres by Martorell etc. [7]. Recently, we experimentally investigated the SHG in a PC slab consisting of centrosymmetric materials and observed the SHG signal in the transmission direction when the incident laser excited the quasi-waveguide [8]. However further analysis and calculation on this issue has not been carried out in those papers.

In this paper we propose a calculation model to investigate the SHG in PCs consisting of centrosymmetric materials. The model is based on the EQ interaction in PCs and Maxwell equations in complex dielectric. The distribution of electromagnetic field and the SHG within a special PC designed by ourselves are analyzed by resorting to Finite Difference–Time Domain (FDTD) algorithm.

## 2. Theoretical model

The configuration described in this paper is shown in Fig. 1. It consists of PC of air columns, PC of dielectric columns and several air waveguides. The units of the PCs are triangular lattice. The waveguides are spaced by dielectric columns. All of them are 2-D structure formed by the same material with inversion symmetry. The incident wave is TEz mode and wave vector lies on the xy plane. Based on the Maxwell’s equations and FDTD algorithm [9], we are able to get the distribution of electromagnetic field in the structure. The intensity of nonlinear polarization originating from EQ mechanism can be described in the following mode[10]

where, *Q*
_{ijkl} is the susceptibility of EQ polarization. E_{j} and E_{l} are electric fields. If we select Si crystal of class m3m, then the four-order tensors *Q*
_{ijkl} are as follows,

where 1, 2 and 3 corresponds to x, y and z, respectively. In the configuration of Fig. 1, the electromagnetic wave along *x*-direction is affected by PBG and modulated by intermediate dielectric columns. The variation is acute leading to the very large spatial gradient. Whereas along *y*-direction, the gradient is much less than that of *x*-direction, so it can be ignored. Consequently Eq. (1) can be simplified as

Equation (3) is the source term of SHG in PC consisting of Si material. When a pumping field is transmitted in the defects of PC, it interacts with material Si through susceptibility *Q*
_{ijkl}, thus a second-order nonlinear polarization vector is generated, and SHG is excited when the vector continuously varies with time. The energy will be transferred back and forth between the pumping wave and SHG. The process leads to a continuous enhancement of SHG. When dielectric is lossless for the pumping wave, the excited SH fulfill the following equations [2,10]

It is clear that the 6 component equations can be divided into two groups. One group consists of 3 equations of Eqs. (a), (b) and (c), which only includes Ex, Ey and Hz, i.e., TEz wave. Another group consists of 3 component equations of Eqs. (c), (d) and (e), which only includes Ez, Hx and Hy, i.e. TMz wave. Among them the TEz equations are active equations. The originating term is just nonlinear polarization intensity determined by Eq. (4). The TMz equations are passive equations, which are independent of the generation of SHG. According to the standard Yee algorithm of FDTD[12], the average value of intensity of SHG can be obtained.

## 3. Calculation and results analysis

From Eq. (3), in order to obtain enhanced SHG, the electric field or the gradient must be enlarged violently. In PCs, because of the localization of the light field, the electric field can be partly enlarged. However, the enhancement is insufficient for exciting obvious SHG. It is necessary to enhance the gradient of the electric field greatly through some special ways accordingly. When the frequency of an incident wave is near the band edge, the maximal DOM and the minimal group velocity will lead to the maximum variation of light field [5,13]. Therefore we have designed the special PC (Fig. 1) with a band edge just near the pumping frequency, which can enlarge the electric field gradient to the utmost. In addition, the asymmetrical structure itself will also enlarge light field gradient. That is why the two PCs are different and the waveguides are partly filled in the structure. The parameters aiming at a CO2 laser with a wavelength of 10.6 *µm* (the pumping source) are as follows: material is Si, the dielectric permittivity *ε* is 11.9, the lattice constant *a* is 3.2µm, the width of the waveguide is 1.6*a*, the fill factor f is 0.6, the radius of air columns *r*1 of PC1 (triangular air columns) is 0.4*a*, the radius of dielectric columns *r*2 of PC2 (triangular dielectric columns) is 0.2*a*. When the incident wave is TE polarization, the PBG of PC1 and PC2, calculated using the plane wave expansion method, are shown in Fig. 2. It is obvious that the fundamental harmonic (FH) frequency, corresponding to *ωa*/2*πc*=0.3 in Fig. 3, is exactly within the PBG of PC1 and near the PBG edge of PC2. The spatial sampling step is Δ*x*=Δ*y*=0.11*µm*, and the time sampling step is Δt=0.2*fs* used in the FDTD algorithm. The distribution of electromagnetic field in this configuration is shown in Fig. 3. The (a), (b) and (c) are amplitude distributions of Hz, Ex and Ey in *xy* plane, respectively. The exciting source, regarded as a plane wave, applies at the position of x=35*µm*. The incident angle is 15° and lasting time is 300*fs*. From Fig. 3 we can see that the electromagnetic fields are mostly localized in the area of the waveguide and dielectric columns between the two PCs for the effect of PBG. The dielectric columns with two lines are insufficient to form PBG, so it can only modulate electromagnetic fields and make the energy coupling or transferring among waveguides. All the modes of electromagnetic field are forbidden in the PBG of PCs, so there is no energy entering PC1. However, since the FH frequency is near the edge of the PBG of PC2, the electromagnetic wave can be transmitted back and forth within PC2 and the area of the waveguides, as well as in dielectric columns, just like a resonant cavity. In order to see the huge variation of electric field more obviously, the cross section distribution at *x*=5*µm* along y direction is plotted in Fig. 3(d). The gradient is plotted in Fig. 3(e). The large gradient of electromagnetic is obtained, especially at air-Silicon interface.

The condition of phase matching can be achieved by Quasi Phase Matching (QPM) [14] or birefringence phase matching techniques [15,16]. In our configuration, the length of waveguide is only 40*µm*, so the phase mismatching can be ignored. We calculated the SH intensity when the effective susceptibility of electric quadrupole polarization is 10^{-16} esu, which is one magnitude less than the susceptibility of calcite estimated by Terhune [4]. The variation curve of SHG along *y*-direction is shown in Fig. 4. Since the exciting source is applied at x=35*µm* position, the SH intensity increases with the decrease of coordinate values. The variation trend is consistent with quadratic curve as shown with dotted line in Fig. 4, which is identical with the SHG in typical nonlinear bulk materials. The intensity variation of SHG at the exit position of the waveguides is shown in Fig. 5. The electric field intensity of the pumping wave is 100kV/mm, which corresponds to a beam intensity S_{in}=1.3mW/mm^{2}. In order to show the intensity of the output harmonic wave more accurately, we have set 100 observing points with an equal gap at the exit position. The average value of 100 observing points is used as the output intensity of SHG. From Fig. 5, we can see that the intensity of SHG is enhanced gradually with the passing time, and the trend of increasing becomes gentle from 420fs. It shows that after a nonlinear energy coupling process for some time, the energy conversion between FH and SH has attained equilibrium. The average value of the intensity curve in balance state is used as the intensity of output SH. The efficiency for power conversion from FH to SH can be defined by

From Fig. 5, *η* is 0.268%. Compared with the conversion efficiency of a typical nonlinear material under the case of perfect phase matching, this efficiency is a little bit lower, but it is comparable to the conversion efficiency of the photonic crystal slabs in Ref. [7].

## 4. Conclusion

In summary, we have discussed a calculation model of SHG in PCs consisting of centro-symmetric dielectric based on the electric quadrupole interaction. The parameters of a special silicon PC have been presented. Light field distribution and SHG intensity in this configuration are calculated. The SHG intensity increases with the length of the waveguides. The trend of SHG increment is consistent with quadratic curve as same as the proceeding in nonlinear bulk materials. By designing the PCs with centrosymmetric materials, SHG originated from quadrupole effect can be optimized for applications.

## References and links

**1. **Y. R. Shen, *The Principles of Nonlinear Optics* (John Wiley & Sons, New York, 1984).

**2. **N. Bloembergen, “Light wave at the boundary of nonlinear media,” Phys. Rev. **128**, 606–608 (1962). [CrossRef]

**3. **R. W. Terhune and P. Maker, “Optical harmonic generation in Calcite,” Phys. Rev. Lett. **8**, 21–24 (1962) [CrossRef]

**4. **E. Yablonovitch, “Inhibited spontaneous emission in solidstate physics,” Phys. Rev. Lett. **58**, 2059–2064 (1987) [CrossRef] [PubMed]

**5. **X. Luo, J. Shi, H. Wang, and G. Yu, “Surface plasmon polariton radiation from metallic photonic crystal slabs breaking the diffraction: Nano-storage and Nano-fabrication, M,”. Phys. Lett. B **18**, 945–954 (2004)

**6. **G. D’Aguanno, M. Centini, and C. Sibilia et al, “Ehencement of *χ*^{(2)} cascading processes in 1-D photonic bandgap structures,” Opt. Lett. **24**, 1663–1668 (1999) [CrossRef]

**7. **J. Martorell, R. Vilaseca, and R. Corbalan, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. **70**, 702 –704(1997) [CrossRef]

**8. **X. Luo and T. Ishihara, “Engineered Second harmonic generation in photonic crystal slabs consisted of centrosymmetric materials,” Advanced Dunction Materials , **14**, 905–912 (2004) [CrossRef]

**9. **Allen Taflove, *Computational Electrodynamics* (Artech House, Boston London, 1995)

**10. **P. S. Persan, “Nonlinear optical properties of solid: energy consideration,” Phys. Rev. **130**, 919–923(1963) [CrossRef]

**11. **N. Bloembergen, N*onlinear Optics* (W. A. Benjamin, Inc., 1977).

**12. **KANE S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transaction on antennas and propagation , **AP-14**, 302–307 (1966)

**13. **G. D’Aguanno, M. Centini, and C. Sibilia et al, “Photonic band gap edge effects in finite structures and application to*χ*^{(2)} interactions,” Phy. Rev. E **64**, 016609(2001) [CrossRef]

**14. **Gary D. Landry and Theresa A. Maldonado, “Counter propagating quasi-phase matching: a generalized analysis,” J. Opt. Soc. Am. B **21**, 1509–1521(2004) [CrossRef]

**15. **F. Genereux, S. W. Lconard, and H. M. van Driel, “Large birefringence in 2-D silicon photonic crystals,” Phys. Rev. B **63**, 161101–161108(2001) [CrossRef]

**16. **P. K. Kashkarov, L. A. Golovan, and A. B. Fedotov et al., “Photonic bandgap materials and birefringence layers anisotropically nanostructured silicon,” J. Opt. Soc. Am. B **19**, 2273–2278(2002) [CrossRef]