## Analysis of optical SHG in photonic crystal consisting of centro-symmetric dielectric

Optics Express, Vol. 12, Issue 22, pp. 5307-5313 (2004)

http://dx.doi.org/10.1364/OPEX.12.005307

Acrobat PDF (317 KB)

### 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

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

*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

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]

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]

## 2. Theoretical model

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

*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,

*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

*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

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

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

## 3. Calculation and results analysis

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]

*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.

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]

*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

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

*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

*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

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

## 4. Conclusion

## References and links

1. | Y. R. Shen, |

2. | N. Bloembergen, “Light wave at the boundary of nonlinear media,” Phys. Rev. |

3. | R. W. Terhune and P. Maker, “Optical harmonic generation in Calcite,” Phys. Rev. Lett. |

4. | E. Yablonovitch, “Inhibited spontaneous emission in solidstate physics,” Phys. Rev. Lett. |

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 |

6. | G. D’Aguanno, M. Centini, and C. Sibilia et al, “Ehencement of |

7. | J. Martorell, R. Vilaseca, and R. Corbalan, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. |

8. | X. Luo and T. Ishihara, “Engineered Second harmonic generation in photonic crystal slabs consisted of centrosymmetric materials,” Advanced Dunction Materials , |

9. | Allen Taflove, |

10. | P. S. Persan, “Nonlinear optical properties of solid: energy consideration,” Phys. Rev. |

11. | N. Bloembergen, N |

12. | KANE S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transaction on antennas and propagation , |

13. | G. D’Aguanno, M. Centini, and C. Sibilia et al, “Photonic band gap edge effects in finite structures and application to |

14. | Gary D. Landry and Theresa A. Maldonado, “Counter propagating quasi-phase matching: a generalized analysis,” J. Opt. Soc. Am. B |

15. | F. Genereux, S. W. Lconard, and H. M. van Driel, “Large birefringence in 2-D silicon photonic crystals,” Phys. Rev. B |

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 |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(190.2620) Nonlinear optics : Harmonic generation and mixing

(230.4320) Optical devices : Nonlinear optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 8, 2004

Revised Manuscript: October 13, 2004

Published: November 1, 2004

**Citation**

Jianping Shi, Xiangang Luo, Xunan Chen, and Chunlei Du, "Analysis of optical SHG in photonic crystal consisting of centro-symmetric dielectric," Opt. Express **12**, 5307-5313 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5307

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### References

- Y. R.Shen, The Principles of Nonlinear Optics (John Wiley & Sons, New York, 1984).
- N. Bloembergen, �??Light wave at the boundary of nonlinear media,�?? Phys. Rev. 128, 606-608 (1962). [CrossRef]
- R. W. Terhune and P. Maker, �??Optical harmonic generation in Calcite,�?? Phys. Rev. Lett. 8, 21-24 (1962) [CrossRef]
- E. Yablonovitch, �??Inhibited spontaneous emission in solidstate physics,�?? Phys. Rev. Lett. 58, 2059-2064 (1987) [CrossRef] [PubMed]
- X. Luo, J. Shi, H. Wang, 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)
- G. D�??Aguanno, M. Centini, C. Sibilia et al, �??Ehencement of �?(2) cascading processes in 1-D photonic bandgap structures,�?? Opt. Lett. 24, 1663-1668 (1999) [CrossRef]
- J. Martorell, R. Vilaseca, and R. Corbalan, �??Second harmonic generation in a photonic crystal,�?? Appl. Phys. Lett. 70, 702 -704(1997) [CrossRef]
- 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]
- Allen Taflove, Computational Electrodynamics (Artech House, Boston London, 1995)
- P. S. Persan, �??Nonlinear optical properties of solid: energy consideration,�?? Phys. Rev. 130, 919-923(1963) [CrossRef]
- N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Inc., 1977).
- 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)
- G. D�??Aguanno, M. Centini, C. Sibilia et al, �??Photonic band gap edge effects in finite structures and application to �?(2) interactions,�?? Phy. Rev. E 64, 016609(2001) [CrossRef]
- 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]
- 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]
- P. K. Kashkarov, L. A. Golovan, A. B. Fedotov et al., �??Photonic bandgap materials and birefringence layers anisotropically nanostructured silicon,�?? J. Opt. Soc. Am. B 19, 2273-2278(2002) [CrossRef]

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