## Numerical simulation of nonlinear field distributions in two-dimensional optical superlattices |

Optics Express, Vol. 20, Issue 2, pp. 1261-1267 (2012)

http://dx.doi.org/10.1364/OE.20.001261

Acrobat PDF (1421 KB)

### Abstract

A finite difference method in real space is presented for calculating nonlinear optical processes in two-dimensional optical superlattices. The focused second-harmonic generation under the local quasi-phase-matched condition is calculated as an example. The field distribution of both the fundamental and the harmonic wave can be simulated well using this method, and the result agrees well with previous theoretical predictions and experimental studies. It is shown that this method is a simple and rapid technique to analysis nonlinear processes in optical superlattices.

© 2012 OSA

## 1. Introduction

1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. **127**(6), 1918–1939 (1962). [CrossRef]

2. A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, “Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO_{4},” Opt. Commun. **142**(4-6), 265–268 (1997). [CrossRef]

5. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

7. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. **81**(19), 4136–4139 (1998). [CrossRef]

9. P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ^{(2)} photonic crystal: a hexagonally poled LiTaO_{3} crystal,” Phys. Rev. Lett. **93**(13), 133904 (2004). [CrossRef] [PubMed]

10. J. J. Chen and X. F. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A **80**(1), 013801 (2009). [CrossRef]

11. C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. **33**(7), 720–722 (2008). [CrossRef] [PubMed]

13. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. **35**(10), 1581–1583 (2010). [CrossRef] [PubMed]

14. M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B. Djafari-Rouhani, “Theory of acoustic band structure of periodic elastic composites,” Phys. Rev. B Condens. Matter **49**(4), 2313–2322 (1994). [CrossRef] [PubMed]

15. F. R. Montero de Espinosa, E. Jimenez, and M. Torres, “Ultrasonic band gap in a periodic two-dimensional composite,” Phys. Rev. Lett. **80**(6), 1208–1211 (1998). [CrossRef]

16. C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B Condens. Matter **51**(23), 16635–16642 (1995). [CrossRef] [PubMed]

20. A. Massaro, V. Tasco, M. T. Todaro, T. Stomeo, R. Cingolani, M. De Vittorio, and A. Passaseo, “FEM design and modeling of**27**, 4262–4268 (2009). [CrossRef]

## 2. Method

*x*axis. Under the slowly varying envelope approximation, the evolution of the harmonics can be described by the paraxial wave equations [6]:where

*K*is the coupling coefficient;

*f(x, y)*is the structure function of the 2D optical superlattice;

*k*,

_{1}, k_{2}*A*,

_{1}*A*are the wave vectors and amplitudes of the fundamental wave and second-harmonic wave respectively; is the phase mismatch between the fundamental and harmonic wave.

_{2}*A*,

_{1}*A*are the field values of the fundamental and the second harmonic respectively;

_{2}*A*is a constant;

*y*is the center of the fundamental wave and

_{0}*r*is the waist radius of the beam. The boundary condition for our method is also quite simple, which can be directly set to be zero:where

_{0}*y = 0*and

*y = L*are the boundaries in

*y*-direction of the calculation area.

*x*-axis should be small. This error is reflected in the dispersion relation of the differential equation, where the dispersion relation of Eq. (1) is:

*x*axis is small, which means that the calculation is nearly accurate under this condition. Actually it was calculated that the angel should be smaller than 20 degrees.

## 3. Numerical results & discussion

_{3}OSL. Based on the Huygens-Fresnel principle for nonlinear interactions, the local QPM method is proposed by our group in 2008 [8

8. Y. Q. Qin, C. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. **100**(6), 063902 (2008). [CrossRef] [PubMed]

*x, y*) and SH beam can be focused at pre-designed point (

*X*,

_{i}*Y*). To make the SH wave in phase at the focused point, the structure function (corresponding to the direction of domain polarization) of the optical superlattice to generate focused SHG can be determined:

_{i}*k*,

_{1}*k*are the wave vectors of the fundamental and SH wave respectively. The length and width of the OSL are 5

_{2}*mm*and 3

*mm*, and the focused point was designed to be the center of the exit surface. The expression

*μm*, and it propagated along the

*x*axis of the OSL, with the incident point at the center of the OSL. The wavelength was set to be 1064

*nm*, and the working temperature was 180°C. Figure 2(c) shows the field distribution of the SH wave, which was focused at the center of exit OSL surface. The propagation and diffraction properties are well resolved in the simulation. We can clearly see that the SH wave was focused gradually with the propagating of the fundamental wave and the focused point was at the exit surface of the OSL.

8. Y. Q. Qin, C. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. **100**(6), 063902 (2008). [CrossRef] [PubMed]

21. R. Drezek, A. Dunn, and R. Richards-Kortum, “A pulsed finite-difference time-domain (FDTD) method for calculating light scattering from biological cells over broad wavelength ranges,” Opt. Express **6**(7), 147–157 (2000). [CrossRef] [PubMed]

_{3}domain-inverted structure using present method and FDTD method, respectively. Actually it is almost impossible to calculate the structure shown in Fig. 2(a) using FDTD method, its computational time is enormous, reaching about 10 to the power of 8 seconds. Figure 4(a)-(b) indicate the calculation results. The results are almost the same. However, the computational time in FD method is only one ten-thousandth of that in FDTD method.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

2. | A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, “Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO |

3. | S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, “Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO |

4. | I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes,” J. Opt. Soc. Am. B |

5. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science |

6. | R. W. Boyd, |

7. | V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. |

8. | Y. Q. Qin, C. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. |

9. | P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ |

10. | J. J. Chen and X. F. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A |

11. | C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. |

12. | T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics |

13. | I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. |

14. | M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B. Djafari-Rouhani, “Theory of acoustic band structure of periodic elastic composites,” Phys. Rev. B Condens. Matter |

15. | F. R. Montero de Espinosa, E. Jimenez, and M. Torres, “Ultrasonic band gap in a periodic two-dimensional composite,” Phys. Rev. Lett. |

16. | C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B Condens. Matter |

17. | Y. Tanaka, Y. Tomoyasu, and S. I. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B |

18. | M. M. Sigalas and N. Garcia, “Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: The aluminum in mercury case,” Appl. Phys. Lett. |

19. | C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, |

20. | A. Massaro, V. Tasco, M. T. Todaro, T. Stomeo, R. Cingolani, M. De Vittorio, and A. Passaseo, “FEM design and modeling of |

21. | R. Drezek, A. Dunn, and R. Richards-Kortum, “A pulsed finite-difference time-domain (FDTD) method for calculating light scattering from biological cells over broad wavelength ranges,” Opt. Express |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(200.0200) Optics in computing : Optics in computing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 2, 2011

Revised Manuscript: December 18, 2011

Manuscript Accepted: December 20, 2011

Published: January 5, 2012

**Citation**

Ming-shuai Zhou, Jun-chao Ma, Chao Zhang, and Yi-qiang Qin, "Numerical simulation of nonlinear field distributions in two-dimensional optical superlattices," Opt. Express **20**, 1261-1267 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1261

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### References

- J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962). [CrossRef]
- A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, “Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4,” Opt. Commun.142(4-6), 265–268 (1997). [CrossRef]
- S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, “Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO4.,” Opt. Lett.23(24), 1883–1885 (1998). [CrossRef] [PubMed]
- I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes,” J. Opt. Soc. Am. B14(12), 3368 (1997). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science278(5339), 843–846 (1997). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Elsevier Science, 2003).
- V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998). [CrossRef]
- Y. Q. Qin, C. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett.100(6), 063902 (2008). [CrossRef] [PubMed]
- P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett.93(13), 133904 (2004). [CrossRef] [PubMed]
- J. J. Chen and X. F. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A80(1), 013801 (2009). [CrossRef]
- C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett.33(7), 720–722 (2008). [CrossRef] [PubMed]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3, 395–398 (2009). [CrossRef]
- I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett.35(10), 1581–1583 (2010). [CrossRef] [PubMed]
- M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B. Djafari-Rouhani, “Theory of acoustic band structure of periodic elastic composites,” Phys. Rev. B Condens. Matter49(4), 2313–2322 (1994). [CrossRef] [PubMed]
- F. R. Montero de Espinosa, E. Jimenez, and M. Torres, “Ultrasonic band gap in a periodic two-dimensional composite,” Phys. Rev. Lett.80(6), 1208–1211 (1998). [CrossRef]
- C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B Condens. Matter51(23), 16635–16642 (1995). [CrossRef] [PubMed]
- Y. Tanaka, Y. Tomoyasu, and S. I. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62(11), 7387–7392 (2000). [CrossRef]
- M. M. Sigalas and N. Garcia, “Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: The aluminum in mercury case,” Appl. Phys. Lett.76(16), 2307 (2000). [CrossRef]
- C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic,χ(2)χ(3) ” J. Lightwave Technol.24(1), 624–634 (2006). [CrossRef]
- A. Massaro, V. Tasco, M. T. Todaro, T. Stomeo, R. Cingolani, M. De Vittorio, and A. Passaseo, “FEM design and modeling ofχ(2),” J. Lightwave Technol.27, 4262–4268 (2009). [CrossRef]
- R. Drezek, A. Dunn, and R. Richards-Kortum, “A pulsed finite-difference time-domain (FDTD) method for calculating light scattering from biological cells over broad wavelength ranges,” Opt. Express6(7), 147–157 (2000). [CrossRef] [PubMed]

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