## Multi-directional Čerenkov second harmonic generation in two-dimensional nonlinear photonic crystal |

Optics Express, Vol. 20, Issue 4, pp. 3948-3953 (2012)

http://dx.doi.org/10.1364/OE.20.003948

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### Abstract

We study Čerenkov-type second-harmonic generation in a two-dimensional quasi-periodically poled LiNbO_{3} crystal. We employ a new geometry of interaction to observe simultaneous emission of multi-directional nonlinear Čerenkov radiation with comparable intensities. This opens a way to control the angle of Čerenkov emission by tailoring the nonlinearity of the material, which is otherwise intrinsically defined by dielectric constants of the medium and their dispersion.

© 2012 OSA

## 1. Introduction

*ν*) that is greater than the speed of the optical second-harmonic (SH) wave (

*ν*

_{0}) in the same medium. Therefore, it can radiate SH signal at an angle determined by cos

*θ*

_{0}=

*ν*

_{0}

*/ν*=

*n*

_{1}/

*n*

_{2}, where

*n*

_{1},

*n*

_{2}is the refractive index of the medium at fundamental and SH frequencies, respectively. As evident from this formula, the Čerenkov SH angle is completely determined by the refractive indices of medium and their dispersion, which is fixed for a given wavelength and material.

*α*) of the fundamental beam [2

2. S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling process,” Opt. Lett. **34**, 848–850 (2009). [CrossRef] [PubMed]

*θ*=

*n*

_{1}cos

*α/n*

_{2}. This enables one to change continuously the Čerenkov angle by varying

*α*. The approach works efficiently for increasing the Čerenkov angle (

*θ*≥

*θ*

_{0}). However, it fails when the ČSHG is expected to be tailed to smaller angles due to the fact cos

*α*≤ 1. Moreover, the oblique incidence of the fundamental beam leads to very asymmetric intensity distribution of the ČSHG [3

3. W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. **35**, 3790–3792 (2010). [CrossRef] [PubMed]

5. Y. Sheng, V. Roppo, Q. Kong, K. Kalinowski, Q. Wang, C. Cojocaru, J. Trull, and W. Krolikowski, “Tailoring Čerenkov second-harmonic generation in bulk nonlinear photonic crystal,” Opt. Lett **36**, 2593–2595 (2011). [CrossRef] [PubMed]

6. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. **100**, 163904 (2008). [CrossRef] [PubMed]

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

8. A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. **4**, 355–373 (2010). [CrossRef]

11. D. Akamatsu, M. Yasami, T. Kohno, A. Onae, and F. Hong, “A compact light source at 461 nm using a periodically poled LiNbO_{3} waveguide for strontium magneto-optical trapping,” Opt. Express **19**, 2046–2051 (2011). [CrossRef] [PubMed]

5. Y. Sheng, V. Roppo, Q. Kong, K. Kalinowski, Q. Wang, C. Cojocaru, J. Trull, and W. Krolikowski, “Tailoring Čerenkov second-harmonic generation in bulk nonlinear photonic crystal,” Opt. Lett **36**, 2593–2595 (2011). [CrossRef] [PubMed]

*k*

_{2}cos

*θ*= 2

*k*

_{1}+

*G*, with

*k*

_{1},

*k*

_{2}being the wave vector of the fundamental and SH waves, respectively and

*G*being reciprocal lattice vector (RLV) of the NPC representing spatial oscillation of the nonlinearity. It is seen then that the Čerenkov SH emission angle can be altered by controlling the nonlinearity modulation. Moreover, with a properly designed nonlinearity distribution, one should be able to observe multi-directional nonlinear Čerenkov radiations, thanks to presence of multiple RLVs with different magnitudes and orientations [5

5. Y. Sheng, V. Roppo, Q. Kong, K. Kalinowski, Q. Wang, C. Cojocaru, J. Trull, and W. Krolikowski, “Tailoring Čerenkov second-harmonic generation in bulk nonlinear photonic crystal,” Opt. Lett **36**, 2593–2595 (2011). [CrossRef] [PubMed]

_{3}crystal with decagonal rotation symmetry [12

12. Y. Sheng, J. Dou, B. Cheng, and D. Zhang, “Effective generation of red-green-blue laser in a two-dimensional decagonal photonic superlattice,” Appl. Phys. B **87**, 603–606 (2007). [CrossRef]

13. R. T. Bratfalean, A. C. Peacock, N. G. R. Broderick, K. Gallo, and R. Lewen, “Harmonic generation in a two-dimensional nonlinear quasi-crystal,” Opt. Lett. **30**, 424–426 (2005). [CrossRef] [PubMed]

6. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. **100**, 163904 (2008). [CrossRef] [PubMed]

6. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. **100**, 163904 (2008). [CrossRef] [PubMed]

## 2. Experiment

_{3}wafer. To visualize the reversed domain pattern, we used a scanning ČSHG microscopy [14

14. Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express **18**, 16539–16545 (2010). [CrossRef] [PubMed]

*a*= 13.19

*μ*m). The average diameter of the inverted domains located at the vertices of the rhombi is about 5.2

*μ*m. The length of the poled sample is 8.8 mm.

*x*-axis to utilize the maximal nonlinear coefficient

*d*

_{33}of LiNbO

_{3}crystal (see Fig. 2). The beam was loosely focused and its diameter at the input facet of the sample was about 100

*μ*m. A CCD camera was used to record the spatial intensity distribution of the SH waves projected on a screen behind the crystal.

## 3. Results and discussions

*λ*

_{1}= 1.174

*μ*m, which is representative for the whole investigated frequency range. In fact, similar SH emission patterns were also observed for other wavelengths. However, in that case the emission angles were slightly changed as predicted by the phase matching condition. To image clearly the SHG spots with large emission angle but low intensity (e.g. the SHG labeled as A and B), we took the picture with the screen located quite close to the NPC. However, as shown in Fig. 3(a), this caused the overlapping and saturation of the other SH spots. Therefore, we show also another image obtained after moving the screen further away from the sample such that the SH spots near the center are clearly resolved [Fig. 3(b)]. In Fig. 3(c), we display the intensity profile of the generated SH spots. The measured emission angles of the SHG are listed in Table 1, together with the respective phase matching conditions, and the calculated emission angles as discussed below. For the sake of briefness, we index the SHG by capital letters [see Figs. 3(a), 3(b) and 3(c)].

**k**

_{2}) in Fig. 4. It is seen that the QPM condition is satisfied at the positions where the ring intersects with RLVs. In this case, the SHG is governed by the vectorial condition

**k**

_{2}= 2

**k**

_{1}+

**G**. For the decagonal NPC with tile side length of 13.19

*μ*m and using reported refractive index [16

16. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. **16**, 373–375 (1984). [CrossRef]

*θ*= 1.12°, 3.6°, 4.81°, 5.78°, and 7.51°, with

_{m}**G**

_{1},

**G**

_{7},

**G**

_{8},

**G**

_{9}, and

**G**

_{10}being involved, respectively. These angles agree well with the experimental values, as listed in Table 1 for SHG indexed from H to L.

*θ*= 24.33° (indexed as C), for which RLV in magnitude of 4.42

*μ*m

^{−1}and oriented with angle of 86.5° against

*x*-axis is required to satisfy the standard QPM condition. However, as shown in Figs. 1(b) and 1(c), no such RLV can be found in the reciprocal space. This problem is solved when we consider only the longitudinal phase-matching condition, i.e. to project all the vectors onto the beam propagation direction [

**k**

_{1}, see Fig. 4(b)]. The projection arms of 2|

**k**

_{1}| + |

**G**

_{4}| cos (3

*π*/10) = 23.5

*μ*m

^{−1}, fitting very well the projection |

**k**

_{2}| cos

*θ*= 23.3

*μ*m

^{−1}. This indicates that the observed SH may be generated via Čerenkov-type phase matching. We have extended this analysis for other emission angles at which noncollinear SHG was observed (indexed from A to G). As can be seen from Table 1, in all cases, we were able to find a RLV at which the longitudinal phase matching condition can be satisfied, i.e. |

**k**

_{2}| cos

*θ*= 2|

**k**

_{1}| ± |

**G**|cos

*γ*(

*γ*defines angle between

**G**and the fundamental wave vector

**k**

_{1}). It is seen that for a given wavelength (i.e. for fixed |

**k**

_{1}| and |

**k**

_{2}|), the condition can be satisfied simultaneously along different directions (

*θ*), thanks to the presence of RLVs with different magnitudes (|

**G**|) and orientations (

*γ*) in the 2D quasi-periodic NPC [see Fig. 4(c)].

*x*axis) of the quasi-periodic domain structure, the same reasoning applies to the other side and consequently the harmonic radiation will appear symmetrically with respect to the fundamental beam. In addition, there are many RLVs that have the same longitudinal component and hence will participate in the Cerenkov emission. For example, the horizontal line in Fig. 4(b) indicates such case for vectors

**G**

_{4},

**G**

_{11}and

**G**

_{12}. In the analysis above, we considered only the emission due to the vector representing the strongest Fourier component, i.e.,

**G**

_{4}. However as all of these RLVs participate in the SHG emission their contributions will affect the total intensity of generated second harmonic.

^{2}. Firstly, the strength of the multi-directional ČSHG is proportional to the Fourier coefficient of the RLV that is responsible for its generation (e.g.

**G**

_{4}for SHG indexed as C) [5

**36**, 2593–2595 (2011). [CrossRef] [PubMed]

**36**, 2593–2595 (2011). [CrossRef] [PubMed]

## 4. Conclusion

*χ*

^{(2)}modulation. Owing to the simultaneous modulation of quadratic nonlinearity in both transverse and longitudinal directions, we demonstrate multiple ČSHGs along different directions. This provides experimental evidence of control of the emission angle of the ČSHG by tailoring the nonlinearity of materials.

## Acknowledgments

## References and links

1. | A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from nonlinear optical polarization,” Opt. Quantum Electron. |

2. | S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling process,” Opt. Lett. |

3. | W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. |

4. | H. Ren, X. Deng, and X. Chen, “Coherently superposed efficient second harmonic generation by domain wall series in ferroelectrics,” arXiv:1010.1593v1 (2010). |

5. | Y. Sheng, V. Roppo, Q. Kong, K. Kalinowski, Q. Wang, C. Cojocaru, J. Trull, and W. Krolikowski, “Tailoring Čerenkov second-harmonic generation in bulk nonlinear photonic crystal,” Opt. Lett |

6. | Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. |

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

8. | A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. |

9. | K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, “Twin-beam optical parametric generation in |

10. | Y. Sheng, D. Ma, M. Ren, W. Chai, Z. Li, K. Koynov, and W. Krolikowski, “Broadband second harmonic generation in one-dimensional randomized nonlinear photonic crystal,” Appl. Phys. Lett. |

11. | D. Akamatsu, M. Yasami, T. Kohno, A. Onae, and F. Hong, “A compact light source at 461 nm using a periodically poled LiNbO |

12. | Y. Sheng, J. Dou, B. Cheng, and D. Zhang, “Effective generation of red-green-blue laser in a two-dimensional decagonal photonic superlattice,” Appl. Phys. B |

13. | R. T. Bratfalean, A. C. Peacock, N. G. R. Broderick, K. Gallo, and R. Lewen, “Harmonic generation in a two-dimensional nonlinear quasi-crystal,” Opt. Lett. |

14. | Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express |

15. | M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar, and R. M. De La Rue, “Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals,” J. Phys.: Condens. Matter |

16. | G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. |

17. | Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B (to be published). |

**OCIS Codes**

(160.2260) Materials : Ferroelectrics

(160.3730) Materials : Lithium niobate

(160.4330) Materials : Nonlinear optical materials

(190.2620) Nonlinear optics : Harmonic generation and mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 5, 2012

Revised Manuscript: January 29, 2012

Manuscript Accepted: January 29, 2012

Published: February 1, 2012

**Citation**

Yan Sheng, Vito Roppo, Mingliang Ren, Ksawery Kalinowski, Crina Cojocaru, Jose Trull, Zhiyuan Li, Kaloian Koynov, and Wieslaw Krolikowski, "Multi-directional Čerenkov second harmonic generation in two-dimensional nonlinear photonic crystal," Opt. Express **20**, 3948-3953 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3948

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### References

- A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from nonlinear optical polarization,” Opt. Quantum Electron.1, 64–66 (1969).
- S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling process,” Opt. Lett.34, 848–850 (2009). [CrossRef] [PubMed]
- W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett.35, 3790–3792 (2010). [CrossRef] [PubMed]
- H. Ren, X. Deng, and X. Chen, “Coherently superposed efficient second harmonic generation by domain wall series in ferroelectrics,” arXiv:1010.1593v1 (2010).
- Y. Sheng, V. Roppo, Q. Kong, K. Kalinowski, Q. Wang, C. Cojocaru, J. Trull, and W. Krolikowski, “Tailoring Čerenkov second-harmonic generation in bulk nonlinear photonic crystal,” Opt. Lett36, 2593–2595 (2011). [CrossRef] [PubMed]
- Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett.100, 163904 (2008). [CrossRef] [PubMed]
- V. Berger, “Nonlinear photonic crystal,” Phys. Rev. Lett.81, 4136–4139 (1998). [CrossRef]
- A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev.4, 355–373 (2010). [CrossRef]
- K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, “Twin-beam optical parametric generation in χ(2) nonlinear photonic crsytal,” Appl. Phys. Lett.98, 161113 (2011). [CrossRef]
- Y. Sheng, D. Ma, M. Ren, W. Chai, Z. Li, K. Koynov, and W. Krolikowski, “Broadband second harmonic generation in one-dimensional randomized nonlinear photonic crystal,” Appl. Phys. Lett.99, 031108 (2011). [CrossRef]
- D. Akamatsu, M. Yasami, T. Kohno, A. Onae, and F. Hong, “A compact light source at 461 nm using a periodically poled LiNbO3 waveguide for strontium magneto-optical trapping,” Opt. Express19, 2046–2051 (2011). [CrossRef] [PubMed]
- Y. Sheng, J. Dou, B. Cheng, and D. Zhang, “Effective generation of red-green-blue laser in a two-dimensional decagonal photonic superlattice,” Appl. Phys. B87, 603–606 (2007). [CrossRef]
- R. T. Bratfalean, A. C. Peacock, N. G. R. Broderick, K. Gallo, and R. Lewen, “Harmonic generation in a two-dimensional nonlinear quasi-crystal,” Opt. Lett.30, 424–426 (2005). [CrossRef] [PubMed]
- Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express18, 16539–16545 (2010). [CrossRef] [PubMed]
- M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar, and R. M. De La Rue, “Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals,” J. Phys.: Condens. Matter1310459–10470 (2001). [CrossRef]
- G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron.16, 373–375 (1984). [CrossRef]
- Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B (to be published).

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