## Simultaneous generation of second to fifth harmonic conical beams in a two dimensional nonlinear photonic crystal |

Optics Express, Vol. 20, Issue 28, pp. 29940-29948 (2012)

http://dx.doi.org/10.1364/OE.20.029940

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### Abstract

Broadly tunable multiple high-harmonic conical beams have been generated by means of a multistep χ^{(2)} cascade processes in a two dimensional nonlinear photonic crystal. The nonlinear structure consists of a square lattice of inverted hexagonal domains with diameters and distances between domains as low as 1 μm. The large number of reciprocal lattice vectors provided by both the square nonlinear structure and the hexagonal shaped domains, along with imperfections on the size and shape of the individual domains make possible the simultaneous generation of second up to fifth harmonic conical beams in a single nonlinear structure by using different types of phase matching geometries. The frequency response can be tuned in an extremely large spectral range, and continuous generation of nonlinear conical beams covering the whole visible spectral region can be achieved. Further, the same photon energy can be generated at different orders, so that concentrically emitted conical beams with angular dispersion as large as *Δθ =* 50° can be observed. The results highlight the significance of highly controlled engineered 2D nonlinear structures to generate advanced multi-photon devices with large spatial and spectral tunable response.

© 2012 OSA

## 1. Introduction

2. 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]

6. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO_{3},” J. Opt. Soc. Am. B **12**(11), 2102–2106 (1995). [CrossRef]

^{(2)}modulation – the so-called two dimensional nonlinear photonic crystals (2DNLPC), expanded the access to several reciprocal lattice vectors within the same crystal, increasing not only the frequency range suitable to be converted but also the directions at which the nonlinear processes take place [7

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

8. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics **3**(7), 395–398 (2009). [CrossRef]

11. K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, “Twin-beam optical parametric generation in χ^{(2)} nonlinear photonic crystals,” Appl. Phys. Lett. **98**(16), 161113 (2011). [CrossRef]

12. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. **100**(18), 183601 (2008). [CrossRef] [PubMed]

14. X. C. Yao, T. X. Wang, P. Xu, H. Lu, G. S. Pan, X. H. Bao, C. Z. Peng, C. Y. Lu, Y. A. Chen, and J. W. Pan, “Observation of eight photon entanglement,” Nat. Photonics **6**(4), 225–228 (2012). [CrossRef]

*f*= 0.35 (Fig. 1(a) ), to demonstrate what we believe to be the highest tunable multistep cascaded χ

^{(2)}conical harmonic generation up to date in a solid state system. Such a short domain periodicity in LiNbO

_{3}is the result of a technological improvement on ferroelectric domain engineering provided by the use of direct electron beam writing (DEBW) as a tool to reverse the spontaneous polarization. From SHG up to fifth harmonic generation (FiHG) are simultaneously generated in a conical geometry by means of a single fundamental beam. The harmonic waves are generated at different conical angles and an extremely large tunability range (Δλ>1000 nm), only limited by the experimental set-up, is demonstrated without any angle or thermal tunning of the nonlinear structure. Further, we show how the short poling period along with the extensive spatial control of the fabricated nonlinear structure allows the PM condition to be fulfilled at numerous

*m,n*diffraction orders. As a result, the generation of multiple high harmonic conical waves exhibiting angular dispersion values as large as

*Δθ =*50° for a fixed frequency converted wavelength is achieved. Together, all these results highlight the significance of highly controlled engineered 2D nonlinear structures to generate broadly tunable multi-photon devices with marked dispersive properties in extremely large spectral ranges (400-1000 nm).

## 2. Experimental

_{3}a 100nm Al film was deposited on the + z face, which acted as a ground electrode.The irradiation process was performed without any mask. The acceleration voltage was 15 kV and the applied charge density 1500 µC/cm

^{2}. The inverted domains grew along the polar axis of the crystal (

*z*axis) and crossed the whole thickness of the sample. The diameter of the inverted domains in the

*xy*plane was 1 µm and the lattice parameter of the two dimensional square lattice was Λ = 2 μm. The average filling factor,

*f,*defined as the ratio of the total inverted area to the original polarization area was

*f =*0.35 in both. The spatial extension of the patterns was 0.5x0.5 mm

^{2}. The inverted ferroelectric domain structures were revealed after a selective chemical etching in a 2:1 solution of HNO

_{3}:HF. For the high harmonic generation experiments, the sample was polished up to optical quality. An ultrafast optical parametric amplifier (OPerA-Solo (COHERENT)) generating 140 fs pulses at a repetition rate of 1 kHz was used as tunable excitation source in the 1200-2400 nm spectral range. The average output power was 100 mW. The laser beam was linearly polarized. All the frequency conversion experiments were performed with the fundamental beam travelling parallel to the ferroelectric axis. The multicolour concentric circular rings generated by the conical beams were projected on a screen. Half wave-plates were used to control the laser polarization. The pictures of the generated conical radiation were obtained by means of Nikon D90 digital camera.

## 3. Results and discussion

^{(2)}-cascade high harmonic generation depending whether the sum frequency mixing (SM) occurs via transverse or longitudinal PM conditions. The nonlinear Cerenkov radiation represents the type of nonlinear interaction in which only the longitudinal PM condition is fulfilled. In this case, the conical angle is defined by the material refractive index dispersion and multiple high harmonic generation is obtained by SHG and successive sum frequency conversion processes between the longitudinal component of the generated harmonics and the fundamental beam. This process is known as Type I Cerenkov harmonic generation. Besides, the longitudinal PM condition can also be fulfilled by sum frequency mixing processes between the collinearly generated SHG and THG and the fundamental beam. This process is labelled as type II Cerenkov harmonic generation. The phase matching conditions for both types of χ

^{(2)}-Cerenkov cascade generation can be written as: where

*k*and

_{ω}*k*are the fundamental and the

_{iω}*i*-th order harmonic wave vector respectively, and

*θ*the internal conical angle of the

_{i}*i*-th order Cerenkov harmonic generated wave. Superscripts I and II refer to the Cerenkov-type process. Figure 2(a) shows the schematic diagrams of Type I and Type II multistep cascaded χ

^{(2)}–Cerenkov harmonic generation.

_{3}. The theoretical fittings to Eqs. (3) and 4 associated with the different Cerenkov harmonic generation processes (from 2th to 5th) are plotted as solid lines according to the expressions:

*et al*[21

21. D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide–doped lithium niobate,” J. Opt. Soc. Am. B **14**(12), 3319–3322 (1997). [CrossRef]

22. Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, “Cerenkov third-harmonic generation in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett. **98**(24), 241114 (2011). [CrossRef]

24. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett. **99**(24), 241109 (2011). [CrossRef]

^{(2)}square modulation and the ferroelectric domain walls along with the additional random components arising from the imperfections on the size and shape of the hexagonal individual domains, make possible to compensate the mismatch of the Cerenkov type harmonic generation in a quasi-continuous set of directions in the XY plane [19

19. S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Cerenkov-type second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. **45**(11), 1465–1472 (2009). [CrossRef]

25. A. Pasquazi, A. Busacca, S. Stivala, R. Morandotti, and G. Assanto, “Nonlinear disorder mapping through three-wave mixing,” IEEE Photon. J. **2**(1), 18–28 (2010). [CrossRef]

_{3}[19

19. S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Cerenkov-type second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. **45**(11), 1465–1472 (2009). [CrossRef]

20. P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. **96**(26), 261111 (2010). [CrossRef]

22. Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, “Cerenkov third-harmonic generation in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett. **98**(24), 241114 (2011). [CrossRef]

*x*and

*y*axis of the crystal. So, the interaction of a circularly polarized fundamental beam would results into symmetrical conical nonlinear diffraction patterns with a homogenous intensity distribution across the whole ring [20

20. P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. **96**(26), 261111 (2010). [CrossRef]

## 4. Summary and conclusions

_{3}has allowed lateral periodic arrangements to be fabricated at pitches beyond the state of the art. This has demonstrated a step forward in the range of harmonic generation with currently available broadband sources in the VIS-NIR. Broadly tunable multiple high-harmonic conical beams have been generated by means of a multistep χ

^{(2)}cascade processes in a single 2DNLPC with a square lattice of inverted domains with periodicity around 2 μm. The large density of ferroelectric domain walls at the irradiated areas combined with the small size and dispersion of the inverted individual domains have made possible the access to a large number of QPM conditions. As a result, the simultaneous generation of multiple annular harmonics (from second up to fifth harmonic conical beams) in a single nonlinear structure is demonstrated to be an efficient phase matched process. The frequency response can be tuned in an extremely large spectral range and continuous generation of nonlinear conical beams covering the whole visible spectral region is demonstrated. Further, the same wavelength can be generated at multiple orders hence leading to concentrically emitted conical beams exhibiting angular dispersion values as large as

*Δθ =*50°. To the best of our knowledge this is the first report in which on a solid state system up to fifth harmonic generated conical beams are obtained by simultaneously phase match both longitudinal and transverse geometries. The generation of conical beams has recently shown a broad number of potential applications in a wide variety of fields including optical deflection, symmetry studies, high resolution optical microscopy, or even photon entanglement [27

27. T. Ellenbogen, A. Ganany-Padowicz, and A. Arie, “Nonlinear photonic structures for all-optical deflection,” Opt. Express **16**(5), 3077–3082 (2008). [CrossRef] [PubMed]

30. A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. **102**(15), 153902 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | P. Ferraro, S. Grilli, and P. De Natale, |

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

3. | C. Canalias and V. Pasiskevicius, “Mirrorles optical parametric oscillator,” Nat. Photonics |

4. | J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, “traffic signal lights”, by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO |

5. | C. Canalias, V. Pasiskevicius, M. Fokine, and F. Laurell, “Backward quasi-phase matched second harmonic generation in sub-micrometer periodically poled flux-grown KTiOPO |

6. | L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO |

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

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

9. | N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, “Temperature and wavelength tuning of second-, third-, and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B |

10. | R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. |

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

12. | M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. |

13. | C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M. Fejer, “All-optical signal processing using χ |

14. | X. C. Yao, T. X. Wang, P. Xu, H. Lu, G. S. Pan, X. H. Bao, C. Z. Peng, C. Y. Lu, Y. A. Chen, and J. W. Pan, “Observation of eight photon entanglement,” Nat. Photonics |

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

16. | S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear bragg diffraction,” Phys. Rev. Lett. |

17. | N. An, H. Ren, Y. Zheng, X. Deng, and X. Chen, “Cherenkov high-order harmonic generation by multistep cascading in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett. |

18. | P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium barium niobate as a multifunctional two-dimensional nonlinear “photonic glass”,” Adv. Funct. Mater. |

19. | S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Cerenkov-type second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. |

20. | P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. |

21. | D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide–doped lithium niobate,” J. Opt. Soc. Am. B |

22. | Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, “Cerenkov third-harmonic generation in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett. |

23. | H. X. Li, S. Y. Mu, P. Xu, M. L. Zhong, C. D. Chen, X. P. Hu, W. N. Cui, and S. N. Zhu, “Multicolor Cerenkov conical beams generation by cascaded- chi((2)) processes in radially poled nonlinear photonic crystals,” Appl. Phys. Lett. |

24. | M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett. |

25. | A. Pasquazi, A. Busacca, S. Stivala, R. Morandotti, and G. Assanto, “Nonlinear disorder mapping through three-wave mixing,” IEEE Photon. J. |

26. | Y. Sheng, W. Wang, R. Shiloh, V. Roppo, A. Arie, and W. Krolikowski, “Third-harmonic generation via nonlinear Raman-Nath diffraction in nonlinear photonic crystal,” Opt. Lett. |

27. | T. Ellenbogen, A. Ganany-Padowicz, and A. Arie, “Nonlinear photonic structures for all-optical deflection,” Opt. Express |

28. | L. Mateos, P. Molina, L. E. Bausa, and M. O. Ramirez, “Second harmonic conical waves for symmetry studies in chi((2)) nonlinear photonic crystals,” Appl. Phys. Express |

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

30. | A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. |

**OCIS Codes**

(130.3730) Integrated optics : Lithium niobate

(160.2260) Materials : Ferroelectrics

(160.4330) Materials : Nonlinear optical materials

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4160) Nonlinear optics : Multiharmonic generation

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 12, 2012

Revised Manuscript: December 8, 2012

Manuscript Accepted: December 17, 2012

Published: December 21, 2012

**Citation**

Luis Mateos, Pablo Molina, Juan Galisteo, Cefe López, Luisa E. Bausá, and Mariola O Ramírez, "Simultaneous generation of second to fifth harmonic conical beams in a two dimensional nonlinear photonic crystal," Opt. Express **20**, 29940-29948 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29940

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### References

- P. Ferraro, S. Grilli, and P. De Natale, Ferroelectric Crystals for Photonic Applications (Springer Series in Materials Science 2009).
- 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]
- C. Canalias and V. Pasiskevicius, “Mirrorles optical parametric oscillator,” Nat. Photonics1(8), 459–462 (2007). [CrossRef]
- J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, “traffic signal lights”, by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett.83(2), 228–230 (2003). [CrossRef]
- C. Canalias, V. Pasiskevicius, M. Fokine, and F. Laurell, “Backward quasi-phase matched second harmonic generation in sub-micrometer periodically poled flux-grown KTiOPO4,” Appl. Phys. Lett.86(18), 181105 (2005). [CrossRef]
- L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B12(11), 2102–2106 (1995). [CrossRef]
- V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998). [CrossRef]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3(7), 395–398 (2009). [CrossRef]
- N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, “Temperature and wavelength tuning of second-, third-, and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B19(9), 2263–2272 (2002). [CrossRef]
- R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95(13), 133901 (2005). [CrossRef] [PubMed]
- K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, “Twin-beam optical parametric generation in χ(2) nonlinear photonic crystals,” Appl. Phys. Lett.98(16), 161113 (2011). [CrossRef]
- M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100(18), 183601 (2008). [CrossRef] [PubMed]
- C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M. Fejer, “All-optical signal processing using χ2 nonlinearities in guided –wave devices,” J. Lightwave Technol.24(7), 2579–2592 (2006). [CrossRef]
- X. C. Yao, T. X. Wang, P. Xu, H. Lu, G. S. Pan, X. H. Bao, C. Z. Peng, C. Y. Lu, Y. A. Chen, and J. W. Pan, “Observation of eight photon entanglement,” Nat. Photonics6(4), 225–228 (2012). [CrossRef]
- Y. Sheng, A. Best, H. J. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Cerenkov-type second harmonic generation,” Opt. Express18(16), 16539–16545 (2010). [CrossRef] [PubMed]
- S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear bragg diffraction,” Phys. Rev. Lett.100(10), 103902 (2008). [CrossRef] [PubMed]
- N. An, H. Ren, Y. Zheng, X. Deng, and X. Chen, “Cherenkov high-order harmonic generation by multistep cascading in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett.100(22), 221103 (2012). [CrossRef]
- P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium barium niobate as a multifunctional two-dimensional nonlinear “photonic glass”,” Adv. Funct. Mater.18(5), 709–715 (2008). [CrossRef]
- S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Cerenkov-type second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron.45(11), 1465–1472 (2009). [CrossRef]
- P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett.96(26), 261111 (2010). [CrossRef]
- D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide–doped lithium niobate,” J. Opt. Soc. Am. B14(12), 3319–3322 (1997). [CrossRef]
- Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, “Cerenkov third-harmonic generation in chi((2)) nonlinear photonic crystal,” Appl. Phys. Lett.98(24), 241114 (2011). [CrossRef]
- H. X. Li, S. Y. Mu, P. Xu, M. L. Zhong, C. D. Chen, X. P. Hu, W. N. Cui, and S. N. Zhu, “Multicolor Cerenkov conical beams generation by cascaded- chi((2)) processes in radially poled nonlinear photonic crystals,” Appl. Phys. Lett.100(10), 101101 (2012). [CrossRef]
- M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett.99(24), 241109 (2011). [CrossRef]
- A. Pasquazi, A. Busacca, S. Stivala, R. Morandotti, and G. Assanto, “Nonlinear disorder mapping through three-wave mixing,” IEEE Photon. J.2(1), 18–28 (2010). [CrossRef]
- Y. Sheng, W. Wang, R. Shiloh, V. Roppo, A. Arie, and W. Krolikowski, “Third-harmonic generation via nonlinear Raman-Nath diffraction in nonlinear photonic crystal,” Opt. Lett.36(16), 3266–3268 (2011). [CrossRef] [PubMed]
- T. Ellenbogen, A. Ganany-Padowicz, and A. Arie, “Nonlinear photonic structures for all-optical deflection,” Opt. Express16(5), 3077–3082 (2008). [CrossRef] [PubMed]
- L. Mateos, P. Molina, L. E. Bausa, and M. O. Ramirez, “Second harmonic conical waves for symmetry studies in chi((2)) nonlinear photonic crystals,” Appl. Phys. Express4(8), 082202 (2011). [CrossRef]
- Y. Sheng, A. Best, H. J. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express18(16), 16539–16545 (2010). [CrossRef] [PubMed]
- A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102(15), 153902 (2009). [CrossRef] [PubMed]

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