## Unified approach to Čerenkov second harmonic generation |

Optics Express, Vol. 21, Issue 22, pp. 25715-25726 (2013)

http://dx.doi.org/10.1364/OE.21.025715

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### Abstract

We discuss the effect of second harmonic generation via the Čerenkov-like process in nonlinear bulk media and waveguides. We show that in both schemes the Čerenkov harmonic emission represents in fact a nonlinear Bragg diffraction process. It is therefore possible, for the first time, to describe the bulk and waveguide Čerenkov emission uniformly by considering the spatial modulation of the second-order nonlinear polarization. This is also experimentally illustrated by studying the Čerenkov second harmonic generation at the boundary of a nonlinear quadratic medium via the total internal reflection inside the nonlinear crystal.

© 2013 Optical Society of America

## 1. Introduction

1. A. Zembrod, H. Puell, and J. A. Giordmaine, “Surface Radiation from Non-linear Optical Polarisation,” Opto-electron. **1**, 64–66 (1969). [CrossRef]

2. E. Mathieu, “Conditions for quasi Cerenkov radiation, generated by optical second harmonic polarisation in a nonlinear cristal,” Z. Angew. Math. Phys. **20**, 433–439 (1969). [CrossRef]

3. S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic Bessel beams by transverse phase-matching in annular periodically poled structures,” Japan. J. Appl. Phys. **47**, 6777–6783 (2009). [CrossRef]

4. 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]

5. J. Chen and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express **18**, 15597–15602 (2010). [CrossRef] [PubMed]

*k*

_{1}=

*ω*

_{1}

*n*(

*ω*

_{1})/

*c*(

*n*is refractive index and

*c*is speed of light), the emitted Čerenkov Second Harmonic (SH) propagates with the wave vector

*k*

_{2}=

*ω*

_{2}

*n*(

*ω*

_{2})/

*c*(at the frequency

*ω*

_{2}= 2

*ω*

_{1}) at an angle cos

*θ*=

_{C}*n*(

*ω*

_{1})/

*n*(

*ω*

_{2}). Graph in Fig. 1(a) illustrates emission of Čerenkov SH signal as a cone in as-grown strontium barium niobate crystal with fundamental beam propagating along its optical axis.

6. P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. **17**, 447–450 (1970). [CrossRef]

7. M. J. Li, M. De Micheli, Q. He, and D. B. Ostrowsky, “Cerenkov configuration second harmonic generation in proton-exchanged Lithium niobate guides,” IEEE J. Quantum Electron. **26**, 1384–1393 (1990). [CrossRef]

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

9. X. W. Deng, H. J. Ren, H. Y. Lao, and X. F. Chen, “Research on Cherenkov second-harmonic generation in periodically poled lithium niobate by femtosecond pulses,” J. Opt. Soc. Am. B **27**, 1475–1480 (2010). [CrossRef]

7. M. J. Li, M. De Micheli, Q. He, and D. B. Ostrowsky, “Cerenkov configuration second harmonic generation in proton-exchanged Lithium niobate guides,” IEEE J. Quantum Electron. **26**, 1384–1393 (1990). [CrossRef]

10. K. Hayata, K. Yanagawa, and M. Koshiba, “Enhancement of the guided-wave second-harmonic generation in the form of Cerenkov radiation,” Appl. Phys. Lett. **56**, 206–208 (1989). [CrossRef]

13. R. Reinisch and G. Vitrant, “Phase matching in Cerenkov second-harmonic generation: A leaky-mode analysis,” Opt. Lett. **22**, 760–762 (1997). [CrossRef] [PubMed]

14. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. **70**, 1401–1404 (1993). [CrossRef] [PubMed]

*longitudinal*and

*transverse*projections of the general vectorial phase matching relation, respectively. Hence, by definition, the nonlinear Čerenkov process is always

*longitudinally*phase matched [Eq.(1)]. As this appears to ensure an efficient energy flow from the fundamental to the SH wave, little attention to the role of the

*transverse*phase-matching condition in the Čerenkov emission [Eq. (2)] has been paid so far. However, the requirement of the momentum conservation in the Čerenkov second harmonic generation requires that the full (vectorial) phase matching condition must be satisfied.

## 2. Čerenkov second harmonic generation in bulk materials

*z*-axis in our case), a strong emission of Čerenkov SH is observed in the direction

*θ*=

_{C}*cos*

^{−1}(2

*k*

_{1}/

*k*

_{2}) [see Fig. 2] [4

4. 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]

5. J. Chen and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express **18**, 15597–15602 (2010). [CrossRef] [PubMed]

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

9. X. W. Deng, H. J. Ren, H. Y. Lao, and X. F. Chen, “Research on Cherenkov second-harmonic generation in periodically poled lithium niobate by femtosecond pulses,” J. Opt. Soc. Am. B **27**, 1475–1480 (2010). [CrossRef]

15. Y. Sheng, W. J. Wang, R. Shiloh, V. Roppo, Y. F. Kong, A. Arie, and W. Krolikowski, “Čerenkov third-harmonic generation in χ^{(2)}nonlinear photonic crystal,” Appl. Phys. Lett. **98**, 241114 (2011). [CrossRef]

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

*G*which are associated with Fourier components of the

*χ*

^{(2)}modulation in the transverse direction. In the particular case of a periodic modulation, |

*G*| = 2

*mπ*/Λ, where Λ is the periodicity of the modulation and

*m*is an integer representing the so called Raman-Nath nonlinear diffraction orders [22

22. 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]

*G*) available to exactly compensate the phase mismatch in the transverse direction.

## 3. Čerenkov SHG as a nonlinear Bragg diffraction

25. Y. Sheng, V. Roppo, K. Kalinowski, and W. Krolikowski, “Role of a localized modulation of χ^{(2)}in Cerenkov second-harmonic generation in nonlinear bulk medium,” Opt. Lett. **37**, 3864–3866 (2012). [CrossRef] [PubMed]

25. Y. Sheng, V. Roppo, K. Kalinowski, and W. Krolikowski, “Role of a localized modulation of χ^{(2)}in Cerenkov second-harmonic generation in nonlinear bulk medium,” Opt. Lett. **37**, 3864–3866 (2012). [CrossRef] [PubMed]

*any single localized*

*χ*

^{(2)}

*modulation*serves as a source of continuous Fourier components (reciprocal wave vectors), enabling to phase match the nonlinear Čerenkov process in the transverse direction and consequently leading to efficient Čerenkov SHG. This has been further confirmed by varying the steepness of the non linearity jump. Calculations showed that the sharper is the

*χ*

^{(2)}modulation, the stronger are the Fourier components, and subsequently more efficient the nonlinear Čerenkov emission.

*k*denotes the transverse component of the wave-vector of the Čerenkov second harmonic. It is clear that the amplitude of the SH is given by just a Fourier integral of the nonlinear polarization. Obviously this integral acquires large value only when its kernel exhibits fast spatial modulation which can be achieved either by varying the strength of nonlinearity or by using strongly spatially localized fundamental beam in a homogeneous medium, or both. In fact the appearance of Čerenkov SH signal in [2

_{c}2. E. Mathieu, “Conditions for quasi Cerenkov radiation, generated by optical second harmonic polarisation in a nonlinear cristal,” Z. Angew. Math. Phys. **20**, 433–439 (1969). [CrossRef]

14. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. **70**, 1401–1404 (1993). [CrossRef] [PubMed]

*χ*

^{(2)}nonlinearity (period Λ) in the regime when the smallest reciprocal lattice vector

*G*= 2

*π*/Λ is already larger than the phase mismatch of Čerenkov radiation in the transverse direction. This means that in this system, in principle, there is no reciprocal lattice vectors which would enable efficient Čerenkov emission. However, as we see the Čerenkov SH signal is still generated. At this point it is then clear that the nonlinear Čerenkov radiation does not necessarily require a specific modulation pattern of nonlinearity

*χ*

^{(2)}. Any variation of

*χ*

^{(2)}in the transverse direction can provide non-zero Fourier coefficients of reciprocal wave vector to ensure the full phase matching of the interaction process. In this context the Čerenkov SHG is actually a particular case of a well known nonlinear Bragg diffraction [22

22. 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]

26. V. Berger, “Nonlinear Photonic Crystals,” Phys. Rev. Lett. **81**, 4136–4139 (1998). [CrossRef]

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

*G⃗*representing the spectrum of the spatial modulation of nonlinearity, satisfying vectorially the following relation As a result, any variation of

*χ*

^{(2)}in the transverse direction already constitutes a sufficient condition for its emission. There is no restriction on nonlinearity modulation which can be of any type including periodic, quasi-periodic, or even spatially localized, with the only consequence being the difference in efficiency of the process.

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

## 4. Čerenkov SHG in planar waveguide

*n*

_{0}and width 2

*d*. Refractive index of the surrounding medium is

*n*

_{1}. The fundamental mode propagating along the

*z*-axis has the following form where

*β*is propagation constant and

*E*(

*x*) is of the form: Here

*κ*,

*γ*and

*β*satisfy the following relations: and

*k*

_{0}= 2

*π/λ*, with

*λ*being the fundamental wavelength.

*k*given by Below we will calculate

_{c}*E*for two cases of nonlinearity modulation.

_{SH}*d*) for the case when nonlinearity is located in the guiding layer (red/solid line) and substrate (black/dashed line). The lines represent the simple formulas Eq. (15) and Eq. (17), while the points depict results of the exact numerical simulation of the SH emission process [Eqs. (5)]. The excellent agreement between the exact and simplified approach confirms the simple physical picture of the emission process based on spatial modulation of nonlinearity (or, more generally, the nonlinear polarization).

*χ*

^{(2)}nonlinearity being artificially uniform across the whole structure. The top (bottom) panel depicts near (far) field of the fundamental and second harmonic. The formation of Čerenkov SH wave with the wave vector

*k*≈ ±5

_{c}*μm*

^{−1}is clearly visible (the refractive indices are the same as in Fig. 4). While in this particular case the

*χ*

^{(2)}nonlinearity is constant the nonlinear polarization is still spatially modulated. Therefore the transverse phase matching necessary for Čerenkov generation is ensured solely by the wave-vectors originating from the waveguide-mediated spatial confinement of the fundamental beam, following the formula Eq. (6). The Čerenkov signal is emitted at

*k*= ±5

_{c}*μm*

^{−1}and grows monotonically with propagation distance.

## 5. Čerenkov SHG at the boundary of linear and nonlinear media

*et al.*[30

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

31. N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. **70**, 1429–1436 (1980). [CrossRef]

32. N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. **181**, 1261–1271 (1969). [CrossRef]

_{3}crystal at point O. It is then totally internally reflected at the C1–C2 crystal facet at point A and exits crystal at point B. This situation is a variation of the scheme in Fig. 3(b), where the fundamental beam now propagates at an oblique angle with respect to the domain wall. In addition, the jump of

*χ*

^{(2)}is accompanied by a change of the refractive index. The parameters of the incident laser beam were: fundamental beam wavelength

*λ*= 1200 nm, pulse duration

*τ*=200 fs, repetition rate

*f*=1 kHz and pulse energy

*E*=1

_{n}*μ*J.

33. S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Yu. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. **104**, 083902 (2010). [CrossRef] [PubMed]

34. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation,” Phys. Rev. Lett. **101**, 113905 (2008). [CrossRef] [PubMed]

35. E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express **17**, 3141–3147 (2009). [CrossRef] [PubMed]

*α*

_{FSH}=

*α*

_{FF}.

*θ*= cos

_{C}^{−1}(

*n*

_{1}cos

*α*

_{FF}/

*n*

_{2}).

*α*as a function of the FF incidence angle

_{D}*α*outside the crystal. The red, green and blue solid lines represent the theoretical detector position calculated for the CSH, FSH and VSH respectively, using the phase-matching scheme in Fig. 8(a). The triangles represent experimental data. The agreement between theory and experiment further confirms the correct physical origin of all generated second harmonic waves and in particular the Čerenkov beam.

_{S}## 6. Conclusions

## Acknowledgments

## References and links

1. | A. Zembrod, H. Puell, and J. A. Giordmaine, “Surface Radiation from Non-linear Optical Polarisation,” Opto-electron. |

2. | E. Mathieu, “Conditions for quasi Cerenkov radiation, generated by optical second harmonic polarisation in a nonlinear cristal,” Z. Angew. Math. Phys. |

3. | S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic Bessel beams by transverse phase-matching in annular periodically poled structures,” Japan. J. Appl. Phys. |

4. | 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 |

5. | J. Chen and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express |

6. | P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. |

7. | M. J. Li, M. De Micheli, Q. He, and D. B. Ostrowsky, “Cerenkov configuration second harmonic generation in proton-exchanged Lithium niobate guides,” IEEE J. Quantum Electron. |

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

9. | X. W. Deng, H. J. Ren, H. Y. Lao, and X. F. Chen, “Research on Cherenkov second-harmonic generation in periodically poled lithium niobate by femtosecond pulses,” J. Opt. Soc. Am. B |

10. | K. Hayata, K. Yanagawa, and M. Koshiba, “Enhancement of the guided-wave second-harmonic generation in the form of Cerenkov radiation,” Appl. Phys. Lett. |

11. | K. Chikuma and S. Umegaki, “Theory of optical second-harmonic generation in crystal-cored fibers based on phase matching of Cerenkov-type radiation,” J. Opt. Soc. Am. B |

12. | G. J. M. Krijnen, W. Tormellas, G. I. Stegeman, H. J. W. M. Hoekstra, and P. V. Lambeck, “Optimization of second harmonic generation and nonlinear phase-shifts in the Cerenkov regime,” IEEE J. Quantum Electron. |

13. | R. Reinisch and G. Vitrant, “Phase matching in Cerenkov second-harmonic generation: A leaky-mode analysis,” Opt. Lett. |

14. | T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. |

15. | Y. Sheng, W. J. Wang, R. Shiloh, V. Roppo, Y. F. Kong, A. Arie, and W. Krolikowski, “Čerenkov third-harmonic generation in χ |

16. | X. W. Deng, H. J. Ren, H. Lao, and X. F. Chen, “Noncollinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B |

17. | K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, “Wavelength and position tuning of Čerenkov second-harmonic generation in optical superlattice,” Appl. Phys. Lett. |

18. | K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. |

19. | Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Cerenkov-type second harmonic generation in periodically poled ferroelectric crystal,” J. Opt. Soc. Am. B |

20. | H. J. Ren, X. W. Deng, Y. L. Zheng, N. An, and X. F. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. |

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

22. | 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. |

23. | A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in the domain walls of periodically poled KTiOPO |

24. | S. J. Holmgren, C. Canalias, and V. Pasiskevicius, “Ultrashort single-shot pulse characterization with high spatial resolution using localized nonlinearities in ferroelectric domain walls,” Opt. Lett. |

25. | Y. Sheng, V. Roppo, K. Kalinowski, and W. Krolikowski, “Role of a localized modulation of χ |

26. | V. Berger, “Nonlinear Photonic Crystals,” Phys. Rev. Lett. |

27. | N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. |

28. | 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. |

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

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

31. | N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. |

32. | N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. |

33. | S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Yu. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. |

34. | M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation,” Phys. Rev. Lett. |

35. | E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.7220) Nonlinear optics : Upconversion

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 19, 2013

Revised Manuscript: September 23, 2013

Manuscript Accepted: September 24, 2013

Published: October 21, 2013

**Citation**

Vito Roppo, Ksawery Kalinowski, Yan Sheng, Wieslaw Krolikowski, Crina Cojocaru, and Jose Trull, "Unified approach to Čerenkov second harmonic generation," Opt. Express **21**, 25715-25726 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25715

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### References

- A. Zembrod, H. Puell, and J. A. Giordmaine, “Surface Radiation from Non-linear Optical Polarisation,” Opto-electron.1, 64–66 (1969). [CrossRef]
- E. Mathieu, “Conditions for quasi Cerenkov radiation, generated by optical second harmonic polarisation in a nonlinear cristal,” Z. Angew. Math. Phys.20, 433–439 (1969). [CrossRef]
- S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic Bessel beams by transverse phase-matching in annular periodically poled structures,” Japan. J. Appl. Phys.47, 6777–6783 (2009). [CrossRef]
- 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]
- J. Chen and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express18, 15597–15602 (2010). [CrossRef] [PubMed]
- P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett.17, 447–450 (1970). [CrossRef]
- M. J. Li, M. De Micheli, Q. He, and D. B. Ostrowsky, “Cerenkov configuration second harmonic generation in proton-exchanged Lithium niobate guides,” IEEE J. Quantum Electron.26, 1384–1393 (1990). [CrossRef]
- S. M. Saltiel, Y. Sheng, N. Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum. Electron.45, 1465–1472 (2009). [CrossRef]
- X. W. Deng, H. J. Ren, H. Y. Lao, and X. F. Chen, “Research on Cherenkov second-harmonic generation in periodically poled lithium niobate by femtosecond pulses,” J. Opt. Soc. Am. B27, 1475–1480 (2010). [CrossRef]
- K. Hayata, K. Yanagawa, and M. Koshiba, “Enhancement of the guided-wave second-harmonic generation in the form of Cerenkov radiation,” Appl. Phys. Lett.56, 206–208 (1989). [CrossRef]
- K. Chikuma and S. Umegaki, “Theory of optical second-harmonic generation in crystal-cored fibers based on phase matching of Cerenkov-type radiation,” J. Opt. Soc. Am. B9, 1083–1091 (1992). [CrossRef]
- G. J. M. Krijnen, W. Tormellas, G. I. Stegeman, H. J. W. M. Hoekstra, and P. V. Lambeck, “Optimization of second harmonic generation and nonlinear phase-shifts in the Cerenkov regime,” IEEE J. Quantum Electron.32, 729–738 (1996). [CrossRef]
- R. Reinisch and G. Vitrant, “Phase matching in Cerenkov second-harmonic generation: A leaky-mode analysis,” Opt. Lett.22, 760–762 (1997). [CrossRef] [PubMed]
- T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett.70, 1401–1404 (1993). [CrossRef] [PubMed]
- Y. Sheng, W. J. Wang, R. Shiloh, V. Roppo, Y. F. Kong, A. Arie, and W. Krolikowski, “Čerenkov third-harmonic generation in χ(2)nonlinear photonic crystal,” Appl. Phys. Lett.98, 241114 (2011). [CrossRef]
- X. W. Deng, H. J. Ren, H. Lao, and X. F. Chen, “Noncollinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B100, 755–758 (2010). [CrossRef]
- K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, “Wavelength and position tuning of Čerenkov second-harmonic generation in optical superlattice,” Appl. Phys. Lett.99, 181128 (2011). [CrossRef]
- K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett.37, 1832–1834 (2012). [CrossRef] [PubMed]
- Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Cerenkov-type second harmonic generation in periodically poled ferroelectric crystal,” J. Opt. Soc. Am. B29, 312–318 (2012). [CrossRef]
- H. J. Ren, X. W. Deng, Y. L. Zheng, N. An, and X. F. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett.108, 223901 (2012). [CrossRef] [PubMed]
- W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Cerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett.35, 3790–3792 (2010). [CrossRef] [PubMed]
- 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]
- A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in the domain walls of periodically poled KTiOPO4,” App. Phys. Lett.85, 375–377 (2004). [CrossRef]
- S. J. Holmgren, C. Canalias, and V. Pasiskevicius, “Ultrashort single-shot pulse characterization with high spatial resolution using localized nonlinearities in ferroelectric domain walls,” Opt. Lett.32, 1545–1547 (2007). [CrossRef] [PubMed]
- Y. Sheng, V. Roppo, K. Kalinowski, and W. Krolikowski, “Role of a localized modulation of χ(2)in Cerenkov second-harmonic generation in nonlinear bulk medium,” Opt. Lett.37, 3864–3866 (2012). [CrossRef] [PubMed]
- V. Berger, “Nonlinear Photonic Crystals,” Phys. Rev. Lett.81, 4136–4139 (1998). [CrossRef]
- N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84, 4345–4348 (2000). [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, 103902 (2008). [CrossRef] [PubMed]
- A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser and Photon. Rev.4, 355–373 (2010). [CrossRef]
- J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127, 1918–1939 (1962). [CrossRef]
- N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am.70, 1429–1436 (1980). [CrossRef]
- N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev.181, 1261–1271 (1969). [CrossRef]
- S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Yu. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett.104, 083902 (2010). [CrossRef] [PubMed]
- M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation,” Phys. Rev. Lett.101, 113905 (2008). [CrossRef] [PubMed]
- E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express17, 3141–3147 (2009). [CrossRef] [PubMed]

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