## Čerenkov-type second-harmonic spectroscopy in random nonlinear photonic structures |

Optics Express, Vol. 21, Issue 7, pp. 8220-8230 (2013)

http://dx.doi.org/10.1364/OE.21.008220

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### Abstract

We report on experimental and theoretical studies of Čerenkov-type second-harmonic generation in random nonlinear photonic structures with different two-dimensional distributions of the ferroelectric domains. We utilize Čerenkov-type second-harmonic generation spectroscopy to estimate the average ferroelectric domain size by analyzing the spatial Fourier spectrum of the domain patterns. This is measured by scanning the Čerenkov second-harmonic signal for different angles of incidence and light polarizations of the fundamental wave. By comparing the experimental results with numerically simulated Fourier spectra, the corresponding domain patterns are retrieved, which are in a good correspondence with images obtained by Čerenkov-type second-harmonic generation microscopy.

© 2013 OSA

## 1. Introduction

*χ*

^{(2)}nonlinearity [1

1. S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needle-like ferroelectric domains in Sr_{0.6}Ba_{0.4}Nd_{2}O_{6} single crystals,” Appl. Phys. Lett. **73**, 768–770 (1998) [CrossRef] .

6. K. Kalinowski, V. Roppo, T. Lukasiewicz, M. Swirkowicz, Y. Sheng, and W. Krolikowski, “Parametric wave interaction in one-dimensional nonlinear photonic crystal with randomized distribution of second-order nonlinearity,” Appl. Phys. B **109**, 557–566 (2012) [CrossRef] .

7. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. **89**, 191105(1)–191105(3) (2006) [CrossRef] .

9. M. Ayoub, J. Imbrock, and C. Denz, “Second harmonic generation in multi-domain *χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

10. A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. **90**, 243901(1)–243901(4) (2003) [CrossRef] .

13. V. Vaičaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. **209**, 485–490 (2002) [CrossRef] .

14. Y. Zhang, X. P. Hu, G. Zhao, S. N. Zhu, and M. Xiao, “Cerenkov second-harmonic arc from a hexagonally poled LiTaO_{3} planar waveguide,” J. Phys. D **42**, 215103–215107 (2009) [CrossRef] .

*χ*

^{(2)}nonlinearity [4

4. K. A. Kuznetsov, G. K. Kitaeva, A. V. Shevlyuga, L. I. Ivleva, and T. R. Volk, “Second harmonic generation in a strontium barium niobate crystal with a random domain structure,” JETP Letters **87**, 98–102 (2008) [CrossRef] .

13. V. Vaičaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. **209**, 485–490 (2002) [CrossRef] .

15. 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**, 1465–1472 (2009) [CrossRef] .

16. X. Deng, H. Ren, H. Lao, and X. 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. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. **89**, 191105(1)–191105(3) (2006) [CrossRef] .

*v*. If this velocity is higher than the phase velocity of light

_{p}*v*, the conical emission due to constructive interference at a certain angle

_{s}*γ*given by cos

*γ*=

*v*/

_{s}*v*can be detected. The physical origin of the existence of this phenomenon in random nonlinear photonic crystals is still ambiguous, being attributed either to reciprocal grating vectors of the

_{p}*χ*

^{(2)}structure [10

10. A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. **90**, 243901(1)–243901(4) (2003) [CrossRef] .

12. P. Molina, M. de la O Ramírez, and L. E. Bausá, “Strontium barium niobate as a multifunctional two-dimensional nonlinear photonic glass,” Adv. Funct. Mater. **18**, 709–715 (2008) [CrossRef] .

*χ*

^{(2)}at domain walls as suggested in [18]. However, Čerenkov-type second-harmonic generation can also be observed in an extreme case where a single boundary between two inversely oriented ferroelectric domains is illuminated [19

19. X. Deng and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express **18**, 15599–15602 (2010) [CrossRef] .

20. 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 **18**, 16539–16545 (2010) [CrossRef] [PubMed] .

21. J. Bravo-Abad, X. Vidal, J. L. D. Jurez, and J. Martorell, “Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains,” Opt. Express **18**, 14202–14211 (2010) [CrossRef] [PubMed] .

22. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Domain-shape-based modulation of Čerenkov second-harmonic generation in multidomain strontium barium niobate,” Opt. Lett. **36**, 4371–4373 (2011) [CrossRef] [PubMed] .

15. 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**, 1465–1472 (2009) [CrossRef] .

23. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett. **99**, 241109(1)–241109(3) (2011) [CrossRef] .

12. P. Molina, M. de la O Ramírez, and L. E. Bausá, “Strontium barium niobate as a multifunctional two-dimensional nonlinear photonic glass,” Adv. Funct. Mater. **18**, 709–715 (2008) [CrossRef] .

26. 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] .

20. 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 **18**, 16539–16545 (2010) [CrossRef] [PubMed] .

*χ*

^{(2)}nonlinearity at the domain walls [27

27. A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in the domain walls of periodically poled KTiOPO_{4},” Appl. Phys. Lett. **85**, 375–377 (2004) [CrossRef] .

9. M. Ayoub, J. Imbrock, and C. Denz, “Second harmonic generation in multi-domain *χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

**Če**renkov-type

**s**econd-harmonic

**s**pectroscopy (ČESS), we explain the process of Čerenkov-type second-harmonic generation emission and its relation to the spatial domain distribution in random nonlinear photonic structures.

26. 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] .

## 2. Principle of Čerenkov-type second-harmonic spectroscopy

*χ*

^{(2)}nonlinearity [26

26. 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] .

28. S. M. Russell, P. E. Powers, M. J. Missey, and K. L. Schepler, “Broadband mid-infrared generation with two-dimensional quasi-phase-matched structures,” J. Quantum Electronics **37**, 877–887 (2001) [CrossRef] .

29. A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser and Photon. Rev. **4**, 355373 (2010) [CrossRef] .

*χ*

^{(2)}structure modulated in the

*xy*-plane the SH intensity in far-field is where

*I*

_{FW}is the intensity of the fundamental wave propagating along the

*z*-axis,

*L*is the domain length, and

_{z}*S*(Δ

*k*, Δ

_{x}*k*) is the Fourier spectrum of the domain structure with the transverse phase mismatch vectors Δ

_{y}*k*, Δ

_{x}*k*in the

_{y}*x*- and

*y*-direction [see Figs. 1(a) and 1(b)]. Figure 1(b) shows the corresponding phase-matching diagram of Čerenkov emission in the

*xz*-plane, for which Δ

*k*=

_{x}*k*

_{2}

*sin*

_{ω}*θ*− 2

*k*sin

_{ω}*α*, Δ

*k*= 0, and Δ

_{y}*k*= 0 =

_{z}*k*

_{2}

*cos*

_{ω}*θ*− 2

*k*cos

_{ω}*α*are the phase mismatch vector components and

*α*is the internal angle of incidence of the fundamental wave and

*θ*the internal second-harmonic emission angle.

*k*,

_{ω}*k*

_{2}

*denote the wave vectors of the FW and the SH waves.*

_{ω}*d*

_{eff}is the effective second-order nonlinear coefficient of the employed configuration. At the Čerenkov angle, Δ

*k*= 0 is fulfilled, i.e. the process is phase-matched for the length of a domain. In a homogeneously poled crystal

_{z}*S*is nearly zero and Čerenkov light can hardly be measured in contrast to a periodically poled crystal which features a discrete Fourier spectrum [26

**37**, 1832–1834 (2012) [CrossRef] [PubMed] .

*S*(Δ

*k*, Δ

_{x}*k*)| will be replaced with its average 〈|

_{y}*S*(Δ

*k*, Δ

_{x}*k*)|〉, calculated over many domains distributions [30

_{y}30. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum. Electron. **28**, 2631–2654 (1992) [CrossRef] .

31. Y. Le Grand, D. Rouede, C. Odin, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH_{2}PO_{4} ferroelectrics,” Opt. Commun. **200**, 249–260 (2001) [CrossRef] .

*I*

_{SH}is measured for each angular position

*α*

_{ex}of the sample. In this way, |

*S*(Δ

*k*, Δ

_{x}*k*)| can be scanned and compared with Fourier spectra numerically calculated for different domain distributions. In Figs. 1(c) and 1(e), simulated domain structures of a square shape with rounded corners for two different averaged domain sizes are depicted. From the corresponding Fourier spectrum in Figs. 1(d) and 1(f), it is expected that varying the angle of incidence of the fundamental wave leads to a strong angular characteristic asymmetry in the Čerenkov signal intensity in the far-field. Thus, the Čerenkov intensity will increase or decrease continuously, corresponding to the Fourier coefficients considered |

_{y}*S*(Δ

*k*, 0)|, which are located on the

_{x}*x*-axis in our treatment here. These are marked with a dashed line [cf. Figs. 1(d) and 1(f)]. We also note that in case of disordered structures for normal incidence of the fundamental wave, the Čerenkov-type second-harmonic generation can be described even by a single-domain model [9

9. M. Ayoub, J. Imbrock, and C. Denz, “Second harmonic generation in multi-domain *χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

21. J. Bravo-Abad, X. Vidal, J. L. D. Jurez, and J. Martorell, “Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains,” Opt. Express **18**, 14202–14211 (2010) [CrossRef] [PubMed] .

## 3. Experimental setup and samples with random nonlinear photonic structures

*τ*= 80 fs, and the pulse energy amounts to 80 μJ at the wavelength 1200 nm which is used in the experiments. Collimated Gaussian laser pulses with a beam diameter of about 1.5 mm are propagating in the

_{p}*xz*-plane of the SBN crystal. The light is always linearly polarized, and the polarization orientation of the entering beam is controlled by a proper

*λ*/2 wave plate for the used wavelength. A Sr

_{0.61}Ba

_{0.39}Nb

_{2}O

_{6}sample with the dimensions 6.6 × 6.6 × 1.6 mm

^{3}is employed. The large surfaces perpendicular to the

*c*-axis are polished to optical quality. As shown in [9

*χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

32. U. Voelker, U. Heine, C. Gödecker, and K. Betzler, “Domain size effects in a uniaxial ferroelectric relaxor system: The case of Sr* _{x}*Ba

_{1−x}Nb

_{2}O

_{6},” J. Appl. Phys.

**102**, 114112(1)–114112(5) (2007) [CrossRef] .

*α*

_{ex}. The Čerenkov second-harmonic signal is emitted on a cone with the opening angle

*θ*

_{Ce}(

*λ*,

*α*

_{ex}). The SH intensity is measured using a silicon photo diode mounted on another rotation stage centered around the crystal in the

*xz*-plane [see the insets in Figs. 3(a) and 3(b)]. Both rotation stages have the same rotating axis. In preparation, the crystal was first heated up above the Curie temperature T

*≈ 70 °C, and then cooled down without applying an electric field to produce a state similar to an as-grown sample. In order to influence the domain size and form, the sample was subsequently poled and repoled stepwise at room temperature with the same procedure shown in [9*

_{c}*χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

32. U. Voelker, U. Heine, C. Gödecker, and K. Betzler, “Domain size effects in a uniaxial ferroelectric relaxor system: The case of Sr* _{x}*Ba

_{1−x}Nb

_{2}O

_{6},” J. Appl. Phys.

**102**, 114112(1)–114112(5) (2007) [CrossRef] .

20. 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 **18**, 16539–16545 (2010) [CrossRef] [PubMed] .

*xy*-plane by a pair of galvanometric mirrors. The intensity of the generated second-harmonic signal is measured by a photomultiplier as a function of the focus position, using the standard transmission mode of the microscope. As shown in [20

**18**, 16539–16545 (2010) [CrossRef] [PubMed] .

*d*= 0.36 μm and standard deviation

*σ*= 0.18 μm. The repoled case obviously shows larger domains of a mean value 3.01 μm and standard deviation 2.6 μm. These two size scales of the ferroelectric domains lead to different Čerenkov SH emissions in the far-field. Figures 3(a) and 3(b) show the typical Čerenkov SH emissions at normal incidence of the fundamental wavelength 1200 nm. Clearly, no modulation is observed on the Čerenkov ring for the unpoled case [cf. Fig. 3(a)] [22

22. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Domain-shape-based modulation of Čerenkov second-harmonic generation in multidomain strontium barium niobate,” Opt. Lett. **36**, 4371–4373 (2011) [CrossRef] [PubMed] .

22. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Domain-shape-based modulation of Čerenkov second-harmonic generation in multidomain strontium barium niobate,” Opt. Lett. **36**, 4371–4373 (2011) [CrossRef] [PubMed] .

*xz*-plane for each angular position

*α*

_{ex}of the sample. The sample is rotated in an angular range of −60° ≤

*α*

_{ex}≤ 60°, corresponding to transverse phase mismatches of |Δ

*k*| = 1.9 – 21 μm

_{x}^{−1}and Δ

*k*= 0. Since the domain shape is symmetric, no significant changes are expected when rotating the sample 90° around the

_{y}*z*-axis. Measuring the SH intensity in the

*xz*-plane is almost equivalent to the one-dimensional case, represented in [26

**37**, 1832–1834 (2012) [CrossRef] [PubMed] .

## 4. Retrieving the averaged domain sizes by ČESS

*c*-axis and this angle increases while rotating the crystal. At the same time, the SH Čerenkov intensity becomes asymmetric and increases with increasing transverse phase mismatch Δ

*k*or equivalently with increasing negative tilting angle. This means that the largest Fourier coefficients are on a ring in the Fourier space with a radius larger than the one for normal incidence as clearly shown in Fig. 1(d). The experiment has been repeated for a repoled sample, prepared in the same manner as explained above, corresponding to the domain image depicted in Fig. 2(b). The result is shown in Fig. 4(b). In this case, in contrast to an unpoled SBN crystal, the Čerenkov intensity increases with decreasing transverse phase mismatch, i.e. increasing positive tilting angle. This indicates large Fourier coefficients close to the origin of the Fourier space [see Figs. 1(e) and 1(f)], what in turn means large domains as shown in the domain image in Fig. 2(b). The line perpendicular to the fundamental wave is due to the reflection at the sample’s edges.

_{x}*xy*-plane with a domain width described by Gaussian distribution [12

12. P. Molina, M. de la O Ramírez, and L. E. Bausá, “Strontium barium niobate as a multifunctional two-dimensional nonlinear photonic glass,” Adv. Funct. Mater. **18**, 709–715 (2008) [CrossRef] .

31. Y. Le Grand, D. Rouede, C. Odin, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH_{2}PO_{4} ferroelectrics,” Opt. Commun. **200**, 249–260 (2001) [CrossRef] .

*d*and standard deviation

*σ*and a square domain shape [22

**36**, 4371–4373 (2011) [CrossRef] [PubMed] .

*z*-direction is supposed at this point. Moreover, the Frensel losses have also been taken into account for each polarization case. For a vertical polarization of E

*, the effective nonlinear coefficient is given by*

_{ω}*d*

_{eff}=

*d*

_{31}sin

*θ*. The calculated Čerenkov intensity as a function of the tilting and emission angles is depicted in Figs. 4(c) and 4(d) for the unpoled (small domains) and repoled (large domains) sample, respectively. For the unpoled case [Figs. 4(a) and 4(c)], the best agreement with the measured intensity and the microscopic domain image [cf. Fig. 2(a)] is with the chosen parameters for the averaged domain

*d*= 0.25 μm with a standard deviation of

*σ*= 0.07 μm, averaged over possible 200 realizations of domain structure. For the repoled case [Figs. 4(b) and 4(d)], the averaged domain width is larger and

*d*= 3.5 μm with

*σ*=0.7 μm, which fits well to Fig.2(b) and previous experimental results [9

*χ*^{2} media: from disorder to order,” Opt. Express **19**, 11340–11354 (2011) [CrossRef] [PubMed] .

*d*

_{eff}for p-polarized light. In the case of normal incidence,

*d*

_{eff}in the

*xz*-plane is described only by

*y*-axis,

*d*

_{eff}takes the form (

*d*

_{31}cos

^{2}

*α*+

*d*

_{33}sin

^{2}

*α*)sin

*θ*+ 2

*d*

_{15}cos

*α*sin

*α*cos

*θ*. The total SH signal consists of these three contributions and depends on whether contributions from both of these processes add coherently or not. The SH intensity behavior for a p-polarized fundamental wave indicates coherent superposition of these three SH contributions in this configuration, where the longitudinal phase-mismatch is zero and the phase velocities are matched over the whole domain length. This characteristic trend of the SH Čerenkov intensity for this polarization is calculated with the corresponding effective nonlinear coefficient. It is worth to mention here that the position of the intensity minimum depends on the ratios of the involved nonlinear tensor elements

*d*

_{33}/

*d*

_{15},

*d*

_{33}/

*d*

_{31}which are chosen to be about 2 [34

34. M. Horowitz, A. Bekker, and B. Fischer, “Broadband second-harmonic generation in Sr* _{x}*Ba

_{1x}Nb

_{2}O

_{6}by spread spectrum phase matching with controllable domain gratings,” Appl. Phys. Lett.

**62**, 2619–2621 (1993) [CrossRef] .

*S*on the intensity traces because of the dominant effect of the effective nonlinear coefficient. This can be explained by the extraordinary polarization component in the effective nonlinear coefficient for p-polarization, which is naturally larger than the ordinary one in the case of s-polarization.

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needle-like ferroelectric domains in Sr |

2. | E. Y. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. |

3. | M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature |

4. | K. A. Kuznetsov, G. K. Kitaeva, A. V. Shevlyuga, L. I. Ivleva, and T. R. Volk, “Second harmonic generation in a strontium barium niobate crystal with a random domain structure,” JETP Letters |

5. | P. Molina, S. Álvarez García, M. O. Ramírez, J. García-Solé, L. E. Bausá, H. Zhang, W. Gao, J. Wang, and M. Jiang, “Nonlinear prism based on the natural ferroelectric domain structure in calcium barium niobate,” Appl. Phys. Lett. |

6. | K. Kalinowski, V. Roppo, T. Lukasiewicz, M. Swirkowicz, Y. Sheng, and W. Krolikowski, “Parametric wave interaction in one-dimensional nonlinear photonic crystal with randomized distribution of second-order nonlinearity,” Appl. Phys. B |

7. | R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. |

8. | W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Y. Kivshar, “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Optics Express |

9. | M. Ayoub, J. Imbrock, and C. Denz, “Second harmonic generation in multi-domain |

10. | A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. |

11. | X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. |

12. | P. Molina, M. de la O Ramírez, and L. E. Bausá, “Strontium barium niobate as a multifunctional two-dimensional nonlinear photonic glass,” Adv. Funct. Mater. |

13. | V. Vaičaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. |

14. | Y. Zhang, X. P. Hu, G. Zhao, S. N. Zhu, and M. Xiao, “Cerenkov second-harmonic arc from a hexagonally poled LiTaO |

15. | 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. |

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

17. | J. V. Jelley, |

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

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

20. | 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 |

21. | J. Bravo-Abad, X. Vidal, J. L. D. Jurez, and J. Martorell, “Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains,” Opt. Express |

22. | M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Domain-shape-based modulation of Čerenkov second-harmonic generation in multidomain strontium barium niobate,” Opt. Lett. |

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

24. | F. Sibbers, J. Imbrock, and C. Denz, “Sum-frequency generation in disordered quadratic nonlinear media,” Proc. of SPIE |

25. | 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- |

26. | 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. |

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

28. | S. M. Russell, P. E. Powers, M. J. Missey, and K. L. Schepler, “Broadband mid-infrared generation with two-dimensional quasi-phase-matched structures,” J. Quantum Electronics |

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

30. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum. Electron. |

31. | Y. Le Grand, D. Rouede, C. Odin, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH |

32. | U. Voelker, U. Heine, C. Gödecker, and K. Betzler, “Domain size effects in a uniaxial ferroelectric relaxor system: The case of Sr _{1−x}Nb_{2}O_{6},” J. Appl. Phys. 102, 114112(1)–114112(5) (2007) [CrossRef] . |

33. | L. Tian, D. A. Scrymgeour, and V. Gopalan, “Real-time study of domain dynamics in ferroelectric Sr |

34. | M. Horowitz, A. Bekker, and B. Fischer, “Broadband second-harmonic generation in Sr _{1x}Nb_{2}O_{6} by spread spectrum phase matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993) [CrossRef] . |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 5, 2012

Revised Manuscript: February 15, 2013

Manuscript Accepted: March 3, 2013

Published: March 28, 2013

**Citation**

Mousa Ayoub, Philip Roedig, Kaloian Koynov, Jörg Imbrock, and Cornelia Denz, "Čerenkov-type second-harmonic spectroscopy in random nonlinear photonic structures," Opt. Express **21**, 8220-8230 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8220

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### References

- S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needle-like ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett.73, 768–770 (1998). [CrossRef]
- E. Y. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett.73, 647–650 (2001). [CrossRef]
- M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004). [CrossRef] [PubMed]
- K. A. Kuznetsov, G. K. Kitaeva, A. V. Shevlyuga, L. I. Ivleva, and T. R. Volk, “Second harmonic generation in a strontium barium niobate crystal with a random domain structure,” JETP Letters87, 98–102 (2008). [CrossRef]
- P. Molina, S. Álvarez García, M. O. Ramírez, J. García-Solé, L. E. Bausá, H. Zhang, W. Gao, J. Wang, and M. Jiang, “Nonlinear prism based on the natural ferroelectric domain structure in calcium barium niobate,” Appl. Phys. Lett.94, 071111(1)–071111(3) (2009). [CrossRef]
- K. Kalinowski, V. Roppo, T. Lukasiewicz, M. Swirkowicz, Y. Sheng, and W. Krolikowski, “Parametric wave interaction in one-dimensional nonlinear photonic crystal with randomized distribution of second-order nonlinearity,” Appl. Phys. B109, 557–566 (2012). [CrossRef]
- R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89, 191105(1)–191105(3) (2006). [CrossRef]
- W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Y. Kivshar, “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Optics Express17, 20117–20123 (2009). [CrossRef] [PubMed]
- M. Ayoub, J. Imbrock, and C. Denz, “Second harmonic generation in multi-domain χ2 media: from disorder to order,” Opt. Express19, 11340–11354 (2011). [CrossRef] [PubMed]
- A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett.90, 243901(1)–243901(4) (2003). [CrossRef]
- X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97, 013902(1)–013902(4) (2006). [CrossRef]
- P. Molina, M. de la O Ramírez, and L. E. Bausá, “Strontium barium niobate as a multifunctional two-dimensional nonlinear photonic glass,” Adv. Funct. Mater.18, 709–715 (2008). [CrossRef]
- V. Vaičaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun.209, 485–490 (2002). [CrossRef]
- Y. Zhang, X. P. Hu, G. Zhao, S. N. Zhu, and M. Xiao, “Cerenkov second-harmonic arc from a hexagonally poled LiTaO3 planar waveguide,” J. Phys. D42, 215103–215107 (2009). [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, 1465–1472 (2009). [CrossRef]
- X. Deng, H. Ren, H. Lao, and X. Chen, “Research on Cherenkov second-harmonic generation in periodically poled lithium niobate by femtosecond pulses,” J. Opt. Soc. Am. B27, 1475–1480 (2010). [CrossRef]
- J. V. Jelley, Čerenkov Radiation and its Application (Pergamon Press, 1958).
- H. Ren, X. Deng, and X. Chen, “Coherently superposed efficient second harmonic generation by domain wall series in ferroelectrics,” arxiv:1010.1593v1 (2010).
- X. Deng and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express18, 15599–15602 (2010). [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, 16539–16545 (2010). [CrossRef] [PubMed]
- J. Bravo-Abad, X. Vidal, J. L. D. Jurez, and J. Martorell, “Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains,” Opt. Express18, 14202–14211 (2010). [CrossRef] [PubMed]
- M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Domain-shape-based modulation of Čerenkov second-harmonic generation in multidomain strontium barium niobate,” Opt. Lett.36, 4371–4373 (2011). [CrossRef] [PubMed]
- M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett.99, 241109(1)–241109(3) (2011). [CrossRef]
- F. Sibbers, J. Imbrock, and C. Denz, “Sum-frequency generation in disordered quadratic nonlinear media,” Proc. of SPIE7728, 77280Y–1 (2010).
- 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-χ(2) processes in radially poled nonlinear photonic crystals,” Appl. Phys. Lett.100, 101101(1)–101101(4) (2012).
- 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]
- A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett.85, 375–377 (2004). [CrossRef]
- S. M. Russell, P. E. Powers, M. J. Missey, and K. L. Schepler, “Broadband mid-infrared generation with two-dimensional quasi-phase-matched structures,” J. Quantum Electronics37, 877–887 (2001). [CrossRef]
- A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser and Photon. Rev.4, 355373 (2010). [CrossRef]
- M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum. Electron.28, 2631–2654 (1992). [CrossRef]
- Y. Le Grand, D. Rouede, C. Odin, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectrics,” Opt. Commun.200, 249–260 (2001). [CrossRef]
- U. Voelker, U. Heine, C. Gödecker, and K. Betzler, “Domain size effects in a uniaxial ferroelectric relaxor system: The case of SrxBa1−xNb2O6,” J. Appl. Phys.102, 114112(1)–114112(5) (2007). [CrossRef]
- L. Tian, D. A. Scrymgeour, and V. Gopalan, “Real-time study of domain dynamics in ferroelectric Sr0.61Ba0.39Nb2O6,” J. Appl. Phys.97, 114111(1)–114111(7) (2005).
- M. Horowitz, A. Bekker, and B. Fischer, “Broadband second-harmonic generation in SrxBa1xNb2O6 by spread spectrum phase matching with controllable domain gratings,” Appl. Phys. Lett.62, 2619–2621 (1993). [CrossRef]

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