## Enhancement of nonlinear Raman-Nath diffraction in two-dimensional optical superlattice |

Optics Express, Vol. 21, Issue 16, pp. 18671-18679 (2013)

http://dx.doi.org/10.1364/OE.21.018671

Acrobat PDF (1091 KB)

### Abstract

We study second harmonic generation via nonlinear Raman-Nath diffraction in an optical superlattice that maintains a periodic modulation of the second-order nonlinear coefficient *χ*^{(2)} in transverse direction but undergoes random modulation in longitudinal direction. We show that the random *χ*^{(2)} modulation offers a continuous set of reciprocal lattice vectors to compensate for the phase mismatch of nonlinear Raman-Nath diffraction in the longitudinal direction, leading to more efficient harmonic generation for a wide range of wavelengths. We also characterize the intensity dependence of nonlinear Raman-Nath diffraction on the degree of randomness of the optical supperlattice.

© 2013 OSA

## 1. Introduction

1. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO_{3}waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. **62**, 435–437 (1993) [CrossRef] .

4. R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. **95**, 133901 (2005) [CrossRef] [PubMed] .

*k⃗*

_{2}= 2

*k⃗*

_{1}+

*lG⃗*

_{0}[5

5. I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett. **21**, 1404–1406 (1968) [CrossRef] .

6. S. M. Saltiel, D. N. Neshev, R. Fischer, W. K. 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] .

*k⃗*

_{1},

*k⃗*

_{2}represent the wave vectors of the fundamental and SH beams,

*G⃗*

_{0}is the basic reciprocal lattice vector of the optical superlattice,

*l*refers to the diffraction order, respectively. In a general situation any multiples of the vector

*G⃗*

_{0}will match exactly the phase-matching relation. In this case the SH will be characterized by the

*partially*phase-matched emissions: 1) nonlinear Raman-Nath diffraction, i.e. SH emission at multiple angles

*α*(

_{m}*m*= ±1, ±2..), satisfying only the transverse phase-matching condition

*k⃗*

_{2}sin

*α*=

_{m}*mG⃗*

_{0}[7

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

8. X. Deng, H. Ren, H. Lao, and X. Chen, “Non-collinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B **100**, 755–758 (2010) [CrossRef] .

12. 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] .

*k⃗*

_{2}cos

*θ*= 2

*k⃗*

_{1}[see Fig. 1(d)]. While nonlinear Bragg and Raman-Nath diffractions represent nonlinear analogues of the well-known linear diffraction of waves on a dielectric grating [13], nonlinear Čerenkov diffraction has no analog in linear diffraction.

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

11. 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**, 3266–3268 (2011) [CrossRef] [PubMed] .

14. S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. K. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. **100**, 103902 (2008) [CrossRef] [PubMed] .

15. A. Shapira and A. Arie, “Phase-matched nonlinear diffraction,” Opt. Lett. **36**, 1933–1935 (2011) [CrossRef] [PubMed] .

*χ*

^{(2)}is engineered to enhance nonlinear Raman-Nath diffraction. For this purpose, we use two dimensional nonlinear structures with spatial modulation of the

*χ*

^{(2)}such that it is periodic modulation in transverse direction but randomized in longitudinal direction. The combination of such periodic and nonlinearity modulation allows for a complete fulfillment of the vectorial phase-matching conditions for some Raman-Nath emission peaks for a broad range of wavelengths. Consequently these nonlinear Raman-Nath diffraction signals keep growing monotonically throughout the crystal, leading to a higher conversion efficiency compared to the case with fully periodic

*χ*

^{(2)}modulation. We also consider the efficiency dependence of nonlinear Raman-Nath diffraction on the degree of randomness of optical superlattice.

16. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) **432**, 374–376 (2004) [CrossRef] .

18. 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,” Opt. Express **17**, 20117–20123 (2009) [CrossRef] [PubMed] .

*noncollinear*nonlinear diffraction. This approach can find applications in nonlinear frequency conversion, such as ultra-short pulse conversion and reconstruction and monitoring of femtosecond pulses [19

19. J. Trull, S. Saltiel, V. Roppo, C. Cojocaru, D. Dumay, W. Krolikowski, D.N. Neshev, R. Vilaseca, K. Staliunas, and Y.S. Kivshar, “Characterization of femtosecond pulses via transverse second-harmonic generation in random nonlinear media,” Appl. Phys. B **95**, 609–615 (2009) [CrossRef] .

## 2. Theory

*g*(

*x*,

*y*) can be generally expressed as the following Fourier series where

*G*

_{0}= 2

*π*/Λ and

*q*is the spatial frequency along the

*x*direction. In Eq. (1) we explicitly assumed periodic

*χ*

^{(2)}modulation in the transverse (

*y*) direction (with period Λ and duty cycle

*D*), while the term

*g̃*(

*q*,

*mG*

_{0}) represents the spatial Fourier spectrum of the distribution along the longitudinal (

*x*) direction.

*A*

_{2}(

*x*,

*y*), is governed by the equation where Δ

*k*=

*k*

_{2}− 2

*k*

_{1}and

*A*

_{1}(

*y*) is the amplitude of the fundamental field. For the input Gaussian beam we have

*A*

_{1}(

*y*) =

*A*

_{0}

*e*

^{−y2/w2}with

*A*

_{0}being the maximum amplitude of the beam and

*w*the the beam width.

*n*

_{2}denotes the refractive index at the SH frequency. Fourier transforming both sides of Eq. (2) along the

*y*coordinate, we get Using the expression in Eq. (1), and substituting

*g̃*(

*x*,

*mG*

_{0}) as in Eq. (1) and integrating the resulting equation we arrive at

*x*) = sin(

*x*)/

*x*. At this point we will introduce the spectral density of the SH field

*S̃*

_{2}(

*x*,

*k*) = |

_{y}*Ã*

_{2}(

*x*,

*k*)|

_{y}^{2}, which is given by with

*m*-th order Raman-Nath diffraction, i.e.

*k*= −

_{y}*mG*

_{0}=

*k*

_{2}sin

*α*. Assuming the paraxial approximation (

_{m}*α*is generally of the order of few degrees), argument of the sinc function in Eq. (6) can be expressed as which describes the phase mismatch in the longitudinal direction for the

_{m}*m*-th order of the Raman-Nath diffraction. Therefore, for fixed transverse periodicity (namely

*G*

_{0}), the right hand side term of the Eq. (6) will peak at and the SH emission will be then proportional to the Fourier coefficient

*g̃*(

*q*,

*mG*

_{0}). It is clear that if one wants the SH emission to be efficient for multiple incident wavelengths, one needs a broad spectrum of reciprocal lattice vectors in the longitudinal direction enabling to satisfy the longitudinal phase matching conditions in Eq. (9) for a wide range of wavelengths. As we show below, this can be achieved by introducing a proper random modulation of the

*χ*

^{(2)}nonlinearity along the propagation

*x*-axis.

## 3. Results and discussions

*χ*

^{(2)}coefficient. Even if different random configurations are still possible, in what follow we present the study of a practical

*χ*

^{(2)}pattern. The sample consists of areas where the sign of nonlinearity has been inverted in an otherwise homogeneous background. These areas form a pattern which is periodic in the transverse (

*y*) direction with period Λ and duty cycle

*D*, but is randomized along the longitudinal(

*x*) direction, as schematically shown in Fig. 2(a). We model the random modulation by using normal distribution with

*ρ*

_{0}denoting mean value of the domain width and

*σ*representing its dispersion. Then the 2D nonlinear modulation

*g*(

*x*,

*y*) can be written as where Π

*(*

_{a,b}*c*) is a rectangle function that equals to one when

*a*≤

*c*≤

*b*and zero elsewhere. If by

*x*we denote the coordinate of the

_{k}*k*interface between the positive and negative

_{th}*χ*

^{(2)}regions along the

*x*direction and

*y*the coordinate of the

_{l}*l*interface along

_{th}*y*direction, then we have where

*N*(

_{k}*ρ*

_{0},

*σ*) denotes a random number with the normal distribution.

*χ*

^{(2)}modulation along

*y*direction to be Λ = 8

*μm*and duty cycle

*D*= 0.5. The random

*χ*

^{(2)}distribution along the

*x*direction is characterized by a mean value

*ρ*

_{0}=

*l*(

_{c}*l*is the coherence length of the first-order nonlinear Raman-Nath emission,

_{c}*m*= ±1) and the dispersion

*σ*. For the incident fundamental beam at wavelength

*λ*

_{0}=1.4

*μ*m that propagates along the

*x*direction, the corresponding coherence length is

*l*=

_{c}*π*/(

*k*

_{2}cos

*α*

_{1}− 2

*k*

_{1}) = 5.8

*μm*, where

*α*

_{1}denotes the emission angle [see Fig. 1(c)]. Here we use the refractive index of strontium barium niobate (SBN) [20

20. Th. Woike, T. Granzow, U. Dörfler, Ch. Poesch, M. Wöhlecke, and R. Pankrath, “Refractive indices of congruently melting Sr_{0.61}Ba_{0.39}Nb_{2}O_{6},” Phys. Status Solidi A **186**, R13–R15 (2001) [CrossRef] .

*σ*= 0

*μm*, the structure is fully periodic and then the longitudinal phase-matching condition of the first-order Raman-Nath diffraction is fulfilled for fundamental wavelength

*λ*

_{0}. The nonzero

*σ*reflects disorder in the

*χ*

^{(2)}modulation along the

*x*direction. The larger

*σ*, the stronger the randomness. In order to fully characterize the performance of the randomized

*χ*

^{(2)}pattern

*g*(

*x*,

*y*) as a quasi-phase matching nonlinear photonic structure we use the spatial Fourier spectrum of the

*χ*

^{(2)}structure. Such spectrum represents the domain of reciprocal vectors that determine the direction and efficiency of the quadratic nonlinear process. In Figs. 2(b) and 2(c) we depict the modulus of Fourier spectrum of the

*χ*

^{(2)}structures with different dispersion values:

*σ*= 0

*μm*and

*σ*= 1.2

*μm*. It is clear that different domain structures lead to different distributions of Fourier spectrum. Compared to the periodic case (

*σ*= 0

*μm*), the Fourier spectrum broadens in the longitudinal direction after the structure randomness is introduced (

*σ*= 1.2

*μm*). The distribution of Fourier components remains the same in the other (

*y*) transverse direction as the

*χ*

^{(2)}patterns have the same period in this direction. For a quantitative illustration of the effect of structure randomness on the Fourier spectrum distribution, we depict the profile of the Fourier components that correspond to the first-order nonlinear Raman-Nath diffraction along the longitudinal direction in Figs. 2(d)–2(f), where the dispersion varies as

*σ*= 0

*μm*,

*σ*= 0.6

*μm*, and

*σ*= 1.2

*μm*, respectively. It is evident that the width of the Fourier spectrum varies strongly with

*σ*, leaving its position virtually unchanged. That is, the

*χ*

^{(2)}structures with large (small) dispersion provides broader (narrower) distribution of the reciprocal vectors with its center determined by the mean value of

*ρ*

_{0}.

*q*are available in the longitudinal direction. This will dramatically affects the Raman-Nath diffraction by enabling the fulfillment of the phase matching conditions in this direction. In this case, the longitudinal phase matching can be always satisfied for a broad range of incident wavelengths by selecting the appropriate reciprocal vector

*q*

_{0}=

*k*

_{2}cos

*α*

_{1}− 2

*k*

_{1}, which maximizes the

*sinc*function in Eq. (6). On the other hand, it is also seen from Eq. (6) that the intensity of nonlinear Raman-Nath diffraction is dependent on the integral across all the reciprocal vectors

*q*. In the randomized structures with broadened Fourier spectrum, the contribution of other reciprocal vectors (instead of

*q*

_{0}) cannot be neglected. However, the longer the propagation distance is, the weaker the effect of other reciprocal vectors becomes, as the longer propagation distance leads to the narrower spectrum width

*δq*of

*sinc*function centered at

*q*

_{0}.

## 4. Numerical calculations

*χ*

^{(2)}distribution on the nonlinear Raman-Nath diffraction we resort to numerical simulation of the wave interaction in 2D domain structures by solving Eq. (4) using fast Fourier transform beam propagation method. The form of the Fourier coefficient

*g̃*(

*q*,

*G*

_{0}) in Eq. (6) depends on the actual realization of nonlinear modulation along the

*x*direction.

*χ*

^{(2)}structures with same mean value

*ρ*

_{0}=

*l*but different dispersion values:

_{c}*σ*= 0

*μm*,

*σ*=0.6

*μ*m and

*σ*= 1.2

*μm*, respectively. We carry out calculations for a broad range of incident wavelengths from

*λ*= 1.30

*μm*to

*λ*= 1.50

*μm*, which involve reciprocal vectors ranging from

*q*= 0.67

*μm*

^{−1}to

*q*= 0.45

*μm*

^{−1}in the fulfillment of longitudinal phase matching condition. The main results are shown in Fig. 3. The dashed and dot-dashed line represent the average intensity of the first-order Raman-Nath beam generated in randomized domain structures with

*σ*=1.2

*μ*m and

*σ*=0.6

*μ*m, respectively, while the solid line corresponds to the fully periodic case (

*σ*= 0

*μm*). It is clearly seen that the introduction of the longitudinal randomness decreases the signal at the central (resonance) wavelength

*λ*=1.40

*μ*m, it leads to a substantial enhancement of the signal across the broad range of fundamental wavelengths.

*σ*= 0

*μm*(top row) and

*σ*= 1.2

*μm*(bottom row), respectively. Graphs in the left and right column of Fig. 4 correspond to the fundamental wavelength of

*λ*=1.35

*μ*m and

*λ*=1.45

*μ*m, respectively. The emission angle of the first-order nonlinear Raman-Nath signal is determined by the transverse phase matching condition, which gives

*α*

_{1}=

*sin*

^{−1}(

*G*

_{0}/

*k*

_{2}), and changes with the wavelength of the fundamental beam, such as, e.g.

*α*

_{1}= 2.10° for

*λ*=1.35

*μ*m, and

*α*

_{1}=2.27° for

*λ*=1.45

*μ*m. For the periodic case of

*σ*= 0

*μm*, the SH intensity oscillates with the propagation distance for both incident wavelengths, as shown in Figs. 4(a) and 4(b), what indicates the phase-mismatch in the longitudinal direction. In fact, in the periodic case, the phase matching condition can only be fulfilled at

*λ*= 1.40

*μm*, as depicted in Fig. 2(b). However, in the randomized nonlinear structure (

*σ*= 1.2

*μm*) the second harmonic keeps growing with propagation distance, a feature indicative of longitudinally phase-matched nonlinear process. For different incident wavelengths, the spatial harmonic intensity distribution varies differently which results from different longitudinal reciprocal vectors

*q*participating in the interaction process.

*χ*

^{(2)}modulations and few different wavelengths of the fundamental beam

*λ*= 1.40

*μm*[Fig. 5(a)] and

*λ*= 1.35

*μm*[Fig. 5(b)]. These graphs confirm the well known effect of randomness on the frequency conversion, namely the linear dependence of the average intensity with propagation distance [16

16. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) **432**, 374–376 (2004) [CrossRef] .

*σ*which maximizes the conversion efficiency. For almost periodic structure (small

*σ*) the conversion efficiency is limited by the coherence length (phase mismatch) of the SH generation process. Moderate disorder enables one to satisfy the full phase matching leading to enhanced frequency conversion. However, for strong disorder (large

*σ*) the effective nonlinearity is too weak to ensure significant energy transfer between fundamental and second harmonics over a reasonable propagation distance. Therefore the degree of structure randomness should be selected properly depending on the required conversion efficiency, operational bandwidth and the interaction length.

## 5. Conclusion

*χ*

^{(2)}nonlinearity in two orthogonal directions. We have shown that the introducing of the randomness in the longitudinal direction provides a broad set of reciprocal vectors which enable the fulfillment of the phase matching conditions of nonlinear Raman-Nath emission at multiple incident wavelengths. The same approach can be used for enhancement of cascading effect of nonlinear Raman-Nath diffraction in a single nonlinear photonic crystal.

## Acknowledgments

## References and links

1. | M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO |

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

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

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

5. | I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett. |

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

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

8. | X. Deng, H. Ren, H. Lao, and X. Chen, “Non-collinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B |

9. | N. An, H. Ren, Y Zheng, X. Deng, and X. Chen, “Cherenkov high-order harmonic generation by multistep cascading in |

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

11. | 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. |

12. | 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. |

13. | M. Born and E. Wolf, |

14. | S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. K. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. |

15. | A. Shapira and A. Arie, “Phase-matched nonlinear diffraction,” Opt. Lett. |

16. | M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) |

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

18. | 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,” Opt. Express |

19. | J. Trull, S. Saltiel, V. Roppo, C. Cojocaru, D. Dumay, W. Krolikowski, D.N. Neshev, R. Vilaseca, K. Staliunas, and Y.S. Kivshar, “Characterization of femtosecond pulses via transverse second-harmonic generation in random nonlinear media,” Appl. Phys. B |

20. | Th. Woike, T. Granzow, U. Dörfler, Ch. Poesch, M. Wöhlecke, and R. Pankrath, “Refractive indices of congruently melting Sr |

**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: May 1, 2013

Revised Manuscript: June 27, 2013

Manuscript Accepted: July 8, 2013

Published: July 30, 2013

**Citation**

Wenjie Wang, Yan Sheng, Vito Roppo, Zhihui Chen, Xiaoying Niu, and Wieslaw Krolikowski, "Enhancement of nonlinear Raman-Nath diffraction in two-dimensional optical superlattice," Opt. Express **21**, 18671-18679 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-18671

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### References

- M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO3waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett.62, 435–437 (1993). [CrossRef]
- L. E. Myers, R. C. Eckardt, M. M. Fejer, and R. L. Byer, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B12, 2012–2116 (1995). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science278, 843–846 (1997). [CrossRef]
- R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005). [CrossRef] [PubMed]
- I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett.21, 1404–1406 (1968). [CrossRef]
- S. M. Saltiel, D. N. Neshev, R. Fischer, W. K. 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]
- S. M. Saltiel, D. N. Neshev, W. K. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett.34, 848–850 (2009). [CrossRef] [PubMed]
- X. Deng, H. Ren, H. Lao, and X. Chen, “Non-collinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B100, 755–758 (2010). [CrossRef]
- N. An, H. Ren, Y Zheng, X. Deng, and X. Chen, “Cherenkov high-order harmonic generation by multistep cascading in χ(2)nonlinear photonic crystal,” Appl. Phys. Lett.100, 221103 (2012). [CrossRef]
- Y. Sheng, A. Best, H-J. Butt, W. Krolikowksi, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express18, 16539–16545 (2010). [CrossRef] [PubMed]
- 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, 3266–3268 (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]
- M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Chap. 12.
- S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. K. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett.100, 103902 (2008). [CrossRef] [PubMed]
- A. Shapira and A. Arie, “Phase-matched nonlinear diffraction,” Opt. Lett.36, 1933–1935 (2011). [CrossRef] [PubMed]
- M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)432, 374–376 (2004). [CrossRef]
- A. Pasquazi, A. Busacca, S. Stivala, R. Morandotti, and G. Assanto, “Nonlinear disorder mapping through three-wave mixing,” IEEE Photonics J2, 18–28 (2010). [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,” Opt. Express17, 20117–20123 (2009). [CrossRef] [PubMed]
- J. Trull, S. Saltiel, V. Roppo, C. Cojocaru, D. Dumay, W. Krolikowski, D.N. Neshev, R. Vilaseca, K. Staliunas, and Y.S. Kivshar, “Characterization of femtosecond pulses via transverse second-harmonic generation in random nonlinear media,” Appl. Phys. B95, 609–615 (2009). [CrossRef]
- Th. Woike, T. Granzow, U. Dörfler, Ch. Poesch, M. Wöhlecke, and R. Pankrath, “Refractive indices of congruently melting Sr0.61Ba0.39Nb2O6,” Phys. Status Solidi A186, R13–R15 (2001). [CrossRef]

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