## Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains

Optics Express, Vol. 18, Issue 13, pp. 14202-14211 (2010)

http://dx.doi.org/10.1364/OE.18.014202

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### Abstract

We show that the weak second harmonic light generated from a random distribution of nonlinear domains of transparent Strontium Barium Niobate crystals can display a particularly intense generation in the forward direction. By using a theoretical model able to analyze the optical response of arbitrary distributions of three-dimensional nonlinear volumes of any shape, we found that the physical origin of this observation can be explained in terms of the scattering of light by a single nonlinear domain.

© 2010 Optical Society of America

## 1. Introduction

1. R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. **134**, A1313–A1319 (1964). [CrossRef]

2. C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. **26**, 442–444 (1975). [CrossRef]

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

4. X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. **97**, 013902 (2006). [CrossRef] [PubMed]

5. 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]

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

7. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express **15**, 15868–15877 (2007). [CrossRef] [PubMed]

5. 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]

## 2. Experimental results

5. 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]

*µ*m. In addition, from these images, we also observed that some of the nonlinear domains have circular section, while there are others which section resembles more that of a square with rounded corners. These square shapes appear to be all oriented in the same direction, following the cubic symmetry of the SBN crystal.

*λ*

_{1}= 1200 nm) propagating along a direction parallel and perpendicular, respectively, to the c-axis of the crystal. Figure 1(b) clearly displays the expected ring-shaped SHG at

*λ*

_{2}= 600 nm. Note however, the appearance of another ring of weaker intensity inscribed in the larger one. This secondary ring is more clearly visualized in inset of Fig. 1(b), which shows the total intensity along one diameter of the ring as measured by moving a detector in a plane perpendicular to the incident beam.

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

4. X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. **97**, 013902 (2006). [CrossRef] [PubMed]

9. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. **8**, 21–22 (1962). [CrossRef]

## 3. Theoretical results and discussion

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

7. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express **15**, 15868–15877 (2007). [CrossRef] [PubMed]

10. J.-X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

11. E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express **14**, 1167–1174 (2006). [CrossRef] [PubMed]

*V*surrounded with a linear material of the same refractive index (which takes values

_{NL}*n*

_{1}and

*n*

_{2}at the fundamental and second harmonic frequencies, respectively. In all the calculations shown in this manuscript we assume

*n*

_{1}= 2.2795 and

*n*

_{2}= 2.3479, which correspond to the values for the refractive index of SBN at

*λ*

_{1}= 1.2

*µ*m and

*λ*

_{2}= 0.6

*µ*m, respectively). Assuming all the fields to be time harmonic, the second harmonic E-field,

**E**

^{(2ω)}(

**r**), can be expressed in terms of the corresponding electric dyadic Green’s function

*Ĝ*(

**r**,

**r′**) as

**P**

^{(2ω)}(

**r**) stands for the induced nonlinear polarization at the second-harmonic frequency 2

*ω*. The Cartesian components of

**P**

^{(2ω)}(

**r**) are defined by

*P*

^{(2ω)}

_{i}=

*ε*

_{0}∑

_{j,k}χ^{(2)}

_{ijk}E^{(ω)}

_{j}E^{(ω)}

_{k}(with

*χ*

^{(2)}

_{ijk}being the corresponding second-order nonlinear susceptibility tensor, which is assumed to be compatible with the symmetry of the non-centrosymmetric ferroelectric phase of SBN; the indexes {

*i,j,k*} label the coordinates {

*x,y,z*}, respectively, whereas

**E**

^{(ω)}(

**r**) represents the fundamental E-field). Notice also that the integral of Eq. (1) is performed over the nonlinear volume under study. For nonlinear volumes with simple external shapes (e.g., a cylinder or a rectangular parallelepiped) this integral can be performed analytically; otherwise it can be computed numerically.

*Ĝ*(

**r**,

**r′**) can be obtained from the scalar Green’s function of a three-dimensional homogeneous medium [12]. In addition, as we are interested in the far-field response, we can use the asymptotic expansion of

*Ĝ*(

**r**,

**r′**) in the limit |

**r**| >> |

**r′**|. Thus, if we assume a form for the fundamental E-field given by

**E**

^{(ω)}(

**r**) =

*E*

^{(ω)}

_{0}exp(

*i*

**k**

^{(ω)}

**r**)

**u**

_{E}(where

**u**

_{E}is a unitary vector defining the polarization of the fundamental E-field,

*E*

^{(ω)}

_{0}stands for the corresponding E-field amplitude, and |

**k**

^{(ω)}| =

*ωn*

_{1}/

*c*), Eq. (1) can be recast as

*r*denotes the distance from the center of the considered nonlinear volume to the observation point, while

_{obs}*ϕ*and

_{obs}*θ*are the azimuthal and polar angles, respectively, defining the observation direction. The matrix

_{obs}*Î*

_{Δk}is defined as

**k**= |

**k**

^{(2ω)}|

**u**

_{r}− 2

**k**

^{(ω)}(

**u**

_{r}denotes a unit vector pointing along the observation direction and |

**k**

^{(2ω)}| = 2

*ωn*

_{2}/

*c*). Notice that this Δ

**k**governs the phase-matching between the fundamental and second-harmonic fields propagating inside the considered nonlinear volume. On the other hand, the matrix

*M̂*is defined as

*M̂*(

*ϕ*,

_{obs}*θ*) =

_{obs}*Î*−

**u**

_{r}⊗

**u**

_{r}(the operator ‘⊗’ denotes a tensor product). Finally,

**p**

^{(2ω)}= 2

*d̂*⊗

**u**

_{E}⊗

**u**

*, where*

_{E}*d̂*represents the contracted second-order nonlinear coefficient whose components are obtained from the tensor components

*χ*

^{(2)}

_{ijk}in the usual manner [13]. Once

**E**

^{(2ω)}(

**r**) has been computed from Eq. (2), the far-field intensity distribution at 2

*ω*,

*I*

^{(2ω)}(

**r**), can be obtained simply by using the fact that

*I*

^{(2ω)}(

**r**) = (1/2)(

*cε*

_{0}

*n*

_{2})|

**E**

^{(2ω)}(

**r**)|

^{2}.

*V*is a cylinder of diameter

_{NL}*d*and height

_{c}*h*[see Fig. 2(a)]. The effect of having a 3D parallelepiped with the same height

*h*but different cross section, such as a square with rounded corners, will be discussed later. After some algebra, from Eq. (2) one can show that the angular second-harmonic intensity distribution emitted by the cylinder can be written analytically as

*I*

^{(ω)}is the intensity of the fundamental E-field incident onto the sample,

*I*

^{(ω)}= (1/2)(

*cε*

_{0}

*n*

_{1})|

*E*

^{(ω)}

_{0}|

^{2}.

*d*stands for the corresponding effective second-order nonlinear coefficient. The function Δ

_{eff}*k*(

_{z}*ϕ*) denotes the projection of the phase-matching wavevector Δ

_{obs},θ_{obs}**k**onto the

*z*axis, whereas Δ

*k*

_{‖}(

*ϕ*) stands for the magnitude of the projection of Δ

_{obs},θ_{obs}**k**onto the

*xy*plane Δ

**k**

_{‖}= |Δ

**k**

_{‖}|. Finally,

*J*

_{1}is the first-order Bessel function.

14. D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. **190**, 1–9 (1995). [CrossRef]

15. S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. **92**, 207401 (2004). [CrossRef] [PubMed]

*xy*plane shown in Fig. 1(c) emerges when looking at the conditions that maximize the expression for

*I*

^{(2ω)}(

**r**) given in Eq. (3). Specifically, noticing that for

*θ*=

_{obs}*π*/2 [see definition of polar and azimuthal angles in schematics of Figs. 2(a) and 2(b)], Δ

*k*= 0 and

_{z}*I*

^{(2ω)}(

**r**) peaks at

*ϕ*= 0. In fact, since one can show that this value of

_{obs}*ϕ*corresponds to Δ

_{obs}*k*= 0, it is also clear that it is precisely the phase-matching in the direction perpendicular to the propagation direction (

_{y}*y*-axis in the considered configuration) the main mechanism behind the appearance of the observed dramatic increase of SHG. We note that in the case of illumination perpendicular to the c-axis, the above-discussed enhancement of SHG signal is invariant with respect to the in-plane rotation of the reference frame (i.e., for the considered cylindrically-shaped domain, the SHG enhancement is always found in the forward direction with respect to the fundamental beam). Thus, without any loss of generality, in our calculations we have assumed the

*x*-axis to be parallel to the direction of propagation of the fundamental beam, whereas the

*y*-axis is defined along the direction perpendicular to both the propagation direction and the c-axis.

*d*=13

_{c}*µ*m and

*h*=26

*µ*m, computed for the case in which the fundamental beam propagates along the

*x*-axis. Importantly, since for this configuration the minimum value for Δ

*k*

_{‖}is different from zero, the efficiency of generation oscillates with the radius of the cylinder, as shown in the inset of Fig. 2(c). A relative maximum is seen at

*d*≈ 13

_{c}*µ*m, which, remarkably, coincides with the experimentally determined average size of the domains. In addition, by analyzing numerically different 3D external shapes of the nonlinear volume

*V*, we have found that the above described phase-matching, responsible for the observed dramatic increase of SHG, seems to be rather independent of the shape of the domain. In particular, Fig. 2(e) renders the computed results for the case of a nonlinear parallelepiped of height

_{NL}*h*= 26

*µ*m and whose cross section is a square with rounded corners (which, as mentioned, resembles some of the domains observed in the actual SBN crystal used in our experiments). The cross section area of this rounded parallelepiped is set to be equal to the cross section area of the cylinder considered in Fig. 2(c). Notice that the peak of SHG that can be observed in Fig. 2(e) at

*ϕ*= 0 is about two orders of magnitude larger than the value defined by the plateaus surrounding this maximum, which agrees with the relative height of the maximum of

_{obs}*I*

^{(2ω)}(

**r**) with respect to the scattered background observed experimentally [see Fig. 1(c)].

*I*

^{(2ω)}(

**r**) computed for the case in which the nonlinear cylinder and the parallelepiped with rounded cross sections considered above are illuminated by a fundamental beam propagating along the c-axis. Remarkably, both panels show clearly that the far-field distribution of SHG from a finite single domain is very similar to that found experimentally in a random distribution of domains. The physical origin of the ring-shaped emission from a single domain shown in Figs. 2(d) and 2(f), can also be unveiled by inspection of Eq. (3) as follows. In the configuration in which fundamental beam propagates in a direction parallel to the c-axis [see Fig. 2(b)], the projection of the phase-matching wavevector Δ

**k**along the c-axis is given by Δ

*k*= (4

_{z}*π*/

*λ*

_{1})(

*n*

_{2}cos

*θ*−

_{obs}*n*

_{1}). Noticing that the condition Δ

*k*= 0 maximizes

_{z}*I*

^{(2ω)}(

**r**) for this configuration, we predict the existence of a ring-shaped emission with an angular aperture given by

*θ*= acos(

_{a}*n*

_{1}/

*n*

_{2}). This result is in agreement with the value of

*θ*found in our experiments, as well as with the value for this magnitude reported in the literature [6

_{a}6. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. **89**, 191105 (2006). [CrossRef]

7. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express **15**, 15868–15877 (2007). [CrossRef] [PubMed]

*d*〉 = 13

*µ*m, while their average height is given by 〈

*h*〉 = 2〈

*d*〉. The standard deviation of the distribution is chosen to be

*σ*= 2.6

*µ*m. Figure 3(a) shows that, when the structure is illuminated along the c-axis, the far-field distribution of second-harmonic light is dominated by the appearance of a ring-shaped distribution of angles in which the SHG is more intense. As shown in left panel of Fig. 3(a), the aperture of far-field cone defined by this ring (

*θ*= 15.5°) is in excellent agreement wih the one obtained experimentally [

_{a}*θ*= 13.8°, see Fig. 1(c)]. Note that, in addition, our model does predict the existence of a secondary internal ring. It is important to notice that the peak of generation in the forward direction predicted by the single domain model survives, as shown in Fig. 3(b), when a random distribution of multiple domains is considered, confirming the experimental observation shown in Fig. 1(c). This peak of generation in the forward direction is more clearly visualized in the top inset of Fig. 3(b), in which the average second-harmonic intensity is plotted in linear scale as a function of the observation angle

_{a}*ϕ*. Furthermore, our simulations also confirm the linear growth of the second-harmonic intensity as a function of the number of nonlinear volumes included in our simulations [see bottom inset of Fig. 3(b)].

_{obs}*ϕ*= ±14°. These two peaks lead to a small increase of second harmonic light intensity as we move the observation angle away from the incident direction. The position of the two side maxima observed experimentally at

_{obs}*ϕ*≈ ±5° does not correspond to the prediction of the theory mostly because a correct effective observation at infinity is only achievable for small angles, for which the lens distortion is minimal.

_{obs}## 4. Conclusions

## Acknowledgements

## References and links

1. | R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. |

2. | C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. |

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

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

5. | 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. |

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

7. | J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express |

8. |
See for instance, A. Ishimaru, |

9. | P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. |

10. | J.-X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B |

11. | E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express |

12. |
See for instance, L. Novotny and B. Hecht, |

13. | R. W. Boyd, |

14. | D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. |

15. | S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 24, 2010

Revised Manuscript: June 10, 2010

Manuscript Accepted: June 12, 2010

Published: June 16, 2010

**Citation**

Jorge Bravo-Abad, Xavier Vidal, Jorge Luis Domínguez Juárez, and Jordi Martorell, "Optical second-harmonic scattering from a non-diffusive random distribution of nonlinear domains," Opt. Express **18**, 14202-14211 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14202

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### References

- R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964). [CrossRef]
- C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975). [CrossRef]
- M. Baudrier-Raybaut, R. Hadar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004). [CrossRef] [PubMed]
- X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006). [CrossRef] [PubMed]
- S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998). [CrossRef]
- R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006). [CrossRef]
- J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007). [CrossRef] [PubMed]
- See for instance, A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).
- P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962). [CrossRef]
- J.-X. Cheng, and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002). [CrossRef]
- E. Yew, and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006). [CrossRef] [PubMed]
- See for instance, L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, Cambridge, 2006), pp. 25–30.
- R. W. Boyd, Nonlinear Optics, 2nd Ed. (Academic, San Diego, 2003), pp. 37–38.
- D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995). [CrossRef]
- S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004). [CrossRef] [PubMed]

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