## Cubic parametric frequency generation in rutile single crystal

Optics Express, Vol. 14, Issue 24, pp. 11715-11720 (2006)

http://dx.doi.org/10.1364/OE.14.011715

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### Abstract

We experimentally investigated third harmonic generation in TiO_{2} rutile single crystal, including phase-matching and cubic nonlinearity. We refined the dispersion equations of rutile and we demonstrated that this crystal allows angular non critical phase-matching at useful wavelengths, with a figure of merit 7.5 times that of KTiOPO_{4}. The measured cubic non linear coefficient and the corresponding Miller coefficients are : χ_{16}=5.0×10^{-21} m^{2}/V^{2} at 613.2 nm and Δ_{16}=3.5×10^{-24} m^{2}/V^{2}. These results are used to predict the phase-matching conditions and the efficiency of triple photon generation in rutile.

© 2006 Optical Society of America

_{4}(KTP): a THG conversion efficiency of 2.5% was obtained [1

1. J. P. Fève, B. Boulanger, and Y. Guillien, “Efficient energy conversion for cubic third-harmonic generation that is phase-matched in KTiOPO_{4},” Opt. Lett. **25**, 1373–1375 (2000). [CrossRef]

^{13}triple photons/pulse [2

2. J. Douady and B. Boulanger, “Experimental demonstration of a pure third-order optical parametric downconversion process,” Opt. Lett. **29**, 2794–2796 (2004). [CrossRef] [PubMed]

1. J. P. Fève, B. Boulanger, and Y. Guillien, “Efficient energy conversion for cubic third-harmonic generation that is phase-matched in KTiOPO_{4},” Opt. Lett. **25**, 1373–1375 (2000). [CrossRef]

3. J. Douady and B. Boulanger, “Calculation of quadratic cascading contributions associated with a phasematched cubic frequency difference generation in a KTiOPO_{4},” J. Opt. A: Pure and Applied Optics **7**, 467–471 (2005). [CrossRef]

^{(3)}is not so high : χ

_{11}=2.32×10

^{-21}m

^{2}/V

^{2}and χ

_{22}=1.96×10

^{-21}m

^{2}/V

^{2}at 1064 nm measured by z-scan experiments [4

4. R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. **18**, 194–196 (1993). [CrossRef] [PubMed]

_{24}=1.46×10

^{-21}m

^{2}/V

^{2}at 539 nm and χ

_{16}=0.80×10

^{-21}m

^{2}/V

^{2}at 491 nm determined by THG [5

5. B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearities of KTiOPO_{4},” J. Phys. B :At. Mol. Opt. Phys. **32**, 475–488 (1999). [CrossRef]

_{2}rutile. It belongs to the centrosymmetric tetragonal crystal class 4/mmm and to the positive uniaxial optical class,

*i.e.*n

_{e}>n

_{o}where no and ne denote the principal ordinary and extraordinary refractive indices. The χ

^{(3)}third order electric susceptibility tensor of rutile has the four following independent elements under Kleinman symmetry [6]:

_{ij}correspond to the contracted notation according to the standard convention. The third order effective coefficient magnitude ranges between 13.0×10

^{-21}m

^{2}/V

^{2}and 16.0×10

^{-21}m

^{2}/V

^{2}according to nonlinear index measurements performed by degenerate four-wave mixing (DFWM) or by nearly degenerate three-wave mixing (TWM) methods respectively [7, 8

8. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B **39**, 3337–3350 (1989). [CrossRef]

9. T. Hashimoto, T. Yoko, and S. Sakka, “Sol-Gel preparation and third-order nonlinear optical properties of TiO_{2} thin films,” Bull. Chem. Soc. Jpn **67**, 653–660 (1994). [CrossRef]

_{ω}=1900 nm [10

10. T. Hashimoto and T. Yoko, “Phase matching of rutile single crystal,” Appl. Phys. Lett. **68**, 2478–2479 (1996). [CrossRef]

_{2}rutile single crystal involving two ordinary waves (o) and two extraordinary waves (e) : the THG [ω(e)+ω(e)+ω(o)→3ω(o)] and the TPG [ω

_{4}(o)→ω

_{1}(e)+ω

_{2}(e)+ω

_{3}(o)]. Collinear phase-matching corresponds to Δk=0 where the mismatch parameter Δk, is defined as:

_{a}, ω

_{b}, ω

_{c}, ω

_{d})=(3ω, ω, ω, ω) for THG and (ω

_{a}, ω

_{b}, ω

_{c}, ω

_{d})=(ω

_{4}, ω

_{1}, ω

_{2}, ω

_{3}) for TPG; k

_{o,e}(ω

_{i})=[ω

_{i}/c] n

_{o,e}(ω

_{i}) are the wave vectors where no,e correspond to the ordinary and extraordinary refractive indices in the considered direction of propagation. The associated effective coefficient is given by [6]:

^{2}) and tuneable from 400 nm to 2400 nm, is used as the fundamental beam for the THG experiments. The beam is focused with a 10-mm-focal length in the rutile plate. The optimization of the THG conversion efficiency is made by adjusting the polarization of the fundamental wave with an achromatic half wave plate, and by translating the crystal with respect to the fundamental beam waist. A collecting lens that is placed behind the crystal allows the emergent beams to be focused on a prism. The third harmonic (TH) beam is separated from the non converted fundamental by the prism and a long-wavelength rejection filter. The TH power is measured with a silicon detector and the corresponding wavelength is determined by using a Chromex 250SM monochromator. The THG fundamental wavelength, corresponding to the maximum of the THG conversion efficiency, is measured at

11. J. R. DeVore, “Refractive indexes of rutile and spharelite,” J. Opt. Soc. Am. **41**, 416 (1951). [CrossRef]

_{ω}=0.47nm.cm, as shown in Fig. 1 where the normalized TH intensity I(ξ)/I(ξ

^{PM}) is plotted as a function of the fundamental wavelength ξ=λ

_{ω}from either side of the phasematching fundamental wavelength ξ

^{PM}=

*i.e.*the x-axis with the spherical coordinates θ=90° and ϕ=0°: the measured angular acceptance is equal to L.Δθ=0.42°.cm as shown in Fig. 2; it is close to the theoretical value calculated with relation (5), where ξ=θ and ξ

^{PM}=90°, and with the dispersion Eq. (4). Note that L.Δϕ is infinite because rutile is an uniaxial crystal.

_{ω}=1618 nm. In the two cases, there is not spatial walk-off because the waves propagate along a principal axis of the dielectric frame. The THG energy conversion efficiencies, η

_{THG}=u

_{3ω}/u

_{ω}, of KTP and TiO

_{2}are measured by using the same experimental setup; they are given in Fig. 3 as a function of the incident fundamental intensity, I

_{ω}. The measured efficiencies are weak because the considered fundamental intensities are very low, below 50 MW/cm

^{2}. For both crystals, η

_{THG}versus I

_{ω}exhibit a quadratic behavior as expected by the theory.

_{THG}under the plane-wave limit and the undepleted pump approximation, in the case of temporally and spatially Gaussian beams that are phase-matched in a direction without spatial and temporal walk-off gives:

_{a}coefficients correspond to the Fresnel transmissions of the interacting waves, L is the crystal length, and F

_{OM}is the figure of merit expressed as:

_{eff}is the THG effective coefficient that reduces to χ

_{16}(3ω) for TiO

_{2}and χ

_{24}(3ω) for KTP according to (1) and (3) for the considered phase-matching type and direction of propagation. The corresponding refractive indices are (n

_{1}, n

_{2}, n

_{3}, n

_{4})≡(n

_{e}, n

_{e}, n

_{o}, n

_{o}) for TiO

_{2}and (n

_{1}, n

_{2}, n

_{3}, n

_{4})≡(n

_{z}, n

_{z}, n

_{y}, n

_{y}) for KTP.

_{2}is found to be 7.5 larger than that of KTP according to Fig. 3 and Eq. (5). Then from Eq. (6) and the magnitude of χ

_{24}of KTP [5

5. B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearities of KTiOPO_{4},” J. Phys. B :At. Mol. Opt. Phys. **32**, 475–488 (1999). [CrossRef]

_{16}=5.0×10

^{-21}m

^{2}/V

^{2}for TiO

_{2}at λ

_{3ω}=613.2 nm. This value is smaller than those previously determined by nonlinear index measurements [7–9], which can be explained in part by the fact that the involved coefficients of the χ

^{(3)}tensor are different. The calculation of χ

_{16}at any wavelength can be done by considering the cubic Miller coefficient Δ

_{16}, which is a non dispersive parameter given by [14

14. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” App. Phys. Lett.5, 17–19 (1964). [CrossRef]

_{16}=3.5×10

^{-24}m

^{2}/V

^{2}from Eq. (7) and the magnitude of χ

_{16}measured at λ

_{3ω}=613.2nm.

_{OM}and dispersion equations show us that TiO

_{2}rutile is a very promising crystal for cubic frequency conversion when compared with the previous performances of KTP [1

1. J. P. Fève, B. Boulanger, and Y. Guillien, “Efficient energy conversion for cubic third-harmonic generation that is phase-matched in KTiOPO_{4},” Opt. Lett. **25**, 1373–1375 (2000). [CrossRef]

2. J. Douady and B. Boulanger, “Experimental demonstration of a pure third-order optical parametric downconversion process,” Opt. Lett. **29**, 2794–2796 (2004). [CrossRef] [PubMed]

^{2}and a crystal length of 10 mm. Such high conversion efficiency should be enough to perform statistical properties measurements of both fundamental and TH beams. Note that this expected efficiency would be very close to that obtained with cascading quadratic processes in KTP: [λ

_{ω}+λ

_{ω}→λ

_{2ω}] coupled with [λ

_{ω}+λ

_{2ω}→λ

_{3ω}]. Secondly, the calculation from Eq. (2), by setting Δk=0, and the dispersion Eq. (4) indicates interesting situations of phase-matching for the TPG [

*i.e.*(

^{-1}=(

^{-1}-(

^{-1}-(

^{-1}. The two phase-matching areas are formed by a framework of curves, each of them corresponding to a given phasematching angle. The locations where

2. J. Douady and B. Boulanger, “Experimental demonstration of a pure third-order optical parametric downconversion process,” Opt. Lett. **29**, 2794–2796 (2004). [CrossRef] [PubMed]

_{2}can be done at a single wavelength,

*i.e.*at 2940 nm, in the ordinary and extraordinary polarizations states. In this situation and according to the magnitude of F

_{OM}of TiO

_{2}compared with that of KTP, it is possible to expect more than 10

^{14}triple photons per pulse for few tenths of GW/cm

^{2}and a crystal length of 10 mm, which is suited to correlation measurements [15

15. I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. **57** (20), 2516–2519 (1986). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. P. Fève, B. Boulanger, and Y. Guillien, “Efficient energy conversion for cubic third-harmonic generation that is phase-matched in KTiOPO |

2. | J. Douady and B. Boulanger, “Experimental demonstration of a pure third-order optical parametric downconversion process,” Opt. Lett. |

3. | J. Douady and B. Boulanger, “Calculation of quadratic cascading contributions associated with a phasematched cubic frequency difference generation in a KTiOPO |

4. | R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. |

5. | B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearities of KTiOPO |

6. | B. Boulanger, B., and J. Zyss, in |

7. | V. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses |

8. | R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B |

9. | T. Hashimoto, T. Yoko, and S. Sakka, “Sol-Gel preparation and third-order nonlinear optical properties of TiO |

10. | T. Hashimoto and T. Yoko, “Phase matching of rutile single crystal,” Appl. Phys. Lett. |

11. | J. R. DeVore, “Refractive indexes of rutile and spharelite,” J. Opt. Soc. Am. |

12. | J. Rams, A. Tejeda, and J. M. Cabrera, “Refractive indices of rutile as a function of temperature and wavelength,” J. Appl. Phys. |

13. | Data from Almaz Optics, Inc., http://www.almazoptics.com/TiO2.htm. |

14. | R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” App. Phys. Lett.5, 17–19 (1964). [CrossRef] |

15. | I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4400) Nonlinear optics : Nonlinear optics, materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 21, 2006

Revised Manuscript: September 15, 2006

Manuscript Accepted: September 20, 2006

Published: November 27, 2006

**Citation**

Fabien Gravier and Benoît Boulanger, "Cubic parametric frequency generation in rutile single crystal," Opt. Express **14**, 11715-11720 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11715

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### References

- J. P. Fève, B. Boulanger, and Y. Guillien, "Efficient energy conversion for cubic third-harmonic generation that is phase-matched in KTiOPO4," Opt. Lett. 25, 1373-1375 (2000). [CrossRef]
- J. Douady and B. Boulanger, "Experimental demonstration of a pure third-order optical parametric downconversion process," Opt. Lett. 29, 2794-2796 (2004). [CrossRef] [PubMed]
- J. Douady and B. Boulanger, "Calculation of quadratic cascading contributions associated with a phase-matched cubic frequency difference generation in a KTiOPO4," J. Opt. A: Pure and Applied Optics 7, 467-471 (2005). [CrossRef]
- R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan and E. W. Van Stryland, "Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals," Opt. Lett. 18, 194-196 (1993). [CrossRef] [PubMed]
- B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, "Cubic optical nonlinearities of KTiOPO4," J. Phys. B : At. Mol. Opt. Phys. 32, 475-488 (1999). [CrossRef]
- B. Boulanger, B. and J. Zyss, in International Tables for Crystallography, Vol. D : Physical Properties of Crystals, A. Authier, ed., (International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 2003), Chap. 1.8, pp. 178-219.
- V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).
- R. Adair, L. L. Chase, and S. A. Payne, "Nonlinear refractive index of optical crystals," Phys. Rev. B 39, 3337-3350 (1989). [CrossRef]
- T. Hashimoto, T. Yoko and S. Sakka, "Sol-Gel preparation and third-order nonlinear optical properties of TiO2 thin films," Bull. Chem. Soc. Jpn 67, 653-660 (1994). [CrossRef]
- T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996). [CrossRef]
- J. R. DeVore, "Refractive indexes of rutile and spharelite," J. Opt. Soc. Am. 41, 416 (1951). [CrossRef]
- J. Rams, A. Tejeda, and J. M. Cabrera, "Refractive indices of rutile as a function of temperature and wavelength," J. Appl. Phys. 82, 994 (1997). [CrossRef]
- Data from Almaz Optics, Inc., http://www.almazoptics.com/TiO2.htm.
- R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17-19 (1964). [CrossRef]
- I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, "Direct observation of the second-order coherence of parametrically generated light," Phys. Rev. Lett. 57, 2516-2519 (1986). [CrossRef] [PubMed]

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