Abstract

We experimentally investigated third harmonic generation in TiO2 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 KTiOPO4. The measured cubic non linear coefficient and the corresponding Miller coefficients are : χ16=5.0×10-21 m2/V2 at 613.2 nm and Δ16=3.5×10-24 m2/V2. These results are used to predict the phase-matching conditions and the efficiency of triple photon generation in rutile.

© 2006 Optical Society of America

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  1. 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]
  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]
  3. 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]
  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]
  5. 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]
  6. 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.
  7. V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).
  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 TiO2 thin films," Bull. Chem. Soc. Jpn 67, 653-660 (1994).
    [CrossRef]
  10. T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996).
    [CrossRef]
  11. J. R. DeVore, "Refractive indexes of rutile and spharelite," J. Opt. Soc. Am. 41, 416 (1951).
    [CrossRef]
  12. 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]
  13. Data from Almaz Optics, Inc., http://www.almazoptics.com/TiO2.htm.
  14. R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. 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. 57, 2516-2519 (1986).
    [CrossRef] [PubMed]

2005

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]

2004

2000

1999

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]

1997

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]

1996

T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996).
[CrossRef]

1994

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]

1993

1991

V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).

1989

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

1986

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]

1964

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17-19 (1964).
[CrossRef]

1951

Abram, I.

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]

Adair, R.

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

Boulanger, B.

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]

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. 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]

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]

Cabrera, J. M.

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]

Chase, L. L.

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

Delarue, P.

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]

DeSalvo, R.

DeVore, J. R.

Dolique, G.

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]

Douady, J.

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]

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

Fève, J. P.

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]

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]

Guillien, Y.

Hagan, D. J.

Hashimoto, T.

T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996).
[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]

Krol, D. M.

V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).

Marnier, G.

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]

Miller, R. C.

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17-19 (1964).
[CrossRef]

Oudar, J. L.

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]

Payne, S. A.

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

Raj, R. K.

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]

Rams, J.

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]

Rousseau, I.

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]

Said, A. A.

Sakka, S.

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]

Sheik-Bahae, M.

Tejeda, A.

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]

Van Stryland, E. W.

Vogel, V.

V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).

Weber, M. J.

V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).

Yoko, T.

T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996).
[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]

Appl. Phys. Lett.

T. Hashimoto and T. Yoko, "Phase matching of rutile single crystal," Appl. Phys. Lett. 68, 2478-2479 (1996).
[CrossRef]

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17-19 (1964).
[CrossRef]

Bull. Chem. Soc. Jpn

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]

J. Appl. Phys.

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]

J. Opt. A: Pure and Applied Optics

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]

J. Opt. Soc. Am.

J. Phys. B : At. Mol. Opt. Phys.

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]

Opt. Lett.

Phys. Chem. Glasses

V. Vogel, M. J. Weber, and D. M. Krol, "Nonlinear optical phenomena in glass," Phys. Chem. Glasses 32, 231 (1991).

Phys. Rev. B

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

Phys. Rev. Lett.

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]

Other

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

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.

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Figures (4)

Fig. 1.
Fig. 1.

Normalized generated third harmonic intensity as a function of the fundamental wavelength around phase-matching. The squares are the experimental data and the line is the calculation from the refined dispersion equations of the principal refractive indices.

Fig. 2.
Fig. 2.

Normalized generated third harmonic intensity as a function of the direction of propagation : θ is the angle between the z-axis and the x-axis of the dielectric frame (x,y,z). The squares are the experimental data and the line is the calculation from the refined dispersion equations of the principal refractive indices.

Fig. 3.
Fig. 3.

Third harmonic energy conversion efficiency as a function of the fundamental intensity in a 1-mm-long TiO2 crystal and a 1-mm-long KTP crystal phase-matched along the x-axis. The squares are the experimental data and the lines correspond to the calculation.

Fig. 4.
Fig. 4.

Calculated phase-matching curves of TPG [λ4oλ1e+λ2e+λ3o] in TiO2 rutile. The indices o and e refer to the ordinary and extraordinary polarizations respectively. The three dotted lines correspond to the situations where two wavelengths are equal. Point D corresponds to the degeneracy in wavelength, i.e. λ1e=λ2e=λ3o. Each continuous line refers to phasematching directions of equal θ angles. The points A and B are defined by the intersections between the phase-matching curves at θ=90° and the locations where λ1e=λ3o and λ2e=λ3o.

Equations (12)

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χ xxxx = χ yyyy ( χ 11 )
χ xxzz = χ xzxz = χ xzzx = χ yyzz = χ yzyz = χ yzzy = χ zyyz = χ zyzy = χ zzyy = χ zxxz = χ zxzx = χ zzxx ( χ 16 )
χ xxyy = χ xyxy = χ xyyx = χ yxxy = χ yxyx = χ yyxx ( χ 18 )
χ zzzz ( χ 33 )
Δ k = k o ( ω a ) k e ( ω b ) k e ( ω c ) k o ( ω c )
χ eff = i , j , k , l [ e i o ( ω a ) . e j e ( ω b ) . e k e ( ω c ) . e l o ( ω d ) ] χ i j k l ( ω a )
n o 2 ( λ ) = 6.0125 + 0.20136 λ 2 0.10146 0.05072 λ 2
n e 2 ( λ ) = 7.07355 + 0.29834 λ 2 0.07957
I 3 ω ( ξ ) I 3 ω ( ξ P M ) = sinc 2 ( Δ k ( ξ ) L 2 )
η T H G = 4 π 2 9 3 T 1 ( ω ) T 2 ( ω ) T 3 ( ω ) T 4 ( 3 ω ) λ ω 2 F O M I ω 2 L 2
F O M = χ eff 2 n 1 ( ω ) n 2 ( ω ) n 3 ( ω ) n 4 ( 3 ω )
Δ 16 = χ 16 ( 3 ω ) ( n e 2 ( ω ) 1 ) ( n e 2 ( ω ) 1 ) ( n o 2 ( ω ) 1 ) ( n o 2 ( 3 ω ) 1 )

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