Abstract

A theoretical model is presented to explain the refractive index change in the ion-implanted KTP waveguide, which includes respective contributions of spontaneous polarization, molar polarization and molar volume, and photoelastic effect. Numerical calculations of refractive indices along different crystalline orientations (X, Y, and Z) as a function of the lattice damage level, determined by Rutherford back-scattering/channeling technique, are performed based on the results from a set of z-cut KTP crystals implanted by 300 keV He+ ions in doses ranging from 4×1016 to 8×1016ions/cm2. The theoretical results show consistency with the experimental data. To our knowledge, this is the first model to comprehensively describe the ion-implanted KTP planar waveguide.

© 2012 Optical Society of America

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    [CrossRef]
  3. B. Agate, E. U. Rafailov, W. Sibbett, S. M. Saltiel, P. Battle, T. Fry, and E. Noonan, “Highly efficient blue-light generation from a compact, diode-pumped femtosecond laser by use of a periodically poled KTP waveguide crystal,” Opt. Lett. 28, 1963–1965 (2003).
    [CrossRef]
  4. L. Ma, O. Slattery, T. Chang, and X. Tang, “Non-degenerated sequential time-bin entanglement generation using periodically poled KTP waveguide,” Opt. Express 17, 15799–15807 (2009).
    [CrossRef]
  5. B. Brecht, A. Eckstein, and C. Silberhorn, “Controlling the correlations in frequency upconversion in PPLN and PPKTP waveguides,” Phys. Status Solidi C 8, 1235–1238 (2011).
    [CrossRef]
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    [CrossRef]
  9. K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
    [CrossRef]
  10. K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
    [CrossRef]
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    [CrossRef]
  13. T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
    [CrossRef]
  14. F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
    [CrossRef]
  15. F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
    [CrossRef]
  16. K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
    [CrossRef]
  17. F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
    [CrossRef]
  18. F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
    [CrossRef]
  19. F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
    [CrossRef]
  20. L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
    [CrossRef]
  21. D. T. Y. Wei, W. W. Lee, and L. R. Bloom, “Large refractive index change induced by ion implantation in lithium niobate,” Appl. Phys. Lett. 25, 329–331 (1974).
    [CrossRef]
  22. V. V. Atuchin, “Causes of refractive indices changes in He-implanted LiNbO3 and LiTaO3 waveguides,” Nucl. Instrum. Methods Phys. Res. Sect. B 168, 498–520 (2000).
    [CrossRef]
  23. H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
    [CrossRef]
  24. Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
    [CrossRef]
  25. J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
    [CrossRef]
  26. S. H. Wemple and M. DiDomenico, “Theory of the elasto-optic effect in nonmetallic crystals,” Phys. Rev. B 1, 193–202 (1970).
    [CrossRef]
  27. Z. W. Hu, P. A. Thomas, and P. Q. Huang, “High-resolution x-ray diffraction and topographic study of ferroelectric domains and absolute structural polarity of KTiOPO4 via anomalous scattering,” Phys. Rev. B 56, 8559–8565 (1997).
    [CrossRef]
  28. G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
    [CrossRef]
  29. E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
    [CrossRef]
  30. D. Shilo, E. Lakin, and E. Zolotoyabko, “Comprehensive strain analysis in thin films based on high-resolution x-ray diffraction: Application to implanted LiNbO3,” Phys. Rev. B 63, 205420 (2001).
    [CrossRef]
  31. D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
    [CrossRef]
  32. P. Bernasconi, M. Zgonik, and P. Günter, “Temperature dependence and dispersion of electro-optic and elasto-optic effect in perovskite crystals,” J. Appl. Phys. 78, 2651–2658 (1995).
    [CrossRef]
  33. M. C. Gupta and J. Ballato, The Handbook of Photonics(Second Ed.) (CRC Press, 2007).
  34. M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
    [CrossRef]
  35. S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
    [CrossRef]
  36. I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
    [CrossRef]
  37. H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
    [CrossRef]
  38. M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
    [CrossRef]
  39. L. C. Feldman and J. W. Rodgers, “Depth profiles of the lattice disorder resulting from ion bombardment of silicon single crystals,” J. Appl. Phys. 41, 3776–3782 (1970).
    [CrossRef]
  40. R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
    [CrossRef]

2011 (2)

B. Brecht, A. Eckstein, and C. Silberhorn, “Controlling the correlations in frequency upconversion in PPLN and PPKTP waveguides,” Phys. Status Solidi C 8, 1235–1238 (2011).
[CrossRef]

A. Eckstein1, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef]

2010 (1)

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

2009 (2)

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

L. Ma, O. Slattery, T. Chang, and X. Tang, “Non-degenerated sequential time-bin entanglement generation using periodically poled KTP waveguide,” Opt. Express 17, 15799–15807 (2009).
[CrossRef]

2008 (1)

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
[CrossRef]

2007 (3)

F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
[CrossRef]

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

2005 (1)

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

2003 (3)

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

B. Agate, E. U. Rafailov, W. Sibbett, S. M. Saltiel, P. Battle, T. Fry, and E. Noonan, “Highly efficient blue-light generation from a compact, diode-pumped femtosecond laser by use of a periodically poled KTP waveguide crystal,” Opt. Lett. 28, 1963–1965 (2003).
[CrossRef]

2002 (1)

F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
[CrossRef]

2001 (4)

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
[CrossRef]

D. Shilo, E. Lakin, and E. Zolotoyabko, “Comprehensive strain analysis in thin films based on high-resolution x-ray diffraction: Application to implanted LiNbO3,” Phys. Rev. B 63, 205420 (2001).
[CrossRef]

2000 (2)

V. V. Atuchin, “Causes of refractive indices changes in He-implanted LiNbO3 and LiTaO3 waveguides,” Nucl. Instrum. Methods Phys. Res. Sect. B 168, 498–520 (2000).
[CrossRef]

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

1999 (2)

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

X. Gu, M. Makarov, Y. J. Ding, J. B. Khurgin, and W. P. Risk, “Backward second-harmonic and third-harmonic generation in a periodically poled potassium titanyl phosphate waveguide,” Opt. Lett. 24, 127–129 (1999).
[CrossRef]

1998 (3)

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

1997 (3)

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Z. W. Hu, P. A. Thomas, and P. Q. Huang, “High-resolution x-ray diffraction and topographic study of ferroelectric domains and absolute structural polarity of KTiOPO4 via anomalous scattering,” Phys. Rev. B 56, 8559–8565 (1997).
[CrossRef]

G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
[CrossRef]

1996 (1)

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

1995 (2)

P. Bernasconi, M. Zgonik, and P. Günter, “Temperature dependence and dispersion of electro-optic and elasto-optic effect in perovskite crystals,” J. Appl. Phys. 78, 2651–2658 (1995).
[CrossRef]

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

1994 (1)

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

1993 (1)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

1989 (1)

1985 (1)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

1974 (1)

D. T. Y. Wei, W. W. Lee, and L. R. Bloom, “Large refractive index change induced by ion implantation in lithium niobate,” Appl. Phys. Lett. 25, 329–331 (1974).
[CrossRef]

1970 (2)

S. H. Wemple and M. DiDomenico, “Theory of the elasto-optic effect in nonmetallic crystals,” Phys. Rev. B 1, 193–202 (1970).
[CrossRef]

L. C. Feldman and J. W. Rodgers, “Depth profiles of the lattice disorder resulting from ion bombardment of silicon single crystals,” J. Appl. Phys. 41, 3776–3782 (1970).
[CrossRef]

Agate, B.

Åhlfeldt, H.

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

Argiolas, N.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Atuchin, V. V.

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

V. V. Atuchin, “Causes of refractive indices changes in He-implanted LiNbO3 and LiTaO3 waveguides,” Nucl. Instrum. Methods Phys. Res. Sect. B 168, 498–520 (2000).
[CrossRef]

Avrahami, Y.

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Bachmann, T.

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

Bakhru, H.

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

Bakhru, S.

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

Ballato, J.

M. C. Gupta and J. Ballato, The Handbook of Photonics(Second Ed.) (CRC Press, 2007).

Baolin, W.

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

Battle, P.

Bazzan, M.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Beeker, C.

F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
[CrossRef]

Bentini, G. G.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Bernasconi, P.

P. Bernasconi, M. Zgonik, and P. Günter, “Temperature dependence and dispersion of electro-optic and elasto-optic effect in perovskite crystals,” J. Appl. Phys. 78, 2651–2658 (1995).
[CrossRef]

Biaggio, I.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Bianconi, M.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Bierlein, J. D.

Bindner, P.

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

Bloom, L. R.

D. T. Y. Wei, W. W. Lee, and L. R. Bloom, “Large refractive index change induced by ion implantation in lithium niobate,” Appl. Phys. Lett. 25, 329–331 (1974).
[CrossRef]

Boudrioua, A.

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

Brecht, B.

B. Brecht, A. Eckstein, and C. Silberhorn, “Controlling the correlations in frequency upconversion in PPLN and PPKTP waveguides,” Phys. Status Solidi C 8, 1235–1238 (2011).
[CrossRef]

Cerutti, A.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
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D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
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F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
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S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
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F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
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Hou, D. C.

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
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Hu, H.

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
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H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
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K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
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J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
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Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
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Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
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S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
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K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

Liebertz, J.

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

Liu, X. D.

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

Liu, Y. G.

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Loulergue, J. C.

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

Lu, F.

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
[CrossRef]

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Lu, Q. M.

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
[CrossRef]

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

Luping, S.

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

Ma, H. J.

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

Ma, L.

Ma, Y. J.

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

Makarov, M.

Mazzoldi, P.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Meng, M. Q.

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Metzger, T. H.

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Ming, X. B.

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

Moretti, P.

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

Mosley, P. J.

A. Eckstein1, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef]

Noonan, E.

Opfermann, T.

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

Oron, M.

G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
[CrossRef]

Osgood, R. M.

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

Peisl, J.

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Pennestrì, G.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Pestrykov, E. V.

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

Rafailov, E. U.

Risk, W. P.

Rodgers, J. W.

L. C. Feldman and J. W. Rodgers, “Depth profiles of the lattice disorder resulting from ion bombardment of silicon single crystals,” J. Appl. Phys. 41, 3776–3782 (1970).
[CrossRef]

Rosenman, G.

G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
[CrossRef]

Roth, R. M.

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

Rottschalk, M.

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

Rui, N.

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

Ruske, J. P.

F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
[CrossRef]

Sada, C.

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Saltiel, S. M.

Sapozhnikov, V. K.

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

Sauer, W.

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Schlesser, R.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Schrempel, F.

F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
[CrossRef]

F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
[CrossRef]

Shen, D. Y.

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
[CrossRef]

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

Shi, B. R.

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

Shilo, D.

D. Shilo, E. Lakin, and E. Zolotoyabko, “Comprehensive strain analysis in thin films based on high-resolution x-ray diffraction: Application to implanted LiNbO3,” Phys. Rev. B 63, 205420 (2001).
[CrossRef]

Sibbett, W.

Silberhorn, C.

B. Brecht, A. Eckstein, and C. Silberhorn, “Controlling the correlations in frequency upconversion in PPLN and PPKTP waveguides,” Phys. Status Solidi C 8, 1235–1238 (2011).
[CrossRef]

A. Eckstein1, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef]

Skliar, A.

G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
[CrossRef]

Slattery, O.

Tan, Y.

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
[CrossRef]

Tang, X.

Teat, S. J.

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

Thomas, P. A.

Z. W. Hu, P. A. Thomas, and P. Q. Huang, “High-resolution x-ray diffraction and topographic study of ferroelectric domains and absolute structural polarity of KTiOPO4 via anomalous scattering,” Phys. Rev. B 56, 8559–8565 (1997).
[CrossRef]

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

Tian, Y. H.

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

Townsen, P. D.

P. D. Townsen, P. J. Chandler, and L. Zhang, Optical Effects of Ion Implantation (Cambridge University, 1994).

Tscheny, J.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Vanherzeele, H.

Voit, E.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Wang, K. M.

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40, 3759–3761 (2001).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Wang, L.

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
[CrossRef]

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

Wang, L. L.

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

Wang, W.

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Wang, X.

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Wang, X. L.

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

Wang, X. M.

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

Webjörn, J.

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

Wei, D. T. Y.

D. T. Y. Wei, W. W. Lee, and L. R. Bloom, “Large refractive index change induced by ion implantation in lithium niobate,” Appl. Phys. Lett. 25, 329–331 (1974).
[CrossRef]

Weis, R. S.

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Welch, D.

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

Wemple, S. H.

S. H. Wemple and M. DiDomenico, “Theory of the elasto-optic effect in nonmetallic crystals,” Phys. Rev. B 1, 193–202 (1970).
[CrossRef]

Wesch, W.

F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
[CrossRef]

F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
[CrossRef]

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

Wostrack, A.

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

Xu, T. B.

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

Yin, J. J.

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

Zgonik, M.

P. Bernasconi, M. Zgonik, and P. Günter, “Temperature dependence and dispersion of electro-optic and elasto-optic effect in perovskite crystals,” J. Appl. Phys. 78, 2651–2658 (1995).
[CrossRef]

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Zhang, L.

P. D. Townsen, P. J. Chandler, and L. Zhang, Optical Effects of Ion Implantation (Cambridge University, 1994).

Zhu, P. R.

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

Zolotoyabko, E.

D. Shilo, E. Lakin, and E. Zolotoyabko, “Comprehensive strain analysis in thin films based on high-resolution x-ray diffraction: Application to implanted LiNbO3,” Phys. Rev. B 63, 205420 (2001).
[CrossRef]

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Zubrinov, I. I.

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Appl. Phys. B (1)

L. L. Wang, K. M. Wang, Q. M. Lu, and H. J. Ma, “Enhanced refractive index well-confined planar and channel waveguides in KTiOPO4 produced by MeV C3+ ion implantation with low dose,” Appl. Phys. B 94, 295–299 (2009).
[CrossRef]

Appl. Phys. Lett. (4)

D. T. Y. Wei, W. W. Lee, and L. R. Bloom, “Large refractive index change induced by ion implantation in lithium niobate,” Appl. Phys. Lett. 25, 329–331 (1974).
[CrossRef]

D. Djukic, R. M. Roth, R. M. Osgood, K. Evans-Lutterodt, H. Bakhru, S. Bakhru, and D. Welch, “X-ray microbeam probing of elastic strains in patterned He+ implanted single-crystal LiNbO3,” Appl. Phys. Lett. 91, 112908 (2007).
[CrossRef]

P. Bindner, A. Boudrioua, J. C. Loulergue, and P. Moretti, “Formation of planar optical waveguides in potassium titanyl phosphate by double implantation of protons,” Appl. Phys. Lett. 79, 2558–2560 (2001).
[CrossRef]

M. Bianconi, N. Argiolas, M. Bazzan, G. G. Bentini, M. Chiarini, A. Cerutti, P. Mazzoldi, G. Pennestrì, and C. Sada, “On the dynamics of the damage growth in 5 MeV oxygen-implanted lithium niobate,” Appl. Phys. Lett. 87, 072901 (2005).
[CrossRef]

Appl. Surf. Sci. (2)

F. Chen, Y. Tan, L. Wang, D. C. Hou, and Q. M. Lu, “Optical channel waveguides with trapezoidal-shaped cross sections in KTiOPO4 crystal fabricated by ion implantation,” Appl. Surf. Sci. 254, 1822–1824 (2008).
[CrossRef]

F. Chen, H. Hu, Q. M. Lu, K. M. Wang, F. Lu, B. R. Shi, and D. Y. Shen, “Refractive index profiles of MeV phosphor ion implanted planar waveguide in KTP,” Appl. Surf. Sci. 183, 39–42 (2001).
[CrossRef]

Cryst. Res. Technol. (1)

S. Haussühl, S. Luping, W. Baolin, W. Jiyang, J. Liebertz, A. Wostrack, and C. Fink, “Physical properties of single crystals of KTiPO4, KxRb1−xTiOPO4 (x=0.85; 0.75), KGeOPO4 and KTiOAsO4,” Cryst. Res. Technol. 29, 583–589 (1994).
[CrossRef]

J. Appl. Phys. (5)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscheny, and P. Günter, “Materials constants of KNbO3 relevant for electro-and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

H. Åhlfeldt, J. Webjörn, P. A. Thomas, and S. J. Teat, “Structural and optical properties of annealed proton-exchanged waveguides in z-cut LiTaO3,” J. Appl. Phys. 77, 4467–4476 (1995).
[CrossRef]

P. Bernasconi, M. Zgonik, and P. Günter, “Temperature dependence and dispersion of electro-optic and elasto-optic effect in perovskite crystals,” J. Appl. Phys. 78, 2651–2658 (1995).
[CrossRef]

J. J. Yin, F. Lu, X. B. Ming, Y. J. Ma, and M. B. Huang, “Theoretical modeling of refractive index in ion implanted LiNbO3 waveguides,” J. Appl. Phys. 108, 033105 (2010).
[CrossRef]

L. C. Feldman and J. W. Rodgers, “Depth profiles of the lattice disorder resulting from ion bombardment of silicon single crystals,” J. Appl. Phys. 41, 3776–3782 (1970).
[CrossRef]

J. Opt. Soc. Am. B (1)

Mater. Lett. (2)

F. Chen, X. L. Wang, K. M. Wang, B. R. Shi, Q. M. Lu, D. Y. Shen, and N. Rui, “Analysis of refractive index profile in a silicon ion-implanted KTiOPO4 waveguide,” Mater. Lett. 57, 1197–1201 (2003).
[CrossRef]

E. Zolotoyabko, Y. Avrahami, W. Sauer, T. H. Metzger, and J. Peisl, “Strain profiles in He-implanted waveguide layers of LiNbO3 crystals,” Mater. Lett. 27, 17–20 (1996).
[CrossRef]

Mater. Sci. Eng. B (1)

K. M. Wang, M. Q. Meng, F. Lu, X. D. Liu, T. B. Xu, P. R. Zhu, D. Y. Shen, and Y. H. Tian, “Waveguide investigation in Er-implanted KTiOPO4 by MeV He ion implantation,” Mater. Sci. Eng. B 52, 8–11 (1998).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. Sect. B (5)

P. Bindner, A. Boudrioua, P. Moretti, and J. C. Loulergue, “Refractive index behaviors of helium implanted optical planar waveguides in LiNbO3, KTiOPO4 and Li2B4O7,” Nucl. Instrum. Methods Phys. Res. Sect. B 142, 329–337 (1998).
[CrossRef]

T. Opfermann, T. Bachmann, W. Wesch, and M. Rottschalk, “He+ implantation for waveguide fabrication in KTP and Rb:KTP,” Nucl. Instrum. Methods Phys. Res. Sect. B 148, 710–714 (1999).
[CrossRef]

F. Schrempel, C. Beeker, J. Fick, and W. Wesch, “Waveguide barriers with adjustable refractive index produced in KTP by irradiation with He- and Li-ions,” Nucl. Instrum. Methods Phys. Res. Sect. B 257, 484–487 (2007).
[CrossRef]

F. Schrempel, T. Höche, J. P. Ruske, U. Grusemann, and W. Wesch, “Depth dependence of radiation damage in Li-implanted KTiOPO4,” Nucl. Instrum. Methods Phys. Res. Sect. B 191, 202–207 (2002).
[CrossRef]

V. V. Atuchin, “Causes of refractive indices changes in He-implanted LiNbO3 and LiTaO3 waveguides,” Nucl. Instrum. Methods Phys. Res. Sect. B 168, 498–520 (2000).
[CrossRef]

Opt. Commun. (2)

K. M. Wang, F. Lu, H. Hu, F. Chen, B. R. Shi, D. Y. Shen, X. M. Wang, and Y. G. Liu, “Planar waveguide formation in KTiOPO4 by MeV Ni+ ion implantation,” Opt. Commun. 182, 357–360 (2000).
[CrossRef]

K. M. Wang, M. Q. Meng, F. Lu, X. Wang, W. Wang, P. J. Ding, and Y. G. Liu, “Analysis of refractive index profile in KTiOPO4 waveguide formed by 3.0 MeV He+ implantation,” Opt. Commun. 134, 55–58 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. D: Appl. Phys. (1)

G. Rosenman, A. Skliar, M. Oron, and M. Katz, “Polarization reversal in KTiOPO4 crystals,” Phys. D: Appl. Phys. 30, 277–282 (1997).
[CrossRef]

Phys. Rev. B (4)

S. H. Wemple and M. DiDomenico, “Theory of the elasto-optic effect in nonmetallic crystals,” Phys. Rev. B 1, 193–202 (1970).
[CrossRef]

Z. W. Hu, P. A. Thomas, and P. Q. Huang, “High-resolution x-ray diffraction and topographic study of ferroelectric domains and absolute structural polarity of KTiOPO4 via anomalous scattering,” Phys. Rev. B 56, 8559–8565 (1997).
[CrossRef]

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in lithium niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75, 195101 (2007).
[CrossRef]

D. Shilo, E. Lakin, and E. Zolotoyabko, “Comprehensive strain analysis in thin films based on high-resolution x-ray diffraction: Application to implanted LiNbO3,” Phys. Rev. B 63, 205420 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

A. Eckstein1, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[CrossRef]

Phys. Status Solidi C (1)

B. Brecht, A. Eckstein, and C. Silberhorn, “Controlling the correlations in frequency upconversion in PPLN and PPKTP waveguides,” Phys. Status Solidi C 8, 1235–1238 (2011).
[CrossRef]

Proc. SPIE (1)

I. I. Zubrinov, V. K. Sapozhnikov, E. V. Pestrykov, and V. V. Atuchin, “Elastic and elastooptic properties of KTiOPO4,” Proc. SPIE 5129, 249–254 (2003).
[CrossRef]

Solid State Commun. (1)

K. M. Wang, W. Li, F. Lu, M. Q. Meng, B. R. Shi, X. Wang, D. Y. Shen, and Y. G. Liu, “Double barred waveguides in KTiOPO4 formed by MeV He ion implantation,” Solid State Commun. 106, 173–175 (1998).
[CrossRef]

Other (2)

P. D. Townsen, P. J. Chandler, and L. Zhang, Optical Effects of Ion Implantation (Cambridge University, 1994).

M. C. Gupta and J. Ballato, The Handbook of Photonics(Second Ed.) (CRC Press, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Rutherford backscattering/channeling spectrum for 300 keV He+ ions implanted into KTP with doses of 4, 6, 8, and 10×1016ions/cm2. The four leading edges in random indicate Ti, K, P, and O elements of KTP, respectively.

Fig. 2.
Fig. 2.

The simulated damage profiles of 150 and 300 keV He+ ions implanted into KTP and the extracted damage profile of 300 keV with a dose of 4×1016ions/cm2.

Fig. 3.
Fig. 3.

Refractive indices of nx, ny, and nz profiles as a function of the lattice damage ratio in x-cut KTP crystal.

Fig. 4.
Fig. 4.

Refractive indices of nx, ny, and nz profiles as a function of the lattice damage ratio in y-cut KTP crystal.

Fig. 5.
Fig. 5.

Refractive indices of nx, ny, and nz profiles as a function of the lattice damage ratio in z-cut KTP crystal.

Fig. 6.
Fig. 6.

Relative intensity of light as a function of effective refractive index nz for KTP waveguide formed with 300 keV He+ ions to the dose of 4, 6, 8, and 10×1016ions/cm2 at room temperature.

Fig. 7.
Fig. 7.

Dependence of surface refractive indices versus implanted doses. The error bars are estimated from the maximum variation between the surface effective refractive indices and the average indices.

Tables (2)

Tables Icon

Table 1. Calculated Values of the Quadratic Electro-Optic Coefficients and the Photoelastic Coefficients

Tables Icon

Table 2. Beam Dose, the Damage Ratio at Sample Surface, and the Surface Effective Refractive Indices

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

1nij21nij,02=k,l(gijklPkPl+pijklSkl)i,j,k,l=1,2,3,
1nij21nij,02=gij3Ps2+pijlSli,j,l=1,2,3.
rijk=2gijklPlε0(εkk1)i,j,k,l=1,2,3,
rijT=rijS+pikdjk1k6,
gij,ρ=(1ρ)gij,Ps,ρ2=(1ρ)Ps2,pijl,ρ=(1ρ)pijl.
1nij21nij,02=(1ρ)2gij3Ps2+(1ρ)pijlSli,j,l=1,2,3.
Δn(M)ij=ρ(nij,0niso),
Sl,ρ=ρSmaxl.
(n21)(n2+2)=αMVM,
Δn(M)=(n21)(n2+2)6n(ΔαMαMΔVMVM).
1nij21(nij,0Δn(M)ij)2=(1ρ)2gij3Ps2+ρ(1ρ)pijlSmaxli,j,l=1,2,3,
ρ=1exp[(ϕϕC)γ],
nij=nij,012nij,03(1ρ)2gij3Ps212nij,03ρ(1ρ)pijlSmaxlΔn(M)ij.
Δn(Ps)ij=12nij,03(1ρ)2gij3Ps2,
Δn(S)ij=12nij,03ρ(1ρ)pijlSmaxl,
Δnij=Δn(Ps)ij+Δn(S)ij+Δn(M)ij
nij=nij,oΔnij,

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