Abstract

The losses of ion-implanted potassium niobate (KNbO3) waveguides are evaluated theoretically and experimentally in dependence on wavelength, irradiation dose, waveguide thickness, and waveguide width. Irradiation-induced absorption and tunneling are identified as the main sources of loss. The contributions from surface scattering and intrinsic material absorption are shown to be small. The attenuation due to irradiation-induced absorption and tunneling is calculated from the experimentally determined complex refractive-index profile. The optical loss is minimum in the red part of the spectrum and increases toward the blue because of absorption and toward the infrared because of tunneling. A minimum loss of less than 0.2 cm-1 (1 dB cm-1) was measured in an ion-implanted KNbO3 waveguide at a wavelength of 0.633 μm. On the basis of a theoretical model we give guidelines for the formation of optimized waveguides for specific applications, e.g., second-harmonic generation.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
    [CrossRef]
  30. T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
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  36. S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
    [CrossRef] [PubMed]

1997 (1)

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

1996 (2)

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

1995 (5)

D. Kip, S. Aulkemeyer, and P. Moretti, “Low-loss planar optical waveguides in strontium barium niobate crystals formed by ion-beam implantation,” Opt. Lett. 20, 1256¿1258 (1995).
[CrossRef] [PubMed]

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
[CrossRef] [PubMed]

D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, “Optical losses in ferroelectric oxide thin films: Is there light at the end of the tunnel?” Mater. Res. Soc. Symp. Proc. 361, 155¿166 (1995).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

1994 (2)

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

H. Mabuchi, E. S. Polzik, and H. J. Kimble, “Blue-light-induced infrared absorption in KNbO3,” J. Opt. Soc. Am. B 11, 2023¿2029 (1994).
[CrossRef]

1993 (2)

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

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

1992 (6)

I. Biaggio, P. Kerkoc, L.-S. Wu, P. Günter, and B. Zysset, “Refractive indices of orthorhombic KNbO3. II. Phase-matching configurations for nonlinear-optical interactions,” J. Opt. Soc. Am. B 9, 507¿517 (1992).
[CrossRef]

U. Ellenberger, R. Weber, J. E. Balmer, D. Ellgehausen, and G. J. Mizell, “Pulsed optical damage threshold of potassium niobate,” Appl. Opt. 31, 7563¿7569 (1992).
[CrossRef] [PubMed]

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

B. Zysset, I. Biaggio, and P. Günter, “Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence,” J. Opt. Soc. Am. B 9, 380¿386 (1992).
[CrossRef]

1991 (2)

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

1990 (2)

L. Zhang, P. J. Chandler, and P. D. Townsend, “Detailed index profiles of ion implanted waveguides in KNbO3,” Ferroelectr. Lett. 11, 89¿97 (1990).
[CrossRef]

P. J. Chandler, L. Zhang, and P. D. Townsend, “High temperature annealing of He+ ion-implanted quartz optical waveguides,” Nucl. Instrum. Methods B 46, 69¿73 (1990).
[CrossRef]

1988 (1)

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

1985 (3)

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57¿R78 (1985).
[CrossRef]

J. M. Naden and B. L. Weiss, “Optical properties of planar waveguides formed by He+ implantation in LiNbO3,” J. Lightwave Technol. LT-3, 855¿859 (1985).
[CrossRef]

R. Regener and W. Sohler, “Loss in low-finesse Ti:LiNbO3 optical waveguide resonators,” Appl. Phys. B 36, 143¿147 (1985).
[CrossRef]

1984 (1)

1980 (1)

J. P. Biersack and L. G. Haggmark, “A Monte Carlo computer program for the transport of energetic ions in amorphous targets,” Nucl. Instrum. Methods 174, 257¿269 (1980).
[CrossRef]

1979 (1)

1974 (3)

E. Wiesendanger, “Dielectric, mechanical and optical properties of orthorhombic KNbO3,” Ferroelectrics 6, 263¿281 (1974).
[CrossRef]

P. Günter, “Electro-optical properties of KNbO3,” Opt. Commun. 11, 285¿290 (1974).
[CrossRef]

Y. Uematsu, “Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase,” Jpn. J. Appl. Phys. 13, 1362¿1368 (1974).
[CrossRef]

1971 (1)

Armani-Leplingard, F.

D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, “Optical losses in ferroelectric oxide thin films: Is there light at the end of the tunnel?” Mater. Res. Soc. Symp. Proc. 361, 155¿166 (1995).
[CrossRef]

Aulkemeyer, S.

Balmer, J. E.

Bauer, St.

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

Beckers, L.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

Biaggio, I.

Biersack, J. P.

J. P. Biersack and L. G. Haggmark, “A Monte Carlo computer program for the transport of energetic ions in amorphous targets,” Nucl. Instrum. Methods 174, 257¿269 (1980).
[CrossRef]

Binder, B.

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

Bremer, T.

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Brülisauer, S.

Buchal, C.

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

Buchal, Ch.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

Busse, L. E.

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

Chandler, P. J.

P. J. Chandler, L. Zhang, and P. D. Townsend, “High temperature annealing of He+ ion-implanted quartz optical waveguides,” Nucl. Instrum. Methods B 46, 69¿73 (1990).
[CrossRef]

L. Zhang, P. J. Chandler, and P. D. Townsend, “Detailed index profiles of ion implanted waveguides in KNbO3,” Ferroelectr. Lett. 11, 89¿97 (1990).
[CrossRef]

Ellenberger, U.

Ellgehausen, D.

Fleuster, M.

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

Fluck, D.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
[CrossRef] [PubMed]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

Fork, D. K.

D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, “Optical losses in ferroelectric oxide thin films: Is there light at the end of the tunnel?” Mater. Res. Soc. Symp. Proc. 361, 155¿166 (1995).
[CrossRef]

Godefroy, G.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

Goldberg, L.

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

Günter, P.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
[CrossRef] [PubMed]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

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

I. Biaggio, P. Kerkoc, L.-S. Wu, P. Günter, and B. Zysset, “Refractive indices of orthorhombic KNbO3. II. Phase-matching configurations for nonlinear-optical interactions,” J. Opt. Soc. Am. B 9, 507¿517 (1992).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

B. Zysset, I. Biaggio, and P. Günter, “Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence,” J. Opt. Soc. Am. B 9, 380¿386 (1992).
[CrossRef]

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

P. Günter, “Electro-optical properties of KNbO3,” Opt. Commun. 11, 285¿290 (1974).
[CrossRef]

Haegele, K. H.

Haggmark, L. G.

J. P. Biersack and L. G. Haggmark, “A Monte Carlo computer program for the transport of energetic ions in amorphous targets,” Nucl. Instrum. Methods 174, 257¿269 (1980).
[CrossRef]

Heiland, W.

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Hellermann, B.

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Hertel, P.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Hesse, H.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

Irmscher, R.

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

Jackel, J. L.

Jundt, D. H.

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

Kerkoc, P.

Kimble, H. J.

Kingston, J. J.

D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, “Optical losses in ferroelectric oxide thin films: Is there light at the end of the tunnel?” Mater. Res. Soc. Symp. Proc. 361, 155¿166 (1995).
[CrossRef]

Kip, D.

Kollewe, D.

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Krätzig, E.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

Küpfer, M.

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

Looser, H.

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

Mabuchi, H.

Mizell, G.

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

Mizell, G. J.

Moretti, P.

D. Kip, S. Aulkemeyer, and P. Moretti, “Low-loss planar optical waveguides in strontium barium niobate crystals formed by ion-beam implantation,” Opt. Lett. 20, 1256¿1258 (1995).
[CrossRef] [PubMed]

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

Naden, J. M.

J. M. Naden and B. L. Weiss, “Optical properties of planar waveguides formed by He+ implantation in LiNbO3,” J. Lightwave Technol. LT-3, 855¿859 (1985).
[CrossRef]

Pliska, T.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
[CrossRef] [PubMed]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

Polzik, E. S.

Regener, R.

R. Regener and W. Sohler, “Loss in low-finesse Ti:LiNbO3 optical waveguide resonators,” Appl. Phys. B 36, 143¿147 (1985).
[CrossRef]

Rytz, D.

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

Schlesser, R.

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

Seaton, C. T.

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57¿R78 (1985).
[CrossRef]

Sohler, W.

R. Regener and W. Sohler, “Loss in low-finesse Ti:LiNbO3 optical waveguide resonators,” Appl. Phys. B 36, 143¿147 (1985).
[CrossRef]

Solcia, C.

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

S. Brülisauer, D. Fluck, C. Solcia, T. Pliska, and P. Günter, “Nondestructive waveguide loss-measurement method using self-pumped phase conjugation for optimum end-fire coupling,” Opt. Lett. 20, 1773¿1775 (1995).
[CrossRef] [PubMed]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57¿R78 (1985).
[CrossRef]

Stritzker, B.

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

Strohkendl, F. P.

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

Surette, M. R.

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

Thevenard, P.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

Tien, P. K.

Townsend, P. D.

L. Zhang, P. J. Chandler, and P. D. Townsend, “Detailed index profiles of ion implanted waveguides in KNbO3,” Ferroelectr. Lett. 11, 89¿97 (1990).
[CrossRef]

P. J. Chandler, L. Zhang, and P. D. Townsend, “High temperature annealing of He+ ion-implanted quartz optical waveguides,” Nucl. Instrum. Methods B 46, 69¿73 (1990).
[CrossRef]

Tscherry, J.

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

Uematsu, Y.

Y. Uematsu, “Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase,” Jpn. J. Appl. Phys. 13, 1362¿1368 (1974).
[CrossRef]

Ulrich, R.

Veselka, J. J.

Voit, E.

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

Weber, R.

Weiss, B. L.

J. M. Naden and B. L. Weiss, “Optical properties of planar waveguides formed by He+ implantation in LiNbO3,” J. Lightwave Technol. LT-3, 855¿859 (1985).
[CrossRef]

Wiesendanger, E.

E. Wiesendanger, “Dielectric, mechanical and optical properties of orthorhombic KNbO3,” Ferroelectrics 6, 263¿281 (1974).
[CrossRef]

Wirl, K.

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

Wu, L.-S.

Zgonik, M.

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

Zhang, L.

L. Zhang, P. J. Chandler, and P. D. Townsend, “Detailed index profiles of ion implanted waveguides in KNbO3,” Ferroelectr. Lett. 11, 89¿97 (1990).
[CrossRef]

P. J. Chandler, L. Zhang, and P. D. Townsend, “High temperature annealing of He+ ion-implanted quartz optical waveguides,” Nucl. Instrum. Methods B 46, 69¿73 (1990).
[CrossRef]

Zysset, B.

Appl. Opt. (3)

Appl. Phys. B (1)

R. Regener and W. Sohler, “Loss in low-finesse Ti:LiNbO3 optical waveguide resonators,” Appl. Phys. B 36, 143¿147 (1985).
[CrossRef]

Appl. Phys. Lett. (1)

D. Fluck, T. Pliska, P. Günter, St. Bauer, L. Beckers, and Ch. Buchal, “Blue-light second-harmonic generation in ion-implanted KNbO3 channel waveguides of new design,” Appl. Phys. Lett. 69, 4133¿4135 (1996).
[CrossRef]

Ferroelectr. Lett. (2)

T. Bremer, W. Heiland, B. Hellermann, P. Hertel, E. Krätzig, and D. Kollewe, “Waveguides in KNbO3 by He+ implantation,” Ferroelectr. Lett. 9, 11¿14 (1988).
[CrossRef]

L. Zhang, P. J. Chandler, and P. D. Townsend, “Detailed index profiles of ion implanted waveguides in KNbO3,” Ferroelectr. Lett. 11, 89¿97 (1990).
[CrossRef]

Ferroelectrics (2)

P. Moretti, P. Thevenard, K. Wirl, P. Hertel, H. Hesse, E. Krätzig, and G. Godefroy, “Proton implanted waveguides in LiNbO3, KNbO3 and BaTiO3,” Ferroelectrics 128, 13¿18 (1992).
[CrossRef]

E. Wiesendanger, “Dielectric, mechanical and optical properties of orthorhombic KNbO3,” Ferroelectrics 6, 263¿281 (1974).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Fluck, T. Pliska, P. Günter, L. Beckers, and C. Buchal, “Cerenkov-type second-harmonic generation in KNbO3 channel waveguides,” IEEE J. Quantum Electron. 32, 905¿916 (1996).
[CrossRef]

J. Appl. Phys. (8)

L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Absorption losses in MgO-doped and undoped potassium niobate,” J. Appl. Phys. 75, 1102¿1110 (1994).
[CrossRef]

F. P. Strohkendl, P. Günter, Ch. Buchal, and R. Irmscher, “Formation of optical waveguides in KNbO3 by low-dose MeV He+ implantation,” J. Appl. Phys. 69, 84¿88 (1991).
[CrossRef]

D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Low-loss optical channel waveguides in KNbO3 by multiple energy ion implantation,” J. Appl. Phys. 72, 1671¿1675 (1992).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57¿R78 (1985).
[CrossRef]

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

T. Pliska, D. H. Jundt, D. Fluck, P. Günter, D. Rytz, M. Fleuster, and Ch. Buchal, “Low-temperature annealing of ion-implanted KNbO3 waveguides for second-harmonic generation,” J. Appl. Phys. 77, 6114¿6120 (1995).
[CrossRef]

T. Pliska, C. Solcia, D. Fluck, P. Günter, L. Beckers, and Ch. Buchal, “Radiation damage profiles of the refractive indices of He+ ion-implanted KNbO3 waveguides,” J. Appl. Phys. 81, 1099¿1102 (1997).
[CrossRef]

D. Fluck, D. H. Jundt, P. Günter, M. Fleuster, and Ch. Buchal, “Modeling of refractive index profiles of He+ ion-implanted KNbO3 waveguides based on the irradiation parameters,” J. Appl. Phys. 74, 6023¿6031 (1993).
[CrossRef]

J. Lightwave Technol. (1)

J. M. Naden and B. L. Weiss, “Optical properties of planar waveguides formed by He+ implantation in LiNbO3,” J. Lightwave Technol. LT-3, 855¿859 (1985).
[CrossRef]

J. Opt. Soc. Am. B (3)

Jpn. J. Appl. Phys. (1)

Y. Uematsu, “Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase,” Jpn. J. Appl. Phys. 13, 1362¿1368 (1974).
[CrossRef]

Mater. Res. Soc. Symp. Proc. (2)

D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, “Optical losses in ferroelectric oxide thin films: Is there light at the end of the tunnel?” Mater. Res. Soc. Symp. Proc. 361, 155¿166 (1995).
[CrossRef]

R. Irmscher, D. Fluck, Ch. Buchal, B. Stritzker, and P. Günter, “Measured lattice damage and optical index change in KNbO3,” Mater. Res. Soc. Symp. Proc. 201, 399¿403 (1991).
[CrossRef]

Nucl. Instrum. Methods (1)

J. P. Biersack and L. G. Haggmark, “A Monte Carlo computer program for the transport of energetic ions in amorphous targets,” Nucl. Instrum. Methods 174, 257¿269 (1980).
[CrossRef]

Nucl. Instrum. Methods B (1)

P. J. Chandler, L. Zhang, and P. D. Townsend, “High temperature annealing of He+ ion-implanted quartz optical waveguides,” Nucl. Instrum. Methods B 46, 69¿73 (1990).
[CrossRef]

Opt. Commun. (3)

D. Fluck, B. Binder, M. Küpfer, H. Looser, Ch. Buchal, and P. Günter, “Phase-matched second harmonic blue light generation in ion implanted KNbO3 planar waveguides with 29% conversion efficiency,” Opt. Commun. 90, 304¿310 (1992).
[CrossRef]

C. Solcia, D. Fluck, T. Pliska, P. Günter, St. Bauer, M. Fleuster, L. Beckers, and Ch. Buchal, “The refractive index distribution nc(z) of ion implanted KNbO3 waveguides,” Opt. Commun. 120, 39¿46 (1995).
[CrossRef]

P. Günter, “Electro-optical properties of KNbO3,” Opt. Commun. 11, 285¿290 (1974).
[CrossRef]

Opt. Lett. (3)

Other (3)

P. D. Townsend, P. J. Chandler, and L. Zhang, Optical Effects of Ion Implantation. P. L. Knight and A. Miller, eds., Vol. 13 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, UK, 1994).

G. Götz, “Optoeletronic materials,” in Ion Beam Modification of Insulators, P. Mazzoldi and G. W. Arnold, eds. (Elsevier, Amsterdam, 1987), pp. 412¿446.

D. Fluck, “Ion-implanted KNbO3 waveguides for blue-light second-harmonic generation,” Ph.D. thesis no. 11225 (ETH Zürich, Zürich, 1995).

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Figures (11)

Fig. 1
Fig. 1

Refractive index profile of nb (percentage decrease, dotted curve) and lattice damage (solid curve) as a function of depth of a He+ ion-implanted KNbO3 waveguide formed with an ion dose of 1.5×1015 cm-2 and an ion energy of 2 MeV. The curves indicate the field distributions of the fundamental TE mode and TM mode in arbitrary units. Note that the maximum relative-index change and the maximum lattice damage are 9% and 100%, respectively.

Fig. 2
Fig. 2

Absorption profile as a function of depth for wavelengths of 0.43 and 0.86 μm and polarization along the b axis. Waveguide fabrication parameters D=1.5×1015 cm-2, E=2 MeV.

Fig. 3
Fig. 3

Attenuation as a function of waveguide thickness of planar waveguides fabricated with an ion dose of 1.5×1015 cm-2. (a) Attenuation at 0.488 μm. Solid curve, total loss calculated with Eq. (13); dashed curve, irradiation-induced absorption loss; (—), dashed–dotted curve, material absorption. (b) Attenuation at 1.064 μm. Curve, tunneling loss; circles, values measured in waveguides I–IV.

Fig. 4
Fig. 4

Attenuation as a function of ion dose at 0.454 and 0.870 μm of planar waveguides of 4.5 μm thickness. Solid curve, total loss at both wavelengths calculated with Eq. (13); dashed curve, absorption loss at 0.454 μm; dotted curve, tunneling loss at 0.870 μm; circles, measured attenuation of waveguides II, V, and VI.

Fig. 5
Fig. 5

Attenuation as a function of wavelength of a planar waveguide of 4.5 μm thickness formed with an ion dose of 1.5×1015 cm-2. Solid curve, calculated total loss given by Eq. (13); dashed curve, irradiation-induced absorption; dashed curve, tunneling; dotted–dashed curve, material absorption; diamonds, attenuation measured in waveguide II.

Fig. 6
Fig. 6

Attenuation as a function of waveguide width of the trapezoidal-type channel waveguides VII and VIII. The waveguide cross section is schematically shown in the inset. The thickness d is 4.5 μm. (a) Attenuation at λ=0.515 μm measured in waveguides VIIA–VIIG (D=0.75×1015 cm-2). The curve indicates the fit given by Eq. (14). (b) Attenuation at λ=0.870 μm measured in waveguides VIIIA–VIIG (D=1.5 1015 cm-2). The curve indicates the fit with Eq. (14).

Fig. 7
Fig. 7

Wavelength dependence of the optical loss of channel waveguides VIIG, VIIIE, and IX, with d=4.5 μm and w=16 μm for all three waveguides. The curves represent the calculation based on Eq. (15).

Fig. 8
Fig. 8

Wavelength of minimum loss as a function of ion dose calculated from Eq. (15). The data points represent the values measured in channel waveguides VIIG, VIIIE, and IX. The inset shows the attenuation measured at the wavelength of minimum loss as a function of ion dose for the same waveguides.

Fig. 9
Fig. 9

Second-harmonic power as a function of ion dose for channel waveguides VIIG, VIIIE, and IX. The curve indicates the calculation based on Eqs. (16) and (17) with the attenuation given by Eq. (15). The diamonds represent measured values. Measurement and calculation were performed for a fundamental input power of 240 mW.

Fig. 10
Fig. 10

Second-harmonic power as a function of channel width for waveguides VIIIA–VIIIG. The curve indicates the calculation based on Eqs. (16) and (17) with the width dependence of the attenuation shown in Fig. 6. The diamonds represent measured values. Fundamental input power, 240 mW.

Fig. 11
Fig. 11

(a) Peak value σmax of the absorption profile as a function of wavelength for a dose of 1.5×1015 cm-2 as determined from the transmission measurements with samples AI and AII. (b) Peak value of the absorption profile as a function of ion dose for a wavelength of 1.0 μm. The symbols represent data measured with sample AI. The inset shows the configuration for the transmission measurement.

Tables (2)

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Table 1 Loss Mechanisms in Ion-implanted KNbO3 Waveguides and Their Trend with Respect to Wavelength and Guide Fabrication Parameters

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Table 2 He+ Ion-Implanted Waveguides Used

Equations (23)

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αmat=αmat,0(λ/λmat,0)4-1
αsc,sf=4π2m1.5nwg σ2d3,
2Φx2+[k2n¯2(x)-k2N¯2]Φ=0,
n¯(x)=Re{n¯(x)}+i Im{n¯(x)}n(x)+in(x),
N¯=Re{N¯}+i Im{N¯}N+iN
Δn(x)=Δnn(x)+Δnel(x)=Δnn,max1-exp-Gn(x)Gn,0γn+Δnel,0Gel(x)Gel,0γel,
d=0.87+1.08E+0.36E2,
s(x)=1-1+Gn(x)G0,RBSexp-Gn(x)G0,RBS,
n(x)=λ4π σ(x).
σ(x)=σmax s(x)smax,
αabs=αabs,0λλ0nDD0mdd0r,
αtun=αtun,0λλ0νDD0μdd0ρ,
α=αmat+αabs,0λλ0nDD0mdd0r+αtun,0λλ0νDD0μdd0ρ.
α(w)=α0+α1ww0k
α=αmat+αabs,0λλ0nDD0m+αtun,0λλ0νDD0μ,
P2ω(L)=2ω2c30 d2N3 L2Pω2(0)Γh(αω, α2ω, L),
h(αω, α2ω, L)=exp-α2ω2 L-exp(-αωL)2α2ω2-αωL2,
T=exp-0Lxσ(x)dx,
σ(λ, D, x)=a(λ)s(D, x).
σmax=asmax.
ln T=-σmax(λ, D)0Lx s(D, x)smax(D) dx.
σmax(λ, D)=σmax,0λλ0mDD0n,
σ(λ, D, x)=σmax,0λλ0mDD0n s(D, x)smax(D).

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