Abstract

We present a method for recovering the refractive-index profile of an anisotropic graded-index waveguide from the effective indices by using a cubic spline interpolation function. The first and second derivatives of cubic splines are continuous to ensure a smooth index profile, which is consistent with practical graded-index distributions. A straightforward iteration with an exact dispersion equation to verify the interpolated profile makes this method easy and reliable for application. This approach is proved by numerical analysis of several typical index distributions and experimental examples showing that the refractive-index profiles in anisotropy can be reconstructed close to the exact profile. Waveguides with both more modes (more than four guiding modes) and fewer modes (two to four) can be universally profiled with good accuracy.

© 2005 Optical Society of America

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  1. E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).
  2. M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
    [CrossRef]
  3. D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).
  4. V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).
  5. W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
    [CrossRef]
  6. K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
    [CrossRef]
  7. P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
    [CrossRef]
  8. J. M. White, P. F. Heidrich, “Optical waveguide refractive profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
    [CrossRef]
  9. K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. Lt-3, 385–391 (1985).
  10. M. G. Shi, L. Z. Gu, “Interpolation and fitting,” in Scientific and Engineering Calculation, (Academic, 1999), pp. 113–136.
  11. W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
    [CrossRef]
  12. P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  13. H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
    [CrossRef]
  14. M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
    [CrossRef]
  15. S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
    [CrossRef]

2004 (1)

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

2003 (1)

H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

2002 (2)

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

2000 (2)

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

1998 (1)

S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
[CrossRef]

1997 (1)

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

1996 (1)

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

1992 (1)

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

1991 (1)

E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).

1985 (1)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. Lt-3, 385–391 (1985).

1976 (1)

J. M. White, P. F. Heidrich, “Optical waveguide refractive profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
[CrossRef]

1970 (1)

P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

Armenise, M. N.

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

Baldi, P.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Bindner, P.

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

Boudrioua, A.

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

Chao, S.

S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
[CrossRef]

Chen, H. Y.

S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
[CrossRef]

Chen, X. F.

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Chen, Y. C.

S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
[CrossRef]

Chen, Y. H.

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Chen, Y. L.

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Chen, Y. P.

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. Lt-3, 385–391 (1985).

Chou, M. H.

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Chung, P. S.

E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).

De Micheli, M. P.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Digonnet, M. J.F.

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

El Hadi, K.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Fedorov, V. A.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Feigelson, R. S.

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

Fejer, M. M.

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

Gu, L. Z.

M. G. Shi, L. Z. Gu, “Interpolation and fitting,” in Scientific and Engineering Calculation, (Academic, 1999), pp. 113–136.

Heidrich, P. F.

J. M. White, P. F. Heidrich, “Optical waveguide refractive profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
[CrossRef]

Hoekstra, H. J.W.M.

H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Huang, Y. C.

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Kondrat’ev, A. V.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Korkishko, Yu. N.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Kunevaa, M.

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

Liao, W. J.

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Loi, K. K.

E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).

Loulergue, J. C.

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

Marangoni, M.

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

Marshall, A. F.

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

Moretti, P.

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

Nedkovb, I.

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

Nesheva, D.

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

Osellame, R.

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

Ostrowsky, D. B.

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Pashtrapanskaa, M.

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

Passaro, V. M.N.

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

Pun, E. Y.B.

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).

Ramponi, R.

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

Russo, V.

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

Santhanaraghavan, P.

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Savatinova, I. T.

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

Shi, M. G.

M. G. Shi, L. Z. Gu, “Interpolation and fitting,” in Scientific and Engineering Calculation, (Academic, 1999), pp. 113–136.

Tien, P. K.

P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

Toncheva, S.

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

Tsou, D. H.

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Ulrich, R.

P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

Uranus, H. P.

H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Vangroesen, E.

H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

White, J. M.

J. M. White, P. F. Heidrich, “Optical waveguide refractive profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
[CrossRef]

Xia, Y. X.

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Young, W. M.

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

Appl. Opt. (1)

J. M. White, P. F. Heidrich, “Optical waveguide refractive profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
[CrossRef]

Electron. Lett. (1)

E. Y.B. Pun, K. K. Loi, P. S. Chung, “Index profile of proton-exchanged waveguides in lithium niobate using cinnamic acid,” Electron. Lett. 27, 1282–1283 (1991).

J. Appl. Phys. (1)

S. Chao, Y. C. Chen, H. Y. Chen, “Determination of ordinary refractive index profile for a planar waveguide by transmission spectrum analysis,” J. Appl. Phys. 83, 5650–5657 (1998).
[CrossRef]

J. Lightwave Technol. (4)

M. Marangoni, R. Ramponi, R. Osellame, V. Russo, “Accurate determination of the ordinary index profile of proton-exchanged waveguides,” J. Lightwave Technol. 18, 1250–1255 (2000).
[CrossRef]

V. M.N. Passaro, M. N. Armenise, D. Nesheva, I. T. Savatinova, E. Y.B. Pun, “LiNbO3 optical waveguides formed in a new proton source,” J. Lightwave Technol. 20, 71–77 (2002).

W. M. Young, M. M. Fejer, M. J.F. Digonnet, A. F. Marshall, R. S. Feigelson, “Fabrication, characterization and index profile modeling of high-damage resistance Zn-diffused waveguides in congruent and MgO:lithium niobate,” J. Lightwave Technol. 10, 1238–1246 (1992).
[CrossRef]

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. Lt-3, 385–391 (1985).

J. Opt. Soc. Am. (1)

P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

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

W. J. Liao, X. F. Chen, Y. P. Chen, Y. X. Xia, Y. L. Chen, “Explicit analysis for anisotropic planar waveguide by analytical transfer-matrix method,” J. Opt. Soc. Am. A 21, 2196–2204 (2004).
[CrossRef]

Mater. Chem. Phys. (1)

D. H. Tsou, M. H. Chou, P. Santhanaraghavan, Y. H. Chen, Y. C. Huang, “Structural and optical characterization for vapor-phase proton exchanged lithium niobite waveguides,” Mater. Chem. Phys. 78, 474–479 (2002).

Mater. Sci. Semicond. Process. (1)

M. Kunevaa, S. Toncheva, M. Pashtrapanskaa, I. Nedkovb, “Proton exchange in Y-cut LiNbO3,” Mater. Sci. Semicond. Process. 3, 581–583 (2000).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (1)

P. Bindner, A. Boudrioua, J. C. Loulergue, P. Moretti, “Refractive index and anisotropy measurements in He+ implanted KTiOPO4 (KTP) optical waveguides,” Nucl. Instrum. Methods Phys. Res. B 120, 88–92 (1996).
[CrossRef]

Opt. Commun. (1)

K. El Hadi, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, Yu. N. Korkishko, V. A. Fedorov, A. V. Kondrat’ev, “Ordinary and extraordinary waveguides realized by reverse proton exchange on LiTaO3,” Opt. Commun. 140, 23–26 (1997).
[CrossRef]

Opt. Quantum Electron. (1)

H. P. Uranus, H. J.W.M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Other (1)

M. G. Shi, L. Z. Gu, “Interpolation and fitting,” in Scientific and Engineering Calculation, (Academic, 1999), pp. 113–136.

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Figures (9)

Fig. 1
Fig. 1

Graded-index anisotropic planar waveguide with arbitrary index profile.

Fig. 2
Fig. 2

Gaussian distribution and recovered index profiles of Ti-diffused Z-cut lithium niobate waveguides. (a) D = 4.5 μ m , supporting four TM and three TE modes; (b) D = 3 μ m , supporting two TM and two TE modes.

Fig. 3
Fig. 3

Gaussian distribution and recovered index profiles of APE lithium niobate waveguides. (a) Z-cut, four TM modes; (b) X-cut, four TE modes; (c) Z-cut, six TM modes.

Fig. 4
Fig. 4

Error function distribution and recovered index profiles of APE lithium niobate waveguides. (a) Δ n = 0.04 , supporting five TM modes; (b) Δ n = 0.01 , supporting three TM modes.

Fig. 5
Fig. 5

Exponential distribution and recovered index profiles of a hypothetical Z-cut lithium niobate waveguide, supporting four TE and four TM modes.

Fig. 6
Fig. 6

Recovered error function profiles with errors of 1 × 10 4 and 5 × 10 4 added to even-order mode indices.

Fig. 7
Fig. 7

Comparison of retrieved ordinary index profiles of a Z-cut lithium niobate waveguide diffused with 100-nm ZnO film at 1000°C for 45 min. The data are from Ref. [5].

Fig. 8
Fig. 8

Comparison of the recovery of the extraordinary index profile in an APE Y-cut lithium niobate waveguide obtained after proton exchange in benzoic acid at 230°C for 3 min and subsequent annealing at 350°C for 1 h in air.

Fig. 9
Fig. 9

Comparison of the recovery of the extraordinary index profile in an APE Z-cut lithium niobate waveguide proton exchanged in benzoic acid at 160°C for 10 h and subsequently annealed at 330°C for 24 h in air.

Equations (23)

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ε = ε 0 ( n x 2 0 0 0 n y 2 0 0 0 n z 2 ) ,
n x ( x ) = { n s x + Δ n x f 1 ( x D 1 ) ( x 0 ) n c x ( x < 0 ) } ,
n y ( x ) = { n s y + Δ n y f 2 ( x D 2 ) ( x 0 ) n c y ( x < 0 ) } ,
n z ( x ) = { n s z + Δ n z f 3 ( x D 3 ) ( x 0 ) n c z ( x < 0 ) } ,
0 x t κ ( x ) d x + 0 x t p ( x ) f ( x ) p 2 ( x ) + κ 2 ( x ) d [ κ ( x ) f ( x ) ] d x d x = N π + tan 1 [ f 0 f c p c κ 0 ] + π 2 ( N = 0 , 1 , 2 , ) .
κ ( x ) = { κ TM ( x ) = n z ( x ) n x ( x ) [ k 0 2 n x 2 ( x ) β 2 ] 1 2 TM mode κ TE ( x ) = [ k 0 2 n y 2 ( x ) β 2 ] 1 2 TE mode } .
d [ p ( x ) f ( x ) ] d x = κ 2 ( x ) + p 2 ( x ) f 2 ( x ) , p ( x t ) = p t .
f ( x ) = { n z 2 ( x ) TM mode 1 TE mode } .
p c = { p c TM = n c z n c x ( β 2 k 0 2 n c x 2 ) 1 2 TM mode p c TE = ( β 2 k 0 2 n c y 2 ) 1 2 TE mode } ,
f c = { n c z 2 TM mode 1 TE mode } .
n ( x ) = M 0 ( x 2 x ) 3 6 h 1 + M 2 x 3 6 h 1 + ( n 0 M 0 h 1 2 6 ) x 2 x h 1 + ( n 2 M 2 h 1 2 6 ) x h 1 , 0 x x 2 ,
n ( x ) = M i ( x i + 1 x ) 3 6 h i + M i + 1 ( x x i ) 3 6 h i + ( n i M i h i 2 6 ) x i + 1 x h i + ( n i + 1 M i + 1 h i 2 6 ) x x i h i ,
x i x x i + 1 , i = 2 , 3 , , k 1 ,
n ( x ) = n s + b exp ( a x ) , x x k ,
[ 2 1 μ 2 2 γ 2 μ k 1 2 γ k 1 1 2 ] ( M 0 M 2 M k 1 M k ) = ( d 0 d 2 d k 1 d k ) ,
μ i = h i 1 h i 1 + h i , γ i = 1 μ i ( i = 2 , 3 , , k 1 ) ,
d 0 = 6 h 1 [ n 2 n 0 h 1 n ( 0 ) ] ,
d 2 = 6 n 2 n 0 h 1 n 3 n 2 h 2 x 3 ,
d i = 6 n i n i 1 h i 1 n i + 1 n i h i x i 1 x i + 1 , i = 3 , , k 1 ,
d k = 6 h k 1 [ n ( x k ) n k n k 1 h k 1 ] ,
n 1 = M 0 ( x 2 x 1 ) 3 6 h 1 + M 2 x 1 3 6 h 1 + ( n 0 M 0 h 1 2 6 ) x 2 x 1 h 1 + ( n 2 M 2 h 1 2 6 ) x 1 h 1 .
n e ( x ) = 2.2 + Δ n 2 [ erf ( 0.9 + x 3.5 ) + erf ( 0.9 x 3.5 ) ] erf ( 0.9 3.5 ) ,
n o ( x ) = 2.286 0.25 × Δ n 2 [ erf ( 0.9 + x 3.5 ) + erf ( 0.9 x 3.5 ) ] erf ( 0.9 3.5 ) ,

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