Abstract

A new design for an electro-optically induced tilted phase grating inside a waveguide is proposed. The electric field and the refractive-index distribution induced inside a waveguide by voltage applied to two systems of interdigitated electrodes that are shifted with respect to each other are calculated rigorously on the basis of an original technique. The model accounts for the arbitrary electrode shift distance d (0d2l), where l is the electrode spatial period. It is shown that the proper choice of the shift can minimize the structure’s capacitance and consequently its time response. The refractive-index distributions are calculated for various schemes for application of electric potential and electrode position that demonstrate the possibility of switching the direction of the grating wave vector. It is shown how the concept can be use to build electro-optically controllable transmissive (long-period) and reflective (short-period) tilted gratings and couplers in both multilayered (transverse) and planar (lateral) configurations.

© 2001 Optical Society of America

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References

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  1. T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).
  2. D. Sun, C. Zhao, and R. T. Chen, “Intraplane to interplane optical interconnects with a high diffraction efficiency electro-optic grating,” Appl. Opt. 36, 629–634 (1997).
    [CrossRef] [PubMed]
  3. D. Brooks and S. Ruschin, “Integrated electrooptic multielectrode tunable filter,” J. Lightwave Technol. 13, 1508–1513 (1995).
    [CrossRef]
  4. H.-P. Nolting and M. Gravert, “SYNGRAT, an electrooptically controlled tunable filter with a synthesized grating structure,” Opt. Quantum Electron. 27, 887–896 (1995).
    [CrossRef]
  5. E. N. Glytsis and T. Gaylord, “Anisotropic guided-wave diffraction by interdigitated electrode-induced phase gratings,” Appl. Opt. 27, 5031–5050 (1988).
    [CrossRef] [PubMed]
  6. M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. 38, 7356–7363 (1999).
    [CrossRef]
  7. Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
    [CrossRef]
  8. H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
    [CrossRef]
  9. M. Kulishov, “Tunalbe electro-optic microlens array. I. Planar geometry,” Appl. Opt. 39, 2332–2339 (2000).
    [CrossRef]
  10. M. A. Title and S. H. Lee, “Modeling and characterization of embedded electrode performance in transverse electro-optic modulators,” Appl. Opt. 29, 85–98 (1990).
    [CrossRef] [PubMed]
  11. A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
    [CrossRef]
  12. R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
    [CrossRef]
  13. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  14. K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber grating,” Appl. Opt. 39, 1394–1404 (2000).
    [CrossRef]
  15. T. Erdogan, “Cladding-mode resonances in short- and long period fibergrating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  16. R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).
  17. S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
    [CrossRef]
  18. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  19. T. W. Ang and G. T. Reed, “Fabrication of silicon-blazed gratings for couplers,” Fiber Integr. Opt. 19, 25–29 (2000).
    [CrossRef]
  20. J. C. Brazas, L. Li, and A. L. McKeon, “High-efficiency input coupling into optical waveguides using gratings with double-surface corrugations,” Appl. Opt. 34, 604–610 (1995).
    [CrossRef] [PubMed]
  21. I. A. Avrutsky, A. S. Svakhin, and V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1990).
    [CrossRef] [PubMed]
  22. S. Ura, H. Moriguchi, S. Kido, T. Suhara, and H. Nishihara, “Switching of output coupling in grating coupler by diffraction transition to the distributed Bragg reflector regime,” Appl. Opt. 38, 2500–2503 (1999).
    [CrossRef]

2000 (3)

1999 (7)

S. Ura, H. Moriguchi, S. Kido, T. Suhara, and H. Nishihara, “Switching of output coupling in grating coupler by diffraction transition to the distributed Bragg reflector regime,” Appl. Opt. 38, 2500–2503 (1999).
[CrossRef]

M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. 38, 7356–7363 (1999).
[CrossRef]

Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
[CrossRef]

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

1998 (1)

T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).

1997 (3)

1996 (1)

1995 (3)

J. C. Brazas, L. Li, and A. L. McKeon, “High-efficiency input coupling into optical waveguides using gratings with double-surface corrugations,” Appl. Opt. 34, 604–610 (1995).
[CrossRef] [PubMed]

D. Brooks and S. Ruschin, “Integrated electrooptic multielectrode tunable filter,” J. Lightwave Technol. 13, 1508–1513 (1995).
[CrossRef]

H.-P. Nolting and M. Gravert, “SYNGRAT, an electrooptically controlled tunable filter with a synthesized grating structure,” Opt. Quantum Electron. 27, 887–896 (1995).
[CrossRef]

1990 (2)

1988 (1)

Ang, T. W.

T. W. Ang and G. T. Reed, “Fabrication of silicon-blazed gratings for couplers,” Fiber Integr. Opt. 19, 25–29 (2000).
[CrossRef]

Avrutsky, I. A.

Brazas, J. C.

Brooks, D.

D. Brooks and S. Ruschin, “Integrated electrooptic multielectrode tunable filter,” J. Lightwave Technol. 13, 1508–1513 (1995).
[CrossRef]

Chen, A.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Chen, J.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Chen, R. T.

Chou, S. Y.

Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
[CrossRef]

Chuyanov, V.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Dalton, L. R.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

David, C.

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

de Ridder, R. M.

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Diemeer, M. B. J.

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Driessen, A.

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Erdogan, T.

Garner, S.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Gaylord, T.

Glytsis, E. N.

Gobrecht, J.

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

Gravert, M.

H.-P. Nolting and M. Gravert, “SYNGRAT, an electrooptically controlled tunable filter with a synthesized grating structure,” Opt. Quantum Electron. 27, 887–896 (1995).
[CrossRef]

Harper, A. W.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

He, M.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Jaszewski, R. W.

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

Kido, S.

Kim, B. Y.

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

Kim, H. K.

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

Kulishov, M.

Lambeck, P. V.

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Lee, B. W.

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

Lee, K. S.

Lee, S. H.

Lee, S.-S.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Li, L.

Mao, S. S. H.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

McKeon, A. L.

Moriguchi, H.

Nishihara, H.

Nolting, H.-P.

H.-P. Nolting and M. Gravert, “SYNGRAT, an electrooptically controlled tunable filter with a synthesized grating structure,” Opt. Quantum Electron. 27, 887–896 (1995).
[CrossRef]

Ra, Y.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Reed, G. T.

T. W. Ang and G. T. Reed, “Fabrication of silicon-blazed gratings for couplers,” Fiber Integr. Opt. 19, 25–29 (2000).
[CrossRef]

Rikkers, E.

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Ruschin, S.

D. Brooks and S. Ruschin, “Integrated electrooptic multielectrode tunable filter,” J. Lightwave Technol. 13, 1508–1513 (1995).
[CrossRef]

Schablitsky, S. J.

Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
[CrossRef]

Schift, H.

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

Sipe, J. E.

Steier, W. H.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Suhara, T.

Sun, D.

Svakhin, A. S.

Sychugov, V. A.

Title, M. A.

Ura, S.

Wang, F.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Yu, Z.

Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
[CrossRef]

Yun, S. H.

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

Zhang, H.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Zhao, C.

Zhu, J.

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

Z. Yu, S. J. Schablitsky, and S. Y. Chou, “Nanoscale GaAs metal–semiconductor–metal photodetectors fabricated using nanoimprint lithography,” Appl. Phys. Lett. 74, 2381–2383 (1999).
[CrossRef]

Fiber Integr. Opt. (1)

T. W. Ang and G. T. Reed, “Fabrication of silicon-blazed gratings for couplers,” Fiber Integr. Opt. 19, 25–29 (2000).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, “Dynamic erbium-doped fiber amplifier based on active gain flattening with fiber acoustooptic tunable filters,” IEEE Photonics Technol. Lett. 11, 1229–1231 (1999).
[CrossRef]

J. Lightwave Technol. (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

D. Brooks and S. Ruschin, “Integrated electrooptic multielectrode tunable filter,” J. Lightwave Technol. 13, 1508–1513 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

Microelectron. Eng. (1)

H. Schift, R. W. Jaszewski, C. David, and J. Gobrecht, “Nanostructuring of polymers and fabrication of interdigitated electrodes by hot embossing lithography,” Microelectron. Eng. 46, 121–124 (1999).
[CrossRef]

Opt. Eng. (1)

A. Chen, V. Chuyanov, H. Zhang, S. Garner, S.-S. Lee, W. H. Steier, J. Chen, F. Wang, J. Zhu, M. He, Y. Ra, S. S. H. Mao, A. W. Harper, and L. R. Dalton, “dc biased electro-optic polymer waveguide modulators with low half-wave voltage and high thermal stability,” Opt. Eng. 38, 2000–2008 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Mater. (1)

R. M. de Ridder, A. Driessen, E. Rikkers, P. V. Lambeck, and M. B. J. Diemeer, “Design and fabrication of electro-optic polymer modulators and switches, Opt. Mater. 12, 205–214 (1999).
[CrossRef]

Opt. Quantum Electron. (1)

H.-P. Nolting and M. Gravert, “SYNGRAT, an electrooptically controlled tunable filter with a synthesized grating structure,” Opt. Quantum Electron. 27, 887–896 (1995).
[CrossRef]

Photonics Spectra (1)

T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).

Other (1)

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

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Figures (10)

Fig. 1
Fig. 1

Cross-sectional view of the waveguide electro-optic grating.

Fig. 2
Fig. 2

Normalized electric potential distribution inside the electro-optic slab for a/l=0.6,2h=34l,d=34l, and a LiNbO3 wafer (11=85,33=29),=1.

Fig. 3
Fig. 3

Normalized surface charge distribution for 2h=34l,d=34l, and LiNbO3 wafer (11=85,33=29).=1 and a/l=0.5 (solid curve) and a/l=0.25 (dashed curve).

Fig. 4
Fig. 4

Electrode PC as a function of the relative shift between the top and the bottom electrodes for several values of the electrode duty ratio: (a/l=0.35, dotted curves; a/l=0.5, solid curves; a/l=0.6, dashed curves) and for several values of the electrode-distance-to-electrode-period ratio: (2h/l=0.5, crosses; 2h/l=0.75, diamonds; and 2h/l=1, squares). The calculation was made for LiNbO3 wafer (11=85,33=29),=1.

Fig. 5
Fig. 5

Contour plots of the refractive-index distribution inside the waveguide for (a) d=1.7l and (b) d=0.7l. The calculation was made for LiNbO3(11=85,33=29),=1,2h=0.5l,V0=3V,h=1 µm; ne=2.3,r33=30×10-12 m/V, and a/l=0.5. The induced refractive index varies as 2.29805n2.30195.

Fig. 6
Fig. 6

Contour plots of the refractive-index distribution for (a) d=1.5l, (b) d=0.5l, and (c) when all top electrodes have +V0 potential and all bottom electrodes have -V0 potential. The calculation was made for LiNbO3 wafer (11=85,33=29),=1,2h=0.5l,V0=3V,h=1 µm; ne=2.3,r33=30×10-12 m/V, and a/l=0.5. The induced refractive index varies as (a), (b) 2.29801n2.30199; (c) 2.29977n2.30192.

Fig. 7
Fig. 7

(a) The first and (b) the second harmonic of the normal component of the electric field in the middle of the core (z=0) as a function of the relative shift between the top and the bottom electrodes for several values of the electrode-distance-to-electrode-period ratio: (2h/l=0.4, dotted curves; 2h/l=0.5, solid curves; 2h/l=0.75, dashed curves; and 2h/l=1 dotted–dashed curves). The calculation was made for LiNbO3 wafer (11=85,33=29),=1,a/l=0.5.

Fig. 8
Fig. 8

First harmonic of the normal component of the electric field in the middle of the waveguide (z=0) as a function of electrode duty ratios: (2h/l=0.5, squares; 2h/l=0.75, diamonds; 2h/l=1, crosses). The calculation was made for LiNbO3 wafer (11=85,33=29),=1,d/l=1.5.

Fig. 9
Fig. 9

Planar geometry of the tilted electro-optic grating: (a) top view, (b) cross section.

Fig. 10
Fig. 10

Basic structure of our waveguide output coupler with an EO induced tilted grating: ITO, indium tin oxide.

Equations (31)

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

φ(1)(x,z)=V0n=0An exp[-(n+1/2)kz]×cos[(n+1/2)x¯],
φ(2)(x,z)=V0n=0En sinh[(n+1/2)kδ(z+h)]×cos[(n+1/2)x¯]+Dn sinh[(n+1/2)kδ(h-z)]×cos[(n+1/2)(x¯-kd)],
n=0En*Hn cos[(n+1/2)x¯)=1,0x¯a¯,
n=0(n+1/2)En*{cos[(n+1/2)x¯]-Gn
×cos[(n+1/2)(x¯-kd)]}=0,a¯x¯π,
Gn=cosh[2khδ(n+1/2)]+33δ sinh[2khδ(n+1/2)]-1,
Hn=sinh[2khδ(n+1/2)]Gn,En*=En/Gn.
n=0En*(1+Fn)cos[(n+1/2)x¯)
=33δ33δ+,0x¯a¯,
n=0(n+1/2)En*{cos[(n+1/2)x¯]
-Gn cos[(n+1/2)(x¯-kd)]}=0,
a¯x¯π,
n=0 En*{sin[(n+1/2)x¯]-Gn sin[(n+1/2)
×(x¯-kd)]}=C,a¯x¯π,
C=n=0(-1)nEn*{1+Gn cos[(n+1/2)kd]}.
2πKsin ξ2=2π0ξdx¯(cos x¯-cos ξ)1/2,
2πKcos ξ2=2πξπdx¯(cos ξ-cos x¯)1/2,
n=0 En*Pn(cos ξ)=2π 33δ33δ+Ksin ξ2-n=0 En*FnPn(cos ξ),
0ξa¯;
n=0 En*Pn(cos ξ)=2πCKcos ξ2-n=0 En*GnQn(ξ),
a¯<ξπ,
Qn(ξ)=2πξπ sin[(n+1/2)(x¯-kd)]dx¯(cos ξ-cos x¯)1/2.
0π Pn(cos ξ)Pk(cos ξ)sin ξdξ=0nk1k+1/2n=k,
m=0amkFm+βmkGm+δmkm+1/2-2π(-1)kNm×(1+Gm)cos[(m+1/2)kd]Em*
=2π +33δ33δMk,
αmk=0a¯ Pm(cos ξ)Pk(cos ξ)sin ξdξ,
βmk=a¯π Pm(cos ξ)Qk(ξ)sin ξdξ,
Mk=0a¯ Ksin ξ2Pk(cos ξ)sin ξdξ,
Nk=a¯πKcos ξ2Pk(cos ξ)sin ξdξ.
Q=V0(1133)1/2h n=0 En* sin[(n+1/2)ka/2]×{1-Gn cos[(n+1/2)kd]}.
ϑ=arctanl-d2h0dl-arctanl+d2hl<d2l.

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