Abstract

We have designed a tunable, oblique-incidence resonant grating filter that covers the C band as an add-drop device for incident TE-polarized light. We tune the filter by tilting a microelectromechanical systems platform onto which the filter is attached. The fabrication tolerances as well as the role of finite incident-beam size and limited device size were addressed. The maximum achievable efficiency of a finite-area device as well as a scaling law that relates the resonance peak width and the minimum device size is derived. In good agreement with simulations, measurements indicate a negligible change in shape of the resonance peak from 1526 nm at a 45° angle of incidence to 1573 nm at a 53° angle with a full width at half-maximum of 0.4 nm. In this range the shift of the peak wavelength is linear with respect to changes in the angle of incidence.

© 2004 Optical Society of America

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  1. See, for example, J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Guided-mode resonant subwavelength gratings: effect of finite beams and finite size,” J. Opt. Soc. Am. A 18, 1912–1928 (2001), and references therein.
  2. D. Lacour, G. Granet, J. P. Pluemey, A. Mure-Ravaud, “Polarization independence of a one-dimensional grating in conical mounting,” J. Opt. Soc. Am. A 20, 1546–1552 (2003).
    [CrossRef]
  3. A. Mizutani, H. Kukuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
    [CrossRef]
  4. S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filter,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
    [CrossRef]
  5. T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
    [CrossRef]
  6. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 15–79.
  7. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 10, 1385–1392 (1982).
  8. D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
    [CrossRef]
  9. S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
    [CrossRef]
  10. Z. Hegedus, R. Netterfield, “Low sideband guided-mode resonant filter,” Appl. Opt. 39, 1469–1473 (2000).
    [CrossRef]
  11. D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
    [CrossRef]
  12. I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
    [CrossRef]
  13. L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
    [CrossRef]
  14. J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
    [CrossRef]
  15. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods grating,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  16. E. Popov, B. Bozhkov, “Corrugated waveguides as resonance optical filters—advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001).
    [CrossRef]
  17. R. R. Boye, R. K. Kostuk, “Investigation of the effect of finite grating size on the performance of guided-mode resonance filters,” Appl. Opt. 39, 3649–3653 (2000).
    [CrossRef]
  18. D. K. Jacob, S. C. Dunn, M. G. Moharam, “Normally incident grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
    [CrossRef]
  19. D. K. Jacob, S. C. Dunn, M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
    [CrossRef]
  20. F. Lemarchand, S. Sentenac, H. Giovanni, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998).
    [CrossRef]
  21. A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
    [CrossRef]
  22. O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
    [CrossRef]
  23. W. W. Rigrod, D. Marcuse, “Radiation loss coefficient of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. Quantum Electron. QE-12, 673–685 (1976).
    [CrossRef]
  24. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 83–137.
  25. N. M. Lyndin, O. Parriaux, V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sensors Actuators B 41, 23–29 (1997).
    [CrossRef]
  26. M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

2003

D. Lacour, G. Granet, J. P. Pluemey, A. Mure-Ravaud, “Polarization independence of a one-dimensional grating in conical mounting,” J. Opt. Soc. Am. A 20, 1546–1552 (2003).
[CrossRef]

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

2001

2000

1998

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

F. Lemarchand, S. Sentenac, H. Giovanni, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998).
[CrossRef]

1997

1996

O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

1995

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

1989

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

1985

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

1982

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 10, 1385–1392 (1982).

1976

W. W. Rigrod, D. Marcuse, “Radiation loss coefficient of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. Quantum Electron. QE-12, 673–685 (1976).
[CrossRef]

Avrutsky, I. A.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

Bendickson, J. M.

Boye, R. R.

Bozhkov, B.

Brundrett, D. L.

Clerc, P.-A.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

David, C.

M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

de Rooij, N. F.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Dunn, S. C.

Erdogan, T.

Friesem, A. A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Gale, M. T.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

Gaylord, T. K.

Giovanni, H.

Glytsis, E. N.

Granet, G.

Hegedus, Z.

Herzig, H. P.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Hirayama, K.

Iwata, K.

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

A. Mizutani, H. Kukuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
[CrossRef]

Jacob, D. K.

Kikuta, H.

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 15–79.

Kostuk, R. K.

Kukuta, H.

Lacour, D.

Lemarchand, F.

Lyndin, N. M.

N. M. Lyndin, O. Parriaux, V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sensors Actuators B 41, 23–29 (1997).
[CrossRef]

Magnusson, R.

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filter,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

Maldonado, T. A.

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

Marcuse, D.

W. W. Rigrod, D. Marcuse, “Radiation loss coefficient of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. Quantum Electron. QE-12, 673–685 (1976).
[CrossRef]

Mashev, L.

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Mizutani, A.

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

A. Mizutani, H. Kukuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Mure-Ravaud, A.

Nakajima, K.

Netterfield, R.

Niederer, G.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Noell, W.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Noponen, E.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Norton, S. M.

Overstolz, T.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Parriaux, O.

N. M. Lyndin, O. Parriaux, V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sensors Actuators B 41, 23–29 (1997).
[CrossRef]

O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Pluemey, J. P.

Popov, E.

Rigrod, W. W.

W. W. Rigrod, D. Marcuse, “Radiation loss coefficient of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. Quantum Electron. QE-12, 673–685 (1976).
[CrossRef]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Saarinen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Schnieper, M.

M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

Sentenac, S.

Sharon, A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Shin, D.

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

Söchtig, J.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Sychugov, V. A.

N. M. Lyndin, O. Parriaux, V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sensors Actuators B 41, 23–29 (1997).
[CrossRef]

O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

Tamir, T.

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 83–137.

Thiele, H.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

Tibuleac, S.

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filter,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

Tishchenko, V.

O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Turunen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Zschokke, C.

M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

Appl. Opt.

IEEE J. Quantum Electron.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

W. W. Rigrod, D. Marcuse, “Radiation loss coefficient of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. Quantum Electron. QE-12, 673–685 (1976).
[CrossRef]

J. Mod. Opt.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

J. Opt. Soc. Am. A

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods grating,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[CrossRef]

E. Popov, B. Bozhkov, “Corrugated waveguides as resonance optical filters—advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001).
[CrossRef]

D. K. Jacob, S. C. Dunn, M. G. Moharam, “Normally incident grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
[CrossRef]

D. K. Jacob, S. C. Dunn, M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
[CrossRef]

S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
[CrossRef]

See, for example, J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Guided-mode resonant subwavelength gratings: effect of finite beams and finite size,” J. Opt. Soc. Am. A 18, 1912–1928 (2001), and references therein.

D. Lacour, G. Granet, J. P. Pluemey, A. Mure-Ravaud, “Polarization independence of a one-dimensional grating in conical mounting,” J. Opt. Soc. Am. A 20, 1546–1552 (2003).
[CrossRef]

A. Mizutani, H. Kukuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
[CrossRef]

S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filter,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 10, 1385–1392 (1982).

Opt. Commun.

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Opt. Eng.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

D. Shin, S. Tibuleac, T. A. Maldonado, R. Magnusson, “Thin-film optical filters with diffractive elements and waveguides,” Opt. Eng. 37, 2634–2646 (1998).
[CrossRef]

Opt. Lett.

Opt. Rev.

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonant grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

Pure Appl. Opt.

O. Parriaux, V. A. Sychugov, V. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Sensors Actuators B

N. M. Lyndin, O. Parriaux, V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sensors Actuators B 41, 23–29 (1997).
[CrossRef]

Other

M. Schnieper, M. T. Gale, C. Zschokke, C. David, “Application and fabrication of subwavelength gratings,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., technical digest, Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 228–230.

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 83–137.

T. Overstolz, P.-A. Clerc, M. T. Gale, H. P. Herzig, G. Niederer, W. Noell, J. Söchtig, H. Thiele, N. F. de Rooij, “Tilting out-of-plane platform for optical applications,” in International Conference on Optical MEMS, conference digest (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 81–82.
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1979), pp. 15–79.

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Figures (12)

Fig. 1
Fig. 1

RGF consisting of three layers. Thicknesses d are given. The assumed refractive indices are n TiO2 = 2.298, n SiO2 = 1.444, and n Glass = 1.51. The grating is rectangular, with a fill factor of 0.513.

Fig. 2
Fig. 2

λ-θ map based on the grating waveguide equations with M = -1, m = 0, h = 287 nm, n f = 2.298, n s = 1.51, and n c = 1.444. (a) c band; (b) whole range of AOI.

Fig. 3
Fig. 3

Numerical results: (a) Reflectivity λ-θ map for TE polarization, (b) reflection spectra for three AOIs (TE polarization), (c) reflection λ-θ map for TM polarization (different scale).

Fig. 4
Fig. 4

Maximum peak reflectivity at λ = 1.545 μm with a slope of 6 nm/°: (a) fixed linewidth Δλ = 0.5 nm and slope with variable incidence beam waist, (b) fixed beam waist w 0 = 0.5 mm and slope with variable linewidth of the filter.

Fig. 5
Fig. 5

Coupling model. A Gaussian beam with waist w 0 located at a distance l 0 from the grating hits the grating at -(L g z) of the grating, described by complex amplitude q(z) on the grating. The light couples in the backward direction of the waveguide and will be coupled out after some distance. Power efficiency η(z) in the waveguide is the amount of power at z divided by the total incident power between -L g and z.

Fig. 6
Fig. 6

Power efficiency η(z) (solid curve) as a function of z on a finite device for a given incident beam. Dashed curve, normalized intensity of the incident beam |q(z)|2. The simulation parameters are w 0 = 500 μm, Δz = 500 μm, λ = 1545 nm, α = 1.70 mm-1, and θ0 = 45°. Through the truncation of the incident beam we lose 8% of the energy. At z = 0 μm, after 2.5 mm we still have 4.4% of the light in the waveguide.

Fig. 7
Fig. 7

(a) Maximum obtainable efficiency and (b) its corresponding width w (solid curve) and position Δz (dashed curve) as functions of grating length with coupling coefficient α = 1.70 mm-1.

Fig. 8
Fig. 8

Scanning-electron microscopy picture of the fabricated grating.

Fig. 9
Fig. 9

Atomic-force microscopy picture and profile of the grating structure, revealing an average grating depth of 35 ± 2 nm.

Fig. 10
Fig. 10

Measured transmissivity of the device with the shallow (depth, 12 nm) grating measured with a big (2w = 8-mm, solid curve) collimator at an angle of incidence of 49° and a small (2w = 1-mm, dashed curve) collimator at an angle of incidence of 49.3°. The values are normalized to the offset of the Lorentzian curve fit. It is obvious that for the big collimator the hole is deeper and narrower than for the smaller collimator, which, owing to the measurement setup, shows undesirable Fabry-Perot interference fringes.

Fig. 11
Fig. 11

Measured (a) transmission and (b) reflection spectra of the device with 35-nm grating depth. The transmission is normalized to the offset of the Lorentzian curve fit. The fit results are presented in Table 4. The collimator used for measurements is specified to have a beam diameter of 8 mm, and the sample size was 1 in. × 1 in.

Fig. 12
Fig. 12

Comparison of measurements and simulation. The measured values (solid curve) from Fig. 11 for a 45° AOI are compared with the simulated values (dashed curve), with n TiO2 changed to 2.259.

Tables (4)

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Table 1 Tolerancing of Various Design Variablesa

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Table 2 Peak Reflection Efficiency of an Infinite Device as a Function of Absorption Loss Coefficient α of the Waveguide Material

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Table 3 Simulations for a Device with Coupling Coefficient α = 1.70 mm-1 a

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Table 4 Results of a Lorentzian Curve Fit of the Measured Reflection and Transmission Spectra at Several Angles of Incidencea

Equations (20)

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h wg k 0 n wg   cos θ m - ϕ c - ϕ s = m π ,
n e = n f   sin θ m ,
for   TE   h wg λ 1 2 π n f 2 - n s 2 arctan n s 2 - n c 2 n f 2 - n s 2 + 1 2 n f 2 - n s 2 ,
for   TM   h wg λ 1 2 π n f 2 - n s 2 arctan n f 4 n c 4 n s 2 - n c 2 n f 2 - n s 2 + 1 2 n f 2 - n s 2 .
Λ λ 0 n wg   sin θ m - n in   sin θ in = M ,
Λ λ 0 n in   sin θ in + max n in ,   n out .
G p x = FT g x = π   wA 0   exp - π p x w 2 ,
G θ = π   wA 0   exp - π   θ w λ 2 = B 0   exp - π   θ w λ 2 , G norm θ = 2 π w λ exp - π   θ w λ 2 ,
| L θ | = Δ θ / 2 2 θ - θ offset 2 + Δ θ / 2 2 .
η B =   | G θ L θ | 2 d θ   | G θ | 2 d θ =   G norm θ 2 | L θ | 2 d θ .
α e = k 0 n wg 2 sin θ m 2 - n c 2 1 / 2 .
α 0 = k 0 h g / 2 2 2 n wg 2 - n e 2 n e h wg n wg 2 - n s 2 N c N wg 2 + N s N c 2 + n wg 2 - n c 2 cos 2 N wg k 0 h wg N wg 2 + N s N c 2 - n wg 2 - n c 2 n wg 2 - n s 2 cos 2 N wg k 0 h wg ,
N i = n i 2 - n e - λ 0 / Λ 2 1 / 2 ,   i = s ,   c ,   wg ,
α = α 0 f c   exp - 2 α e d sep .
q z = 1 / π w 0 1 + i δ 1 + δ 2 exp - i δ 1 + δ 2 z + Δ z w 0 / cos θ 2   × exp - 1 1 + δ 2 z + Δ z w 0 / cos θ 2 ,
δ = 2 l 0 + z + Δ z sin θ kw 0 2 .
η z = 2 α   cos θ 0 exp - α z - L g z   exp α + i Δ β z q z d z 2 ,
Δ β = k sin   θ 0 - sin   θ .
l B = 1 - - L g 0   q z q * z d z .
η tot = η B 1 - l c - l B .

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