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Design of polarization gratings for broadband illumination

Hanna Lajunen, Jari Turunen, and Jani Tervo

Open Access Open Access

Abstract

Design of broadband diffractive elements is studied. It is shown that dielectric polarization gratings can be made to perform the same optical function over a broad band of wavelengths. Any design of paraxial-domain diffractive elements can be realized as such broadband elements that may, e.g., give constant diffraction efficiencies over the wavelength band while the field propagation after the elements remains wavelength-dependent. Furthermore, elements producing symmetric signals are shown to work with arbitrarily polarized or partially polarized incident planar broadband fields. The performance of the elements is illustrated by numerical examples and some practical issues related to their fabrication are discussed.

©2005 Optical Society of America

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References

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  • Publication
  1. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, Berlin, 1997).
  2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997).
    [Crossref] [PubMed]
  3. G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express 5, 163–168 (1999).
    [Crossref] [PubMed]
  4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
    [Crossref]
  5. Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. 24, 1422–1424 (1999).
    [Crossref]
  6. I. R. Hooper and J. R. Sambles, “Broadband polarization-converting mirror for the visible region of the spectrum,” Opt. Lett. 27, 2152–2154 (2002).
    [Crossref]
  7. D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. 31, 6321–6326 (2003).
    [Crossref]
  8. D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, “Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,” Opt. Lett. 29, 754–756 (2004).
    [Crossref] [PubMed]
  9. C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004).
    [Crossref] [PubMed]
  10. F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
    [Crossref]
  11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
    [Crossref]
  12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
    [Crossref]
  13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
    [Crossref]
  14. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
  15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345.
    [Crossref] [PubMed]
  16. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587–589 (2001).
    [Crossref]
  17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001).
    [Crossref]
  18. F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [Crossref] [PubMed]
  19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004).
    [Crossref] [PubMed]
  20. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.
  21. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [Crossref]
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  23. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
    [Crossref]
  24. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001).
    [Crossref]
  25. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
    [Crossref]
  26. L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963).
    [Crossref]
  27. J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973).
    [Crossref]
  28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
    [Crossref]
  29. J. Turunen“Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.
  30. S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nmOpt. Lett. 23, 552–554 (1998).
    [Crossref]
  31. S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: application to titanium dioxide film,” Appl. Opt. 34, 7355–7360 (1995).
    [Crossref] [PubMed]
  32. H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749–763.

2004 (5)

2003 (3)

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
[Crossref]

D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. 31, 6321–6326 (2003).
[Crossref]

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

2002 (2)

2001 (4)

2000 (2)

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

1999 (3)

1998 (2)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nmOpt. Lett. 23, 552–554 (1998).
[Crossref]

1997 (1)

1995 (1)

1991 (1)

1973 (1)

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973).
[Crossref]

1963 (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963).
[Crossref]

Adachi, J.

Astilean, S.

Biener, G.

Bomzon, Z.

Borghi, R.

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001).
[Crossref]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Bovard, B. G.

Burke, K.

D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. 31, 6321–6326 (2003).
[Crossref]

Cambril, E.

Chavel, P.

Chiao, S.-C.

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Davis, J. A.

Deguzman, P. C.

Di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Fainman, Y.

Fernández-Pousa, C. R.

Friberg, A. T.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Gori, F.

F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Hane, K.

Hasman, E.

Honkanen, M.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

Hooper, I. R.

Iwata, K.

Jin, G.

Jin, G.-F.

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Kanamori, Y.

Kettunen, V.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

Kikuta, H.

Kim, D.

D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. 31, 6321–6326 (2003).
[Crossref]

Kim, H.-C.

Kleiner, V.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

Lajunen, H.

Lalanne, P.

Lalanne, Ph.

Launois, H.

Lee, M.-S. L.

Levy, U.

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

Liu, H.

Liu, H.-T.

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Lu, S.

-Lu, Si

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Macleod, H. A.

Mandel, L.

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

Mondello, A.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
[Crossref]

Moreno, I.

Nordin, G. P.

Ohira, Y.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[Crossref]

Philipp, H. R.

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749–763.

Piquero, G.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
[Crossref]

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001).
[Crossref]

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

Sambles, J. R.

Samson, J. C.

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973).
[Crossref]

Santarsiero, M.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
[Crossref]

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Sasaki, M.

Sauvan, C.

Setälä, T.

Tervo, J.

Tsai, C.-H.

Turunen, J.

H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004).
[Crossref] [PubMed]

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
[Crossref]

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

J. Turunen“Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

Wyrowski, F.

Yan, Y.

Yan, Y.-B.

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Yi, D.

Yi, D.-E.

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Appl. Opt. (3)

Geophys. J. R. Astr. Soc. (1)

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973).
[Crossref]

J. Mod. Opt. (1)

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

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

Opt. Commun. (3)

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998).
[Crossref]

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (12)

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587–589 (2001).
[Crossref]

C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001).
[Crossref]

F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[Crossref] [PubMed]

H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004).
[Crossref] [PubMed]

Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. 24, 1422–1424 (1999).
[Crossref]

I. R. Hooper and J. R. Sambles, “Broadband polarization-converting mirror for the visible region of the spectrum,” Opt. Lett. 27, 2152–2154 (2002).
[Crossref]

D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, “Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,” Opt. Lett. 29, 754–756 (2004).
[Crossref] [PubMed]

C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004).
[Crossref] [PubMed]

F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
[Crossref]

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nmOpt. Lett. 23, 552–554 (1998).
[Crossref]

Proc. Phys. Soc. (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963).
[Crossref]

Other (6)

J. Turunen“Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.

J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, Berlin, 1997).

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749–763.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

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Figures (4)
Fig. 1.
Fig. 1. Geometry of a form-birefringent, y-invariant, sub-wavelength-period grating with period d, modulation depth h, minimum transverse feature size g, and refractive indices n 0, n 1, n 2, and n 3.

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Fig. 2.
Fig. 2. Diffraction efficiencies of the 1→3 beam splitter based on a scalar design, realized as a polarization grating, η 0=η ±1 (solid line), and as a surface-relief phase grating, η 0 (dashed line) and η ±1 (dash-dotted line).

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Diffraction efficiency of a 1→2 beam splitter made of TiO2 using subwavelength grating with parameters d=220 nm, f=0.6, and h=750 nm (dashed line), h=800 nm (solid line), and h=850 nm (dotted line).

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Fig. 4.
Fig. 4. Diffraction efficiency of a 1→2 beam splitter made of TiO2 using subwavelength grating with parameters d=220 nm, h=800 nm, and f=0.55 (dashed line), f=0.60 (solid line), and f=0.65 (dotted line).

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Equations (38)

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

T ( x , y ) = [ t ∥ cos 2 θ ( x , y ) + t ⊥ sin 2 θ ( x , y ) ( t ∥ − t ⊥ ) sin θ ( x , y ) cos θ ( x , y ) ( t ∥ − t ⊥ ) sin θ ( x , y ) cos θ ( x , y ) t ∥ sin 2 θ ( x , y ) + t ⊥ cos 2 θ ( x , y ) ] ,
t ∥ = ∣ t ∥ ∣ exp ( i kn ∥ h ) ,
t ⊥ = ∣ t ⊥ ∣ exp ( i kn ⊥ h ) ,
ϕ ( λ ) = arg t ∥ − arg t ⊥ = kh ( n ∥ − n ⊥ )
J i = α J R + β J L
J R = 1 2 ( 1 i )
J L = 1 2 ( 1 − i ) .
α = ( E i x − i E i y ) ⁄ 2 ,
β = ( E i x + i E i y ) ⁄ 2 ,
E ( x , y ) = T ( x , y ) J i .
E ( x , y ) = α E R ( x , y ) + β E L ( x , y )
E R ( x , y ) = 1 2 ( t ∥ + t ⊥ ) J R + 1 2 ( t ∥ − t ⊥ ) exp [ i 2 θ ( x , y ) ] J L
E L ( x , y ) = 1 2 ( t ∥ + t ⊥ ) J L + 1 2 ( t ∥ − t ⊥ ) exp [ − i 2 θ ( x , y ) ] J R
t ∥ ± t ⊥ = exp ( i arg t ∥ ) [ 1 ± exp ( − i ϕ 0 ) ]
E ( x , y ) = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ J m , n exp [ i 2 π ( mx ⁄ d x + ny ⁄ d y ) ] ,
J m , n = 1 d x d y ∫ 0 d x ∫ 0 d y E ( x , y ) exp [ − i 2 π ( mx ⁄ d x + ny ⁄ d y ) ] d x d y
η m , n = ∥ J m , n ∥ 2
J m , n = T m , n ( 0 ) J i + α T m , n ( 1 ) J L + β T m , n ( − 1 ) J R
T m , n ( 0 ) = 1 2 ( t ∥ + t ⊥ ) δ 0 , 0 ,
T m , n ( ± 1 ) = 1 2 ( t ∥ − t ⊥ ) t m , n ( ± 1 ) ,
t m , n ( ± 1 ) = 1 d x d y ∫ 0 d x ∫ 0 d y exp [ ± i 2 θ ( x , y ) ] exp [ − i 2 π ( mx ⁄ d x + ny ⁄ d y ) ] d x d y .
η m , n = ∣ T m , n ( 0 ) ∣ 2 + ∣ α T m , n ( 1 ) ∣ 2 + ∣ β T m , n ( − 1 ) ∣ 2 + 2 Re [ α * T m , n ( 0 ) * β T m , n ( − 1 ) ] + 2 Re [ β * T m , n ( 0 ) * α T m , n ( 1 ) ]
η m , n = ∣ α T m , n ( 1 ) ∣ 2 + ∣ β T m , n ( − 1 ) ∣ 2
η m , n = ∣ T m , n ( 0 ) ∣ 2 + ∣ T m , n ( 1 ) ∣ 2 = 1 4 ∣ t ∥ + t ⊥ ∣ 2 δ 0 , 0 + 1 4 ∣ t ∥ − t ⊥ ∣ 2 ∣ t m , n ( 1 ) ∣ 2 .
θ ( x , y ) = θ ( x ) = π x ⁄ D ,
η 0 = 1 4 ∣ t ∥ + t ⊥ ∣ 2
η ± 1 = 1 8 ∣ t ∥ − t ⊥ ∣ 2 ( 1 ∓ ∣ E i x E i y ∣ sin ϑ ) ,
η m , n = ( ∣ α ∣ 2 + ∣ β ∣ 2 ) ∣ T m , n ( 1 ) ∣ 2 = ∣ T m , n ( 1 ) ∣ 2
η 0 , 0 = ∣ T 0 , 0 ( 0 ) ∣ 2 + ∣ T 0 , 0 ( 1 ) ∣ 2 + ( ∣ t ∥ ∣ 2 − ∣ t ⊥ ∣ 2 ) Re [ α β * t 0 , 0 ( 1 ) ]
J ( r , ω ) = [ J xx ( r , ω ) J xy ( r , ω ) J yx ( r , ω ) J yy ( r , ω ) ] = [ 〈 E x * ( r , ω ) E x ( r , ω ) 〉 〈 E x * ( r , ω ) E y ( r , ω ) 〉 〈 E y * ( r , ω ) E x ( r , ω ) 〉 〈 E y * ( r , ω ) E y ( r , ω ) 〉 ] ,
J t = T † J i T ,
J ( r , ω ) = ∑ n = 1 2 λ n ( ω ) ϕ n ( r , ω ) ϕ n † ( r , ω )
J t = T † ( λ 1 J 1 + λ 2 J 2 ) T = λ 1 T † J 1 T + λ 2 T † J 2 T ,
η m , n = λ 1 η 1 m , n + λ 2 η 2 m , n ,
η m , n = ( λ 1 + λ 2 ) η 1 m , n = η 1 m , n .
t ( x , λ ) = exp [ i Φ ( λ ) sin ( 2 π x ⁄ D ) ] ,
Φ ( λ ) = 2 π ( n 2 − n 1 ) h 0 ⁄ λ
η m ( λ ) = J m 2 [ Φ ( λ ) ⁄ 2 ] ,

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