Abstract

A method is developed to enhance the amplitudes of the non-propagating evanescent orders of resonant dielectric gratings. The origin of these resonances is analyzed in detail. The method relies on interactions between stacked gratings with different periods, and so a formalism is developed to model such stacks mathematically. In addition, a theoretical approach is developed to design gratings that enhance or blaze desired orders. These orders, controlled independently by incident fields from different angles, interfere and are optimized to produce steerable sub-Rayleigh field concentrations on a surface. These spots may function as a virtual scanning probe for non-invasive sub-Rayleigh microscopy. Optimization is conducted using a Monte Carlo Markov chain, and spots are generated which are both 1 order of magnitude narrower than the free space Rayleigh limit and robust to noise in the incident fields.

© 2010 Optical Society of America

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  8. A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
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  9. G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  20. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
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  23. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” J. Mod. Opt. 33, 607–619 (1986).
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  24. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–418 (1873).
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  25. R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
    [CrossRef]
  26. A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” Opt. Acta 34, 511–538 (1988).
  27. J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
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  28. P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences (Cambridge U. Press, 2005).
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  29. F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2010 (2)

R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
[CrossRef]

C. J. Handmer, C. M. de Sterke, R. C. McPhedran, L. C. Botten, M. J. Steel, and A. Rahmani, “Blazing evanescent grating orders: A spectral approach to beating the Rayleigh limit,” Opt. Lett. 35, 2846–2848 (2010).
[CrossRef] [PubMed]

2009 (2)

X. Zhuang, “Nano-imaging with STORM,” Nat. Photonics 3, 365–367 (2009).
[CrossRef] [PubMed]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (1)

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
[CrossRef]

2006 (2)

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Exp. 14, 8247–8256 (2006).
[CrossRef]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

2005 (2)

2004 (1)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 183966 (2000).
[CrossRef]

1997 (1)

1994 (1)

1993 (1)

1988 (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” Opt. Acta 34, 511–538 (1988).

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” J. Mod. Opt. 33, 607–619 (1986).
[CrossRef]

1984 (1)

G. H. Derrick and R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

1982 (1)

M. C. Hutley, Diffraction Gratings (Academic, 1982).

1981 (3)

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

1980 (1)

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
[CrossRef]

1978 (1)

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

1931 (1)

R. de L. Kronig and W. G. Penney, “Quantum mechanics of electrons in crystal lattices,” Proc. R. Soc. London, Ser. A 130, 499–513 (1931).
[CrossRef]

1910 (1)

R. W. Wood, “The echelette grating for the infrared,” Philos. Mag. 20, 770–778 (1910).

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–418 (1873).
[CrossRef]

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–418 (1873).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Alekseyev, L. V.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Exp. 14, 8247–8256 (2006).
[CrossRef]

Andersen, J. B.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

Andrewartha, J.

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Belkebir, K.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

Botten, L. C.

C. J. Handmer, C. M. de Sterke, R. C. McPhedran, L. C. Botten, M. J. Steel, and A. Rahmani, “Blazing evanescent grating orders: A spectral approach to beating the Rayleigh limit,” Opt. Lett. 35, 2846–2848 (2010).
[CrossRef] [PubMed]

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Brolo, A. G.

R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
[CrossRef]

Campbell, S.

Chaumet, P.

A. Sentenac and P. Chaumet, “Subdiffraction light focusing on a grating substrate,” Phys. Rev. Lett. 101, 013901 (2008).
[CrossRef] [PubMed]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

Chaumet, P. C.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

Chen, Y.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

de Abajo, F. J. G.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
[CrossRef]

de Sterke, C. M.

Derrick, G. H.

G. H. Derrick and R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

Drsek, F.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Giovannini, H.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Girard, J.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Gordon, R.

R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
[CrossRef]

Gregory, P.

P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences (Cambridge U. Press, 2005).
[CrossRef]

Handmer, C. J.

Hell, S. W.

Huang, F. M.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, 1982).

Jacob, Z.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Exp. 14, 8247–8256 (2006).
[CrossRef]

Kavanagh, K. L.

R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
[CrossRef]

Konan, D.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Kronig, R. de L.

R. de L. Kronig and W. G. Penney, “Quantum mechanics of electrons in crystal lattices,” Proc. R. Soc. London, Ser. A 130, 499–513 (1931).
[CrossRef]

Li, L.

Maire, G.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” J. Mod. Opt. 33, 607–619 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” J. Mod. Opt. 33, 607–619 (1986).
[CrossRef]

McPhedran, R. C.

C. J. Handmer, C. M. de Sterke, R. C. McPhedran, L. C. Botten, M. J. Steel, and A. Rahmani, “Blazing evanescent grating orders: A spectral approach to beating the Rayleigh limit,” Opt. Lett. 35, 2846–2848 (2010).
[CrossRef] [PubMed]

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” Opt. Acta 34, 511–538 (1988).

G. H. Derrick and R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The finitely-conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, R. C. McPhedran, J. L. Adams, J. Andrewartha, and M. S. Craig, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Mosk, A.

E. van Putten, I. Vellekoop, and A. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008).
[CrossRef] [PubMed]

M. Vellekoop and A. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef] [PubMed]

Narimanov, E.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Exp. 14, 8247–8256 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 183966 (2000).
[CrossRef]

Penney, W. G.

R. de L. Kronig and W. G. Penney, “Quantum mechanics of electrons in crystal lattices,” Proc. R. Soc. London, Ser. A 130, 499–513 (1931).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” J. Mod. Opt. 33, 607–619 (1986).
[CrossRef]

Rahmani, A.

Roberts, A.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” Opt. Acta 34, 511–538 (1988).

Sentenac, A.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

A. Sentenac and P. Chaumet, “Subdiffraction light focusing on a grating substrate,” Phys. Rev. Lett. 101, 013901 (2008).
[CrossRef] [PubMed]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

Sinton, D.

R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4, 311–335 (2010).
[CrossRef]

Solodukhov, V. V.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

Steel, M. J.

Talneau, A.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

van Putten, E.

Vellekoop, I.

Vellekoop, M.

M. Vellekoop and A. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef] [PubMed]

Wichmann, J.

Wood, R. W.

R. W. Wood, “The echelette grating for the infrared,” Philos. Mag. 20, 770–778 (1910).

Zheludev, N. I.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9, S285–S288 (2007).
[CrossRef]

Zhuang, X.

X. Zhuang, “Nano-imaging with STORM,” Nat. Photonics 3, 365–367 (2009).
[CrossRef] [PubMed]

Appl. Opt. (1)

Arch. Mikrosc. Anat. Entwicklungsmech. (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–418 (1873).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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Figures (19)

Fig. 1
Fig. 1

Central three plane-wave orders generated by a grating demonstrating propagation and evanescence. The grating occupies the region y < 0 and has period d = λ / 2 . Wavelength λ = 1 , angle of incidence θ 0 = 30 ° .

Fig. 2
Fig. 2

Diagram of a unit cell of a lamellar grating, showing geometrical factors and regions of different refractive indices.

Fig. 3
Fig. 3

Plot of R [ u n ( x ) exp ( i β n y ) ] for n = { 1 , 2 , , 6 } or the fields of the first six grating modes across period. As before the grating occupies y < 0 , has a period d = 1 , and a duty cycle c 1 / d = 0.1 . The thicker solid line delineates the boundary between n = 1.0 and n = 3.61 in this particular example. Numbers above each graph give { γ 1 , γ 2 , β } for each mode, emphasizing the propagating nature of the upper three modes and the evanescent nature of the lower three modes.

Fig. 4
Fig. 4

Density plot of the amplitude of the transmitted propagating ( m = 0 ) and first evanescent ( m = 1 ) orders as a function of incident angle and primary grating thickness (measured here in multiples of the grating period d). Chosen optimal thickness is t g = 4.177 d = 3.08 μ m . The other grating parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Line plot showing amplitude in transmission of central orders for the stacked grating of fixed thickness shown in Fig. 7 as a function of incident angle θ 0 . The solid line represents the propagating transmitted amplitude ( m = 0 ) , which trends downward through grazing incidence, interrupted by grating anomalies in the form of Fano resonances, signaling the enhancement of corresponding evanescent orders ( m = 1 , 1 , 2 ) . The grating has period d = 0.738 μ m , thickness t g = 3.08 μ m , and duty cycle of 10% for the low index component.

Fig. 6
Fig. 6

Absolute value of the incident (inset) and transmitted spectrum from the Si / SiO 2 grating shown at right. The grating has period d = 0.738 μ m , thickness t g = 3.03 μ m , and duty cycle of 10%.

Fig. 7
Fig. 7

Absolute value of the incident, reflected, and transmitted spectrum from the Si / SiO 2 grating shown at the right. The long period grating has period d = 0.738 μ m , thickness t g = 3.08 μ m , and duty cycle of 10%. Each scattering grating layer has thickness t s = 0.0148 μ m .

Fig. 8
Fig. 8

Diagram showing the interleaving technique used to construct scattering matrices for denser sets of orders (or a larger period) than the short period gratings naturally produce. The upper curve represents | χ m | = | k y m | ; propagating orders exist within the semicircle only.

Fig. 9
Fig. 9

Diagram of the configuration of fields and gratings used to derive the stacked grating recursion formula.

Fig. 10
Fig. 10

Correspondences between spectral and spatial spot characteristics due to the Fourier transform. The width of the spectrum is inversely proportional to the width of the spot, while the width of each peak within the spectrum corresponds to the width of the spatial envelope. Spectral peak spacing is inversely proportional to spatial peak spacing.

Fig. 11
Fig. 11

Amplitude of the spectrum of the leftmost spot in Fig. 13.

Fig. 12
Fig. 12

Details of 30 parallel optimizing processes over 1000 steps. The left figure shows the temperature of each chain as it converges on a local maximum. The right figure shows the progress of the merit function as it trends from a low initial position to an optimized position.

Fig. 13
Fig. 13

Energy density of a series of optimized spots across a single period d = 0.738 μ m below the grating shown in Fig. 7.

Fig. 14
Fig. 14

Diagram showing a set of spots where the previously normalized optimized incident fields in Fig. 13 have been subjected to 10% random variation or blurring.

Fig. 15
Fig. 15

(a) Spectrum of incident field; unit amplitude in every incident angle calculated. (b) Spectrum in transmission of grating shown in (d). (c) Spectrum in transmission of grating shown in (d) with a 1 mm superstrate above the grating. (d) Diagram of a stacked grating thickness t g = 2.87 μ m , period d = 0.738 μ m , with a duty cycle of 25%. Sub-period ratio (1:2) and scattering grating thickness t s = 0.148 μ m . The data in this figure alone were calculated with n H = 3.61 .

Fig. 16
Fig. 16

Diagram showing energy density for a set of spots optimized with only 18 (instead of 54) incident angles.

Fig. 17
Fig. 17

Plot of the left and right hand sides of the Kronig–Penney equation as a function of β 2 . Points of equality or intersection are marked with crosses, corresponding to the modes that exist under a given angle of plane-wave incidence. Varying incident angle varies the height of the intermediate horizontal line between its two extrema.

Fig. 18
Fig. 18

Diagram showing the relative directions and positions of orthonormal mode/order amplitude vectors.

Fig. 19
Fig. 19

Diagram showing the interaction of modes ( c ± ) and orders ( δ , r , t ) within a complete grating formulation.

Equations (65)

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k x p = k 0   sin   θ 0 + 2 π p / d ,     k y p = ( k 0 2 k x p 2 ) 1 / 2 ,
A = χ 1 / 2 J β 1 / 2 ,
B = β 1 / 2 J χ 1 / 2 ,
R 12 = ( A B + I ) 1 ( A B I ) ,
R 21 = ( B A + I ) 1 ( I B A ) ,
T 12 = 2 B ( A B + I ) 1 ,
T 21 = 2 A ( B A + I ) 1 .
R = R 12 + T 21 P R 21 P ( I R 21 P R 21 P ) 1 T 12 ,
T = T 21 P ( I R 21 P R 21 P ) 1 T 12 .
r = R n δ + T n P ¯ s c ¯ + ,
c ¯ = T n δ + R n P ¯ s c ¯ + ,
c ¯ + = R s P ¯ s c ¯ ,
t = T s P ¯ s c ¯ ,
R n + 1 = R n + T n P ¯ s R s P ¯ s ( I R n P ¯ s R s P ¯ s ) 1 T n ,
T n + 1 = T s P ¯ s ( I R n P ¯ s R s P ¯ s ) 1 T n .
R n + 1 = R s + T s P ¯ s R n P ¯ s ( I R s P ¯ s R n P ¯ s ) 1 T s ,
T n + 1 = T n P ¯ s ( I R s P ¯ s R n P ¯ s ) 1 T s .
R 1 R 1 = R ,
T 1 T 1 = T ,
R s = Q 1 R Q ,
T s = Q 1 T Q ,
E = n a n f n ( x ) e i β n y z ,
θ ( 0 ) = x ψ ( 0 ) ψ ( 0 ) = 1 ,
ψ ( 0 ) = x θ ( 0 ) θ ( 0 ) = 0.
θ ( x ) = { cos   γ 1 x if   0 < x < c 1 cos   γ 1 c 1   cos   γ 2 ( x c 1 ) γ 1 γ 2   sin   γ 1 c 1   sin   γ 2 ( x c 1 ) if   c 1 < x < d , }
ψ ( x ) = { 1 γ 1 sin   γ 1 x if   0 < x < c 1 1 γ 1 sin   γ 1 c 1   cos   γ 2 ( x c 1 ) + 1 γ 2 cos   γ 1 c 1   sin   γ 2 ( x c 1 ) if   c 1 < x < d . }
( θ ( d ) ψ ( d ) θ ( d ) ψ ( d ) ) ( A B ) = τ ( θ ( 0 ) ψ ( 0 ) θ ( 0 ) ψ ( 0 ) ) ( A B ) = τ ( 1 0 0 1 ) ( A B ) = τ ( A B ) ,
( θ ( d ) τ ψ ( d ) θ ( d ) ψ ( d ) τ ) ( A B ) = 0 ,
θ ( d ) ψ ( d ) ψ ( d ) θ ( d ) ( θ ( d ) + ψ ( d ) ) τ + τ 2 = 0.
θ ( d ) + ψ ( d ) = 1 + τ 2 τ = 2   cos   α 0 d ,
2   cos   γ 1 c 1   cos   γ 2 ( d c 1 ) ( γ 1 γ 2 + γ 2 γ 1 ) sin   γ 1 c 1   sin   γ 2 ( d c 1 ) = 2   cos   α 0 d .
u n ( x ) = A n θ n ( x ) + B n ψ n ( x ) ,
e p ( x ) = d 1 / 2 e i α p x ,
0 d u n ( x ) u ¯ m ( x ) d x = δ n m ,
0 d e p ( x ) e ¯ q ( x ) d x = δ p q .
J p n = 0 d e ¯ p ( x ) u n ( x ) d x = 0 d d 1 / 2 e i α p x u n ( x ) d x ,
u n ( x ) = p J p n e p ( x ) .
E pw ( x , y ) = p χ p 1 / 2 ( r p e i χ p y + δ p e i χ p y ) e p ( x ) ,
E mode ( x , y ) = n β n 1 / 2 c n e i β n y u n ( x ) ,
χ 1 / 2 ( r + δ ) = J β 1 / 2 c ,
J χ 1 / 2 ( r δ ) = β 1 / 2 c ,
δ + r = χ 1 / 2 J β 1 / 2 β 1 / 2 J χ 1 / 2 ( δ r ) .
A = χ 1 / 2 J β 1 / 2 ,
B = β 1 / 2 J χ 1 / 2 ,
r = ( A B + I ) 1 ( A B I ) δ = R 12 δ ,
R 12 = ( A B + I ) 1 ( A B I ) .
c = B ( δ r ) = B ( δ R 12 δ ) = B ( I R 12 ) δ ,
T 12 = B ( I R 12 ) = 2 B ( A B + I ) 1 .
R 12 = ( A B + I ) 1 ( A B I ) ,
R 21 = ( B A + I ) 1 ( I B A ) ,
T 12 = 2 B ( A B + I ) 1 ,
T 21 = 2 A ( B A + I ) 1 ,
A = χ 1 / 2 J β 1 / 2 ,
B = β 1 / 2 J χ 1 / 2 .
r = R 12 δ + T 21 P c + ,
c = T 12 δ + R 21 P c + ,
c + = R 21 P c ,
t = T 21 P c ,
P = e i β t g ,     β = diag { β n } ,
r = [ R 12 + T 21 P R 21 P ( I R 21 P R 21 P ) 1 T 12 ] δ ,
c = ( I R 21 P R 21 P ) 1 T 12 δ ,
c + = R 21 P ( I R 21 P R 21 P ) 1 T 12 δ ,
t = T 21 P ( I R 21 P R 21 P ) 1 T 12 δ .
R = R 12 + T 21 P R 21 P ( I R 21 P R 21 P ) 1 T 12 ,
T = T 21 P ( I R 21 P R 21 P ) 1 T 12 .

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