Abstract

We theoretically and numerically investigate resonant optical properties of composite structures consisting of several subwavelength resonant diffraction gratings separated by homogeneous layers. Using the scattering matrix formalism, we demonstrate that the composite structure comprising N gratings has a multiple transmittance zero of the order N. We show that at the distance between the gratings satisfying the Fabry–Pérot resonance condition, an (N – 1)-degenerate bound state in the continuum (BIC) is formed. The results of rigorous numerical simulations fully confirm the theoretically predicted formation of multiple zeros and BICs in the composite structures. Near the BICs, an effect very similar to the electromagnetically induced transparency is observed. We show that by making the proper choice of the thicknesses of the layers separating the gratings, nearly rectangular reflectance or transmittance peaks with steep slopes and virtually no sidelobes can be obtained. In particular, one of the presented examples demonstrates the possibility of obtaining an approximately rectangular transmittance peak with a significantly subnanometer width. The presented results may find application in the design of optical filters, sensors and devices for optical differentiation and transformation of optical signals.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

2017 (4)

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
[Crossref]

K. Yamada, K. J. Lee, Y. H. Ko, J. Inoue, K. Kintaka, S. Ura, and R. Magnusson, “Flat-top narrowband filters enabled by guided-mode resonance in two-level waveguides,” Opt. Lett. 42(20), 4127–4130 (2017).
[Crossref]

2016 (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

2014 (1)

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

2013 (2)

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31(5), 793–801 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
[Crossref]

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

2009 (1)

2008 (1)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

2005 (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

2003 (1)

2002 (4)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002).
[Crossref]

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

1996 (1)

1995 (3)

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4(21), 396–402 (1902).
[Crossref]

Basset, G.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Bogdanov, A. A.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Bykov, D. A.

Chang-Hasnain, C. J.

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Doskolovich, L. L.

Dunn, S. C.

Fan, S.

Fehrembach, A.-L.

Gallinet, B.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Gaylord, T. K.

Giessen, H.

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
[Crossref]

Grann, E. B.

Hermannsson, P. G.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Inoue, J.

Iorsh, I. V.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Jacob, D. K.

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Kintaka, K.

Kivshar, Y. S.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Ko, Y. H.

Koshelev, K. L.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Kristensen, A.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Lavrinenko, A. V.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Lee, K. J.

Li, L.

Limonov, M. F.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Liu, H.-C.

Liu, V.

Magnusson, R.

Malureanu, R.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Marinica, D. C.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Martin, O. J. F.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

Maystre, D.

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref]

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Ndangali, R. F.

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

Nevière, M.

Poddubny, A. N.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Pommet, D. A.

Popov, E.

Povinelli, M.

Qiao, P.

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Quaranta, G.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Reinisch, R.

Rybin, M. V.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Sadrieva, Z. F.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Samusev, A.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Sentenac, A.

Shabanov, S. V.

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Sinev, I. S.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Smith, C. L. C.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Soifer, V. A.

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Suh, W.

Takayama, O.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

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S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
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S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
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ACS Photonics (1)

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Adv. Opt. Photonics (1)

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
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N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
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Nat. Photonics (1)

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
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Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Nature (1)

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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[Crossref]

Phys. Rev. B (3)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

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D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Geometry of a composite structure containing two identical diffraction gratings with period $\Lambda $, height h, and refractive indices ${n_1}$ and ${n_2}$ separated by a dielectric layer with the thickness l and refractive index n. The structure is located in a symmetric dielectric environment with refractive index ${n_{env}}$. The arrows denote the incident, reflected and transmitted waves.
Fig. 2.
Fig. 2. Spectra $R(\omega )= {|{r(\omega )} |^2}$, $T(\omega )= {|{t(\omega )} |^2}$ of the subwavelength resonant diffraction grating with the following parameters: period $\Lambda = 300\,\textrm{nm}$, grating height $h = 130\,\textrm{nm}$, refractive indices of the grating materials ${n_1} = 2.1$ and ${n_2} = 1.9$, width of the region with the refractive index ${n_1}$ $w = {\Lambda \mathord{\left/ {\vphantom {\Lambda 2}} \right.} 2}$, refractive index of the surrounding medium ${n_{env}} = 1.52$. The normally incident plane wave is TE-polarized. The inset shows the geometry of the grating.
Fig. 3.
Fig. 3. (a) Reflectance ${R_3}({\omega ,l} )$ and transmittance ${T_3}({\omega ,l} )$ of the composite structure at $N = 3$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.
Fig. 4.
Fig. 4. (a) Reflectance ${R_4}({\omega ,l} )$ and transmittance ${T_4}({\omega ,l} )$ of the composite structure at $N = 4$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.
Fig. 5.
Fig. 5. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures with $N = 3$ (a) and $N = 4$ (b) at $l = 948\,\textrm{nm}$. Dashed lines show the transmittance (black) and reflectance (red) of the initial DG.
Fig. 6.
Fig. 6. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures at $N = 4$ and ${l_1} = {l_3} = 952\,\textrm{nm}$, ${l_2} = 1037\,\textrm{nm}$ (a) and at $N = 6$ and ${l_1} = {l_5} = 1126\,\textrm{nm}$, ${l_2} = {l_4} = 1050\,\textrm{nm}$, ${l_3} = 943\,\textrm{nm}$ (b). Dashed lines show the spectra of the initial DG. Dash-dot lines show the functions ${D_N}(\omega )$ and ${P_N}(\omega )$ at $N = 4$ (a) and $N = 6$ (b).
Fig. 7.
Fig. 7. Transmittance spectra (solid blue lines) of the composite structure with $N = 4$ at ${l_1} = {l_2} = {l_3} = 1033\,\textrm{nm}$ (a) and of the optimized structure with ${l_1} = {l_3} = 1027\,\textrm{nm}$ and ${l_2} = 950\,\textrm{nm}$ (b). Red dash-dot lines show the desired peak shape ${P_4}(\omega )$. Black dashed lines show the transmittance spectrum of the initial DG. The inset shows the transmittance spectrum of the optimized structure in a wider frequency range.

Equations (27)

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( T R ) = S ( I u I d ) ,
S 1 ( ω ) = ( t 1 ( ω ) r d , 1 ( ω ) r u , 1 ( ω ) t 1 ( ω ) ) .
S 2 ( ω ) = S 1 ( ω ) L ( ω ) S 1 ( ω ) ,
( a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ) ( b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ) = = 1 1 a 1 , 2 b 2 , 1 ( b 1 , 1 a 1 , 1 b 1 , 2 a 1 , 2 ( b 1 , 2 b 2 , 1 b 1 , 1 b 2 , 2 ) a 2 , 1 b 2 , 1 ( a 1 , 2 a 2 , 1 a 1 , 1 a 2 , 2 ) a 2 , 2 b 2 , 2 ) ,
ψ ( ω ) = l ( n ω / ω c c ) 2 k x 2 ( ω ) ,
L ( ω ) = exp { i ψ ( ω ) } E ,
S 2 ( ω ) = ( t 2 ( ω ) r d , 2 ( ω ) r u , 2 ( ω ) t 2 ( ω ) ) = 1 1 exp { 2 i ψ } r u , 1 r d , 1 ( exp { i ψ } t 1 2 r d , 1 [ 1 exp { 2 i ψ } ( r u , 1 r d , 1 t 1 2 ) ] r u , 1 [ 1 exp { 2 i ψ } ( r u , 1 r d , 1 t 1 2 ) ] exp { i ψ } t 1 2 ) .
t 1 ( ω ) = τ ( ω ω 0 ) + O ( ω ω 0 ) 2 , r u , 1 ( ω ) = exp { i φ u } + ρ u ( ω ω 0 ) + O ( ω ω 0 ) 2 , r d , 1 ( ω ) = exp { i φ d } + ρ d ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
exp { i ψ ( ω ) } = exp { i ψ 0 } + γ ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
t 2 ( ω ) = exp { i ψ } τ 2 1 exp { 2 i ( ψ 0 + φ ) } ( ω ω 0 ) 2 + O ( ω ω 0 ) 3 , r ( u , d ) , 2 ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
ψ 0 + φ = π m , m N
t 2 ( ω ) = exp { i φ } τ 2 2 exp { 3 i φ } γ + ( 1 ) m ( exp { i φ d } ρ u + exp { i φ u } ρ d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
S N + 1 ( ω ) = S N ( ω ) L ( ω ) S 1 ( ω )
t N ( ω ) = exp { i ( N 1 ) ψ } τ N [ 1 exp { 2 i ( ψ 0 + φ ) } ] N 1 ( ω ω 0 ) N + O ( ω ω 0 ) N + 1 , r ( u , d ) , N ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
t N + 1 ( ω ) = exp { i N ψ } τ N + 1 [ 1 exp { 2 i ( ψ 0 + φ ) } ] N ( ω ω 0 ) N + 1 + O ( ω ω 0 ) N + 2 , r ( u , d ) , N + 1 ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
r u , 1 ( ω ) = exp { i φ u } i Im ω p ω ω p , r d , 1 ( ω ) = exp { i φ d } i Im ω p ω ω p , t 1 ( ω ) = exp { i φ } ω Re ω p ω ω p ,
L = exp { i ψ 0 } E = exp { i ω 0 c l n 2 ( n env sin θ ) 2 } E .
t 2 ( ω ) = exp { i ( ψ 0 + 2 φ ) } ( ω Re ω p ) 2 ( ω ω p , 1 ) ( ω ω p , 2 ) ,
ω p , 1 = Re ω p + i [ 1 exp { i ( ψ 0 + φ ) } ] Im ω p , ω p , 2 = Re ω p + i [ 1 + exp { i ( ψ 0 + φ ) } ] Im ω p .
t 3 ( ω ) = exp { i ( 2 ψ 0 + 3 φ ) } ( ω Re ω p ) 3 ( ω ω p , 1 ) ( ω ω p , 2 ) ( ω ω p , 3 ) ,
ω p , 1 = Re ω p + i ( 1 σ ) Im ω p , ω p , 2 = ω p + i σ σ ( 8 + σ ) 2 Im ω p , ω p , 3 = ω p + i σ + σ ( 8 + σ ) 2 Im ω p ,
ω p , 1 = ω p , 2 = Re ω p , ω p , 3 = Re ω p + 3 i Im ω p .
t N ( ω ) = exp { i ξ } ( ω Re ω p ) N m = 1 N ( ω ω p , m ) ,
t N ( ω ) = ( 1 ) N 1 exp { i φ } ω Re ω p ω Re ω p + i N Im ω p ,
ψ 0 + φ = π ( m 1 / 1 2 2 ) , m N .
D N ( ω ) = [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N 1 + [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N ,
P N ( ω ) = 1 D N ( ω ) = 1 1 + [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N ,

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