Abstract

Resonant properties of composite structures consisting of several identical resonant structures (e.g. multilayer thin-film structures or guided-mode resonance gratings) separated by phase-shift layers are investigated theoretically. Using the scattering matrix formalism, we analytically demonstrate that, at properly chosen thicknesses of the phase-shift layers, the composite structures comprising two or four resonant diffractive structures with a Lorentzian transmittance profile optically implement the Butterworth filters of the order two or three, respectively, and enable achieving flat-top transmission spectra with steep slopes and low sidebands. In addition, we show that the composite structures consisting of three or four second-order Butterworth filters can accurately approximate the fourth- or fifth-order Butterworth filters, respectively. The presented theoretical results are confirmed by rigorous numerical simulations of composite structures consisting of the so-called W-structures (simple three-layer resonant structures comprising a high-index core layer and two low-index cladding layers in a high-index dielectric environment). The simulation results confirm the formation of flat-top transmittance peaks, the shape of which fully agrees with the derived theoretical description. Moreover, we demonstrate an exceptionally simple mechanism of controlling the transmittance peak width, which consists in changing the thicknesses of the cladding layers of the initial W-structure and enables generating flat-top transmission peaks with a significantly subnanometer width.

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

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References

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  1. H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1989).
  2. P. Vincent and M. Nevière, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20(4), 345–351 (1979).
    [Crossref]
  3. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993).
    [Crossref]
  4. L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
    [Crossref]
  5. W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004).
    [Crossref]
  6. M. Niraula, J. W. Yoon, and R. Magnusson, “Single-layer optical bandpass filter technology,” Opt. Lett. 40(21), 5062–5065 (2015).
    [Crossref]
  7. M. Niraula, J. W. Yoon, and R. Magnusson, “Mode-coupling mechanisms of resonant transmission filters,” Opt. Express 22(21), 25817–25829 (2014).
    [Crossref]
  8. S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34(14), 2414–2420 (1995).
    [Crossref]
  9. R. Magnusson and S. S. Wang, “Transmission bandpass guided-mode resonance filters,” Appl. Opt. 34(35), 8106–8109 (1995).
    [Crossref]
  10. S. Tibuleac and R. Magnusson, “Narrow-linewidth bandpass filters with diffractive thin-film layers,” Opt. Lett. 26(9), 584–586 (2001).
    [Crossref]
  11. R. Zengerle and O. Leminger, “Phase-shifted Bragg grating filter with improved transmission characteristics,” J. Lightwave Technol. 13(12), 2354–2358 (1995).
    [Crossref]
  12. Y. H. Ko and R. Magnusson, “Flat-top bandpass filters enabled by cascaded resonant gratings,” Opt. Lett. 41(20), 4704–4707 (2016).
    [Crossref]
  13. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58–106 (2009).
    [Crossref]
  14. Y. Suematsu and K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23(1), 170–175 (1975).
    [Crossref]
  15. N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
    [Crossref]
  16. L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
    [Crossref]
  17. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
    [Crossref]
  18. D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
    [Crossref]
  19. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996).
    [Crossref]
  20. 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]
  21. 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]
  22. 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]
  23. H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
    [Crossref]
  24. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995).
    [Crossref]
  25. 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]
  26. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
    [Crossref]

2018 (2)

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[Crossref]

L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
[Crossref]

2017 (1)

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]

2016 (1)

2015 (1)

2014 (1)

2013 (2)

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

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]

2012 (1)

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

2011 (1)

2009 (1)

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58–106 (2009).
[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]

2004 (1)

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004).
[Crossref]

2002 (2)

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

2001 (1)

1996 (1)

1995 (4)

1993 (1)

1985 (1)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

1979 (1)

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20(4), 345–351 (1979).
[Crossref]

1975 (1)

Y. Suematsu and K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23(1), 170–175 (1975).
[Crossref]

Bezus, E. A.

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[Crossref]

L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
[Crossref]

Bykov, D. A.

L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
[Crossref]

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[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]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

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]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

Doskolovich, L. L.

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[Crossref]

L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
[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]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

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]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

Fan, S.

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004).
[Crossref]

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

Furuya, K.

Y. Suematsu and K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23(1), 170–175 (1975).
[Crossref]

Gaylord, T. K.

Gippius, N. A.

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, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[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]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

Grann, E. B.

Hu, J.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58–106 (2009).
[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]

Joannopoulos, J. D.

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

Ko, Y. H.

Leminger, O.

R. Zengerle and O. Leminger, “Phase-shifted Bragg grating filter with improved transmission characteristics,” J. Lightwave Technol. 13(12), 2354–2358 (1995).
[Crossref]

Li, L.

Liu, H.-C.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1989).

Magnusson, R.

Mashev, L.

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

Menyuk, C. R.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58–106 (2009).
[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]

Nevière, M.

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20(4), 345–351 (1979).
[Crossref]

Niraula, M.

Pommet, D. A.

Popov, E.

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

Soifer, V. A.

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

Suematsu, Y.

Y. Suematsu and K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23(1), 170–175 (1975).
[Crossref]

Suh, W.

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004).
[Crossref]

Tibuleac, S.

Tikhodeev, S. G.

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]

Vincent, P.

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20(4), 345–351 (1979).
[Crossref]

Wang, S. S.

Yablonskii, A. L.

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]

Yariv, A.

Yoon, J. W.

Zengerle, R.

R. Zengerle and O. Leminger, “Phase-shifted Bragg grating filter with improved transmission characteristics,” J. Lightwave Technol. 13(12), 2354–2358 (1995).
[Crossref]

Adv. Opt. Photonics (1)

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58–106 (2009).
[Crossref]

Appl. Opt. (3)

Appl. Phys. (1)

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20(4), 345–351 (1979).
[Crossref]

Appl. Phys. Lett. (1)

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004).
[Crossref]

Comput. Opt. (1)

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]

IEEE Trans. Microwave Theory Tech. (1)

Y. Suematsu and K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23(1), 170–175 (1975).
[Crossref]

J. Exp. Theor. Phys. (2)

N. V. Golovastikov, L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “An optical differentiator based on a three-layer structure with a W-shaped refractive index profile,” J. Exp. Theor. Phys. 127(2), 202–209 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

J. Lightwave Technol. (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]

R. Zengerle and O. Leminger, “Phase-shifted Bragg grating filter with improved transmission characteristics,” J. Lightwave Technol. 13(12), 2354–2358 (1995).
[Crossref]

J. Opt. (1)

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

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

Opt. Commun. (1)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Photonics Res. (1)

L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photonics Res. 6(1), 61–65 (2018).
[Crossref]

Phys. Rev. B (3)

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

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]

Other (1)

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1989).

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Figures (5)

Fig. 1.
Fig. 1. Three-layer W-structure (a), refractive index profile of the W-structure (b), and a composite structure consisting of two W-structures (c).
Fig. 2.
Fig. 2. Locations of the poles of the designed filters (crosses) and of the Butterworth filters of orders 1–5 (small circles). Dashed lines show the circles with radii ${\Delta _1} = {\mathop{\rm Im}\nolimits}\ {\omega _p},{\Delta _2},{\Delta _4},{\Delta _6}$, and ${\Delta _8}$.
Fig. 3.
Fig. 3. Reflectance (а) and transmittance (b) spectra of the W-structure at different thicknesses of the cladding layers: ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$ (dashed blue line), ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$ (dash-dotted red line), and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$ (solid yellow line).
Fig. 4.
Fig. 4. Reflectance (red) and transmittance (black) of the composite W-structures with (solid lines) at $N = 2$ (а), $N = 4$ (b), $N = 6$ (c), and $N = 8$ (d). Dotted lines show the spectra of the initial W-structures. Dashed green lines show the squared moduli of the transfer functions of the corresponding Butterworth filters.
Fig. 5.
Fig. 5. Transmittance spectra of the composite W-structures with $N = 4$ (a) and $N = 6$ (b) calculated for different thicknesses of the cladding layers: ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$ (dashed blue line), ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$ (dash-dotted red line), and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$ (solid yellow line).

Equations (26)

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r 1 ( ω ) = exp { i φ } ω Re   ω p ω ω p = exp { i φ } + exp { i φ } i Im   ω p ω ω p , t 1 ( ω ) = ± exp { i φ } i Im   ω p ω ω p ,
S 1 ( ω ) = ( t 1 ( ω ) r 1 ( ω ) r 1 ( ω ) t 1 ( ω ) ) ,
( t ( ω ) r ( ω ) ) = S 1 ( ω ) ( i u ( ω ) i d ( ω ) ) .
S 2 ( ω ) = S 1 ( ω ) L ( l 1 ) 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 ) ,
L ( l 1 ) = exp { i ψ ( l 1 ) } E ,
ψ ( l 1 ) = l 1 ( n ω 0 / n ω 0 c c ) 2 k x 2 ,
S 2 ( ω ) = ( t 2 ( ω ) r 2 ( ω ) r 2 ( ω ) t 2 ( ω ) ) = = 1 1 exp { 2 i ψ } r 1 2 ( exp { i ψ } t 1 2 r 1 [ 1 exp { 2 i ψ } ( r 1 2 t 1 2 ) ] r 1 [ 1 exp { 2 i ψ } ( r 1 2 t 1 2 ) ] exp { i ψ } t 1 2 ) .
r 2 ( ω ) = exp { i φ } ( ω Re   ω p ) [ ω p ω + σ ( l 1 ) ( ω ω p ) ] ( ω ω p ) 2 + σ ( l 1 ) ( ω Re   ω p ) 2 , t 2 ( ω ) = σ ( l 1 ) exp { i ψ } ( Im   ω p ) 2 ( ω ω p ) 2 + σ ( l 1 ) ( ω Re   ω p ) 2 ,
ψ ( l 1 ) = π 2 φ + π m 1 , m 1 Z ,
r 2 ( ω ) = exp { i φ } ( ω Re   ω p ) 2 ( ω ω p , 1 ) ( ω ω p , 2 ) , t 2 ( ω ) = ( 1 ) m 1 + 1 i exp { i φ } ( Im   ω p ) 2 2 ( ω ω p , 1 ) ( ω ω p , 2 ) ,
ω p , 1 = Re   ω p + i 1 2 Im   ω p , ω p , 2 = Re   ω p + i + 1 2 Im   ω p .
T B W , M ( ω ) = 1 1 + [ ( ω ω 0 ) / ( ω ω 0 ) Δ Δ ] 2 M ,
T 2 ( ω ) = T B W , 2 ( ω ) = 1 1 + [ ( ω Re   ω p ) / ( ω Re   ω p ) Δ 2 Δ 2 ] 4 ,
S 4 ( ω ) = S 2 ( ω ) L ( l 2 ) S 2 ( ω ) ,
ψ ( l 2 ) = π φ + π m 2 , m 2 Z ,
r 4 ( ω ) = exp { i φ } ( ω Re   ω p ) 3 ( ω ω p , 1 ) ( ω ω p , 2 ) ( ω ω p , 3 ) , t 4 ( ω ) = ( 1 ) m 2 i exp { i φ } ( Im   ω p ) 3 8 ( ω ω p , 1 ) ( ω ω p , 2 ) ( ω ω p , 3 ) .
ω p , 1 = Re   ω p + i + 3 4 Im   ω p ,   ω p , 2 = Re   ω p + i 3 4 Im   ω p , ω p , 3 = Re   ω p + i 2 Im   ω p .
T 4 ( ω ) = T B W , 3 ( ω ) = 1 1 + [ ( ω Re   ω p ) / ( ω Re   ω p ) Δ 4 Δ 4 ] 6 ,
S 6 ( ω ) = S 2 ( ω ) L ( l 2 ) S 2 ( ω ) L ( l 2 ) S 2 ( ω ) = S 4 ( ω ) L ( l 2 ) S 2 ( ω ) .
t 6 ( ω ) = ( 1 ) m 1 + 1 i exp { i φ } ( Im   ω p ) 4 32 i = 1 4 ( ω ω p , i ) ,
ω p , 1 = Re   ω p + 1 8 ( 1 + 2i ( 1 + i ) 2 5 i / 5 i 2 2 ) Im   ω p , ω p , 2 = Re   ω p + 1 8 ( 1 + 2i + ( 1 + i ) 2 5 i / 5 i 2 2 ) Im   ω p , ω p , 3 = Re   ω p + 1 8 ( 1 + 2i ( 1 i ) 2 + 5 i / 5 i 2 2 ) Im   ω p , ω p , 4 = Re   ω p + 1 8 ( 1 + 2 i + ( 1 i ) 2 + 5 i / 5 i 2 2 ) Im   ω p .
T 6 ( ω ) T B W , 4 ( ω ) = 1 1 + [ ( ω Re   ω p ) / ( ω Re   ω p ) Δ 6 Δ 6 ] 8 ,
S 8 ( ω ) = S 6 ( ω ) L ( l 2 ) S 2 ( ω ) .
t 8 ( ω ) = ( 1 ) m 2 i exp { i φ } ( Im   ω p ) 5 128 i = 1 5 ( ω ω p , i ) ,
ω p , 1 = Re   ω p + ( 0.43036 + 0.08805 i ) Im   ω p , ω p , 2 = Re   ω p + ( 0 .43036 + 0.08805 i ) Im   ω p , ω p , 3 = Re   ω p + 0.25 ( 1 + i ) Im   ω p , ω p , 4 = Re   ω p + 0.25 ( 1 + i ) Im   ω p , ω p , 5 = Re   ω p + 0.32390 i Im   ω p .