Abstract

We investigate the diffraction of the guided modes of a dielectric slab waveguide on a simple integrated structure consisting of a single dielectric ridge on the surface of the waveguide. Numerical simulations based on aperiodic rigorous coupled-wave analysis demonstrate the existence of sharp resonant features and bound states in the continuum (BICs) in the reflectance and transmittance spectra occurring at the oblique incidence of a transverse-electric (TE)-polarized guided mode on the ridge. Using the effective index method, we explain the resonances by the excitation of cross-polarized modes of the ridge. Formation of the BICs are confirmed using a theoretical model based on coupled-wave theory. The model suggests that the BICs occur due to the coupling of quasi-TE and quasi-transverse-magnetic modes of the structure. Simple analytical expressions for the angle of incidence and the ridge width predicting the location of the BICs are obtained. The existence of high-Q resonances and BICs enables using the considered integrated structure for sensing, transformation of optical signals, and enhancing nonlinear light–matter interactions. Due to the Lorentzian line shape of the resonances near the BICs, the structure is also promising for filtering applications.

© 2018 Chinese Laser Press

Full Article  |  PDF Article
OSA Recommended Articles
Optical bound states in slotted high-contrast gratings

Yifei Wang, Jiming Song, Liang Dong, and Meng Lu
J. Opt. Soc. Am. B 33(12) 2472-2479 (2016)

Spatial integration and differentiation of optical beams in a slab waveguide by a dielectric ridge supporting high-Q resonances

Evgeni A. Bezus, Leonid L. Doskolovich, Dmitry A. Bykov, and Victor A. Soifer
Opt. Express 26(19) 25156-25165 (2018)

Propagating bound states in the continuum in dielectric gratings

E. N. Bulgakov, D. N. Maksimov, P. N. Semina, and S. A. Skorobogatov
J. Opt. Soc. Am. B 35(6) 1218-1222 (2018)

References

  • View by:
  • |
  • |
  • |

  1. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  2. T. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26, 414–416 (2001).
    [Crossref]
  3. G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
    [Crossref]
  4. S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
    [Crossref]
  5. C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
    [Crossref]
  6. C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
    [Crossref]
  7. X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (2013).
    [Crossref]
  8. L. L. Doskolovich, E. A. Bezus, and D. A. Bykov, “Two-groove narrowband transmission filter integrated into a slab waveguide,” Photon. Res. 6, 61–65 (2018).
    [Crossref]
  9. L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25, 22328–22340 (2017).
    [Crossref]
  10. M. Hammer, A. Hildebrandt, and J. Förstner, “How planar optical waves can be made to climb dielectric steps,” Opt. Lett. 40, 3711–3714 (2015).
    [Crossref]
  11. M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semiguided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34, 997–1005 (2016).
    [Crossref]
  12. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
    [Crossref]
  13. W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
    [Crossref]
  14. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
    [Crossref]
  15. Z. F. Sadrieva and A. A. Bogdanov, “Bound state in the continuum in the one-dimensional photonic crystal slab,” J. Phys. Conf. Ser. 741, 012122 (2016).
  16. 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 Photon. 4, 723–727 (2017).
    [Crossref]
  17. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
    [Crossref]
  18. 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, 188–191 (2013).
    [Crossref]
  19. E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
    [Crossref]
  20. F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19, 093011 (2017).
    [Crossref]
  21. C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
    [Crossref]
  22. S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
    [Crossref]
  23. L. Yuan and Y. Y. Lu, “Propagating Bloch modes above the lightline on a periodic array of cylinders,” J. Phys. B 50, 05LT01 (2017).
    [Crossref]
  24. C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
    [Crossref]
  25. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
    [Crossref]
  26. I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
    [Crossref]
  27. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
    [Crossref]
  28. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
    [Crossref]
  29. S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
    [Crossref]
  30. C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
    [Crossref]
  31. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Low-scattering surface plasmon refraction with isotropic materials,” Opt. Express 22, 13547–13554 (2014).
    [Crossref]
  32. G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).
  33. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17, 24112–24129 (2009).
    [Crossref]
  34. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [Crossref]
  35. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [Crossref]
  36. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
    [Crossref]
  37. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
    [Crossref]
  38. C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill, 2003).
  39. S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I — mathematical formulations,” IEEE Trans. Microw. Theory Tech. 29, 843–855 (1981).
    [Crossref]
  40. A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
    [Crossref]
  41. 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, 793–801 (2013).
    [Crossref]
  42. D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigen modes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
    [Crossref]
  43. N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
    [Crossref]
  44. V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36, 1704–1706 (2011).
    [Crossref]
  45. R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
    [Crossref]
  46. R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
    [Crossref]

2018 (2)

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

I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
[Crossref]

2017 (4)

F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19, 093011 (2017).
[Crossref]

L. Yuan and Y. Y. Lu, “Propagating Bloch modes above the lightline on a periodic array of cylinders,” J. Phys. B 50, 05LT01 (2017).
[Crossref]

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25, 22328–22340 (2017).
[Crossref]

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 Photon. 4, 723–727 (2017).
[Crossref]

2016 (6)

M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semiguided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34, 997–1005 (2016).
[Crossref]

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

Z. F. Sadrieva and A. A. Bogdanov, “Bound state in the continuum in the one-dimensional photonic crystal slab,” J. Phys. Conf. Ser. 741, 012122 (2016).

C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
[Crossref]

2015 (3)

D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigen modes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
[Crossref]

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

M. Hammer, A. Hildebrandt, and J. Förstner, “How planar optical waves can be made to climb dielectric steps,” Opt. Lett. 40, 3711–3714 (2015).
[Crossref]

2014 (4)

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Low-scattering surface plasmon refraction with isotropic materials,” Opt. Express 22, 13547–13554 (2014).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

2013 (5)

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (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, 188–191 (2013).
[Crossref]

X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (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, 793–801 (2013).
[Crossref]

2012 (2)

2011 (3)

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36, 1704–1706 (2011).
[Crossref]

2010 (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

2009 (3)

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17, 24112–24129 (2009).
[Crossref]

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

2008 (2)

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

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

2005 (2)

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[Crossref]

2001 (2)

1996 (1)

1995 (1)

1981 (2)

S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I — mathematical formulations,” IEEE Trans. Microw. Theory Tech. 29, 843–855 (1981).
[Crossref]

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

Alù, A.

F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19, 093011 (2017).
[Crossref]

Babin, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Barnard, E. S.

Bezus, E. A.

Blanchard, C.

C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
[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 Photon. 4, 723–727 (2017).
[Crossref]

Z. F. Sadrieva and A. A. Bogdanov, “Bound state in the continuum in the one-dimensional photonic crystal slab,” J. Phys. Conf. Ser. 741, 012122 (2016).

Borisov, A. G.

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

Brongersma, M. L.

Bugrov, A.

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

Bykov, D. A.

Cabrini, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Calafiore, G.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

Calo, C.

Cao, Q.

Chadha, A.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Chang-Hasnain, C. J.

Chase, C.

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, 188–191 (2013).
[Crossref]

Chua, S.-L.

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Chuwongin, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Cui, J. M.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Debernardi, P.

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
[Crossref]

Dhuey, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Doskolovich, L. L.

Dreisow, F.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Fan, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Förstner, J.

Gaylord, T. K.

Gippius, N. A.

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

Golovastikov, N. V.

Goltsov, A.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Grann, E. B.

Guo, G. C.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Hammer, M.

Han, Z. F.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Harteneck, B.

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

He, J. J.

X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (2013).
[Crossref]

Heinrich, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Hildebrandt, A.

Hryciw, A. C.

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, 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, 188–191 (2013).
[Crossref]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Hsu, T. I.

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

Hugonin, J. P.

Hugonin, J.-P.

C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
[Crossref]

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[Crossref]

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 Photon. 4, 723–727 (2017).
[Crossref]

Ivonin, I.

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

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, 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, 188–191 (2013).
[Crossref]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Johnson, S. G.

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (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, 188–191 (2013).
[Crossref]

Karagodsky, V.

Kazanskiy, N. L.

Keil, R.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Kekatpure, R. D.

Kivshar, Y. S.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Kley, E.-B.

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Kopyatev, S.

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

Koshelev, A.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

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 Photon. 4, 723–727 (2017).
[Crossref]

Lalanne, P.

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 Photon. 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, 188–191 (2013).
[Crossref]

Li, L.

Li, M.

X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (2013).
[Crossref]

Lifante, G.

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).

Liu, V.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Lu, Y. Y.

L. Yuan and Y. Y. Lu, “Propagating Bloch modes above the lightline on a periodic array of cylinders,” J. Phys. B 50, 05LT01 (2017).
[Crossref]

Ma, X.

X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (2013).
[Crossref]

Ma, Z.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Maksimov, D. N.

I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
[Crossref]

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 Photon. 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, 183902 (2008).
[Crossref]

Miroshnichenko, A. E.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Moharam, M. G.

Molina, M. I.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref]

Monticone, F.

F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19, 093011 (2017).
[Crossref]

Mossberg, T.

Nolte, S.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Oliner, A. A.

S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I — mathematical formulations,” IEEE Trans. Microw. Theory Tech. 29, 843–855 (1981).
[Crossref]

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

Orta, R.

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
[Crossref]

Peleg, O.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Peng, S.-T.

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I — mathematical formulations,” IEEE Trans. Microw. Theory Tech. 29, 843–855 (1981).
[Crossref]

Peroz, C.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Plotnik, Y.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Pollock, C. R.

C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill, 2003).

Pommet, D. A.

Sadreev, A. F.

I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[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 Photon. 4, 723–727 (2017).
[Crossref]

Z. F. Sadrieva and A. A. Bogdanov, “Bound state in the continuum in the one-dimensional photonic crystal slab,” J. Phys. Conf. Ser. 741, 012122 (2016).

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 Photon. 4, 723–727 (2017).
[Crossref]

Sanchez, A.

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

Sasorov, P.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

Sauvan, C.

C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
[Crossref]

Schmidt, H.

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Segev, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

Seo, J.-H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Shabanov, S. V.

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

Shipman, S. P.

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Shuai, Y.-C.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Silberstein, E.

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 Photon. 4, 723–727 (2017).
[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, 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, 188–191 (2013).
[Crossref]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (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, 16048 (2016).
[Crossref]

Sukhorukov, A. A.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Sun, F. W.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Szameit, A.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

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 Photon. 4, 723–727 (2017).
[Crossref]

Tibaldi, A.

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
[Crossref]

Tikhodeev, S. G.

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

Timofeev, I. V.

I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
[Crossref]

Tünnermann, A.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Venakides, S.

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Wang, K. X.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Weimann, S.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Xiong, X.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Xu, Y.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

Yang, H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Yankov, V.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

C. Peroz, C. Calo, A. Goltsov, S. Dhuey, A. Koshelev, P. Sasorov, I. Ivonin, S. Babin, S. Cabrini, and V. Yankov, “Multiband wavelength demultiplexer based on digital planar holography for on-chip spectroscopy applications,” Opt. Lett. 37, 695–697 (2012).
[Crossref]

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Yuan, L.

L. Yuan and Y. Y. Lu, “Propagating Bloch modes above the lightline on a periodic array of cylinders,” J. Phys. B 50, 05LT01 (2017).
[Crossref]

Zhao, D.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 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, 188–191 (2013).
[Crossref]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Zhou, W.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zou, C. L.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Zou, X. B.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

ACS Photon. (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 Photon. 4, 723–727 (2017).
[Crossref]

Appl. Phys. Lett. (1)

S. Babin, A. Bugrov, S. Cabrini, S. Dhuey, A. Goltsov, I. Ivonin, E.-B. Kley, C. Peroz, H. Schmidt, and V. Yankov, “Digital optical spectrometer-on-chip,” Appl. Phys. Lett. 95, 041105 (2009).
[Crossref]

Comput. Opt. (1)

D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigen modes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
[Crossref]

IEEE J. Quantum Electron. (2)

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part II: high/low-contrast grating resonators,” IEEE J. Quantum Electron. 52, 6600409 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena — part I: generalized Fabry–Pérot interferometers,” IEEE J. Quantum Electron. 52, 6100508 (2016).
[Crossref]

IEEE Photon. J. (2)

C. Peroz, A. Goltsov, S. Dhuey, P. Sasorov, B. Harteneck, I. Ivonin, S. Kopyatev, S. Cabrini, S. Babin, and V. Yankov, “High-resolution spectrometer-on-chip based on digital planar holography,” IEEE Photon. J. 3, 888–896 (2011).
[Crossref]

X. Ma, M. Li, and J. J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photon. J. 5, 7101307 (2013).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I — mathematical formulations,” IEEE Trans. Microw. Theory Tech. 29, 843–855 (1981).
[Crossref]

A. A. Oliner, S.-T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microw. Theory Tech. 29, 855–869 (1981).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. A (4)

J. Phys. B (1)

L. Yuan and Y. Y. Lu, “Propagating Bloch modes above the lightline on a periodic array of cylinders,” J. Phys. B 50, 05LT01 (2017).
[Crossref]

J. Phys. Conf. Ser. (1)

Z. F. Sadrieva and A. A. Bogdanov, “Bound state in the continuum in the one-dimensional photonic crystal slab,” J. Phys. Conf. Ser. 741, 012122 (2016).

Laser Photon. Rev. (1)

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9, 114–119 (2015).
[Crossref]

Light Sci. Appl. (2)

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014).
[Crossref]

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, 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, 188–191 (2013).
[Crossref]

New J. Phys. (1)

F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19, 093011 (2017).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Photon. Res. (1)

Phys. Rev. A (1)

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

Phys. Rev. B (3)

C. Blanchard, J.-P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near optical bound states in the continuum,” Phys. Rev. B 94, 155303 (2016).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable Q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B 97, 024308 (2018).
[Crossref]

Phys. Rev. E (1)

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Phys. Rev. Lett. (4)

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface Fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111, 240403 (2013).
[Crossref]

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

Phys. Usp. (1)

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

Prog. Quantum Electron. (1)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Rev. Mod. Phys. (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Other (3)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).

C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill, 2003).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. (a) Geometry of the considered integrated structure. I, R, and T denote the incident, reflected, and transmitted TE-polarized guided modes, respectively. (b) 2D effective index model and corresponding refractive indices.
Fig. 2.
Fig. 2. (a) Reflectance and (b) transmittance of the TE-polarized mode versus the angle of incidence θ and ridge width w. Horizontal dashed lines show the cutoff angles θsup, θsub, θTM, and θTM,r. Field distributions in the structure at the points marked with asterisks in (a) are shown in Fig. 3. Dashed red curves in (b) show the dispersion of quasi-TM modes calculated using the approximate Eq. (4). (c) Reflectance and (d) transmittance of a plane wave upon refraction by a thin film (effective index model) are also shown.
Fig. 3.
Fig. 3. Electric field distributions in the structure at resonance conditions: (a) w=326  nm, θ=52.66°; (b) w=470  nm, θ=47.58°.
Fig. 4.
Fig. 4. (a) Magnified fragment of the rigorously calculated reflectance spectrum shown in Fig. 2(a). Red dashed curves show mode dispersion obtained from the model in Section 4 using Eq. (12). (w,θ) points shown with white circles and black crosses were obtained from Eqs. (18) and (16). White circles predict the BIC locations, whereas black crosses correspond to the “low-Q” resonances. (b) Model reflectance spectrum |R|2 calculated using Eq. (8).
Fig. 5.
Fig. 5. Rigorously calculated (a) excitation angle θ and (b) quality factor Q of the mode versus ridge width w. (c) Angular reflectance spectra of the structure at different ridge widths. Red line shows the spectrum of the structure supporting a BIC. Black points in (b) and (c) depict the BIC location.
Fig. 6.
Fig. 6. Plane wave diffraction by a slab supporting two cross-polarized waves coupled at the interfaces.

Equations (20)

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

θsub=arcsin(ϵsub/nTE),
θTM=arcsin(nTM/nTE),
θTM,r=arcsin(nTM,r/nTE).
tan(kw)=2kγk2γ2,
ϕ=wk02nTE,r2ky2,ψ=wk02nTM,r2ky2.
[U1eiϕV1eiψR]=[r1rctrcr2tcttcr]×[U2V2I],
{U1=eiϕ(r1U2+rcV2+tI),V1=eiψ(rcU2+r2V2+tcI),U2=eiϕ(r1U1+rcV1),V2=eiψ(rcU1+r2V1),R=tU2+tcV2+rI.
R=r+ND1D2.
D1=(eiϕ+r1)(eiψ+r2)rc2,
D2=(eiϕr1)(eiψr2)rc2,
N=r1t2e2iψ+2rctcteiϕeiψ+r2tc2e2iϕ(r1r2rc2)(r2t22rctct+r1tc2).
D1D2=0.
det{I([r1rcrcr2]×[eiϕ00eiψ])2}=0.
{D1=0,N=0.
eiϕ=tcrctr1tc,eiψ=trctcr2t.
eiϕ=±r2r1(r1r2rc2),eiψ=r1r2(r1r2rc2).
ϕ=πm+argtcrctr1tc,ψ=πl+argtrctcr2t.
ω=cwϕ2ψ2nTE,r2nTM,r2,ky=1wϕ2nTM,r2ψ2nTE,r2nTE,r2nTM,r2.
w=1k0ϕ2ψ2nTE,r2nTM,r2,θ=arcsin(1nTEϕ2nTM,r2ψ2nTE,r2ϕ2ψ2).
[r1*rc*t*rc*r2*tc*t*tc*r*]=1detS[rr2tc2ttcrrcrctcr2tttcrrcrr1t2rctr1tcrctcr2trctr1tcr1r2rc2].