Abstract

We materialized the isotropic Dirac-cone dispersion relation in the mid-infrared range by fabricating photonic crystal slabs of the C4v symmetry in SOI (silicon-on-insulator) wafers by electron beam lithography. The dispersion relation was examined by the angle-resolved reflection spectra with our home-made high-resolution apparatus, which showed a good agreement with the dispersion relation and the reflection spectra calculated by the finite element method. The reflection spectra also agreed with the selection rules derived from the spatial symmetry of the Dirac-cone modes, which proved to be a powerful tool for the mode assignment.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
    [Crossref]
  2. C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
    [Crossref]
  3. A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
    [Crossref]
  4. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18(26), 27371–27386 (2010).
    [Crossref]
  5. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
    [Crossref]
  6. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
    [Crossref]
  7. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
    [Crossref]
  8. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
    [Crossref]
  9. K. Sakoda, “Photonic Dirac cones and relevant physics,” in Electromagnetic Metamaterials, K. Sakoda, ed. (Springer-Verlag, 2019), Chap. 16.
  10. Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016).
    [Crossref]
  11. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
    [Crossref]
  12. K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
    [Crossref]
  13. K. Sakoda and H. Takeda, “Dirac cones in photonic crystal slabs,” in Program and Abstract Book of 10th International Symposium on Modern Optics and its Applications (2015), pp. 68–69.
  14. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
    [Crossref]
  15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).
  16. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
  17. D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005).
    [Crossref]
  18. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965).
    [Crossref]
  19. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001).
    [Crossref]

2016 (1)

Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016).
[Crossref]

2015 (1)

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

2014 (1)

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

2012 (4)

2011 (1)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

2010 (1)

2005 (1)

D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005).
[Crossref]

2004 (3)

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
[Crossref]

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
[Crossref]

2001 (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001).
[Crossref]

1965 (1)

Amirtharaj, P. M.

D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005).
[Crossref]

Caloz, C.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
[Crossref]

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
[Crossref]

Chan, C. T.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Chandler-Horowitz, D.

D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005).
[Crossref]

Chua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Hsu, C. W.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).

Itoh, T.

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
[Crossref]

Joannopoulos, J. D.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Malitson, I. H.

Mei, J.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001).
[Crossref]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).

Pick, A.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Sakoda, K.

Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016).
[Crossref]

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
[Crossref]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
[Crossref]

K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
[Crossref]

K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18(26), 27371–27386 (2010).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001).
[Crossref]

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).

K. Sakoda, “Photonic Dirac cones and relevant physics,” in Electromagnetic Metamaterials, K. Sakoda, ed. (Springer-Verlag, 2019), Chap. 16.

K. Sakoda and H. Takeda, “Dirac cones in photonic crystal slabs,” in Program and Abstract Book of 10th International Symposium on Modern Optics and its Applications (2015), pp. 68–69.

Sanada, A.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
[Crossref]

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
[Crossref]

Soljacic, M.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Takeda, H.

K. Sakoda and H. Takeda, “Dirac cones in photonic crystal slabs,” in Program and Abstract Book of 10th International Symposium on Modern Optics and its Applications (2015), pp. 68–69.

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).

Wu, Y.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Yao, Y.

Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016).
[Crossref]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Zhou, H.-F.

IEEE Microw. Wirel. Compon. Lett. (1)

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004).
[Crossref]

J. Appl. Phys. (1)

D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Soc. Jpn. (1)

Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016).
[Crossref]

Nat. Mater. (1)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref]

Nature (1)

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Opt. Express (4)

Phys. Rev. A (1)

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

Phys. Rev. B (2)

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001).
[Crossref]

Other (4)

K. Sakoda and H. Takeda, “Dirac cones in photonic crystal slabs,” in Program and Abstract Book of 10th International Symposium on Modern Optics and its Applications (2015), pp. 68–69.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).

K. Sakoda, “Photonic Dirac cones and relevant physics,” in Electromagnetic Metamaterials, K. Sakoda, ed. (Springer-Verlag, 2019), Chap. 16.

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Figures (9)

Fig. 1.
Fig. 1. (a) Dirac cone with an auxiliary flat dispersion surface (dotted lines), and (b) double Dirac cone on the $\Gamma$ point ($\textbf {k} = 0$) of the two-dimensional Brillouin zone materialized by accidental degeneracy of two modes with particular combinations of their spatial symmetries.
Fig. 2.
Fig. 2. Configuration of the incident plane wave for the angle-resolved reflection measurement. $\theta$ and $\phi$ denote the tilt angle from the normal ($z$) direction and the azimuthal angle from the $x$ axis, respectively. The inset shows the three symmetric points of the first Brillouin zone.
Fig. 3.
Fig. 3. Illustration of the specimen structure. Two-dimensional photonic crystal slabs were fabricated in the top silicon layer of SOI wafers, which consisted of a regular-square array of cylindrical air holes. Thickness of the top silicon layer, 400 nm; thickness of the SiO$_2$ layer, 3000 nm; lattice constant $a$, 2270 nm; air-hole radius $R$, 440 - 620 nm; typical air-hole depth $d$, 210 nm.
Fig. 4.
Fig. 4. Dispersion relation of photonic crystal slabs fabricated in SOI wafers. The radius and depth of air cylinders were assumed to be (a) $R=440$ nm , $d=210$ nm and (b) $R=526$ nm, $d=216$ nm. The vertical axis is the frequency of the electromagnetic eigenmodes and the horizontal axis is the wave vector in the first Brillouin zone. The dispersion relation is plotted in the $\Gamma$-to-M and $\Gamma$-to-X directions. M/10, for example, implies that the horizontal axis is magnified by 10 times.
Fig. 5.
Fig. 5. Angle-resolved reflection spectra calculated for an incident light tilted from the normal direction to the (1,0) ($\Gamma$-to-X) direction. The incident light was assumed to be polarized (a) perpendicular and (b) parallel to the incidence plane. In each figure, the upper and lower limits of the reflection spectrum for $\theta = 0^{\circ }$ are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by unity from each other in the vertical direction. The following specimen parameters were assumed to materialize the Dirac cone with a flat band: $a=2270$ nm, $d=216$ nm, $R=526$ nm.
Fig. 6.
Fig. 6. (a) Photo and (b), (c) SEM images of photonic crystal slabs fabricated in the top silicon layer of an SOI wafer. Seven specimens with different air-hole radii were fabricated on the same wafer.
Fig. 7.
Fig. 7. Our home-made optics for a high-resolution angle-resolved reflection spectroscopy fabricated in the sample chamber of an FT-IR spectrometer (JASCO 6800).
Fig. 8.
Fig. 8. Angle-resolved reflection spectra for an incident beam tilted to (a) the $\Gamma$-to-$X$ direction ($\phi =0^{\circ }$) and (b) the $\Gamma$-to-$M$ direction ($\phi =45^{\circ }$) . The upper and lower panels are for the s and p polarizations, respectively. 27 spectra were measured for different incident angles by $0.292^{\circ }$ steps for each panel. In each panel, the upper and lower limits of the lowest reflection spectrum are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by 0.1 from each other in the vertical direction. Sample parameters: $a= 2270$ nm, $R=440$ nm, $d= 210$ nm.
Fig. 9.
Fig. 9. Angle-resolved reflection spectra for the specimen closest to the Dirac-cone dispersion relation. The incident beam was tilted to (a) the $\Gamma$-to-$X$ direction ($\phi =0^{\circ }$) and (b) the $\Gamma$-to-$M$ direction ($\phi =45^{\circ }$). The upper and lower panels are for the s and p polarizations, respectively. DC, FB, and $A_{1}$ denote the reflection peaks of the Dirac cone, the flat band, and the $A_{1}$-symmetric mode. 27 spectra were measured for different incident angles by $0.292^{\circ }$ steps for each panel. In each panel, the upper and lower limits of the lowest reflection spectrum are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by 0.1 from each other in the vertical direction. Sample parameters: $a= 2270$ nm, $R= 530$ nm, $d= 210$ nm.

Tables (1)

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Table 1. Selection rules for reflection peaks.

Equations (6)

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C k = ( 0 0 b k y 0 0 b k x b k y b k x 0 ) ,
ω k = { ω D ( F B : f l a t   b a n d ) , ω D ± | b | c 2 k 2 ω D ( D C : D i r a c   c o n e ) .
H k ( F B ) ( r ) = 1 k e i k r { k x u 01 ( r ) k y u 02 ( r ) } ,
H k ( D C ) ( r ) = 1 2 k e i k r { ± k y e i β u 01 ( r ) ± k x e i β u 02 ( r ) + k u 03 ( r ) } ,
σ y u 01 = u 01 , σ d u 01 = u 02 , σ y u 02 = u 02 , σ d u 02 = u 01 , σ y u 03 = u 03 , σ d u 03 = u 03 .
F o r   ϕ = 0 , σ y H k ( F B ) = H k ( F B ) , σ y H k ( D C ) = H k ( D C ) , F o r   ϕ = 45 , σ d H k ( F B ) = H k ( F B ) , σ d H k ( D C ) = H k ( D C ) .

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