Abstract

The oscillation model of the circular membrane is used to calculate the effective refractive index of the two-dimensional triangular photonic crystal at normal incidence within the second photonic band. Negatively effective refractive indices deduced from this model match those calculated by equifrequency surfaces very well. The result reveals that the field distribution has relation with the effective refractive index at certain frequency regions. Besides, the field distribution described by the Bessel function is more compact than the Fourier series expansion.

© 2012 Optical Society of America

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  1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef]
  2. A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
    [CrossRef]
  3. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
    [CrossRef]
  4. A. L. Pokrovsky and A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901 (2002).
    [CrossRef]
  5. M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
    [CrossRef]
  6. C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E 70, 026608 (2004).
  7. D. Felbacq and G. Bouchitté, “Left-handed media and homogenization of photonic crystals,” Opt. Lett. 30, 1189–1191(2005).
    [CrossRef]
  8. A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005).
    [CrossRef]
  9. I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
    [CrossRef]
  10. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
    [CrossRef]
  11. T.-H. Pei and Y.-T. Huang, “Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method,” AIP Advances 2, 012188(2012).
    [CrossRef]
  12. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [CrossRef]
  13. T.-H. Pei and Y.-T. Huang, “The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region,” J. Appl. Phys. 109, 034504 (2011).
    [CrossRef]
  14. T.-H. Pei and Y.-T. Huang, “The equivalent structure and some optical properties of the periodic-defect photonic crystal,” J. Appl. Phys. 109, 073014 (2011).
    [CrossRef]
  15. T.-H. Pei and Y.-T. Huang, “The heterostructure photonic crystal waveguide splitter,” IEEE Photon. Technol. Lett. 23, 1145–1147 (2011).
    [CrossRef]
  16. A. Taflove and S. C. Hagness, Computational Electrodynamics, 2nd ed. (Artech House, 2000).
  17. G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 6th ed. (Academic, 2005).
  18. M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables, 2nd ed. (McGraw-Hill, 1999).
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  20. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1990).

2012

T.-H. Pei and Y.-T. Huang, “Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method,” AIP Advances 2, 012188(2012).
[CrossRef]

2011

T.-H. Pei and Y.-T. Huang, “The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region,” J. Appl. Phys. 109, 034504 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The equivalent structure and some optical properties of the periodic-defect photonic crystal,” J. Appl. Phys. 109, 073014 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The heterostructure photonic crystal waveguide splitter,” IEEE Photon. Technol. Lett. 23, 1145–1147 (2011).
[CrossRef]

2006

I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

2005

D. Felbacq and G. Bouchitté, “Left-handed media and homogenization of photonic crystals,” Opt. Lett. 30, 1189–1191(2005).
[CrossRef]

A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005).
[CrossRef]

2004

C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E 70, 026608 (2004).

2003

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

2002

A. L. Pokrovsky and A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

2000

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

1998

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

1991

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef]

1987

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 6th ed. (Academic, 2005).

Bouchitté, G.

Bulu, I.

I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
[CrossRef]

Caglayan, H.

I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
[CrossRef]

Decoopman, T.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

Enoch, S.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Felbacq, D.

Garcia, N.

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef]

Genack, A. Z.

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Gralak, B.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics, 2nd ed. (Artech House, 2000).

Harris, F.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 6th ed. (Academic, 2005).

Huang, Y.-T.

T.-H. Pei and Y.-T. Huang, “Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method,” AIP Advances 2, 012188(2012).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region,” J. Appl. Phys. 109, 034504 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The equivalent structure and some optical properties of the periodic-defect photonic crystal,” J. Appl. Phys. 109, 073014 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The heterostructure photonic crystal waveguide splitter,” IEEE Photon. Technol. Lett. 23, 1145–1147 (2011).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1990).

Joannopoulos, J. D.

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Johnson, Steven G.

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Kuo, C.-H.

C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E 70, 026608 (2004).

Liu, J.

M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables, 2nd ed. (McGraw-Hill, 1999).

Martí, J.

A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Martínez, A.

A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Maystre, D.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Ozbay, E.

I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
[CrossRef]

Pei, T.-H.

T.-H. Pei and Y.-T. Huang, “Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method,” AIP Advances 2, 012188(2012).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region,” J. Appl. Phys. 109, 034504 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The equivalent structure and some optical properties of the periodic-defect photonic crystal,” J. Appl. Phys. 109, 073014 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The heterostructure photonic crystal waveguide splitter,” IEEE Photon. Technol. Lett. 23, 1145–1147 (2011).
[CrossRef]

Pendry, J. B.

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

Povinelli, M. L.

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Spiegel, M. R.

M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables, 2nd ed. (McGraw-Hill, 1999).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics, 2nd ed. (Artech House, 2000).

Tayeb, G.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

Weber, H. J.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 6th ed. (Academic, 2005).

Ye, Z.

C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E 70, 026608 (2004).

AIP Advances

T.-H. Pei and Y.-T. Huang, “Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method,” AIP Advances 2, 012188(2012).
[CrossRef]

Appl. Phys. Lett.

M. L. Povinelli, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

T.-H. Pei and Y.-T. Huang, “The heterostructure photonic crystal waveguide splitter,” IEEE Photon. Technol. Lett. 23, 1145–1147 (2011).
[CrossRef]

J. Appl. Phys.

T.-H. Pei and Y.-T. Huang, “The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region,” J. Appl. Phys. 109, 034504 (2011).
[CrossRef]

T.-H. Pei and Y.-T. Huang, “The equivalent structure and some optical properties of the periodic-defect photonic crystal,” J. Appl. Phys. 109, 073014 (2011).
[CrossRef]

J. Phys. Conf. Ser.

I. Bulu, H. Caglayan, and E. Ozbay, “Negative refraction and focusing of electromagnetic waves by photonic crystals,” J. Phys. Conf. Ser. 36, 33–40 (2006).
[CrossRef]

Opt. Lett.

Phys. Rev. B

A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998).
[CrossRef]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

Phys. Rev. E

C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E 70, 026608 (2004).

Phys. Rev. Lett.

A. L. Pokrovsky and A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics, 2nd ed. (Artech House, 2000).

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 6th ed. (Academic, 2005).

M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables, 2nd ed. (McGraw-Hill, 1999).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1990).

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Figures (5)

Fig. 1.
Fig. 1.

(a) PBS of a 2D triangular photonic crystal with εa=12.25, εb=1.0, and r=0.42a. The bold curve denotes second photonic band. (b) EFSs in the first Brillouin zone where frequencies are uniformly between 0.345 and 0.365(c/a) from the outermost curve to the innermost one.

Fig. 2.
Fig. 2.

(a) Ez-field distribution of the TM-polarization Bloch wave with frequency 0.36(c/a) in a 2D triangular PhC. Some air holes are denoted as black circles; (b) Field distribution of the rectangular region in (a). Each unit can be covered by a circle.

Fig. 3.
Fig. 3.

Wavelengths of the Bloch waves from 0.345 to 0.365(c/a) calculated by the FDTD method.

Fig. 4.
Fig. 4.

(a) Effective refractive indices from 0.345 to 0.365(c/a) calculated by Eq. (8) and EFSs, respectively. (b) Field distribution in an elliptic unit at 0.345(c/a).

Fig. 5.
Fig. 5.

Group velocities from 0.3475 to 0.3625(c/a) calculated by data in Fig. 4(a).

Equations (11)

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

(2r2+1rr+1r22ϕ2)Ψ=1vc22t2Ψ,
Tvc2T=αmn2,
ΦΦ=m2,
r2R+rR+(αmn2r2m2)R=0,
Ψ(r,ϕ)=m=0n=1Jm(kmnrrc)[pmncos(mϕ)+qmnsin(mϕ)],
α31vc=K31vc/rc=ωr=2πfc,
vp=fc2πrc/K31,
|n|=11K31a/16πf¯λBloch,
λBloch=Aa/(f¯edgef¯),
|n|=11K31(f¯edgef¯)/16πAf¯.
vg=cn+f(dn/df),

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