Abstract

A Fourier–Bessel (FB) basis is used to solve two-dimensional (2D) cylindrical Maxwell’s equations for localized states within dielectric structures that possess rotational symmetry. The technique is used to determine the wavelengths and profiles of the stationary states supported by the structure and identify the bandgaps. 12-fold quasi-crystals for the TE and TM polarizations are analyzed. Since the FB approach with 2D photonic crystals in this fashion is new, the accuracy of the results is confirmed using finite-difference time-domain simulations.

© 2012 Optical Society of America

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References

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  1. A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E: Low-dimens. Syst. Nanostruct. 42, 1871–1895 (2010).
    [CrossRef]
  2. E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 036502 (2012).
    [CrossRef]
  3. C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
    [CrossRef]
  4. X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
    [CrossRef]
  5. R. Gauthier and K. Mnaymneh, “Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis,” Opt. Express 13, 1985–1998 (2005).
    [CrossRef]
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    [CrossRef]
  8. S. R. Newman and R. C. Gauthier, “Representation of photonic crystals and their localized modes through the use of Fourier-Bessel expansions,” IEEE Photon. J. 3, 1133–1141 (2011).
    [CrossRef]
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  14. S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
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  15. R. C. Gauthier and A. Ivanov, “Production of quasi-crystal template patterns using a dual beam multiple exposure technique,” Opt. Express 12, 990–1003 (2004).
    [CrossRef]
  16. S. R. Newman and R. C. Gauthier, “FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals,” Proc. SPIE 7223, 72230R (2009).
    [CrossRef]
  17. R. Gauthier, “FDTD analysis of out-of-plane propagation in 12-fold photonic quasi-crystals,” Opt. Commun. 269, 395–410 (2007).
    [CrossRef]

2012

E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 036502 (2012).
[CrossRef]

2011

S. R. Newman and R. C. Gauthier, “Representation of photonic crystals and their localized modes through the use of Fourier-Bessel expansions,” IEEE Photon. J. 3, 1133–1141 (2011).
[CrossRef]

2010

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E: Low-dimens. Syst. Nanostruct. 42, 1871–1895 (2010).
[CrossRef]

2009

S. R. Newman and R. C. Gauthier, “FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals,” Proc. SPIE 7223, 72230R (2009).
[CrossRef]

2007

R. Gauthier, “FDTD analysis of out-of-plane propagation in 12-fold photonic quasi-crystals,” Opt. Commun. 269, 395–410 (2007).
[CrossRef]

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

A. Ortega-Monux, J. G. Wanguemert-Perez, and I. Molina-Fernandez, “Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver,” IEEE Photon. Technol. Lett 19, 414–416 (2007).
[CrossRef]

2006

2005

2004

2001

S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
[CrossRef]

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
[CrossRef]

1999

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

1998

1990

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef]

Andres, M.

Andres, P.

Ban, S.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Chan, C.

X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef]

Charlton, M. D. B.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Cheng, B.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Crawford, G. P.

Gauthier, R.

R. Gauthier, “FDTD analysis of out-of-plane propagation in 12-fold photonic quasi-crystals,” Opt. Commun. 269, 395–410 (2007).
[CrossRef]

R. Gauthier and K. Mnaymneh, “Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis,” Opt. Express 13, 1985–1998 (2005).
[CrossRef]

Gauthier, R. C.

S. R. Newman and R. C. Gauthier, “Representation of photonic crystals and their localized modes through the use of Fourier-Bessel expansions,” IEEE Photon. J. 3, 1133–1141 (2011).
[CrossRef]

S. R. Newman and R. C. Gauthier, “FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals,” Proc. SPIE 7223, 72230R (2009).
[CrossRef]

R. C. Gauthier and A. Ivanov, “Production of quasi-crystal template patterns using a dual beam multiple exposure technique,” Opt. Express 12, 990–1003 (2004).
[CrossRef]

Gorkhali, S. P.

Guan, S.

S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, 2000).

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef]

Ivanov, A.

Ivchenko, E. L.

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E: Low-dimens. Syst. Nanostruct. 42, 1871–1895 (2010).
[CrossRef]

Jin, C.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Joannopoulos, J.

Johnson, S.

Lai, C. H.

S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
[CrossRef]

Lee, T.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Li, Z.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Maciá, E.

E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 036502 (2012).
[CrossRef]

Man, B.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Mnaymneh, K.

Molina-Fernandez, I.

A. Ortega-Monux, J. G. Wanguemert-Perez, and I. Molina-Fernandez, “Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver,” IEEE Photon. Technol. Lett 19, 414–416 (2007).
[CrossRef]

Newman, S. R.

S. R. Newman and R. C. Gauthier, “Representation of photonic crystals and their localized modes through the use of Fourier-Bessel expansions,” IEEE Photon. J. 3, 1133–1141 (2011).
[CrossRef]

S. R. Newman and R. C. Gauthier, “FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals,” Proc. SPIE 7223, 72230R (2009).
[CrossRef]

Ortega-Monux, A.

A. Ortega-Monux, J. G. Wanguemert-Perez, and I. Molina-Fernandez, “Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver,” IEEE Photon. Technol. Lett 19, 414–416 (2007).
[CrossRef]

Poddubny, A. N.

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E: Low-dimens. Syst. Nanostruct. 42, 1871–1895 (2010).
[CrossRef]

Qi, J.

Silvestre, E.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef]

Sun, B.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, 2000).

Wanguemert-Perez, J. G.

A. Ortega-Monux, J. G. Wanguemert-Perez, and I. Molina-Fernandez, “Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver,” IEEE Photon. Technol. Lett 19, 414–416 (2007).
[CrossRef]

Wei, G. W.

S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
[CrossRef]

Zhang, D.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Zhang, X.

X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
[CrossRef]

Zhang, Z.-Q.

X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
[CrossRef]

Zoorob, M. E.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Appl. Phys. Lett.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

IEEE Photon. J.

S. R. Newman and R. C. Gauthier, “Representation of photonic crystals and their localized modes through the use of Fourier-Bessel expansions,” IEEE Photon. J. 3, 1133–1141 (2011).
[CrossRef]

IEEE Photon. Technol. Lett

A. Ortega-Monux, J. G. Wanguemert-Perez, and I. Molina-Fernandez, “Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver,” IEEE Photon. Technol. Lett 19, 414–416 (2007).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Commun.

R. Gauthier, “FDTD analysis of out-of-plane propagation in 12-fold photonic quasi-crystals,” Opt. Commun. 269, 395–410 (2007).
[CrossRef]

Opt. Express

Phys. Rev. B

X. Zhang, Z.-Q. Zhang, and C. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63, 1–4 (2001).
[CrossRef]

Phys. Rev. Lett.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef]

Physica D

S. Guan, C. H. Lai, and G. W. Wei, “Fourier-Bessel analysis of patterns in a circular domain,” Physica D 151, 83–98 (2001).
[CrossRef]

Physica E: Low-dimens. Syst. Nanostruct.

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E: Low-dimens. Syst. Nanostruct. 42, 1871–1895 (2010).
[CrossRef]

Proc. SPIE

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

S. R. Newman and R. C. Gauthier, “FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals,” Proc. SPIE 7223, 72230R (2009).
[CrossRef]

Rep. Prog. Phys.

E. Maciá, “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 036502 (2012).
[CrossRef]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, 2000).

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Figures (8)

Fig. 1.
Fig. 1.

Dielectric layouts for the 12-fold PQCs for the TE (top) and TM (bottom) polarizations. The TE structure is an array of air holes (ϵr=1.0000) with radii of 92.7 nm in a silicon (ϵr=12.1104) background. The structure used for the TM polarization is an array of silicon rods with radii of 92.7 nm in an air background.

Fig. 2.
Fig. 2.

Magnitude of the FB expansion coefficients for the hole-based structure (top) and rod-based strcuture (bottom). The expansions were done for a 10 μm diameter circle about the center of the PQCs with 200 Bessel and 200 exponential terms.

Fig. 3.
Fig. 3.

FB expansion (top) and FDTD (bottom) results for the rod-based 12-fold quasi-crystal structure with a TM polarized field.

Fig. 4.
Fig. 4.

Reconstructed TM modes within the rod-based 12-fold PQC that are centrally localized, calculated with the FB approach.

Fig. 5.
Fig. 5.

Reconstructed TM modes for the 12-fold rod-based PQC that are localized at an extended distance from the center, calculated with the FB approach.

Fig. 6.
Fig. 6.

Reconstructed TM modes for the 12-fold rod-based PQC that are either interface or surface states calculated with the FB approach.

Fig. 7.
Fig. 7.

FB expansion (top) and FDTD (bottom) results for the hole-based 12-fold quasi-crystal structure with a TE-polarized field.

Fig. 8.
Fig. 8.

Reconstructed TE modes within the hole-based 12-fold PQC that are centrally localized, calculated with the FB approach.

Equations (15)

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

×1ϵr(r)×H(r)=(ωc)2H(r).
1r(r[rϵr(Hzr)]dϕ[1ϵrrHzϕ])=(ωc)2Hz.
f(r,ϕ)=m,nκm,nJm(ρm,nr)ejmϕ,
J0(r)r=J1(r),
2J0(z)r2=J0(z)1zJ1(z).
κm,n=02π0brf(r,ϕ)J0(ρnr)ejmϕdrdϕπb2J12(ρnb).
Hz=mHz,nκmHz,nHzJo(ρnr)ejmHzϕ,
1ϵr=mϵ,qκmϵ,qϵJo(ρqr)ejmϵϕ.
mHz,n,q(ρn2Sn,n,q+mHzmHzTn,n,qρnρqUn,n,q)κ(mHzmHz),qϵκmHz,nHz=(ωc)2κmHz,nHz,
Sn,n,q=20brJ0(ρnr)J0(ρnr)J0(ρqr)drb2J12(ρnb),
Tn,n,q=2δrb1rJ0(ρnr)J0(ρnr)J0(ρqr)drb2J12(ρnb),
Un,n,q=20brJ0(ρnr)J1(ρnr)J1(ρqr)drb2J12(ρnb).
1ϵr(r)××E(r)=(ωc)2E(r).
Ez=mEz,nκmEz,nEzJo(ρnr)ejmEzϕ.
mEz,q,n(ρn2Sn,n,q+mEz2Tn,n,q)κ(mEzmEz),qϵκmEz,nEz=(ωc)2κmEz,nEZ,

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