Abstract

We determine the optimal thicknesses for which omnidirectional reflection from generalized Fibonacci quasicrystals occurs. By capitalizing on the idea of wavelength- and angle-averaged reflectance, we assess in a consistent way the performance of the different systems. Our results indicate that some of these aperiodic arrangements can largely over-perform the conventional photonic crystals as omnidirectional reflection is concerned.

© 2013 Optical Society of America

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2013 (3)

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics7, 197–204 (2013).
[CrossRef]

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics7, 177–187 (2013).
[CrossRef]

C. H. O. Costa and M. S. Vasconcelos, “Band gaps and transmission spectra in generalized Fibonacci σ(p, q) one-dimensional magnonic quasicrystals,” J. Phys. Condens. Matter25, 286002 (2013).
[CrossRef]

2012 (2)

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariñena, “The transfer matrix: A geometrical perspective,” Phys. Rep.513, 191–227 (2012).
[CrossRef]

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6, 178–218 (2012).
[CrossRef]

2011 (3)

V. Grigoriev and F. Biancalana, “Exact analytical representations for broadband transmission properties of quarter-wave multilayers,” Opt. Lett.36, 3774–3776 (2011).
[CrossRef] [PubMed]

Z. Zhang, P. Tong, J. Gong, and B. Li, “Wave packet dynamics in one-dimensional linear and nonlinear generalized Fibonacci lattices,” Phys. Rev. E83, 056205 (2011).
[CrossRef]

S. Thiem, M. Schreiber, and U. Grimm, “Light transmission through metallic-mean quasiperiodic stacks with oblique incidence,” Philos. Mag.91, 2801–2810 (2011).
[CrossRef]

2010 (2)

S. Thiem and M. Schreiber, “Photonic properties of metallic-mean quasiperiodic chains,” Eur. Phys. J. B76, 339–345 (2010).
[CrossRef]

A. Poddubny and E. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E42, 1871–1895 (2010).
[CrossRef]

2009 (1)

S. Thiem, M. Schreiber, and U. Grimm, “Wave packet dynamics, ergodicity, and localization in quasiperiodic chains,” Phys. Rev. B80, 214203 (2009).
[CrossRef]

2008 (1)

S. V. Zhukovsky and S. V. Gaponenko, “Constraints on transmission, dispersion, and density of states in dielectric multilayers and stepwise potential barriers with an arbitrary layer arrangement,” Phys. Rev. E77, 046602 (2008).
[CrossRef]

2007 (3)

A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe, “Integral merit function for broadband omnidirectional mirrors,” Appl. Opt.46, 2903–2906 (2007).
[CrossRef] [PubMed]

W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D40, R229–R247 (2007).
[CrossRef]

K. Buscha, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegenerd, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007).
[CrossRef]

2006 (3)

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys.69, 397–441 (2006).
[CrossRef]

J. A. Monsoriu, F. R. Villatoro, M. J. Marín, J. Pérez, and L. Monreal, “Quantum fractal superlattices,” Am. J. Phys.74, 831–836 (2006).
[CrossRef]

Y. Chen, X. Yang, Q. Guo, and S. Lan, “Second-harmonic generation in GF(m, 1) ferroelectric superlattices,” J. Phys. Condens. Matter18, 2587–2600 (2006).
[CrossRef]

2005 (1)

A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe, “Comparing omnidirectional reflection from periodic and quasiperiodic one-dimensional photonic crystals,” Opt. Express13, 3913–3920 (2005).
[CrossRef] [PubMed]

2004 (3)

S. V. Zhukovsky, A. V. Lavrinenko, and S. V. Gaponenko, “Spectral scalability as a result of geometrical self-similarity in fractal multilayers,” Europhys. Lett.66, 455–461 (2004).
[CrossRef]

F. Qiu, R. W. Peng, X. Q. Huang, X. F. Hu, M. Wang, A. Hu, S. S. Jiang, and D. Feng, “Omnidirectional reflection of electromagnetic waves on Thue-Morse dielectric multilayers,” Europhys. Lett.68, 658–663 (2004).
[CrossRef]

T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A6, 127–131 (2004).
[CrossRef]

2002 (1)

A. V. Lavrinenko, S. V. Zhukovsky, K. S. Sandomirski, and S. V. Gaponenko, “Propagation of classical waves in nonperiodic media: Scaling properties of an optical cantor filter,” Phys. Rev. E65, 036621 (2002).
[CrossRef]

2001 (2)

E. Cojocaru, “Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,” Appl. Opt.40, 6319–6326 (2001).
[CrossRef]

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun.198, 273–279 (2001).
[CrossRef]

2000 (2)

X. Wang, U. Grimm, and M. Schreiber, “Trace and antitrace maps for aperiodic sequences: Extensions and applications,” Phys. Rev. B62, 14020–14031 (2000).
[CrossRef]

J. Lekner, “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A2, 349–352 (2000).
[CrossRef]

1999 (4)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightwave Technol.17, 2018–2024 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A68, 25–28 (1999).
[CrossRef]

V. W. Spinadel, “The metallic means family and multifractal spectra,” Nonlinear Anal.36, 721–745 (1999).
[CrossRef]

W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt.38, 5464–5467 (1999).
[CrossRef]

1998 (4)

C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B15, 1947–1952 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science282, 1679–1682 (1998).
[CrossRef] [PubMed]

J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?” Science282, 1841–1842 (1998).
[CrossRef]

E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett.23, 1648–1649 (1998).
[CrossRef]

1997 (2)

N.-H. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B55, 3543–3547 (1997).
[CrossRef]

X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, “Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices,” Phys. Rev. B55, 2882–2889 (1997).
[CrossRef]

1996 (2)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep.270, 143–215 (1996).
[CrossRef]

E. Maciá and F. Domínguez-Adame, “Physical nature of critical wave functions in Fibonacci systems,” Phys. Rev. Lett.76, 2957–2960 (1996).
[CrossRef] [PubMed]

1995 (1)

B. A. van Tiggelen, “Transverse diffusion of light in Faraday-active media,” Phys. Rev. Lett.75, 422–424 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

G. Y. Oh and M. H. Lee, “Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices,” Phys. Rev. B48, 12465–12477 (1993).
[CrossRef]

B. Kramer and A. MacKinnon, “Localization: Theory and experiment,” Rep. Prog. Phys.56, 1469–1564 (1993).
[CrossRef]

1992 (3)

C. Godrèche and J. M. Luck, “Indexing the diffraction spectrum of a non-Pisot self-similar structure,” Phys. Rev. B45, 176–185 (1992).
[CrossRef]

J. Bellissard, A. Bovier, and J. M. Ghez, “Gap labelling theorems for one-dimensional discrete Schrödinger operators,” Rev. Math. Phys.4, 1–37 (1992).
[CrossRef]

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B45, 105–114 (1992).
[CrossRef]

1991 (2)

A. Chakrabarti and S. N. Karmakar, “Renormalization-group method for exact Green’s functions of self-similar lattices: Application to generalized Fibonacci chains,” Phys. Rev. B44, 896–899 (1991).
[CrossRef]

M. Kolář, M. K. Ali, and F. Nori, “Generalized Thue-Morse chains and their physical properties,” Phys. Rev. B43, 1034–1047 (1991).
[CrossRef]

1990 (2)

M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B42, 3680–3689 (1990).
[CrossRef]

Z. Cheng and R. Savit, “Structure factor of substitutional sequences,” J. Stat. Phys.60, 383–393 (1990).
[CrossRef]

1989 (6)

M. Severin and R. Riklund, “Using the Fourier spectrum to classify families of generalised extensions of the Fibonaccian lattice,” J. Phys. Condens. Matter1, 5607–5612 (1989).
[CrossRef]

J. A. Dobrowolski, F. C. Ho, A. Belkind, and V. A. Koss, “Merit functions for more effective thin film calculations,” Appl. Opt.28, 2824–2831 (1989).
[CrossRef] [PubMed]

M. Severin, M. Dulea, and R. Riklund, “Periodic and quasiperiodic wavefunctions in a class of one-dimensional quasicrystals: an analytical treatment,” J. Phys. Condens. Matter1, 8851–8858 (1989).
[CrossRef]

J. M. Luck, “Cantor spectra and scaling of gap widths in deterministic aperiodic systems,” Phys. Rev. B39, 5834–5849 (1989).
[CrossRef]

S. Tamura and F. Nori, “Transmission and frequency spectra of acoustic phonons in Thue-Morse superlattices,” Phys. Rev. B40, 9790–9801 (1989).
[CrossRef]

T. Fujiwara, M. Kohmoto, and T. Tokihiro, “Multifractal wave functions on a Fibonacci lattice,” Phys. Rev. B40, 7413–7416 (1989).
[CrossRef]

1988 (2)

G. Gumbs and M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett.60, 1081–1084 (1988).
[CrossRef] [PubMed]

M. Holzer, “Three classes of one-dimensional, two-tile Penrose tilings and the Fibonacci Kronig-Penney model as a generic case,” Phys. Rev. B38, 1709–1720 (1988).
[CrossRef]

1987 (2)

E. Bombieri and J. Taylor, “Quasicrystals, tilings, and algebraic number theory,” Contemp. Math.64, 241–264 (1987).
[CrossRef]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: Quasiperiodic media,” Phys. Rev. Lett.58, 2436–2438 (1987).
[CrossRef] [PubMed]

1986 (2)

F. Nori and J. P. Rodríguez, “Acoustic and electronic properties of one-dimensional quasicrystals,” Phys. Rev. B34, 2207–2211 (1986).
[CrossRef]

E. Bombieri and J. E. Taylor, “Which distributions of matter diffract? an initial investigation,” J. Phys. Colloq.47, 19–28 (1986).
[CrossRef]

1985 (1)

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, and P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett.55, 1768–1770 (1985).
[CrossRef] [PubMed]

1983 (1)

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S. Thiem, M. Schreiber, and U. Grimm, “Light transmission through metallic-mean quasiperiodic stacks with oblique incidence,” Philos. Mag.91, 2801–2810 (2011).
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M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics7, 197–204 (2013).
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M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics7, 197–204 (2013).
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X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, “Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices,” Phys. Rev. B55, 2882–2889 (1997).
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W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D40, R229–R247 (2007).
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M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: Quasiperiodic media,” Phys. Rev. Lett.58, 2436–2438 (1987).
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S. Tamura and F. Nori, “Transmission and frequency spectra of acoustic phonons in Thue-Morse superlattices,” Phys. Rev. B40, 9790–9801 (1989).
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S. Thiem, M. Schreiber, and U. Grimm, “Light transmission through metallic-mean quasiperiodic stacks with oblique incidence,” Philos. Mag.91, 2801–2810 (2011).
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E. Bombieri and J. E. Taylor, “Which distributions of matter diffract? an initial investigation,” J. Phys. Colloq.47, 19–28 (1986).
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J. Phys. Condens. Matter (1)

Y. Chen, X. Yang, Q. Guo, and S. Lan, “Second-harmonic generation in GF(m, 1) ferroelectric superlattices,” J. Phys. Condens. Matter18, 2587–2600 (2006).
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Figures (9)

Fig. 1
Fig. 1

Illustrating different arrangements considered in this work. In the top panel, we have the periodic case (40 letters). In the mid panel, the Olympic-metal family FS (h,1), with (from left to right) h = 1 (golden mean, 34 letters), h = 2 (silver mean, 41 letters), and h = 3 (bronze mean, 43 letters). In the bottom panel, the non-Olympic-metal family FS (1, ), with = 2 (copper mean, 43 letters) and = 3 (nickel mean, 40 letters). The differences can be appreciated at a simple glance.

Fig. 2
Fig. 2

Normalized power spectrum for words up to 1500 letters for different generalized Fibonacci sequences. In the left panel, for the Olympic-metal family FS(h,1), with h = 1, 2, and 3. In the right panel, for the non-Olympic-metal sequences FS(1,2) and FS(1,3).

Fig. 3
Fig. 3

Logarithm (with changed sign) of 1−ℛ̄α [ℛ̄α is the angle-averaged reflectance Eq. (21)] for the optimal thicknesses and several generations of different generalized Fibonacci sequences as a function of the number of layers Nα.

Fig. 4
Fig. 4

Averaged reflectance ℛ̄α as a function of the adimensional thicknesses nLdL and nHdH (at a working wavelength of 0.65 μm) for the periodic system with 218 layers (left panel) and the nickel-mean system with 217 layers (right panel). In both figures, we include a plane of constant reflectance 0.98.

Fig. 5
Fig. 5

Contours of ℛ̄α = 0.98, as a function of the adimensional thicknesses nLdL and nHdH for the systems: periodic (218 layers), Olympic-metal family [gold (233 layers), silver (239 layers), bronze (142 layers)], and non-Olympic-metal family [copper (171 layers) and nickel (217 layers)].

Fig. 6
Fig. 6

Logarithm (with changed sign) of the 1 − ℛ̄α [ℛ̄α is the wavelength- and angle-averaged reflectance (20)] for the optimal thicknesses and the same generations of different generalized Fibonacci sequences as in Fig. 3, as a function of the number of layers.

Fig. 7
Fig. 7

Reflectance α (θ, λ) for the periodic (left panel, 218 layers) and the nickel-mean (right panel, 217 layers) sequences, as a function of the wavelength λ (in μm) and the incidence angle θ (in degrees). We have used the optimal thicknesses (in μm) (dL = 0.1391, dH = 0.0568) for the periodic and (dL = 0.0446, dH = 0.0604) for the nickel sequences.

Fig. 8
Fig. 8

Wavelength-averaged reflectance α (θ) versus the angle of incidence (left panel) and angle-averaged reflectance α (λ) versus the wavelength (right panel), for the same examples as in Fig. 7. Black continuous line for the periodic system and purple broken line for the nickel sequence. For clarity, we have included an inset of the region where α (λ) is greater for the periodic structure.

Fig. 9
Fig. 9

Contour plots of ℛ̄α = 0.90 in the broadband case, for the same arrangements as in Fig. 5.

Tables (1)

Tables Icon

Table 1 Fitting parameters of the narrowband model Eq. (23) and the broadband model Eq. (25). In both cases, we include the Pearson correlation coefficients.

Equations (27)

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L φ 1 ( L , H ) , H φ 2 ( L , H ) ,
L H , H H h L ,
W α + 1 = W α h W α 1 ,
w α + 1 = h w α + w α 1 .
lim α w α w α 1 σ ( h , ) = 1 2 ( h + h 2 + 4 ) .
σ ( h , 1 ) = [ h ¯ ] ,
σ ( 1 , 2 ) = [ 2 , 0 ¯ ] , σ ( 1 , 3 ) = [ 2 , 3 ¯ ] ,
T = ( | φ 1 ( L , H ) | L | φ 2 ( L , H ) | L | φ 1 ( L , H ) | H | φ 2 ( L , H ) | H ) ,
T h , = ( h 1 0 ) ,
τ h , ( ± ) = 1 2 ( h ± h 2 + 4 ) .
W ^ N ( k ) = 1 N j = 1 N 1 W ( j ) exp ( 2 π i j k N ) , k = 1 , 2 , , N ,
F N ( k ) = | W ^ N ( k ) | 2 .
d ν ( k ) = d ν pp ( k ) + d ν ac ( k ) + d ν sc ( k )
k m 1 m 2 = 2 π Λ 0 m 1 τ h , 1 m 2 ,
M L ( θ , λ ) = ( cos β L q L sin β L 1 q L sin β L cos β L ) .
q L | | = n L cos θ cos θ L , q L = cos θ n L cos θ L .
M α + 1 = M α h M α 1 ,
α | | , ( θ , λ ) = 1 4 M α 2 + 2 ,
α ( θ , λ ) = 1 2 [ α | | ( θ , λ ) + α ( θ , λ ) ] .
¯ α = 1 Δ λ λ min λ max [ 2 π 0 π / 2 α ( θ , λ ) d θ ] d λ .
¯ α = 2 π 0 π / 2 α ( θ , λ ) d θ .
n L d L / λ = 1 / 4 , n H d H / λ = 1 / 4 ,
ln ( 1 ¯ α ) = a 0 + a 1 N α .
n H 2 ( λ ) = 4 + 1.9 λ 2 λ 2 ( 0.336 ) 2 ,
ln ( 1 ¯ α ) = b 0 + b 1 [ 1 exp ( b 2 N α ) ] .
α ( θ ) = 1 Δ λ λ min λ max α ( θ , λ ) d λ , α ( λ ) = 2 π 0 π / 2 α ( θ , λ ) d θ ,
B = λ + λ 1 2 ( λ + + λ ) .

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