Abstract

We present analytical and numerical studies of a photonic lattice with short- and long-range harmonic modulations of the refractive index. Such structures can be prepared experimentally with holographic photolithography. In the spectral region of the photonic bandgap of the underlying single-periodic crystal, we observe a series of bands with anomalously small dispersion. The related slow-light effect is attributed to the long-range modulation of the photonic lattice that leads to formation of an array of evanescently coupled high-Q cavities. The band structure of the lattice is studied with several techniques: (i) transfer matrix approach; (ii) an analysis of resonant coupling in the process of band folding; (iii) effective-medium approach based on coupled-mode theory; and (iv) the Bogolyubov–Mitropolsky approach. The latter method, commonly used in the studies of nonlinear oscillators, was employed to investigate the behavior of eigenfunction envelopes and the band structure of the dual-periodic photonic lattice. We show that reliable results can be obtained even in the case of large refractive index modulation.

© 2008 Optical Society of America

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References

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2007 (3)

L. A. Dorado, R. A. Depine, and H. Miguez, “Effect of extinction on the high-energy optical response of photonic crystals,” Phys. Rev. B 75, 241101 (2007).
[CrossRef]

A. Yamilov and M. Bertino, “Disorder-immune coupled resonator optical waveguide,” Opt. Lett. 32, 283-285 (2007).
[CrossRef] [PubMed]

M. F. Bertino, R. R. Gadipalli, L. A. Martin, L. E. Rich, A. Yamilov, B. R. Heckman, N. Leventis, S. Guha, J. Katsoudas, R. Divan, and D. C. Mancini, “Quantum dots by ultraviolet and X-ray lithography,” Nanotechnology 18, 315603 (2007).
[CrossRef]

2006 (4)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

M. Scharrer, A. Yamilov, X. Wu, H. Cao, and R. P. H. Chang, “Ultraviolet lasing in high-order bands of three-dimensional ZnO photonic crystals,” Appl. Phys. Lett. 88, 201103 (2006).
[CrossRef]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer microring coupled-resonator optical waveguides,” J. Lightwave Technol. 24, 1843-1849 (2006).
[CrossRef]

2005 (4)

H. Altug and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

Yu. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

J. Scheuer, G. Paloczi, J. Poon, and A. Yariv, “Toward the slowing and storage of light,” Opt. Photonics News 16, 36-40 (2005).
[CrossRef]

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg grating,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

2004 (4)

M. F. Bertino, R. R. Gadipalli, J. G. Story, C. G. Williams, G. Zhang, C. Sotiriou-Leventis, A. T. Tokuhiro, S. Guha, and N. Leventis, “Laser writing of semiconductor nanoparticles and quantum dots,” Appl. Phys. Lett. 85, 6007-6009 (2004).
[CrossRef]

H. Kitahara, T. Kawaguchi, J. Miyashita, R. Shimada, and M. W. Takeda, “Strongly localized singular bloch modes created in dual-periodic microstrip lines,” J. Phys. Soc. Jpn. 73, 296-299 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21, 1665-1673 (2004).
[CrossRef]

J. F. Galisteo-López and C. López, “High-energy optical response of artificial opals,” Phys. Rev. B 70, 035108 (2004).
[CrossRef]

2003 (2)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikovwski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

2002 (3)

2001 (4)

N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. 89, 815-823 (2001).
[CrossRef]

M. Bayindir, S. Tanriseven, and E. Ozbay, “Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,” Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, S184-S189 (2001).
[CrossRef]

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905-3909 (2001).
[CrossRef]

2000 (1)

1999 (2)

1998 (4)

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Energy spectra in dual-periodic multilayer structures,” J. Phys. Soc. Jpn. 67, 3414-3419 (1998).
[CrossRef]

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E 57, 3502-3509 (1998).
[CrossRef]

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys., Part 2 37, L565-L567 (1998).
[CrossRef]

1996 (1)

J. M. Benedickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

1994 (1)

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Appl. Phys. A (1)

M. Bayindir, S. Tanriseven, and E. Ozbay, “Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,” Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Appl. Phys. Lett. (3)

H. Altug and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

M. F. Bertino, R. R. Gadipalli, J. G. Story, C. G. Williams, G. Zhang, C. Sotiriou-Leventis, A. T. Tokuhiro, S. Guha, and N. Leventis, “Laser writing of semiconductor nanoparticles and quantum dots,” Appl. Phys. Lett. 85, 6007-6009 (2004).
[CrossRef]

M. Scharrer, A. Yamilov, X. Wu, H. Cao, and R. P. H. Chang, “Ultraviolet lasing in high-order bands of three-dimensional ZnO photonic crystals,” Appl. Phys. Lett. 88, 201103 (2006).
[CrossRef]

Europhys. Lett. (1)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

J. Appl. Phys. (2)

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905-3909 (2001).
[CrossRef]

N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. 89, 815-823 (2001).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A, Pure Appl. Opt. (1)

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, S184-S189 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

J. Phys. Soc. Jpn. (2)

H. Kitahara, T. Kawaguchi, J. Miyashita, R. Shimada, and M. W. Takeda, “Strongly localized singular bloch modes created in dual-periodic microstrip lines,” J. Phys. Soc. Jpn. 73, 296-299 (2004).
[CrossRef]

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Energy spectra in dual-periodic multilayer structures,” J. Phys. Soc. Jpn. 67, 3414-3419 (1998).
[CrossRef]

Jpn. J. Appl. Phys., Part 2 (1)

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys., Part 2 37, L565-L567 (1998).
[CrossRef]

Nanotechnology (1)

M. F. Bertino, R. R. Gadipalli, L. A. Martin, L. E. Rich, A. Yamilov, B. R. Heckman, N. Leventis, S. Guha, J. Katsoudas, R. Divan, and D. C. Mancini, “Quantum dots by ultraviolet and X-ray lithography,” Nanotechnology 18, 315603 (2007).
[CrossRef]

Nature (2)

Yu. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Opt. Photonics News (1)

J. Scheuer, G. Paloczi, J. Poon, and A. Yariv, “Toward the slowing and storage of light,” Opt. Photonics News 16, 36-40 (2005).
[CrossRef]

Phys. Rev. B (3)

J. F. Galisteo-López and C. López, “High-energy optical response of artificial opals,” Phys. Rev. B 70, 035108 (2004).
[CrossRef]

L. A. Dorado, R. A. Depine, and H. Miguez, “Effect of extinction on the high-energy optical response of photonic crystals,” Phys. Rev. B 75, 241101 (2007).
[CrossRef]

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

Phys. Rev. E (4)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

J. M. Benedickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg grating,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E 57, 3502-3509 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Other (8)

C.M.Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, 1996).

P. W. Milonni, Fast Light, Slow Light and Left-Handed Light (Institute of Physics, 2005).

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

P. S. Landa, Regular and Chaotic Oscillations (Springer, 2001).

N. N. Bogolyubov and Yu. A. Mitropolsky, Asymptotic Methods in Theory of Nonlinear Oscillations (Nauka, 1974) (in Russian).

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Figures (6)

Fig. 1
Fig. 1

(a) Dependence of the index of refraction in a dual-periodic photonic crystal as defined by Eq. (2). We used ε 0 = 2.25 , Δ ε = 1 , N = 80 , and the modulation parameter γ is equal to 0.25 . (b) Local (position-dependent) photonic bandgap diagram for n ( x ) in (a). A i ( N ) and B i ( N ) mark the frequencies of the foremost photonic bands on the long- and short-wavelength sides of the photonic bandgap of the corresponding single-periodic crystal.

Fig. 2
Fig. 2

(a) Transmission coefficient through a finite segment of length L (one period) of the periodic superstructure defined in Fig. 1. Solid and dashed curves correspond to 0 < x < N a and N a 2 < x < N a 2 segments [shown in the inset of panel (b)], respectively. (b) Solid and dashed thin curves plot the corresponding phase of t ( ω ) . Bold line depicts the Bloch number K ( ω ) × a of the infinite crystal computed using Eq. (5).

Fig. 3
Fig. 3

Left panel shows dispersion of a PhSC ω ( K ) reduced to the first Brillouin zone. The eigenmodes that correspond to the series of flat bands in the vicinity of the parent bandgap of the single periodic crystal are depicted on the right. Calculations were performed for the structure described in Fig. 1.

Fig. 4
Fig. 4

Dispersion relation computed with the transfer matrix formalism for ε 0 = 2.25 , Δ ε = 0.32 , N = 9 (dashed curve), and N = 10 (solid curve). The modulation parameter γ is equal to 0.25 . For this set of parameters, the applicability condition [Eq. (13)] of the resonant approximation is satisfied.

Fig. 5
Fig. 5

(a) Value of the integral in Eq. (20) (solid curve), as a function of frequency is shown. For easy comparison with (b), the plot is transposed so that ω is plotted along the y axis. The circles depict frequencies that satisfy the quantization condition of Eq. (20). The dashed lines denote the actual position of photonic states, as determined by direct numerical analysis of Section 3. (b) Gray-scale plot of Re [ n eff ( x , ω ) ] given by Eq. (19). The solid curve shows the boundary of the region where Im [ n eff ( x , ω ) ] 0 . For comparison we also show the local PBG of Fig. 1b (dashed curve). In both (a) and (b), the parameters of Fig. 1 are adopted.

Fig. 6
Fig. 6

Numerical solutions of Eqs. (23, 24, 25, 26) are shown. Filled circles in panels (a) and (e) denote the spatial position where the particular ϕ ( x ) is equal to m π 2 . At these special points d A ( x ) d x = 0 is denoted by the vertical dashed lines in (b)–(d), (f), and (g). A i ( N ) and B i ( N ) denote the low-dispersion photonic bands as defined in Section 3.

Tables (1)

Tables Icon

Table 1 Results of Resonant Approximation Analysis of Eq. (12) With Dielectric Function Given by Eq. (2) a

Equations (30)

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

ω ( K ) = Ω [ 1 + κ cos ( K L ) ] .
ε ( x ) = ε 0 + Δ ε 2 1 + γ [ 1 + γ cos ( 2 π x L ) ] [ 1 + cos ( 2 π x a ) ] .
M ̂ ( x , x + d x ) = [ cos ( k n ( x ) d x ) n 1 ( x ) sin ( k n ( x ) d x ) n ( x ) sin ( k n ( x ) d x ) cos ( k n ( x ) d x ) ] ,
M ̂ tot = x = 0 L M ̂ ( x , x + d x ) .
cos ( K ( ω ) L ) = Re [ 1 t ( ω ) ] 1 t ( ω ) cos ( ϕ ( ω ) ) ,
t ( ω ) = ( 1 ) N ( Γ 2 ) i ( Γ 2 ) ( ω ω 0 ) ,
ω ( K ) = ω 0 [ 1 ± κ cos ( K L ) ] ,
cos ( K ( ω ) L ) = ( ω ω 0 ) ( 1 ) m t ( ω 0 ) d ϕ ( ω 0 ) d ω .
E ( x ) + ω 2 c 2 δ ε ( x ) E ( x ) = ω 2 c 2 ε ¯ E ( x ) ,
ε ( x ) = m = ε m exp [ i 2 π L m x ] ,
E ( x ) = exp [ i K ( ω ) x ] m = E m exp [ i 2 π L m x ] ,
[ ω 2 c 2 ε ¯ ( K ( ω ) + 2 π L m ) 2 ] E m + ω 2 c 2 m 0 ε m E m m = 0 ,
( N + 1 ) 2 ( ε 1 + ε N ) 4 N ε ¯ + 2 ( N + 1 ) 2 ε N 1 1 .
Δ ε 8 ε ¯ N 1 ,
n ( x ) n 0 = 1 + σ ( x ) + 2 κ ( x ) cos [ 2 k 0 x + φ ( x ) ] ,
σ ( x ) = γ Δ ε 4 1 + γ ε 0 + Δ ε 2 1 + γ cos 2 π L x ,
κ ( x ) = Δ ε 8 1 + γ ε 0 + Δ ε 2 1 + γ ( 1 + γ cos 2 π L x ) ,
φ ( x ) 0 , n 0 = ( ε 0 + Δ ε 2 1 + γ ) 1 2 , k 0 = π a .
δ = ω ω 0 ω 0 1 , ω 0 = k 0 c n 0 ,
d 2 E eff d x 2 + k 0 2 n eff 2 ( x , ω ) E eff = 0 .
n eff = { ( σ ( x ) + Δ ) 2 κ ( x ) 2 } 1 2 ,
I ( ω ) = k 0 x L x R n eff ( x , ω ) d x = ( m + 1 2 ) π ,
E ( x ) = A ( x ) cos ( k 0 x + ϕ ( x ) ) ,
d E ( x ) d x = k 0 A ( x ) sin ( k 0 x + ϕ ( x ) ) ,
d ϕ ( x ) d x = 1 k 0 [ ω 2 c 2 ε ( x ) k 0 2 ] cos 2 ( k 0 x + ϕ ( x ) ) ,
d A ( x ) d x = A ( x ) 2 k 0 [ ω 2 c 2 ε ( x ) k 0 2 ] sin 2 ( k 0 x + ϕ ( x ) ) .
d ϕ ( x ) d x = 1 2 k 0 [ ω 2 c 2 ε 0 k 0 2 + ω 2 c 2 Δ ε 2 1 + γ ( 1 + γ cos 2 π L x ) ( 1 + 1 2 cos 2 ϕ ( x ) ) ] ,
d log A ( x ) d x = 1 2 k 0 ω 2 c 2 Δ ε 2 1 + γ ( 1 + γ cos 2 π L x ) sin 2 ϕ ( x ) .
ϕ ( L ) = ϕ ( 0 ) + m π ,
sin 2 ϕ ( 0 , L 2 , L ) = 0 .

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