Abstract

We present a numerical stable method to accurately solve band structures and the eigenvalues of a multilayer-basis photonic crystal cavity. We derive a set of band-edge equations to determine the band structures rather than use the cosine of the Bloch phase, which is traditionally used but may induce numerical instability. Moreover, two novel formulas are proposed to solve the eigenvalues for the cavity modes. The eigenvalues solved by the method are accurate without including the spurious solutions. Thus, it is not required to eliminate the spurious solutions from the results. Finally, numerical examples of binary and Fibonacci multilayers in each cell are studied to demonstrate that this method has better numerical stability in computing the band structure and cavity modes than traditional methods.

© 2007 Optical Society of America

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2007 (1)

2006 (4)

2005 (5)

M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, "Optical properties of one-dimensional photonic crystals based on multiple-quantum-well structures," Phys. Rev. B 71, 235335 (2005).
[CrossRef]

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method," J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

H. S. Sözüer and K. Sevim, "Robustness of one-dimensional photonic band gaps under random variations of geometrical parameters," Phys. Rev. B 72, 195101 (2005).
[CrossRef]

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, "Optical necklace states in Anderson localized 1D systems," Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef] [PubMed]

F. Szmulowicz, "Tangent formulation of the Kronig-Penney problem for N-period layered systems with application to photonic crystals," Phys. Rev. B 72, 235103 (2005).
[CrossRef]

2004 (5)

2003 (3)

Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

S. Mishra and S. Satpathy, "One-dimensional photonic crystal: the Kronig-Penney model," Phys. Rev. B 68, 045121 (2003).
[CrossRef]

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

2002 (1)

J. M. Cervero and A. Rodriguez, "Infinite chain of N different deltas: a simple model for a quantum wire," Eur. Phys. J. B 30, 239-251 (2002).
[CrossRef]

2001 (1)

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

1999 (1)

1998 (2)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

1997 (4)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

N. H. Liu, "Defect modes of stratified dielectric media," Phys. Rev. B 55, 4097-4101 (1997).
[CrossRef]

F. Ramos-Mendieta and R. Halevi, "Electromagnetic surface modes of a dielectric superlattice: the supercell method," J. Opt. Soc. Am. B 14, 370-381 (1997).
[CrossRef]

P. Tran, "Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect," J. Opt. Soc. Am. B 14, 2589-2595 (1997).
[CrossRef]

1996 (1)

H. Miyazaki, Y. Jimba, and T. Watanabe, "Multiphotonic lattices and Stark localization of electromagnetic fields in one dimension," Phys. Rev. A 53, 2877-2880 (1996).
[CrossRef] [PubMed]

1994 (1)

J. Mendialdua, A. Rodriguez, M. More, A. Akjouj, and L. Dobrzynski, "Bulk and surface phonon polaritons in three-layer superlattices," Phys. Rev. B 50, 14605-14608 (1994).
[CrossRef]

1993 (2)

1991 (1)

Y. Yamamoto, S. Machida, K. Igeta, and G. Bjork, "Microcavity semiconductor laser with enhanced spontaneous emission," Phys. Rev. A 44, 657-668 (1991).
[CrossRef] [PubMed]

1977 (1)

Eur. Phys. J. B (1)

J. M. Cervero and A. Rodriguez, "Infinite chain of N different deltas: a simple model for a quantum wire," Eur. Phys. J. B 30, 239-251 (2002).
[CrossRef]

J. Appl. Phys. (1)

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method," J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (6)

Nature (1)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Opt. Commun. (2)

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (2)

Y. Yamamoto, S. Machida, K. Igeta, and G. Bjork, "Microcavity semiconductor laser with enhanced spontaneous emission," Phys. Rev. A 44, 657-668 (1991).
[CrossRef] [PubMed]

H. Miyazaki, Y. Jimba, and T. Watanabe, "Multiphotonic lattices and Stark localization of electromagnetic fields in one dimension," Phys. Rev. A 53, 2877-2880 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (7)

M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, "Optical properties of one-dimensional photonic crystals based on multiple-quantum-well structures," Phys. Rev. B 71, 235335 (2005).
[CrossRef]

H. S. Sözüer and K. Sevim, "Robustness of one-dimensional photonic band gaps under random variations of geometrical parameters," Phys. Rev. B 72, 195101 (2005).
[CrossRef]

N. H. Liu, "Defect modes of stratified dielectric media," Phys. Rev. B 55, 4097-4101 (1997).
[CrossRef]

S. Mishra and S. Satpathy, "One-dimensional photonic crystal: the Kronig-Penney model," Phys. Rev. B 68, 045121 (2003).
[CrossRef]

J. Mendialdua, A. Rodriguez, M. More, A. Akjouj, and L. Dobrzynski, "Bulk and surface phonon polaritons in three-layer superlattices," Phys. Rev. B 50, 14605-14608 (1994).
[CrossRef]

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, and K. A. Nelson, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

F. Szmulowicz, "Tangent formulation of the Kronig-Penney problem for N-period layered systems with application to photonic crystals," Phys. Rev. B 72, 235103 (2005).
[CrossRef]

Phys. Rev. E (1)

Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, "Optical necklace states in Anderson localized 1D systems," Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef] [PubMed]

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, "Photonic crystal heterostructures and interfaces," Rev. Mod. Phys. 78, 455-481 (2006).
[CrossRef]

Science (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Other (5)

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

W. Mayeda, Graph Theory (Wiley, 1972).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1988).

P. R. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

J. Rarity and C. Weisbuch, Microcavities and Photonic Bandgaps: Physics and Applications (Kluwer, 1996).

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Figures (9)

Fig. 1
Fig. 1

Geometry of a one-dimensional PCC with a cavity. The structure consists of N layers in each cell of DBR and N c layers in the cavity.

Fig. 2
Fig. 2

Graph representation of electromagnetic waves in each layer of cell m for the TE and TM polarizations.

Fig. 3
Fig. 3

Value of cos ( K L ) and the absolute values of J ± 1 , as function of β in the range 0 < β < 4.0 ω c for a bilayer DBR, in which n 1 = 1.0 , n 2 = 4.0 , d 1 = 0.8 μ m , d 2 = 0.2 μ m , and ω = 0.9 ( 2 π c d 1 + d 2 ) .

Fig. 4
Fig. 4

Maximum absolute values of J ± 1 , J 0 , f, g, h, and cos ( K L ) as a function of the normalized frequency ω ¯ in the range 0 < β < 4.0 ω c for a bilayer basis DBR. The normalized frequency is defined by ω ¯ = ω ( d 1 + d 2 ) 2 π c . The parameters of the DBR are n 1 = 1.0 , n 2 = 4.0 , d 1 = 0.8 μ m , d 2 = 0.2 μ m .

Fig. 5
Fig. 5

Maximum absolute values of J c , o , J c , e , and m i , j ( TMM ) , i, j = 1 , 2 as function of ω ¯ in the range 0 < β < 4.0 ω c for a bilayer basis PCC with a uniform cavity. The normalized frequency is defined by ω ¯ = ω ( d 1 + d 2 ) 2 π c . The parameters of the PCC are n 1 = 1.0 , n 2 = n c = 4.0 , d 1 = 0.8 μ m , d 2 = 0.2 μ m , and d c = 0.5 μ m .

Fig. 6
Fig. 6

Maximum absolute values of J ± 1 , J 0 , f, g, h, and cos ( K L ) as function of the order ν of Fibonacci sequence in the range 0 < β < 1.5 ω c for ω = 0.75 ( 2 π c d ) in the Fibonacci-basis DBR. The parameters for the Fibonacci PCCs are n ( A ) = 1.5 , n ( B ) = 1.0 , and d j = d = 1 μ m .

Fig. 7
Fig. 7

Band structures and the cavity modes of a Fibonacci-sequence-basis PCC with a uniform cavity layer, n c , 1 = 1.5 and d c , 1 = 1.4 μ m . The parameters for the basis are n ( A ) = 1.5 , n ( B ) = 1.0 , and d j = 1.0 μ m , ν = 3 , and L = 5.0 μ m . (a) TE polarization. (b) TM polarization.

Fig. 8
Fig. 8

Maximum absolute values of J c , o , J c , e ; and m i , j ( TMM ) , i, j = 1 , 2 as function of ν in the range 0 < β < 1.5 ω c for the νth order of the Fibonacci-based PCC. The parameters for the PCC are n ( A ) = 1.5 , n ( B ) = 1.0 , n c = 1.5 , d j = d = 1 μ m , and d c = 1.4 μ m .

Fig. 9
Fig. 9

Squared amplitude of the fields for (a) the TE cavity mode within the first bandgap at β = 0 , (b) the TE mode below the first allowed band at β = 2 π L , (c) the TM mode within the second bandgap at β = 0 , and (d) the TM mode below the first allowed band at β = 2 π L in a Fibonacci-basis PCC with a cavity. The parameters for the PCC are n ( A ) = 1.5 , n ( B ) = 1.0 , n c = 1.5 , d j = 1 μ m , d c = 1.4 μ m , and ν = 3 .

Tables (1)

Tables Icon

Table 1 Calculated Effective Indices for TE and TM Eigenmodes of a Bilayer PBG with a Cavity, n 1 = 1.0 , n 2 = n c = 4.0 , d 1 = 0.8 μ m , d 2 = 0.2 μ m , d c = 0.5 μ m , and ω = 0.9 ( 2 π c d 1 + d 2 ) a

Equations (40)

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d 2 E y ( m , j , x ) d x 2 + k j 2 E y ( m , j , x ) = 0 .
E y ( m , j , x ) = a m , j exp ( i k j ( x x m , j 1 ) ) + b m , j exp ( i k j ( x x m , j 1 ) ) ,
E y ( m , j , x ) = E y ( x m , j 1 ) sin k j ( x m , j x ) sin ( k j d j ) + E y ( x m , j ) sin k j ( x x m , j 1 ) sin ( k j d j ) .
H z ( m , j , x ) = i k j ω μ o ( a m , j exp ( i k j ( x x m , j 1 ) ) + b m , j exp ( i k j ( x x m , j 1 ) ) ) .
A m , j = f j A m , j 1 + h j B m , j ,
B m , 1 = f j B m , j + g j A m , j 1 ,
A m , N = A m , 0 f + B m , N h ,
B m , 0 = B m , N f + A m , 0 g .
B m , N = B m , 0 exp ( i K L ) ,
A m , 0 = A m , N exp ( i K L ) ,
1 g h + f 2 2 f cos ( K L ) = 0 .
( 1 + f ) 2 g h = 0
( 1 f ) 2 g h = 0 .
1 g h + f 2 = 0 .
cos ( K L ) = 1 g h + f 2 2 f .
S p , q = s = 0 q p i 2 s = p + s q i 2 s 1 = p + s 1 i 2 s 1 i 2 s 1 = p + s 1 i 2 s 1 i 2 s 2 = p + s 2 i 2 s 1 1 i 2 = p + 1 i 3 i 1 = p i 2 1 u = 1 s ( L i 2 u 1 , i 2 u ) ,
L p , q = h p g q j = p q f j 2 .
f = j = 1 N f j S 1 , N .
g = 1 S 1 , N p = 1 N S p , N g p j = 1 p 1 f j 2 .
h = 1 S 1 , N p = 1 N S 1 , p h p j = p + 1 N f j 2 .
A c , j = f c , j A c , j 1 + h c , j B c , j ,
B c , j 1 = f c , j B c , j + g c , j A c , j 1 ,
A c , N c = f c A c , 0 + h c B c , N c ,
B c , 0 = f c B c , N c + g c A c , 0 .
B 1 , 0 = R s A 1 , 0 ,
R s = g 1 f e i K L .
1 h c R s = 0 for odd modes.
1 ( h c f c 2 g c ) R s = 0 for even modes.
f exp ( 2 i K L ) ( 1 g h + f 2 ) exp ( i K L ) + f = 0 .
exp ( i K L ) = 1 g h + f 2 ± [ ( 1 g h + f 2 ) 2 4 f 2 ] 1 2 2 f .
e i K L = ( 1 g h + f 2 ) α ( ( 1 g h + f 2 ) 2 4 f 2 ) 1 2 2 f ,
1 2 g h c 1 + g h f 2 + α [ ( 1 g h + f 2 ) 2 4 f 2 ] 1 2 = 0 for odd modes ,
1 2 g 1 + g h f 2 + α [ ( 1 g h + f 2 ) 2 4 f 2 ] 1 2 ( h c f c 2 g c ) = 0 for even modes.
A 1 = n 2 = 2 2 n 1 = 1 n 2 1 ( L n 1 , n 2 ) = L 1 , 2 .
1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 e 1 e 2 = 0 ,
1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 = 0 ,
cos ( K L ) = c 1 c 2 ( k 1 2 + k 2 2 2 k 1 k 2 ) s 1 s 2 ,
1 ( k 1 t 1 + k 2 t 2 ) t c k c ( ( k 1 2 k 2 2 2 k 1 k 2 ) t 1 t 2 + α { [ 1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 ] 2 e 1 2 e 2 2 } 1 2 ) = 0 for odd modes ,
1 + k 1 t 1 + k 2 t 2 k c t c ( ( k 1 2 k 2 2 2 k 1 k 2 ) t 1 t 2 + α { [ 1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 ] 2 e 1 2 e 2 2 } 1 2 ) = 0 for even modes ,
1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 0 , α = 1 for 1 ( k 1 2 + k 2 2 2 k 1 k 2 ) t 1 t 2 < 0 .

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