Abstract

We present two efficient and accurate models for the analysis and optimization of reflection at the interface of three-dimensional (3D) photonic crystal structures. For the most general photonic crystal interfaces, we develop a rigorous technique based on mode matching at the interface. We also explain a more efficient (yet accurate) model based on effective impedance definition for the analysis of 3D photonic crystals (PC) structures that are highly desired for practical applications. The two techniques are used to model practical 3D PC structures, and the issue of reflection minimization at the interface of such structures is addressed.

© 2007 Optical Society of America

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  1. M. Loncar, T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, D. Deppe, P. Gogna, Y. Qiu, D. Nedeljkovic, and T. P. Pearsall, "Nanophotonics based on planar photonic crystals," in The 15th Annual Meeting of the IEEE LEOS (IEEE, 2002), Vol. 2, pp. 671-672.
  2. S. G. Romanov, P. Ferrand, M. Egen, R. Zentel, J. Ahopelto, N. Goponik, A. Eychmüller, A. Rogach, and C. M. Sotomayor Torres, "Exploring integration prospects of opal-based photonic crystals," Synth. Met. 139, 701-704 (2003).
    [CrossRef]
  3. Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, "Photonic crystal with diamondlike structure fabricated by holographic lithography," Appl. Phys. Lett. 87, 061103 (2005).
    [CrossRef]
  4. J. H. Moon, J. Ford, and S. Yang, "Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography," Int. Symp. Polym. Adv. Technol. 17, 83-93 (2006).
  5. J. H. Moon, S. Yang, and S.-M. Yang, "Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography," Appl. Phys. Lett. 88, 121101 (2006).
    [CrossRef]
  6. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
    [CrossRef] [PubMed]
  7. R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, "Log-pile photonic crystal fabricated by two-photon photopolymerization," J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
    [CrossRef]
  8. M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, "3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing," Opt. Lett. 31, 805-807 (2006).
    [CrossRef] [PubMed]
  9. Y. Zeng, X. Chen, and W. Lu, "Electromagnetic modes in semi-infinite photonic crystals," Physica E (Amsterdam) 30, 55-58 (2005).
    [CrossRef]
  10. T. P. White, C. M. De Sterke, R. C. McPhedran, and L. C. Botten, "Highly efficient wide-angle transmission into uniform rod-type photonic crystals," Appl. Phys. Lett. 87, 111107 (2005).
    [CrossRef]
  11. Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. E 70, 016706 (2004).
    [CrossRef]
  12. Z.-Y. Li and K.-M. Ho, "Light propogation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68, 155101 (2003).
    [CrossRef]
  13. E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
    [CrossRef]
  14. X. Yu and S. Fan, "Anomalous reflections at photonic crystal surfaces," Phys. Rev. E 70, 055601 (2004).
    [CrossRef]
  15. E. Schonbrun, Q. Wu, W. Park, T. Yamashita, and C. J. Summers, "Polarization beam splitter based on a photonic crystal heterostructure," Opt. Lett. 31, 3104-3106 (2006).
    [CrossRef] [PubMed]
  16. X. Ao, L. Liu, L. Wosinski, and S. He, "Polarization beam splitter based on a two-dimensional photonic crystal of pillar type," Appl. Phys. Lett. 89, 171115 (2006).
    [CrossRef]
  17. B. Momeni, A. A. Eftekhar, and A. Adibi, "Effective impedence model for analysis of reflection at the interfaces of photonic crystals," Opt. Lett. 32, 778-780 (2007).
    [CrossRef] [PubMed]

2007 (1)

2006 (5)

E. Schonbrun, Q. Wu, W. Park, T. Yamashita, and C. J. Summers, "Polarization beam splitter based on a photonic crystal heterostructure," Opt. Lett. 31, 3104-3106 (2006).
[CrossRef] [PubMed]

X. Ao, L. Liu, L. Wosinski, and S. He, "Polarization beam splitter based on a two-dimensional photonic crystal of pillar type," Appl. Phys. Lett. 89, 171115 (2006).
[CrossRef]

J. H. Moon, J. Ford, and S. Yang, "Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography," Int. Symp. Polym. Adv. Technol. 17, 83-93 (2006).

J. H. Moon, S. Yang, and S.-M. Yang, "Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography," Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, "3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing," Opt. Lett. 31, 805-807 (2006).
[CrossRef] [PubMed]

2005 (5)

Y. Zeng, X. Chen, and W. Lu, "Electromagnetic modes in semi-infinite photonic crystals," Physica E (Amsterdam) 30, 55-58 (2005).
[CrossRef]

T. P. White, C. M. De Sterke, R. C. McPhedran, and L. C. Botten, "Highly efficient wide-angle transmission into uniform rod-type photonic crystals," Appl. Phys. Lett. 87, 111107 (2005).
[CrossRef]

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, "Photonic crystal with diamondlike structure fabricated by holographic lithography," Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, "Log-pile photonic crystal fabricated by two-photon photopolymerization," J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

2004 (3)

X. Yu and S. Fan, "Anomalous reflections at photonic crystal surfaces," Phys. Rev. E 70, 055601 (2004).
[CrossRef]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. E 70, 016706 (2004).
[CrossRef]

2003 (2)

Z.-Y. Li and K.-M. Ho, "Light propogation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68, 155101 (2003).
[CrossRef]

S. G. Romanov, P. Ferrand, M. Egen, R. Zentel, J. Ahopelto, N. Goponik, A. Eychmüller, A. Rogach, and C. M. Sotomayor Torres, "Exploring integration prospects of opal-based photonic crystals," Synth. Met. 139, 701-704 (2003).
[CrossRef]

Appl. Phys. Lett. (4)

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, "Photonic crystal with diamondlike structure fabricated by holographic lithography," Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

J. H. Moon, S. Yang, and S.-M. Yang, "Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography," Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

T. P. White, C. M. De Sterke, R. C. McPhedran, and L. C. Botten, "Highly efficient wide-angle transmission into uniform rod-type photonic crystals," Appl. Phys. Lett. 87, 111107 (2005).
[CrossRef]

X. Ao, L. Liu, L. Wosinski, and S. He, "Polarization beam splitter based on a two-dimensional photonic crystal of pillar type," Appl. Phys. Lett. 89, 171115 (2006).
[CrossRef]

Int. Symp. Polym. Adv. Technol. (1)

J. H. Moon, J. Ford, and S. Yang, "Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography," Int. Symp. Polym. Adv. Technol. 17, 83-93 (2006).

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

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, "Log-pile photonic crystal fabricated by two-photon photopolymerization," J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Nat. Mater. (1)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. B (2)

Z.-Y. Li and K.-M. Ho, "Light propogation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68, 155101 (2003).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Phys. Rev. E (2)

X. Yu and S. Fan, "Anomalous reflections at photonic crystal surfaces," Phys. Rev. E 70, 055601 (2004).
[CrossRef]

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. E 70, 016706 (2004).
[CrossRef]

Physica E (Amsterdam) (1)

Y. Zeng, X. Chen, and W. Lu, "Electromagnetic modes in semi-infinite photonic crystals," Physica E (Amsterdam) 30, 55-58 (2005).
[CrossRef]

Synth. Met. (1)

S. G. Romanov, P. Ferrand, M. Egen, R. Zentel, J. Ahopelto, N. Goponik, A. Eychmüller, A. Rogach, and C. M. Sotomayor Torres, "Exploring integration prospects of opal-based photonic crystals," Synth. Met. 139, 701-704 (2003).
[CrossRef]

Other (1)

M. Loncar, T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, D. Deppe, P. Gogna, Y. Qiu, D. Nedeljkovic, and T. P. Pearsall, "Nanophotonics based on planar photonic crystals," in The 15th Annual Meeting of the IEEE LEOS (IEEE, 2002), Vol. 2, pp. 671-672.

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Figures (4)

Fig. 1
Fig. 1

(a) Setup for reflection calculation is shown, with α being the angle between the incident wave vector and the normal to the interface ( z ) , and ϕ being the angle between the plane of incidence and x z plane. (b) Schematic of the tetragonal woodpile lattice considered throughout this paper is shown. Lattice constants and filling factors in different directions of this lattice are marked on this figure.

Fig. 2
Fig. 2

Power reflection coefficient at the interface of a tetragonal woodpile PC structure, as shown in Fig. 1a, with ε r = 2.5 , f x = f y = 0.3 , f z = 0.5 , a x = a y = a , and a z = 1.2 a is shown with z 0 = 0 , and the incident planewave at α = 5 ° and ϕ = 0 ° . Two cases with TE and TM polarizations (for the incident plane wave with electric field and magnetic field along the y direction) are considered with the incident plane wave coming either from air ( ε r = 1.0 ) or substrate ( ε r = 2.5 ) .

Fig. 3
Fig. 3

(a) Calculated reflection for a plane wave incident from a dielectric substrate ( ε r = 2.5 ) to a tetragonal woodpile PC with a x = a y = a , a z = 1.2 a , f x = f y = 0.3 , and f z = 0.5 (parameters as defined in Fig. 1) with TM incident polarization (magnetic field along the y direction in Fig. 1) at an angle α = 7 ° is shown. The solid curve corresponds to the direct calculation results, and the star markers are those calculated using the effective impedance model. The interface of the PC is assumed to be at z 0 = 0.75 a z . (b) Calculated effective impedance (normalized to the impedance of the vacuum) of the PC in (a) is shown (marked by stars) and compared with that of the incident region (dashed line). Perfect impedance matching is observed at a λ = 0.337 , and low reflection in the vicinity of that normalized frequency is consistent with the direct reflection calculation results.

Fig. 4
Fig. 4

Calculated reflection for a plane wave incident from air ( ε r = 1.0 ) to the same tetragonal woodpile PC as in Fig. 3 with TM incident polarization at an angle α = 7 ° and ϕ = 0 ° is shown. The interface of the PC is assumed to be at z 0 = 0.25 a z for the results plotted in (a), and the corresponding effective impedance is shown in (b). The calculated reflection and effective impedance for an interface at z 0 = 0.75 a z are shown in (c) and (d), respectively. The dashed lines in (b) and (d) represent the normalized effective impedance of the incident region (i.e., air).

Equations (32)

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ε r ( x , y , z ) = l m n ε ̃ l m n exp [ j ( l K 1 + m K 2 + n K 3 ) r ] ,
H k ( r ) = ε 0 μ 0 l m n U l m n exp [ j ( k + l K 1 + m K 2 + n K 3 ) r ] ,
E k ( r ) = l m n S l m n exp [ j ( k + l K 1 + m K 2 + n K 3 ) r ] ,
k l m n = k + l K 1 + m K 2 + n K 3 .
× E = j ω μ 0 H ,
× H = j ω ε 0 ε r E ,
[ k y ] S z [ k z ] S y = k 0 U x ,
[ k z ] S x [ k x ] S z = k 0 U y ,
[ k x ] S y [ k y ] S x = k 0 U z ,
[ k y ] U z [ k z ] U y = k 0 [ ε ] S x ,
[ k z ] U x [ k x ] U z = k 0 [ ε ] S y ,
[ k x ] U y [ k y ] U x = k 0 [ ε ] S z .
( [ ε ] ) ( l , m , n ) , ( q , r , s ) = ε ̃ ( l q ) ( m r ) ( n s ) .
1 k 0 [ k y ] [ η ] ( [ k x ] U y [ k y ] U x ) [ k z ] S y = k 0 U x ,
[ k z ] S x + 1 k 0 [ k x ] [ η ] ( [ k x ] U y [ k y ] U x ) = k 0 U y ,
1 k 0 [ k y ] ( [ k x ] S y [ k y ] S x ) [ k z ] U y = k 0 [ ε ] S x ,
[ k z ] U x 1 k 0 [ k x ] ( [ k x ] S y [ k y ] S x ) = k 0 [ ε ] S y .
[ M ] ( U x U y S x S y ) = k z ( U x U y S x S y ) ,
[ M ] = ( [ K z ] 0 1 k 0 [ k x ] [ k y ] 1 k 0 [ k x ] 2 k 0 [ ε ] 0 [ K z ] 1 k 0 [ k y ] 2 + k 0 [ ε ] 1 k 0 [ k y ] [ k x ] 1 k 0 [ k x ] [ η ] [ k y ] 1 k 0 [ k x ] [ η ] [ k x ] + k 0 [ K z ] 0 1 k 0 [ k y ] [ η ] [ k y ] k 0 1 k 0 [ k y ] [ η ] [ k x ] 0 [ K z ] ) .
P k = 1 2 E k × H k * ,
P z = 1 2 ω { U y T S x + U x T S y } .
δ l m , 00 U i , 00 x + U r , l m x TE + U r , l m x TM = t = 1 2 N 1 N 2 a t n U t , l m n x exp [ j ( k z t + n K z ) z 0 ] ,
δ l m , 00 U i , 00 y + U r , l m y TE + U r , l m y TM = t = 1 2 N 1 N 2 a t n U t , l m n y exp [ j ( k z t + n K z ) z 0 ] ,
δ l m , 00 S i , 00 x + S r , l m x TE + S r , l m x TM = t = 1 2 N 1 N 2 a t n S t , l m n x exp [ j ( k z t + n K z ) z 0 ] ,
δ l m , 00 S i , 00 y + S r , l m y TE + S r , l m y TM = t = 1 2 N 1 N 2 a t n S t , l m n y exp [ j ( k z t + n K z ) z 0 ] ,
S r , l m TE ( k r , l m x x ̂ + k r , l m y y ̂ ) × z ̂ ,
k 0 U r , l m TE = k r , l m x × S r , l m TE ,
U r , l m TM ( k r , l m x x ̂ + k r , l m y y ̂ ) × z ̂ ,
k 0 n 1 2 S r , l m TM = U r , l m TM × k r , l m x ,
k r , l m z = n 1 2 k 0 2 k r , l m x 2 k r , l m y 2 .
η PC = 2 P n H int 2 ,
R = η PC η i n c η PC + η i n c 2 ,

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