Abstract

We present a robust method for computing the reflection of arbitrarily shaped and sized beams from finite thickness photonic crystals. The method is based on dividing the incident beam into plane waves, each of which can be solved individually using Bloch periodic boundary conditions. This procedure allows us to take a full advantage of the crystal symmetry and also leads to a linear scaling of the computation time with respect to the number of plane waves needed to expand the incident beam. The algorithm for computing the reflection of an individual plane wave is also reviewed. Finally, we find an excellent agreement between the computational results and measurement data obtained from opals that are synthesized using polystyrene and poly(methyl methacrylate) microspheres.

© 2005 Optical Society of America

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References

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    [CrossRef]
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  26. K. Varis and A. R. Baghai-Wadji, �??A Novel 3D Pseudo-Spectral Analysis of Photonic Crystal Slabs,�?? ACES J. 19, 101�??111 (2004).
  27. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434�??8437 (1993).
    [CrossRef]
  28. X. Zhang, �??Image resolution depending on slab thickness and object distance in a two-dimensional photonic-crystal-based superlens,�?? Phys. Rev. B 70, 195, 110 (2004).
    [CrossRef]
  29. A. R. Baghai-Wadji, �??A Symbolic Procedure for the Diagonalization of Linear PDEs in Accelerated Computational Engineering,�?? in Lecture Notes in Computer Science, vol 2630, F.Winkler and U. Langer, eds., pp. 347�??360 (Springer-Verlag, Heidelberg, Germany, 2003).
    [CrossRef]
  30. M. T. Manzuri-Shalmani and A. R. Baghai-Wadji, �??Elemental field distributions in corrugated structures with large-amplitude gratings,�?? Electron. Lett. 39, 1690�??1691 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  35. F. Jonsson, C. M. Sotomayor Torres, J. Seekamp, M. Schniedergers, A. Tiedemann, J. Ye, and R. Zentel, �??Artificially inscribed defects in opal photonic crystals,�?? Microelectr. Eng. (to appear 2005).
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    [CrossRef]
  38. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, �??Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,�?? Phys. Med. Biol. 48, 4165�??4172 (2003).
    [CrossRef]

ACES J. (1)

K. Varis and A. R. Baghai-Wadji, �??A Novel 3D Pseudo-Spectral Analysis of Photonic Crystal Slabs,�?? ACES J. 19, 101�??111 (2004).

Adv. Mater. (2)

D. J. Norris, E. G. Arlinghaus, L. Meng, R. Heiny, and L. E. Scriven, �??Opaline Photonic Crystals: How Does Self-Assembly Work?�?? Adv. Mater. 16, 1393�??1399 (2004).
[CrossRef]

D. J. Norris and Y. A. Vlasov, �??Chemical Approaches to Three-Dimensional Semiconductor Photonic Crystals,�?? Adv. Mater. 13, 371�??376 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, N. Shinya, and Y. Aoyagi, �??Three-dimensional photonic crystals for optical wavelengths assembled by micromanipulation,�?? Appl. Phys. Lett. 81(17), 3122�??3124 (2002).
[CrossRef]

Appl. Surf. Sci. (1)

F. Bresson, C.-C. Chen, G.-C. Chi, and Y.-W. Chen, �??Simplified sedimentation process for 3D photonic thick layers/bulk crystals with a stop-band in the visible range,�?? Appl. Surf. Sci. 217, 281�??288 (2003).
[CrossRef]

Chem. Mater. (4)

P. Jiang, J. F. Bertone, K. S. Hwang, and V. L. Colvin, �??Single-Crystal Colloidal Multilayers of Controlled Thickness,�?? Chem. Mater. 11, 2131�??2140 (1999).
[CrossRef]

Z.-Z. Gu, A. Fujishima, and O. Sato, �??Fabrication of High-Quality Opal Films with Controllable Thickness,�?? Chem. Mater. 14, 760�??765 (2002).
[CrossRef]

M. Egen, R. Voss, B. Griesebock, R. Zentel, S. Romanov, and C. M. Sotomayor Torres, �??Heterostructures of Polymer Photonic Crystal Films,�?? Chem. Mater. 15, 3786�??3792 (2003).
[CrossRef]

M. Müller, R. Zentel, T. Maka, S. G. Romanov, and C. M. Sotomayor Torres, �??Dye-Containing Polymer Beads as Photonic Crystals,�?? Chem. Mater. 12, 2508�??2512 (2000).
[CrossRef]

Electron. Lett. (1)

M. T. Manzuri-Shalmani and A. R. Baghai-Wadji, �??Elemental field distributions in corrugated structures with large-amplitude gratings,�?? Electron. Lett. 39, 1690�??1691 (2003).
[CrossRef]

J. Appl. Phys. (1)

M. Mulot, M. Swillo, M. Qiu, M. Strassner, M. Hede, and S. Anand, �??Investigation of Fabry-Perot cavities based on 2D Photonic crystals fabricated in InP membranes,�?? J. Appl. Phys. 95, 5928�??5930 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Mater. Chem. (1)

M. Bardosova and R. H. Tredgold, �??Ordered layers of monodispersive colloids,�?? J. Mater. Chem. 12, 2835�??2842 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys.: Condens. Matter (1)

N. Stefanou, V. Karathanos, and A. Modinos, �??Scattering of electromagnetic waves by periodic structures,�?? J. Phys.: Condens. Matter 4, 7389�??7400 (1992).
[CrossRef]

J. Vac. Sci. Technol. B (1)

M. Mulot, S. Anand, M. Swillo, M. Qui, B. Jaskorzynska, and A. Talneau, �??Low-loss InP-based photonic-crystal waveguides etched with Ar/Cl2 chemically assisted ion beam ething,�?? J. Vac. Sci. Technol. B 21, 900�??903 (2003).
[CrossRef]

Nature (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, �??A three-dimensional photonic crystal operating at infrared wavelengths,�?? Nature 394, 251�??253 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, �??Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,�?? Phys. Med. Biol. 48, 4165�??4172 (2003).
[CrossRef]

Phys. Rev. (1)

W. G. Spitzer and J. M. Whelan, �??Infrared absorption and electron effective mass in n-type gallium arsenide,�?? Phys. Rev. 114, 59�??63 (1959).
[CrossRef]

Phys. Rev. B (3)

L.-M. Li and Z.-Q. Zhang, �??Multiple-scattering approach to finite-sized photonic band-gap materials,�?? Phys. Rev. B 58, 9587�??9590 (1998).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434�??8437 (1993).
[CrossRef]

X. Zhang, �??Image resolution depending on slab thickness and object distance in a two-dimensional photonic-crystal-based superlens,�?? Phys. Rev. B 70, 195, 110 (2004).
[CrossRef]

Phys. Rev. E (1)

Y. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, �??Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals,�?? Phys. Rev. E 61, 5784�??5793 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, �??Existence of photonic gaps in periodic dielectric structures,�?? Phys. Rev. Lett. 65, 3152�??3155 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, �??Photonic Band Structure: The Face-Centered-Cubic Case Employing Nonspherical Atoms,�?? Phys. Rev. Lett. 67(17), 2295�??2299 (1991).
[CrossRef]

Phys. Rev. Lett. 69 (1)

J. B. Pendry and A. MacKinnon, �??Calculation of Photon Dispersion Relations,�?? Phys. Rev. Lett. 69, 2772�??2775 (1992).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

P. R. Villeneuve and M. Pich, �??Photonic bandgaps in periodic dielectric structures,�?? Prog. Quantum Electron. 18, 153�??200 (1994).
[CrossRef]

Science (1)

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, �??Full Three-Dimensional Photonic Bandgap Crystals at Near-Infrared Wavelengths,�?? Science 289, 604�??605 (2000).
[CrossRef] [PubMed]

Other (6)

K. Sakoda, Optical properties of photonic crystals (Springer-Verlag, Berlin, 2001).

A. R. Baghai-Wadji, �??A Symbolic Procedure for the Diagonalization of Linear PDEs in Accelerated Computational Engineering,�?? in Lecture Notes in Computer Science, vol 2630, F.Winkler and U. Langer, eds., pp. 347�??360 (Springer-Verlag, Heidelberg, Germany, 2003).
[CrossRef]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, USA, 1995).

A. Bjarklev, W. Bogaerts, T. Felici, D. Gallagher, M. Midrio, A. Lavrinenko, D. Mogitlevtsev, T. Søndergaard, D. Taillaert, and B. Tromborg, �??Comparison of strengths/weaknesses of existing numerical tools and outlining of modelling strategy,�?? A public report on Picco project (2001), <a href="http://www.intec.rug.ac.be/picco/reports.asp">http://www.intec.rug.ac.be/picco/reports.asp</a>

F. Jonsson, C. M. Sotomayor Torres, J. Seekamp, M. Schniedergers, A. Tiedemann, J. Ye, and R. Zentel, �??Artificially inscribed defects in opal photonic crystals,�?? Microelectr. Eng. (to appear 2005).

O. Madelung, Data in Science and Technology: Semiconductors-Group IV Elements and III-V Compounds (Springer-Verlag, New York, 1991).

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Figures (5)

Figure 1.
Figure 1.

A schematic representation of the discretization. The unit cell in the slab is limited by planes z=0 and z=h and is periodic with a 1 and a 2. Electric fields are expanded in plane waves on light blue planes and magnetic on dark brown. The plane wave expansions in adjacent planes are related to each other through finite differences. Outside the slab, for z<0 and z>h, the fields are expanded in outgoing eigenvectors. The eigenvector expansion is used to terminate the finite difference grid by relating the electric field at z=-(1/2)Δ to the magnetic field at z=0 and similarly for the other cladding.

Figure 2.
Figure 2.

The wave vectors of the incident and reflected fields. a) A single plane wave with a wave vector (K 0-w 0 u z ) is incident from the homogeneous medium (HM) to the surface of the photonic crystal (PC) and b) its reflection is expressed in terms of the different Bragg orders, both propagating and evanescent (not shown). c) An incident beam is decomposed to plane waves with wave vectors [K m -wm u z ] and d) the reflection is expressed in terms of the corresponding Bragg orders.

Figure 3.
Figure 3.

Measured and simulated reflection spectra from a typical PMMA on silicon opal. Numerical aperture of 0.55 corresponds to 50x magnifying optics with a spot size of about 3 µm and numerical aperture 0.3 corresponds to 10x magnification and a spot size of 15 µm.

Figure 4.
Figure 4.

A typical SEM image of the polystyrene on GaAs opal.

Figure 5.
Figure 5.

Reflection from a polystyrene on GaAs opal for different incident angles: a) ϕ=20°, b) ϕ=30° and c) ϕ=40°. d) The incidence angle and the measurement data is as in c) but the simulated reflection is computed using a random mixture of FCC and HCP lattices. Curves marked with stars are simulated and continuous lines are measured.

Equations (21)

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𝓛 ( x , y , ε , μ , ω ) Ψ = z Ψ ,
𝓛 = [ 0 𝓐 𝓑 0 ] ,
𝓐 = [ x ( j ω ε ) 1 y x ( j ω ε ) 1 x + j ω μ y ( j ω ε ) 1 y j ω μ y ( j ω ε ) 1 x ] .
Ψ K ( r , z ) = exp ( j K · r ) n Ψ n ( z ) exp ( j G n · r ) ,
Ψ K ( r , ( i + 0.5 ) Δ ) = Ψ K ( r , ( i 0.5 ) Δ ) + Δ [ 𝓛 Ψ K ( r , z ) ] z = i Δ ,
Ψ n ( z ) = Ψ n exp ( j λ z ) ,
Ψ K ( r , z ) = exp ( j K · r ) n exp ( j G n · r ) [ ( a K , n + Ψ K , n 1 + + b K , n + Ψ K , n 2 + ) exp ( j w K , n z )
+ ( a K , n Ψ K , n 1 + b K , n Ψ K , n 2 ) exp ( j w K , n z ) ] ,
λ K , n = ± w K , n = ± ( ω 2 ε μ K + G n 2 ) 1 2 ,
Ψ K , n 1 ± = α K , n [ ± w K , n ω μ 0 k x k y k y 2 + ω 2 ε μ ] , Ψ K , n 2 ± = β K , n [ k x k y k y 2 ω 2 ε μ ± w K , n ω ε 0 ] ,
u z · [ 𝓔 ( Ψ K , n 1 ± ) × 𝓗 ( Ψ K , n 2 ± ) * ] = u z · [ 𝓔 ( Ψ K , n 2 ± ) × 𝓗 ( Ψ K , n 1 ± ) * ] = 0 ,
u z · [ 𝓔 ( Ψ K , n l ± ) × 𝓗 ( Ψ K , n l ± ) * ] = { ± 1 , w K , n 2 > 0 0 , w K , n 2 0 ,
M ( ω , K ) f = b ,
Ψ inc ( r , z ) = m exp ( j K m · r j w m z ) ( c m 1 Ψ m 1 + c m 2 Ψ m 2 ) ,
Ψ refl ( r , z ) = m exp ( j K m · r ) ( c m 1 Θ m 1 + c m 2 Θ m 2 ) ,
Θ m l = n exp ( j G n · r + j w m , n z ) ( a m , m l Ψ m , n 1 + + b m , n l + Ψ m , n 2 + ) , l = 1 , 2 .
P = 1 2 { S d 2 r m , m 𝓔 ( c m 1 Θ m 1 + c m 2 Θ m 2 ) × 𝓗 ( c m 1 Θ m 1 + c m 2 Θ m 2 ) * exp [ j ( K m K m ) · r ] } ,
R = Σ m { c m 1 2 R m 1 + [ c m 1 ( c m 2 ) * R m 12 ] + c m 2 2 R m 2 } Σ m ( c m 1 2 + c m 2 2 ) ,
R m l = n δ ( w m , n ) ( a m , n l + 2 + b m , n l + 2 ) , l = 1 , 2 ,
R m 12 = 2 n δ ( w m , n ) [ a m , n 1 + ( a m , n 2 + ) * + b m , n 1 + ( b m , n 2 + ) * ] .
R = Σ m 2 π Δ K ( K m + 0.5 Δ K ) c m 2 R m Σ m 2 π Δ K ( K m + 0.5 Δ K ) c m 2 ,

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