## Reflection of focused beams from opal photonic crystals

Optics Express, Vol. 13, Issue 7, pp. 2653-2667 (2005)

http://dx.doi.org/10.1364/OPEX.13.002653

Acrobat PDF (197 KB)

### 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

## 1. Introduction

2. S. W. Leonard, H. M. van Driel, A. Birner, U. Gsele, and P. R. Villeneuve, “Single-mode transmission in two-dimensional macroporous silicon photonic crystal waveguides,” Opt. Lett. **25**, 1550–1552 (2000). [CrossRef]

7. 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]

8. 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]

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

14. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

16. P. R. Villeneuve and M. Pich, “Photonic bandgaps in periodic dielectric structures,” Prog. Quantum Electron. **18**, 153–200 (1994). [CrossRef]

17. S. Guo, F. Wu, and S. Albin, “Photonic band gap analysis using finite-difference frequency-domain method,” Opt. Express **12**, 1741–1746 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741. [CrossRef] [PubMed]

18. J. B. Pendry and A. MacKinnon, “Calculation of Photon Dispersion Relations,” Phys. Rev. Lett. **69**, 2772–2775 (1992). [CrossRef] [PubMed]

19. J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A **12**, 1765–1771 (1995). [CrossRef]

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

21. 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]

22. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped my microcavities,” J. Opt. Soc. Am. A **14**, 3323–3332 (1997). [CrossRef]

23. 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]

25. 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), http://www.intec.rug.ac.be/picco/download/D8 report.pdf.

18. J. B. Pendry and A. MacKinnon, “Calculation of Photon Dispersion Relations,” Phys. Rev. Lett. **69**, 2772–2775 (1992). [CrossRef] [PubMed]

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]

*i*) Instead of solving a big problem once, one solves a small problem many times, resulting in a linear scaling of computational resources,

*ii*) after the reflection coefficients for individual plane waves have been computed, the reflection coefficient of any arbitrary beam can be obtained with minor effort,

*iii*) the decomposition technique parallelizes trivially since each of the incident plane waves creates an independent problem.

## 2. Computational method

### 2.1. Preliminary considerations and definitions

*z*<0, is homogeneous medium characterized by ε

_{a}. Region 2, 0≤

*z*<

*h*, is a finite thickness photonic crystal defined by ε

_{b}(

**r**,

*z*)=ε

_{b}(

**r**+

*m*

**a**

_{1}+

*n*

**a**

_{2},

*z*), where

**a**

_{1}and

**a**

_{2}are lattice vectors on

*xy*-plane and

*m*and

*n*are integers. Region 3,

*z*≥

*h*, is again homogeneous medium characterized by ε

_{c}. Dielectric functions in all three regions may be dispersive and complex valued.

*z*=0 and

*z*=

*h*and the two transverse lattice vectors

**a**

_{1}and

**a**

_{2}. This should not be confused with the unit cell of a truly three-dimensionally periodic photonic crystal, defined by three lattice vectors. The semi-periodicity in the

*z*-direction, as in finite thickness opals, can be obtained simply by filling the unit cell with a semi-periodic dielectric function. The time dependence is assumed to be harmonic

*f*(

*t*) ~exp(-

*jωt*) throughout this paper and its explicit notation is omitted.

### 2.2. Diagonalized form

*z*since all

*z*-dependency is isolated to the right hand side of the equation [29

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]

*E*

_{x}

*E*

_{y}

*H*

_{x}

*H*

_{y}]

^{T}is a vector consisting of the transversal field components and

*∂*

_{k}denotes differentiation with respect to the given coordinate

*k*=

*x*,

*y*,

*z*. 𝓛 is a matrix operator

*𝓐*is defined below:

*j*is the imaginary unit and parameters

*ε*and

*µ*are the (generally) complex valued and position dependent permittivity and permeability functions, respectively. The remaining sub-operator

*𝓑*can be obtained from

*𝓐*by replacing

*ε*with -

*µ*and vice versa. The obvious consequence of Eq. (1) is that once the transversal field distribution in some

*z*=constant plane is known, the derivative in the normal direction is easily evaluated and the transversal fields are then uniquely determined everywhere in the space. The two remaining field components are redundant and can be computed from the transversal fields a posteriori [26].

### 2.3. Field expansions

*xy*-plane suggests expanding the fields in a plane wave basis

**G**

_{n}] is a truncated set of reciprocal lattice vectors and

**K**is a Bloch wave vector restricted in the first Brillouin zone (BZ). The expansion coefficients Ψ

_{n}(

*z*) are vectors of the

*z*-dependent transversal field components, which will be expanded differently in the photonic crystal medium and the claddings.

*z*=constant layers and use finite differences for relating the fields in adjacent layers to each other. Substituting Eq. (4) into Eq. (1) gives the

*z*-derivative of the transversal fields and by using the standard first order finite difference scheme we get

*i*is the index of the plane. A numerically useful formula is obtained by multiplying both sides of Eq. (5) with exp[-

*j*(

**G**

_{m}+

**K**) ·

**r**] for each plane wave

**G**

_{m}in the basis and integrating over the unit cell. The result is a matrix equation relating the plane wave expansion coefficients in adjacent layers to each other. In the actual implementation, we define electric fields in planes

*z*=(

*i*+0.5)Δ and magnetic fields in

*z*=

*i*Δ, for whole numbers

*i*, since it follows from Eq. (1) that the derivative of electric field depends only on the magnetic field and vice versa. Equation (5) is then correspondingly separated into two parts. Fig. 1 illustrates the discretization scheme.

*z*-dependency:

*N*eigenvalues for a set of

*N*plane waves (due to the four-dimensional vector coefficients) but the system decouples into

*N*independent eigensystems of dimension four due to the position independent material parameters. For a given

**K**, the complete expression for the fields is

*z*-axis. The corresponding four eigenvectors are

*k*

_{x,y}=

**u**

_{x,y}·(

**K**+

**G**

_{n}), with

**u**

_{x,y}denoting the dimensionless unit vector in

*x*- and

*y*-directions. There is a great freedom in the selection of the eigenvectors since any linear combination of eigenvectors corresponding to the same eigenvalue is also an eigenvector. Our preferred choice is such that

*𝓔*evaluates to the electric field part of the argument and

*𝓗*to the magnetic,

**u**

_{z}is the unit vector in

*z*-direction and the asterisk denotes complex conjugation. The normalization coefficients

*α*

_{K,n}and

*β*

_{K,n}are chosen such that

### 2.4. Constructing and solving the system equation

*N*of the total 4

*N*eigenvectors are needed to satisfy the interface conditions for each cladding and we select those which either decay or radiate energy away from the slab. The result is a homogeneous system of equations, whose solutions correspond to guided slab modes in the photonic crystal. Sources such as currents or incident fields can be easily included by replacing the right hand zero vector with the source vector, to obtain an equation of the form

**M**is a matrix containing the eigenvectors and all the finite-difference relations in the slab volume,

**f**is a vector containing all the expansion coefficients for the transverse field components and

**b**describes the excitation.

## 3. Reflection of beams from periodic structures

*et al*. [30

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]

*z*>

*h*, i.e. in the half space of the incident and reflected beams. This is not essential to the method, but for absorptive materials, the incident and reflected intensities are

*z*-dependent which complicates the interpretation of the results.

### 3.1. The incident and reflected fields

*ω*, propagating from

*z*=+∞ towards the photonic crystal surface at

*z*=

*h*. A general expression of such a beam is given by a Fourier integral of the eigenvectors given in Eq. (9) but in the realm of numerical computation we will sample the wave vector and use a series representation

**K**

_{m}as a Bloch vector of the photonic crystal lattice and compute the reflection of each plane wave [exp(

*j*

**K**

_{m}·

**r**-

*jw*

_{m}

*z*)

*l*=1, 2, individually from Eq. (12). The results are then added to give the reflection of the complete beam. Since each of the incident plane waves is a Bloch wave, it suffices to consider only one unit cell of the photonic crystal. Notice that

**K**

_{m}need not be limited to the first BZ of the photonic crystal since we can always write it in the form

**K**

_{m}=

**K′**

_{m}+

**G**

_{n}, where

**K′**

_{m}is a vector in the first BZ and

**G**

_{n}is a suitable reciprocal vector.

*w*

_{m,n}is the eigenvalue corresponding to the transversal wave vector (

**K**

_{m}+

**G**

_{n}) and

*a*

^{l}

^{+}

_{m,n}and

*b*

^{l}

^{+}

_{m,n}are scalar coefficients obtained from the solution of Eq. (12). Qualitatively speaking, summing over

*m*means summing over different incident angles and summing over n means summing over different Bragg orders. The wave vectors of the incident and reflected beams are illustrated in Fig. 2.

### 3.2. Reflection coefficient of the beam

**P**=(1/2)ℜ(

**E**×

**H***) over a surface

*S*. Taking the differential surface vector d

**S**to be parallel to

*z*, the projection d

**S**·

**P**can be written in terms of the transversal field components and the reflected power becomes

*𝓔*and

*𝓗*have the same meaning as in Eq. (10). The reflectance is obtained as a quotient of

*P*and the power in the incident beam.

**K**

_{m}. The aim is to select the vector set [

**K**

_{m}] such that the harmonic basis functions defined by vectors [

**K**

_{m}+

**G**

_{n}] form an orthogonal and unique (that is, no two functions are the same) set over some surface

*S*. The orthogonality can be achieved if the vectors

**K**

_{m}are selected among the reciprocal vectors of a supercell defined by

*L*

_{1}

**a**

_{1}and

*L*

_{2}

**a**

_{2}, where

*L*

_{1}and

*L*

_{2}are integers and

**a**

_{1}and

**a**

_{2}are the lattice vectors of the photonic crystal. The requirement of uniqueness is satisfied if we include only vectors which are inside the first BZ of the photonic crystal. Then [

**K**

_{m}] together with the reciprocal vectors of the photonic crystal [

**G**

_{n}] form the set of reciprocal vectors for the supercell. These restrictions limit the beams which can be expressed using Eq. (13). The introduction of the supercell makes the incident beam (and the reflected field) periodic, which is usually not a problem since adjacent beams can be decoupled by giving

*L*

_{1}and

*L*

_{2}sufficiently large values. A more fundamental restriction is the upper bound on the length of

**K**

_{m}, which defines a cone or a numerical aperture (NA) limiting the possible plane waves in the incident beam. This in turn sets a lower bound on the achievable localization on the crystal surface. Whether this is of practical concern or not, depends on the measurement setup. In our case, the NA of the optics used for the measurement is more restrictive than the NA originating from the limitations on

**K**

_{m}. In any case, if these conditions are too restrictive, the reflected power can always be evaluated directly from Eq. (16), disregarding the simplified formula we are about to develop.

**K**

_{m}|≥

*ω*(

*εµ*)

^{1/2}. Relaxing this condition opens up a possibility that the incident beam excites guided slab modes, which cannot be decoupled from each other simply by increasing the periodicity interval of the sources. These modes should in principle have an infinite amplitude because they are excited by an infinite number of sources and they do not decay, but in practice the amplitude is determined by numerical effects and is more or less random. Even if guided modes are excited, it is not impossible that the reflection coefficient of intensity is computed correctly, since guided modes, by definition, do not contribute to radiation. However, we do not have a proof on this and therefore we avoid computing the reflection in the presence of guided modes.

*m,m′*) and (

*n,n′*) and the double summations cancel to single summations. The numerous vector products can be evaluated using the properties of the eigenvectors given in Eqs. (10) and (11). After a cumbersome but straightforward calculation we can show that the reflection coefficient of a beam defined by the coefficients

*w*

_{m,n})=1, if the eigenvalue

*w*

_{m,n}is real valued and 0 otherwise. The physical origin for the appearance of δ(

*w*

_{m,n}) is that the eigenvectors with an imaginary wave number are evanescent and do not carry energy in a direction parallel to the

*z*-axis.

## 4. PMMA opals for visible light

31. 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]

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

33. 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]

*a*=268 nm, using the modified surfactant free emulsion polymerization technique described elsewhere [34

34. 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]

*v*

_{0}=2.6 mm/h, resulting in films of approximately 18 monolayers, or a thickness of 4.2

*µ*m. After the growth, the samples were sintered at 80° C during 90 minutes.

### 4.1. Comparison of simulations and measurements

*e*- and

*h*-polarizations (

*e*-polarization has electric field parallel to the sample surface and correspondingly for

*h*-polarization) separately and averaged. For a given polarization, the ratio of

*c*

_{m}.

**K**and

*w*such that the angle between the

*z*-axis and the plane wave is given by

*ϕ*=tan

^{-1}(|

**K**|/

*w*) and the angle between

*x*-axis and the plane wave by

*θ*=tan

^{-1}(

*K*

_{y}/

*K*

_{x}), where

*K*

_{x,y}=

**u**

_{x,y}·

**K**. It turns out that for a sufficiently small

*ϕ*, the reflectivity varies only little with

*θ*. Particularly, at the maximum acceptance angle of the collecting optics,

*ϕ*=33.4°, the variation of reflectivity with

*θ*is at most 1.5% for any wavelength considered in the measurement. Therefore, we can take advantage of the rotational symmetry of the incident beam and integrate Eq. (17) over

*θ*with a little loss in accuracy. Then it is sufficient to sample

**K**only over a line instead of a surface, thus reducing the computational burden significantly. The low dependence on

*θ*can be addressed to the high, six fold rotational symmetry of the FCC-lattice about [111]-direction (

*z*-axis) and low refractive index contrast between PMMA and air. However, it should be noted that if

*ω*is high enough to allow more than one diffraction order, then the orientation of

**K**may indeed have a significant effect on the reflectivity. In our samples, the second diffraction order arises at a vacuum wavelength of 464 nm, which is outside the spectral scope of the measurements.

_{K}=|

**K**

_{m}

_{+1}-

**K**

_{m}| is the uniformly spaced difference in the length between two consecutive

**K**-vectors on the sampling line and

*R*

_{m}is the appropriate linear combination of

*ω*-points and

**K**was sampled at 40 points for each value of

*ω*. We assumed PMMA to be transparent and to possess a frequency independent refractive index

*n*

_{PMMA}=1.489. The dispersive and absorptive dielectric constant of the silicon substrate was taken according to Ref. [36]. The shape of the incident beam on the sample was not precisely known but the optics used for collecting light had a smaller NA than the optics used for illumination. Therefore we set

*c*

_{m}=1 for all plane waves that are accepted by the collecting optics, i.e. sin(ϕ

_{m})<NA, and

*c*

_{m}=0 otherwise.

## 5. Polystyrene opals for infrared light

8. 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]

### 5.1. Results

*ϕ*using

*e*-polarized light. Simulations were performed with the same resolution as in the previous section but the incident beam was so wide (about 1 mm in diameter) and slightly focused that we assumed a plane wave excitation. A frequency dependent and complex valued dielectric function was used for both GaAs [37

37. 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]

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]

*M*direction of the triangular lattice on the crystal surface.

11. 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]

## 6. Conclusion

*O*(

*NT*), where

*N*is the number of plane waves needed to expand the incident beam and

*T*is the time required for solving the Bloch periodic problem of a single plane wave.

## Acknowledgments

## References

1. | K. Sakoda, |

2. | S. W. Leonard, H. M. van Driel, A. Birner, U. Gsele, and P. R. Villeneuve, “Single-mode transmission in two-dimensional macroporous silicon photonic crystal waveguides,” Opt. Lett. |

3. | 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 |

4. | 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 |

5. | S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full Three-Dimensional Photonic Bandgap Crystals at Near-Infrared Wavelengths,” Science |

6. | E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case Employing Nonspherical Atoms,” Phys. Rev. Lett. |

7. | 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. |

8. | P. Jiang, J. F. Bertone, K. S. Hwang, and V. L. Colvin, “Single-Crystal Colloidal Multilayers of Controlled Thickness,” Chem. Mater. |

9. | 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. |

10. | G. Subramania, K. Constant, R. Biswas, M. M. Sigalas, and K.-M. Ho, “Optical Photonic Crystals Synthesized from Colloidal Systems of Polystyrene Spheres and Nanocrystalline Titania,” J. Lightwave Technol. |

11. | 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 |

12. | M. Bardosova and R. H. Tredgold, “Ordered layers of monodispersive colloids,” J. Mater. Chem. |

13. | D. J. Norris and Y. A. Vlasov, “Chemical Approaches to Three-Dimensional Semiconductor Photonic Crystals,” Adv. Mater. |

14. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express |

15. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. |

16. | P. R. Villeneuve and M. Pich, “Photonic bandgaps in periodic dielectric structures,” Prog. Quantum Electron. |

17. | S. Guo, F. Wu, and S. Albin, “Photonic band gap analysis using finite-difference frequency-domain method,” Opt. Express |

18. | J. B. Pendry and A. MacKinnon, “Calculation of Photon Dispersion Relations,” Phys. Rev. Lett. |

19. | J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A |

20. | N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter |

21. | L.-M. Li and Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B |

22. | G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped my microcavities,” J. Opt. Soc. Am. A |

23. | 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 |

24. | A. Taflove, |

25. | 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), http://www.intec.rug.ac.be/picco/download/D8 report.pdf. |

26. | K. Varis and A. R. Baghai-Wadji, “A Novel 3D Pseudo-Spectral Analysis of Photonic Crystal Slabs,” ACES J. |

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 |

28. | X. Zhang, “Image resolution depending on slab thickness and object distance in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. B |

29. | A. R. Baghai-Wadji, “A Symbolic Procedure for the Diagonalization of Linear PDEs in Accelerated Computational Engineering,” in |

30. | M. T. Manzuri-Shalmani and A. R. Baghai-Wadji, “Elemental field distributions in corrugated structures with large-amplitude gratings,” Electron. Lett. |

31. | D. J. Norris, E. G. Arlinghaus, L. Meng, R. Heiny, and L. E. Scriven, “Opaline Photonic Crystals: How Does Self-Assembly Work?” Adv. Mater. |

32. | Z.-Z. Gu, A. Fujishima, and O. Sato, “Fabrication of High-Quality Opal Films with Controllable Thickness,” Chem. Mater. |

33. | M. Egen, R. Voss, B. Griesebock, R. Zentel, S. Romanov, and C. M. Sotomayor Torres, “Heterostructures of Polymer Photonic Crystal Films,” Chem. Mater. |

34. | M. Müller, R. Zentel, T. Maka, S. G. Romanov, and C. M. Sotomayor Torres, “Dye-Containing Polymer Beads as Photonic Crystals,” Chem. Mater. |

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). |

36. | O. Madelung, |

37. | W. G. Spitzer and J. M. Whelan, “Infrared absorption and electron effective mass in n-type gallium arsenide,” Phys. Rev. |

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. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(160.5470) Materials : Polymers

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 17, 2005

Revised Manuscript: March 23, 2005

Published: April 4, 2005

**Citation**

Karri Varis, Marco Mattila, Sanna Arpiainen, Jouni Ahopelto, Fredrik Jonsson, Clivia Sotomayor Torres, Marc Egen, and Rudolf Zentel, "Reflection of focused beams from opal photonic crystals," Opt. Express **13**, 2653-2667 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2653

Sort: Journal | Reset

### References

- K. Sakoda, Optical properties of photonic crystals (Springer-Verlag, Berlin, 2001).
- S. W. Leonard, H. M. van Driel, A. Birner, U. Gsele, and P. R. Villeneuve, �??Single-mode transmission in two-dimensional macroporous silicon photonic crystal waveguides,�?? Opt. Lett. 25, 1550�??1552 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- G. Subramania, K. Constant, R. Biswas, M. M. Sigalas, and K.-M. Ho, �??Optical Photonic Crystals Synthesized from Colloidal Systems of Polystyrene Spheres and Nanocrystalline Titania,�?? J. Lightwave Technol. 17, 1970�??1974 (1999). [CrossRef]
- 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]
- M. Bardosova and R. H. Tredgold, �??Ordered layers of monodispersive colloids,�?? J. Mater. Chem. 12, 2835�??2842 (2002). [CrossRef]
- D. J. Norris and Y. A. Vlasov, �??Chemical Approaches to Three-Dimensional Semiconductor Photonic Crystals,�?? Adv. Mater. 13, 371�??376 (2001). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell�??s equations in a plane wave basis,�?? Opt. Express 8, 173�??190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.</a> [CrossRef] [PubMed]
- 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]
- P. R. Villeneuve and M. Pich, �??Photonic bandgaps in periodic dielectric structures,�?? Prog. Quantum Electron. 18, 153�??200 (1994). [CrossRef]
- S. Guo, F. Wu, and S. Albin, �??Photonic band gap analysis using finite-difference frequency-domain method,�?? Opt. Express 12, 1741�??1746 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741.</a> [CrossRef] [PubMed]
- J. B. Pendry and A. MacKinnon, �??Calculation of Photon Dispersion Relations,�?? Phys. Rev. Lett. 69, 2772�??2775 (1992). [CrossRef] [PubMed]
- J. M. Elson and P. Tran, �??Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,�?? J. Opt. Soc. Am. A 12, 1765�??1771 (1995). [CrossRef]
- N. Stefanou, V. Karathanos, and A. Modinos, �??Scattering of electromagnetic waves by periodic structures,�?? J. Phys.: Condens. Matter 4, 7389�??7400 (1992). [CrossRef]
- 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]
- G. Tayeb and D. Maystre, �??Rigorous theoretical study of finite-size two-dimensional photonic crystals doped my microcavities,�?? J. Opt. Soc. Am. A 14, 3323�??3332 (1997). [CrossRef]
- 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]
- 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>
- K. Varis and A. R. Baghai-Wadji, �??A Novel 3D Pseudo-Spectral Analysis of Photonic Crystal Slabs,�?? ACES J. 19, 101�??111 (2004).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.