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Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 3 — Jan. 29, 2001
  • pp: 197–202
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Applications of the layer-KKR method to photonic crystals

A. Modinos, N. Stefanou, and V. Yannopapas  »View Author Affiliations


Optics Express, Vol. 8, Issue 3, pp. 197-202 (2001)
http://dx.doi.org/10.1364/OE.8.000197


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Abstract

A brief introduction of the layer-Korringa-Kohn-Rostoker method for calculations of the frequency band structure of photonic crystals and of the transmission and reflection coefficients of light incident on slabs of such crystals is followed by two applications of the method. The first relates to the frequency band structure of metallodi-electric composites and demonstrates the essential difference between cermet and network topology of such composites at low frequencies. The second application is an analysis of recent measurements of the reflection of light from a slab of a colloidal system consisting of latex spheres in air.

© Optical Society of America

In traditional energy band structure calculations of an electron in an ordinary crystal, or of the frequency band structure of the electromagnetic (EM) field in the case of photonic crystals, one starts with a fixed value of the reduced wavevector k, and by some method or other (in photonic crystals this is usually the plane-wave method, but the Korringa-Kohn-Rostoker (KKR) method has also been used to this effect [1

1. X. D. Wang, X-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electro-magnetic waves,” Phys. Rev. B 47, 4161–4167 (1993). [CrossRef]

, 2

2. A. Moroz, “Inward and outward integral equations and the KKR method for photons,” J. Phys.: Condens. Matter 6, 171–182 (1994). [CrossRef]

, 3

3. A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999). [CrossRef]

]) one solves the eigenvalue problem, for the given k, to obtain all the eigenfrequencies within a very wide frequency range, and the corresponding eigenmodes of the scalar/vector field under consideration. These eigenmodes are, in an infinite crystal, propagating Bloch waves which have the property

ψkα(r+Rn(3))exp[iωa(k)t]=exp(ik·Rn(3))ψkα(r)exp[iωα(k)t],
(1)

where Rn(3) is any vector of the three-dimensional (3D) lattice which defines the periodicity of the infinite crystal; and a is a band index which defines the different frequency bands ωα (k) and the corresponding eigenmodes. For the Schrödinger field (electrons in a crystal) ψ is a scalar quantity; in the case of the EM field ψ is a vector quantity.

On-shell methods [4

4. K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979). [CrossRef]

, 5

5. K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. I. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65, 2265–2275 (1996). [CrossRef]

, 6

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

, 7

7. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). [CrossRef]

, 8

8. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). [CrossRef]

, 9

9. J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992). [CrossRef] [PubMed]

, 10

10. P. M. Bell, J. B. Pendry, L. Martín-Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Commun. 85, 306–322 (1995). [CrossRef]

, 11

11. Y. Qiu, K. M. Leung, L. Cavin, and D. Kralj, “Dispersion curves and transmission spectra of a two-dimensional photonic band-gap crystal-Theory and experiment,” J. Appl. Phys. 77, 3631–3636 (1995). [CrossRef]

, 12

12. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 791–809 (1999).

] proceed differently; the frequency is fixed and one obtains the eigenmodes of the crystal for this frequency. Therefore, these methods are ideally suited for photonic crystals consisting of strongly dispersive materials such as real metals [13

13. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals” Phys. Rev. B 60, 5359–5365 (1999). [CrossRef]

, 14

14. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, C. M. de Sterke, P. A. Robinson, and A. A. Asatryan, “Anomalous absorptance by stacked metallic cylinders,” Opt. Commun. 168, 47–53 (1999). [CrossRef]

]. One views the crystal as a succession of layers (slices) parallel to a given crystallographic plane of the crystal. The layers have the same two-dimensional (2D) periodicity (that of the chosen crystallographic plane) described by a 2D lattice:

Rn=n1a1+n2a2,
(2)

where a1 and a2 are primitive vectors of the said plane (which is taken to be the xy plane), and n 1,n 2=0, ±1, ±2,…. We may number the sequence of layers which constitute the infinite crystal, extending from z=-∞ to z=+∞, as follows: …- 2, -1, 0,1, 2,…. The (N+1)th layer is obtained from the Nth layer by a primitive translation to be denoted by a3. Obviously, a1,a2, and a3 constitute a basis for the 3D space lattice of the infinite crystal.

We define the 2D reciprocal lattice corresponding to Eq. (2):

g=m1b1+m2b2,m1,m2=0,±1,±2±,
(3)

where b j ·a j =2πδij, i,j=1, 2. The reduced (kx , ky )-zone associated with the above, which has the full symmetry of the given crystallographic plane is known as the surface Brillouin zone (SBZ) (see, e.g., Ref. [15

15. A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).

]). We define a corresponding 3D reduced k-zone as follows:

k(kx,ky)withintheSBZ
b32<kzb32,
(4)

where b3=2πa1×a2/[a1·(a2×a3)] is normal to the chosen crystallographic plane. The reduced k-zone defined by Eq. (4) is of course completely equivalent to the commonly used, more symmetrical Brillouin zone (BZ), in the sense that a point in one of them lies also in the other or differs from such one by a vector of the 3D reciprocal lattice.

Let us now assume that we have a photonic crystal consisting of nonoverlapping spherical scatterers in a host medium of different dielectric function and let us look at the structure as a sequence of layers of spheres with the 2D periodicity of Eq. (2). A Bloch wave solution, of given frequency ω and given k‖, of Maxwell’s equations for the given system has the following form in the space between the Nth and the (N+1)th layers (we write down only the electric-field component of the EM wave):

E(r)=g{Eg+(N)exp[iKg+·(rAN)]+Eg(N)exp[iKg·(rAN)]}
(5)

with

Kg±=(k+g,±[q2(k+g)2]12),
(6)

Egi(N)=g'i'Qgi;g'i'IVEg'i'(N+1)+g'i'Qgi;g'i'IIIEg'i'+(N)
Egi+(N+1)=g'i'Qgi;g'i'IEg'i'+(N)+g'i'Qgi;g'i'IIEg'i'(N+1),
(7)

where i=x,y,z, and Q are appropriately constructed transmission/reflection matrices for the layer. For a detailed description of these matrices, which are functions of ω, k‖, the scattering properties of the individual scatterer (sphere) and the geometry of the layer, see Refs.[6

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

, 7

7. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). [CrossRef]

, 8

8. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). [CrossRef]

].

A generalized Bloch wave, by definition, has the property:

Eg±(N+1)=exp(ik·a3)Eg±(N)
k=(k,kz(ω,k))
(8)

where kz may be real or complex. Substituting Eq. (8) into Eq. (7) we obtain:

(QIQII[QIV]1QIIIQI[QIV]1[IQIIIQII])(E+(N)E(N+1))=
exp(ik·a3)(E+(N)E(N+1))
(9)

where E ± are column matrices with elements: Eg1x± , Eg1y± , Eg1z± , Eg2x± , Eg2y± , Eg2z± , …. In practice we keep a finite number of g-vectors (those with |g|<gmax , where gmax is a cutoff parameter) which leads to a solvable system of equations. The number of independent components of the electric field is in fact 2/3 of the above, in view of the zero-divergence of the electric field, but we need not go into technical details here [7

7. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). [CrossRef]

, 8

8. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). [CrossRef]

].

The on-shell method we have described has a number of advantages over the traditional methods. The unit layer along the z-direction may consist not of one plane of spheres (as implied so far) but by a number of planes which may be different (the radii of the spheres and/or their dielectric functions may be different) as long as they have the same 2D periodicity; which allows us to study a variety of heterostructures with relative ease. Moreover, we can easily calculate the transmission, reflection and absorption coefficients of light incident at any angle on a slab of the photonic crystal. For this purpose we combine the Q-matrices of the different layers that make the slab into a final set of matrices which determine the scattering properties of the slab [6

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

, 7

7. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). [CrossRef]

, 8

8. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). [CrossRef]

]. The method applies equally well to nonabsorbing systems which have a well defined frequency band structure, and to absorbing systems (with a complex dielectric function for some of the constituent materials of the crystal) which do not have a well defined frequency band structure (there can be no truly propagating waves in such systems). Finally we can deal with slabs of photonic crystals which contain impurity planes as long as the 2D periodicity parallel to the surface of the slab is retained.

A region of frequency constitutes an absolute (omnidirectional) frequency gap if no propagating wave exists in the infinite crystal over this region whatever the value of k . This can be ascertained from the projection of the frequency band structure on the SBZ of the given surface. In Fig. 1 we show two examples of such band-structure projections. Both examples refer to so-called metallodielectric structures. Nowadays, it is possible to fabricate well-defined ordered metallodielectric composites consisting of tailored mesoscopic building blocks. Such nanocrystals provide opportunities for optimizing optical properties of materials and offer possibilities for observing new, and potentially useful physical phenomena [16

16. M. P. Pileni, “Nanosized particles made in colloidal assemblies,” Langmuir 13, 3266–3276 (1997) [CrossRef]

, 17

17. W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84, 2853–2856 (2000). [CrossRef] [PubMed]

, 18

18. O. D. Velev and E. W. Kaler, “Structured porous materials via colloidal crystal templating: from inorganic oxides to metals,” Adv. Mater. 12, 531–534 (2000). [CrossRef]

]. In both cases considered here the crystal is an fcc one and is viewed as a sequence of planes parallel to the (001) surface. In the first case (an example of cermet topology in relation to the metallic component) the crystal consists of metallic spheres, described by a Drude dielectric function

(ω)=1ωp2ω2,
(10)

in gelatine. In the second case, which is typical of network topology in relation to the metallic component, the crystal consists of nonoverlapping silicon spheres in a metal described by Eq. (10). We have deliberately disregarded absorption by the metallic component, in order to be able to calculate a frequency band structure in an unambiguous manner. The shaded areas in Fig. 1 correspond to allowed regions of frequency; for a given k , these regions extend over those frequencies for which there is at least one propagating wave in the infinite crystal; the blank regions represent frequency gaps for the given k . Obviously an absolute gap exists only when a blank region of frequency is common to all k in the SBZ. As a rule one shows the projection of the band structure for k along symmetry lines of the SBZ as in Fig.1; but one must check that a gap over these lines extends to the whole of the SBZ. In Fig. 1 we indicate the positions of the 2 l -pole plasma resonances (for l=1,2,3) of an individual scatterer (a metallic sphere in gelatine in the first case, and a silicon sphere in a metal in the second case). It is evident that in each case things develop about these resonances; in the second case one clearly obtains the allowed frequency bands about these levels. Evidently absorption will occur over the allowed regions of frequency and to some lesser degree beyond these regions into the frequency gaps, when ε(ω>) of Eq. (10) is replaced by the complex Drude dielectric function: ε(ω)=1-ωp2 /ω(ω+ -1) [13

13. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals” Phys. Rev. B 60, 5359–5365 (1999). [CrossRef]

]. One thing is worth pointing out, and this concerns the very different behavior of the cermet and network topologies at low frequencies. In the cermet topology the composite behaves like a dielectric in the long wavelength limit, allowing the propagation of EM waves through it; in the network topology the composite behaves like a metal (its DC conductivity does not vanish as ω→0) and does not allow the propagation of EM waves through it.

Fig. 1. Projection of the frequency band structure of a fcc crystal of (a): nonabsorbing Drude spheres (f=0.1, ωp a0/c=1, f is the fractional volume occupied by the spheres and a 0 is the first-neighbor distance) in gelatine (ε=2.37) and (b): silicon (ε=11.9) spheres in a nonabsorbing Drude metal (p/c=0.2, ωpa 0/c=1, S is the radius of the spheres), on the SBZ of the fcc (001) surface along the symmetry lines shown in the inset.

Fig. 2. (a) The photonic band structure normal to the (111) surface of a fcc crystal of close-packed PMMA (ε=2.2231) spheres of radius S=243.415 nm in air. The black/red lines refer to doubly degenerate/nondegenerate bands. (b) The calculated reflectance curve for light incident normally on a slab of the above crystal consisting of 32 planes of spheres parallel to the (111) surface. (c) The solid line was calculated for the same crystal as in (b) but now the dielectric constant of the PMMA spheres contains an imaginary part Imε=0.05 and the crystal consists of 4096 planes of spheres. The blue dotted curve represents the experimental data [19].

Acknowledgements

We thank M. Allard and coworkers for sending us a copy of their paper prior to publication. V. Yannopapas is supported by the State Foundation (I. K. Y.) of Greece.

References and links

1.

X. D. Wang, X-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electro-magnetic waves,” Phys. Rev. B 47, 4161–4167 (1993). [CrossRef]

2.

A. Moroz, “Inward and outward integral equations and the KKR method for photons,” J. Phys.: Condens. Matter 6, 171–182 (1994). [CrossRef]

3.

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999). [CrossRef]

4.

K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979). [CrossRef]

5.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. I. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65, 2265–2275 (1996). [CrossRef]

6.

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

7.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). [CrossRef]

8.

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). [CrossRef]

9.

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992). [CrossRef] [PubMed]

10.

P. M. Bell, J. B. Pendry, L. Martín-Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Commun. 85, 306–322 (1995). [CrossRef]

11.

Y. Qiu, K. M. Leung, L. Cavin, and D. Kralj, “Dispersion curves and transmission spectra of a two-dimensional photonic band-gap crystal-Theory and experiment,” J. Appl. Phys. 77, 3631–3636 (1995). [CrossRef]

12.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 791–809 (1999).

13.

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals” Phys. Rev. B 60, 5359–5365 (1999). [CrossRef]

14.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, C. M. de Sterke, P. A. Robinson, and A. A. Asatryan, “Anomalous absorptance by stacked metallic cylinders,” Opt. Commun. 168, 47–53 (1999). [CrossRef]

15.

A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).

16.

M. P. Pileni, “Nanosized particles made in colloidal assemblies,” Langmuir 13, 3266–3276 (1997) [CrossRef]

17.

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84, 2853–2856 (2000). [CrossRef] [PubMed]

18.

O. D. Velev and E. W. Kaler, “Structured porous materials via colloidal crystal templating: from inorganic oxides to metals,” Adv. Mater. 12, 531–534 (2000). [CrossRef]

19.

M. Allard, E. Sargent, E. Kumacheva, and O. Kalinina, “Characterization of internal order of colloidal crystals by optical diffraction,” Opt. Quant. Elec. (to be published).

OCIS Codes
(160.5470) Materials : Polymers
(260.2110) Physical optics : Electromagnetic optics
(260.3910) Physical optics : Metal optics
(290.4210) Scattering : Multiple scattering

ToC Category:
Focus Issue: Photonic bandgap calculations

History
Original Manuscript: November 13, 2000
Published: January 29, 2001

Citation
A. Modinos, N. Stefanou, and Vassilios Yannopapas, "Applications of the layer-KKR method to photonic crystals," Opt. Express 8, 197-202 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-197


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References

  1. X. D. Wang, X-G. Zhang, Q. L. Yu, and B. N. Harmon, "Multiple-scattering theory for electro-magnetic waves," Phys. Rev. B 47, 4161-4167 (1993). [CrossRef]
  2. A. Moroz, "Inward and outward integral equations and the KKR method for photons," J. Phys.: Condens. Matter 6, 171-182 (1994). [CrossRef]
  3. A. Moroz, "Three-dimensional complete photonic-band-gap structures in the visible," Phys. Rev. Lett. 83, 5274-5277 (1999). [CrossRef]
  4. K. Ohtaka, "Energy band of photons and low-energy photon diffraction," Phys. Rev. B 19, 5057-5067 (1979). [CrossRef]
  5. K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. I. Various properties of Bloch electric fields and heavy photons," J. Phys. Soc. Jpn. 65, 2265-2275 (1996). [CrossRef]
  6. N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys.: Condens. Matter 4, 7389-7400 (1992). [CrossRef]
  7. N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: Frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998). [CrossRef]
  8. N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000). [CrossRef]
  9. J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992). [CrossRef] [PubMed]
  10. P. M. Bell, J. B. Pendry, L. Martín-Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995). [CrossRef]
  11. Y. Qiu, K. M. Leung, L. Cavin, and D. Kralj, "Dispersion curves and transmission spectra of a two-dimensional photonic band-gap crystal-Theory and experiment," J. Appl. Phys. 77, 3631-3636 (1995). [CrossRef]
  12. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Ordered and disordered photonic band gap materials," Aust. J. Phys. 52, 791-809 (1999).
  13. V. Yannopapas, A. Modinos and N. Stefanou, "Optical properties of metallodielectric photonic crystals," Phys. Rev. B 60, 5359-5365 (1999). [CrossRef]
  14. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, C. M. de Sterke, P. A. Robinson, and A. A. Asatryan, "Anomalous absorptance by stacked metallic cylinders," Opt. Commun. 168, 47-53 (1999). [CrossRef]
  15. A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).
  16. M. P. Pileni, "Nanosized particles made in colloidal assemblies," Langmuir 13, 3266-3276 (1997) [CrossRef]
  17. W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, "Robust photonic band gap from tunable scatterers," Phys. Rev. Lett. 84, 2853-2856 (2000). [CrossRef] [PubMed]
  18. O. D. Velev and E. W. Kaler, "Structured porous materials via colloidal crystal templating: from inorganic oxides to metals," Adv. Mater. 12, 531-534 (2000). [CrossRef]
  19. M. Allard, E. Sargent, E. Kumacheva, and O. Kalinina, "Characterization of internal order of colloidal crystals by optical diffraction," Opt. Quant. Elec. (to be published).

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