## Applications of the layer-KKR method to photonic crystals

Optics Express, Vol. 8, Issue 3, pp. 197-202 (2001)

http://dx.doi.org/10.1364/OE.8.000197

Acrobat PDF (282 KB)

### 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

**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. 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]

**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

*ω*

_{α}(

**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.

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]

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]

_{1}and a

_{2}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

*N*th layer by a primitive translation to be denoted by a

_{3}. Obviously, a

_{1},a

_{2}, and a

_{3}constitute a basis for the 3D space lattice of the infinite crystal.

**b**

_{j}·

**a**

_{j}=2πδ

*ij*,

*i,j*=1, 2. The reduced (

*k*

_{x},

*k*

_{y})-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]). We define a corresponding 3D reduced

**k**-zone as follows:

_{3}=2πa

_{1}×a

_{2}/[a

_{1}·(a

_{2}×a

_{3})] 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.

**k**‖, of Maxwell’s equations for the given system has the following form in the space between the

*N*th and the (

*N*+1)th layers (we write down only the electric-field component of the EM wave):

*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. 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]

**E**

^{±}are column matrices with elements:

*g*|<

*g*

_{max}, where

*g*

_{max}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. 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]

*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]

**113**, 49–77 (1998). [CrossRef]

**132**, 189–196 (2000). [CrossRef]

**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. 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]

**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-

*ω*(

*ω*+

*iτ*

^{-1}) [13

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

*ω*→0) and does not allow the propagation of EM waves through it.

*et al*. [19], is shown by the dotted curve in Fig. 2c. In Fig. 2a, we show the band structure of the corresponding infinite crystal along the normal to the (111) surface (

**k**

_{‖}=

**0**). Next to the band structure we show the reflectance evaluated for a perfect slab with no absorption. The calculation was done for a slab 32-layers thick, but we note that the reflectance, when absorption is taken into account (see below), does not depend on the thickness of the slab when this exceeds 30 layers or so. The reflectance equals unity over the Bragg gap about λ≈1085 nm. The oscillations on either side of the peak are of the Fabry-Perot type and are due to the finite size of the slab. The fine structure of the reflectance at higher frequencies (λ<600 nm) reflects the complexity of the frequency band structure in this region. Some of these bands (denoted by red lines in Fig. 2a) are nondegenerate, non optically active bands (they do not couple with the incident EM field) and the rest (denoted by black lines) are active, doubly degenerate bands, and it is through them that light flows into the crystal. Evidently, the theoretical reflectance of the perfect nonabsorbing slab relates, at a qualitative level, to the measured reflectance of the imperfect and absorbing (at least to a small degree) slab of the material. At this stage we could take into account absorption and disorder in an approximate manner, by adding an imaginary part independent of frequency to the dielectric constant of the PMMA spheres, which effectively removes light from the coherent beam. The results are shown by the solid line in Fig. 2c, and one can see that with the use of this single parameter one reproduces satisfactorily the essential features of the experimental curve.

## Acknowledgements

## 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 |

2. | A. Moroz, “Inward and outward integral equations and the KKR method for photons,” J. Phys.: Condens. Matter |

3. | A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. |

4. | K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B |

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. |

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

7. | N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. |

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. |

9. | J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. |

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. |

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. |

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. |

13. | V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals” Phys. Rev. B |

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. |

15. | A. Modinos, |

16. | M. P. Pileni, “Nanosized particles made in colloidal assemblies,” Langmuir |

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. |

18. | O. D. Velev and E. W. Kaler, “Structured porous materials via colloidal crystal templating: from inorganic oxides to metals,” Adv. Mater. |

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

Sort: Journal | Reset

### References

- 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]
- A. Moroz, "Inward and outward integral equations and the KKR method for photons," J. Phys.: Condens. Matter 6, 171-182 (1994). [CrossRef]
- A. Moroz, "Three-dimensional complete photonic-band-gap structures in the visible," Phys. Rev. Lett. 83, 5274-5277 (1999). [CrossRef]
- K. Ohtaka, "Energy band of photons and low-energy photon diffraction," Phys. Rev. B 19, 5057-5067 (1979). [CrossRef]
- 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]
- N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys.: Condens. Matter 4, 7389-7400 (1992). [CrossRef]
- N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: Frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998). [CrossRef]
- 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]
- J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992). [CrossRef] [PubMed]
- 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]
- 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]
- 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).
- V. Yannopapas, A. Modinos and N. Stefanou, "Optical properties of metallodielectric photonic crystals," Phys. Rev. B 60, 5359-5365 (1999). [CrossRef]
- 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]
- A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).
- M. P. Pileni, "Nanosized particles made in colloidal assemblies," Langmuir 13, 3266-3276 (1997) [CrossRef]
- 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]
- 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]
- 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).

## 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.