Skip to main content
Log in

Optics of anisotropic nanostructures

  • Published:
Czechoslovak Journal of Physics Aims and scope

Abstract

The analytical formalism of Rokushima and Yamakita [J. Opt. Soc. Am. 73, 901–908 (1983)] treating the Fraunhofer diffraction in planar multilayered anisotropic gratings proved to be a useful introduction to new fundamental and practical situations encountered in laterally structured periodic (both isotropic and anisotropic) multilayer media. These are employed in the spectroscopic ellipsometry for modeling surface roughness and in-depth profiles, as well as in the design of various frequency-selective elements including photonic crystals. The subject forms the basis for the solution of inverse problems in scatterometry of periodic nanostructures including magnetic and magneto-optic recording media. It has no principal limitations as for the frequencies and period to radiation wavelength ratios and may include matter wave diffraction. The aim of the paper is to make this formalism easily accessible to a broader community of students and non-specialists. Many aspects of traditional electromagnetic optics are covered as special cases from a modern and more general point of view, e.g., plane wave propagation in isotropic media, reflection and refraction at interfaces, Fabry-Perot resonator, optics of thin films and multilayers, slab dielectric waveguides, crystal optics, acousto-, electro-, and magneto-optics, diffraction gratings, etc. The formalism is illustrated on a model simulating the diffraction on a ferromagnetic wire grating.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. K. Rokushima and J. Yamakita: Analysis of anisotropic dielectric gratings, J. Opt. Soc. Am. 73 (1983) 901–908. Reprinted in Selected papers on Diffraction Gratings, (Ed. D. Maystre, Gen. Ed. B.J. Thompson), SPIE Milestone Series, Vol. MS 83, SPIE Optical Engineering Press, 1993, pp. 519–526.

    ADS  Google Scholar 

  2. O. Francescangeli, S. Melone, and R. Deleo: Dynamic Diffraction of Guided Electromagnetic Waves by 2-Dimensional Periodic Dielectric Gratings, Phys. Rev. A 43 (1991) 6975–6989.

    Article  ADS  Google Scholar 

  3. D. W. Berreman and A. T. Macrander: Asymmetric X-Ray-Diffraction by Strained Crystal Wafers-8 × 8-Matrix Dynamical Theory, Phys. Rev. B 37 (1988) 6030–6040.

    Article  ADS  Google Scholar 

  4. E. N. Glytsis and T. K. Gaylord: 3-Dimensional (Vector) Rigorous Coupled-Wave Analysis of Anisotropic Grating Diffraction, J. Opt. Soc. Am. A 7 (1990) 1399–1420.

    ADS  Google Scholar 

  5. Š. Višňovský and K. Yasumoto: Multilayer anisotropic bi-periodic diffraction gratings, Czech. J. Phys. B 51 (2001) 229–247.

    Article  ADS  Google Scholar 

  6. Lifeng Li: Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors, J. Optics A: Pure Appl. Opt. 5 (2003) 345–355.

    Article  ADS  Google Scholar 

  7. S. P. Liu: Exact Theories For Light, X-Ray, Electron, and Neutron Diffractions from Planar Media with Periodic Structures, Phys. Rev. B 39 (1989) 10640–10650.

  8. S. Ponti, C. Oldano, and M. Becchi: Bloch wave approach to the optics of crystals, Phys. Rev. E 64 (2001) Art. No. 021704.

    Google Scholar 

  9. M. Becchi, C. Oldano, S. Ponti: Spatial dispersion and optics of crystals, J. Optics A: Pure Appl. Opt. 1 (1999) 713–718.

    Article  ADS  Google Scholar 

  10. D. Ciprian and J. Pištora: Magneto-optic periodic strip structures, Acta Phys. Pol. A 99 (2001) 33–46.

    Google Scholar 

  11. J. M. Jarem, and P. P. Banerjee: Application of the complex Poynting theorem to diffraction gratings, J. Opt. Soc. Am. A 16 (1999) 1097–1107.

    ADS  Google Scholar 

  12. G. Montemezzani and M. Zgonik: Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries, Phys. Rev. E 55 (1997) 1035–1047.

    Article  ADS  Google Scholar 

  13. Y. Ohkawa, Y. Tsuji, and M. Koshiba: Analysis of anisotropic dielectric grating diffraction using the finite-element method, J. Opt. Soc. Am. A 13 (1996) 1006–1012.

    ADS  Google Scholar 

  14. P. Galatola, C. Oldano, and P. B. S. Kumar: Symmetry Properties of Anisotropic Dielectric Gratings, J. Opt. Soc. Am. A 11 (1994) 1332–1341.

    ADS  Google Scholar 

  15. R. A. Depine, V. L. Brudny, and A. Lakhtakia: T-Matrix Approach for Calculating the Electromagnetic-Fields Diffracted by a Corrugated, Anisotropic Grating, J. Mod. Optics 39 (1992) 589–601.

    ADS  Google Scholar 

  16. A. Vial and D. Van Labeke: Diffraction hysteresis loop modelisation in transverse magneto-optical Kerr effect, Opt. Commun. 153 (1998) 125–133.

    Article  ADS  Google Scholar 

  17. Y. Pagani, D. Van Labeke, B. Guizal, A. Vial, and F. Baida: Diffraction hysteresis loop modeling in magneto-optical gratings, Opt. Commun. 209 (2002) 237–244.

    Article  ADS  Google Scholar 

  18. N. Bardou, B. Bartenlian, F. Rousseaux, D. Decanini, F. Carsenac, C. Chappert, P. Veillet, P. Beauvillain, R. Mégy, Y. Suzuki, and J. Ferré: Light diffraction effects in the magneto-optical properties of 2D arrays of magnetic dots of Au/Co/Au(111) films with perpendicular magnetic anisotropy, J. Magn. Magn. Mat. 156 (1995) 293–294.

    Article  Google Scholar 

  19. Y. Suzuki, C. Chapert, P. Bruno, and P. Veillet: Simple model for the magneto-optical Kerr diffraction of a regular array of magnetic dots, J. Magn. Magn. Mat. 165 (1997) 516–519.

    Article  ADS  Google Scholar 

  20. S. Mori, K. Mukai, J. Yamakita, and K. Rokushima: Analysis of Dielectric Lamellar Gratings Coated With Anisotropic Layers, J. Opt. Soc. Am. A 7 (1990) 1661–1665.

    Article  ADS  Google Scholar 

  21. M. Nevière and E. Popov: Light Propagation in Periodic Media, Marcel Dekker, Inc., New York-Basel, 2003.

    Google Scholar 

  22. S. Mitani, K. Takanashi, H. Nakajima, K. Sato, R. Schreiber, P. Grunberg, and H. Fujimori: Structural and magnetic properties of Fe/noble metal monoatomic multilayers equivalent to L1 0 ordered alloys, J. Magn. Magn. Mat. 156 (1996) 7–10.

    Article  ADS  Google Scholar 

  23. K. Machida, T. Tezuka, T. Yamamoto, T. Ishibashi, Y. Morishita, A. Koukitu, and K. Sato: Magnetic structure of cross-shaped permalloy arrays embedded in silicon wafers, J. Magn. Magn. Mat. 290/291 (2005) 779–782.

    Article  ADS  Google Scholar 

  24. K. Yasumoto, H. Toyama, and T. Kushta: Accurate Analysis of Two-Dimensional Electromagnetic Scattering from Multilayered Periodic Arrays of Circular Cylinders using Lattice Sums Technique, IEEE Trans. Antenna Propag. 52 (2004) 2603–2611.

    Article  MathSciNet  ADS  Google Scholar 

  25. Kiyotoshi Yasumoto, Editor: Electromagnetic Theory and Applications for Photonic Crystals, Taylor & Francis, Boca Raton-London-New York, 2006.

    Google Scholar 

  26. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov: Les cristaux photoniques ou la lumière en cage, Collection Téchnique et Scientifique des Té lécommunications, GET et Lavoisier, Paris, 2003; Photonic crystals: towards nanoscale photonic devices, Springer-Verlag, Berlin-Heidelberg-New York, 2005.

    Google Scholar 

  27. D. E. Aspnes: Expanding horizons: new developments in ellipsometry and polarimetry, Thin Solid Films 455/456 (2004) 3–13.

    Article  ADS  Google Scholar 

  28. F. Abelès: Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces, Ann. Phys. Paris 5 (1950) 596–640.

    Google Scholar 

  29. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri, and K. Driss-Khodja: Effective medium equivalent to a symmetric multilayer at oblique incidence, Appl. Opt. 29 (1990) 2460–2465.

    Article  ADS  Google Scholar 

  30. D. Marcuse: Light Transmission Optics, Bell Laboratories Series, Van Nostrand and Rienhold Company, New York, 1972, Chapter 1.

    Google Scholar 

  31. D. Marcuse: Theory of Dielectric Optical Waveguides, Academic Press, New York-London, 1974, Chapter 2.

    Google Scholar 

  32. P. K. Tien: Integrated optics and new wave phenomena in optical waveguides, Rev. Mod. Phys. 49 (1977) 361–420.

    Article  ADS  Google Scholar 

  33. Max Born and Emil Wolf with contributions by A.B. Bhatia, P.C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P.A. Wayman, and W. L. Wilcock: Principles of Optics, Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, Sixth (Corrected) Edition, Cambridge University Press, Cambridge-New York-Melbourne, 1997.

    Google Scholar 

  34. R. M. A. Azzam and N. M. Bashara: Ellipsometry and Polarized Light, North Holland, Elsevier, Amsterdam-Lausanne-New York-Oxford-Shannon-Tokyo, 1987.

    Google Scholar 

  35. A. K. Zvezdin and V.A. Kotov: Modern Magnetooptics and Magnetooptical Materials Institute of Physics Publishing, Bristol-Philadelphia, 1997.

    Book  Google Scholar 

  36. Š. Višňovský: Magneto-optical permittivity tensor in crystals, Czech. J. Phys. B 36 (1986) 1424–1433.

    Article  ADS  Google Scholar 

  37. D. W. Berreman: Optics in Stratified and Anisotropic Media: 4 × 4-Matrix Formulation, J. Opt. Soc. Am. 62 (1972) 502–510.

    ADS  Google Scholar 

  38. D. W. Berreman: Optics in Smoothly Varying Anisotropic Planar Structures: Application to Liquid-Crystal Twist Cells, J. Opt. Soc. Am. 63 (1973) 1374–1380.

    ADS  Google Scholar 

  39. P. Yeh: Electromagnetic Propagation in Birefringent Layered Media, J. Opt. Soc. Am. 69 (1979) 742–756; Optics of anisotropic layered media: a new 4 × 4 matrix algebra, Surf. Sci. 96 (1980) 41–53.

    ADS  Google Scholar 

  40. Š. Višňovský: Magneto-optical ellipsometry, Czech. J. Phys. B 36 (1986) 625–650.

    Article  ADS  Google Scholar 

  41. M. Mansuripur: The Physical Principles of Magneto-optical Recording, Cambridge University Press, London, 1996.

    Google Scholar 

  42. M. Schubert: Polarization-dependent optical parameters of arbitrarily anisotropic homogenous layered systems, Phys. Rev. B 53 (1996) 4265–4274.

    Article  ADS  Google Scholar 

  43. H. Benisty: Photonic crystals — New designs to confine light, Nat. Phys. 1 (2005) 9–10.

    Article  Google Scholar 

  44. C. Jamois, R. B. Wehrspohn, L.C. Andreani, C. Hermann, O. Hess, and U. Gosele: Silicon-based two-dimensional photonic crystal waveguides, Photonics and Nanostructures-Fundamentals and Applications 1 (2003) 1–13.

    Article  ADS  Google Scholar 

  45. Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss: Square lattice photonic-crystal collimator, Photonics and Nanostructures-Fundamentals and Applications 1 (2003) 31–36.

    Article  ADS  Google Scholar 

  46. R.-C. Tyan, P.-C. Sun, A. Scherer, and Y. Fainman: Polarizing beam splitter based on the anisotropic spectral reflectivity characteristic of form-birefringent multilayer gratings, Opt. Lett. 21 (1996) 761–763.

    ADS  Google Scholar 

  47. N. Kono and M. Koshiba: Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides, Opt. Express 13 (2005) 9155–9166.

    Article  ADS  Google Scholar 

  48. A. Figotin and I. Vitebskiy: Electromagnetic unidirectionality in magnetic photonic crystals, Phys. Rev. B 67 (2003) 165210.

    Google Scholar 

  49. N. Kono and Y. Tsuji: A novel finite-element method for nonreciprocal magnetophotonic crystal waveguides, J. Lightwave Technol. 22 (2004) 1741–1747.

    Article  ADS  Google Scholar 

  50. A. Figotin and I. Vitebskiy: Nonreciprocal magnetic photonic crystals, Phys. Rev. E 63 (2001) 066609.

    Google Scholar 

  51. T.V. Murzina, R.V. Kapra, T.V. Dolgova, A.A. Fedyanin, O.A. Aktsipetrov, K. Nishimura, H. Uchida, and M. Inoue: Magnetization-induced second-harmonic generation in magnetophotonic crystals, Phys. Rev. B 70 (2004) 012407.

    Google Scholar 

  52. O. A. Aktsipetrov, T.V. Dolgova, A.A. Fedyanin, T.V. Murzina, M. Inoue, K. Nishimura, and H. Uchida: Magnetization-induced second-and third-harmonic generation in magnetophotonic crystals, J. Opt. Soc. Am. B 22 (2005) 176–186.

    Article  ADS  Google Scholar 

  53. K. Yee: Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE T. Antenn. Propag. AP-14 (1966) 302–307.

    ADS  Google Scholar 

  54. H. Kogelnik: Coupled wave theory for thick hologram gratings, Bell. Syst. Tech. J. 48 (1969) 2909–2947; Integrated Optics, Topics in Applied Physics, Vol. 9, (Ed. T. Tamir), Springer-Verlag, Berlin, 1975.

    Google Scholar 

  55. L. Li and W. Haggans: Convergence of the coupled-wave method for metallic lamellar diffraction gratings, J. Opt. Soc. Am. A 10 (1993) 1184–1189.

    ADS  Google Scholar 

  56. P. Lalanne and G.M. Morris: Highly improved convergence of the coupled-wave method for TM polarization, J. Opt. Soc. Am. A 13 (1996) 779–784.

    ADS  Google Scholar 

  57. G. Granet and B. Guizal: E.cient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization, J. Opt. Soc. Am. A 13 (1996) 1019–1023.

    Article  ADS  Google Scholar 

  58. L. Li: Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13 (1996) 1870–1876.

    ADS  Google Scholar 

  59. L. Li: New formulation of the Fourier modal method for crossed surface-relief gratings, J. Opt. Soc. Am. 14 (1997) 2758–2767.

    ADS  Google Scholar 

  60. L. Li: Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials, J. Mod. Opt. 45 (1998) 1313–1334.

    Article  ADS  Google Scholar 

  61. B. Chernov, M. Nevière, and E. Popov: Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings, Opt. Commun. 194 (2001) 289–297.

    Article  ADS  Google Scholar 

  62. E. Popov and M. Nevière: Grating theory: new equations in Fourier space leading to fast converging results for TM polarization, J. Opt. Soc. Am. A 17 (2000) 1773–1784.

    ADS  Google Scholar 

  63. K. Watanabe, R. Petit, and M. Nevière: Differential theory of gratings made of anisotropic materials, J. Opt. Soc. Am. A 19 (2002) 325–334.

    ADS  Google Scholar 

  64. L. Li: Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors, J. Opt. A: Pure Appl. Opt. 5 (2003) 345–355.

    Article  ADS  Google Scholar 

  65. N. Bonod, E. Popov, and M. Nevière: Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect, Opt. Commun. 244 (2005) 389–398.

    Article  ADS  Google Scholar 

  66. P. Boyer, E. Popov, M. Nevière, and G. Tayeb: Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section, J. Opt. Soc. Am. A 21 (2004) 2146–2153.

    Article  ADS  Google Scholar 

  67. K. Rokushima and J. Yamakita: Analysis of Diffraction in Periodic Liquid-Crystals — The Optics of the Chiral Smectic-C Phase, J. Opt. Soc. Am. A 4 (1987) 27–33.

    ADS  Google Scholar 

  68. R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands: Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth-and first-diffraction orders, Appl. Phys. Lett. 86 (2005) 231101.

    Google Scholar 

  69. J. I. Martin, J. Nogues, K. Liu, J. L. Vicent, and I. K. Schuller: Ordered magnetic nanostructures: fabrication and properties, J. Magn. Magn. Mater. 256 (2003) 449–501.

    Article  ADS  Google Scholar 

  70. M. Grimsditch and P. Vavassori: The diffracted magneto-optic Kerr effect: what does it tell you?, J. Phys.: Condens. Matter. 16 (2004) R275–R294.

    Article  ADS  Google Scholar 

  71. M. R. Freeman: Picosecond pulsed-field probes of magnetic systems (invited), J. Appl. Phys. 75 (1994) 6194–6198.

    Article  ADS  Google Scholar 

  72. R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands: Evaluation of the quality of Permalloy gratings by diffracted magneto-optical spectroscopy, Opt. Express 13 (2005) 4651–4656.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Štefan Višňovský.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rokushima, K., Antoš, R., Mistrík, J. et al. Optics of anisotropic nanostructures. Czech J Phys 56, 665–764 (2006). https://doi.org/10.1007/s10582-006-0128-0

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10582-006-0128-0

Key words

Navigation