Photonic Crystals, or Photonic Bandgap Materials, Headlines 25 Years Old, at least

Photonic Crystals Headlines 40 Years Old, at Least ...

Photonic crystals may allow to control the spontaneous emission
A detailed investigation of the effect of a photonic band gap on the spontaneous emission (SE) of embedded atoms and molecules has been performed by V. P. Bykov [1,2]. For a toy one-dimensional model, he obtained the energy and the decay law of the excited state with transition frequency in the photonic band gap, and calculated the spectrum which accompanies this decay. Bykov's detailed analytic investigation revealed that the SE can be strongly suppressed in volumes much greater than the wavelength. He also discussed importance of the SE suppression for laser applications. On p. 871, right column, paragraph 2, he suggested that the active medium of a laser may have a three-dimensional periodic structure. He speculated that if the propagation of electromagnetic waves were forbidden in all directions except for a narrow cone, a laser with much lower threshold could have been obtained.

References:

  1. V. P. Bykov, Spontaneous emission in a periodic structure, Sov. Phys. JETP 35, 269-273 (1972).

  2. V. P. Bykov, Spontaneous emission from a medium with a band spectrum, Sov. J. Quant. Electron. 4, 861-871 (1975).

Two dimensional distributed feedback devices and lasers
The use of a two-dimensional distributed feedback periodic structures in thin film optical devices and lasers is disclosed in the US patent publication US-3,884,549 [1]. (See especially Figs. 1-2 therein.) Passive (optical filters, modulators, and deflectors) and active (laser) thin-film devices can be made. In contrast to devices with a one-dimensional distributed feedback, which provide an output beam in the form of a sheet of light, the two-dimensional distributed feedback devices can provide a pencil like output beam, and, at the same time, manufacturing tolerances and limits to materials with a relatively broad gain profile can be relaxed. The two-dimensional refractive index variations can either be made direct and permanent, or can be induced by acoustic, magnetic, or electric fields applied to optical waveguide of boundary layers thereof.

References:

  1. S. Wang and S. K. Sheem, Two dimensional distributed feedback devices and lasers, US patent publication US-3,884,549 A (May 20, 1975).
Read here how to download fulltext of any patent publication.

Negative refraction and frequency-tunable beam narrowing, focusing, and expanding in singly and doubly periodic planar waveguides
Work that stands out among many others is that by Remigius Zengerle [1,2]. Long before an ``official discovery" of photonic crystals, and shortly after finishing his PhD thesis at the Max-Planck-Institute for Solid State Physics in Stuttgart (between 1977 and 1979), he presented his work on the optical Bloch waves in (singly and doubly) periodic planar waveguides on the conference on Integrated and Guided-Wave Optics held in Incline Village, NV, USA, on 28-30 Jan. 1980 [1]. The conference contribution [1] discusses interference, transitions to nonperiodic guides, beam steering, negative refraction, and focusing as possible applications. More accessible is later publication [2], which is a condensed form of Zengerle's PhD thesis. He demonstrated, both theoretically and experimentally, negative refraction in the visible in certain areas of the backward bending dispersion relation (see, for instance, photograph 10a of Ref. [2]) and focusing properties of a doubly periodic planar waveguide between unmodulated regions with parallel straight boundaries (see, for instance, photographs in Figs. 13-14 of Ref. [2]). A clear graphical representation of the observable propagation effects was given in wave-vector diagrams, showing the directional dispersion of the elementary waves in periodic structures. Examples of applications of planar periodic (singly and doubly periodic) structures as highly selective frequency filters, optical multiplexers as well as frequency-tunable beam narrowing, focusing and expanding devices (decade later rediscovered as superprisms) were given together with measured data.
(Some other negative refraction, left-handed, negative refractive index material, or metamaterial headlines long before Veselago work and going back as far as to 1905 can be found here.)

References:

  1. R. Ulrich and R. Zengerle, Optical Bloch waves in periodic planar waveguides, (IEEE, New York, 1980), pp. TuB1/1-4 (conference technical digest).

  2. R. Zengerle, Light propagation in singly and doubly periodic planar waveguides, J. Mod. Optics. 34(12), 1589-1617 (1987).

Plasma-like behavior in artificial dielectric composed of periodically spaced lattices of metallic rods and plates

An artificial dielectric composed of a number of conducting plates of proper shape and spacing [1] or of periodically spaced lattices of metallic rods, also known as wire grid, wire mesh, or rodded structure [2-8], can exhibit a plasma-like behaviour [2-6,8] and its index of refraction can be less than unity [1,2].
(For negative refraction headlines see here.)

References:

  1. Winston E. Kock, Metal-Lens Antennas, Proc. IRE 34, 828-836 (1946).

  2. J. Brown, Artificial dielectrics having refractive indices less than unity, Proc. IEEE, Monograph no. 62R, vol. 100, Pt. 4, pp. 51-62 (1953).

  3. J. Brown and W. Jackson, The properties of artificial dielectrics at centimeter wavelengths, Proc. IEEE, paper no. 1699R, vol. 102B, pp. 11-21 (1955).

  4. J. S. Seeley and J. Brown, The use of artificial dielectrics in a beam scanning prism, Proc. IEEE, paper no. 2735R, vol. 105C, pp. 93-102 (1958).

  5. J. S. Seeley, The quarter-wave matching of dispersive materials, Proc. IEEE, paper no. 2736R, vol. 105C, pp. 103-106 (1958).

  6. A. Carne and J. Brown, Theory of reflections from the rodded-type artificial dielectrics, Proc. IEEE, paper no. 2742R, vol. 105C, pp. 107-115 (1958).

  7. A. M. Model, Propagation of plane electromagnetic waves in a space which is filled with plane parallel grids, Radiotekhnika 10, 52-57 (1955) (in Russian).

  8. W. Rotman, Plasma simulation by artificial dielectrics and parallel-plate media, IEEE Trans. Antennas Propag. 10, 82-95 (1962).

Effective medium properties, mean-field description, homogenization, or homogenisation of photonic crystals

Let lambda be the wavelength in the ambient medium, and the respective n and R be refractive index and a characteristic dimension of a particle embedded in the medium. One then speaks of the static limit when both 2*pi*R/lambda << 1 and 2*pi*R*n/lambda << 1. Obviously if the first condition is satisfied then is also the second, provided that n is moderate. However, if n is relatively large (e.g., metallic particles at optical and infrared frequencies) one can find a wavelength range so that 2*pi*R/lambda << 1 but not necessarily 2*pi*R*n/lambda << 1. One then speaks of the quasi-static limit. Note that unlike in the static limit, the effective medium parameters in the quasi-static limit are dispersive even for medium composed of nondispersive components.

In the case of a cubic array of spherical particles, the static case is satisfactorily covered by the Maxwell Garnett theory [1] that can be viewed as a mean field theory. In fact, the Maxwell-Garnett formula effective medium is an exact static solution for the case of coated spheres of all sizes which, without any void, fill in the entire medium (see Sec. IV of [2]). The static limit has also been discussed in depth by D. J. Bergman and K.-J. Dunn [3].

The usual assumptions of the validity of the Maxwell Garnett type of composite geometry have been that:

  1. the inclusions are assumed to be spheres or ellipsoids of a size much smaller than the optical wavelength;

  2. the distance between them is much larger than their characteristic size and

  3. much smaller than the optical wavelength.

Suprizingly enough, the Maxwell Garnett theory works very well even if the hypotheses that 2*pi*R/lambda << 1 and f<< 1, where f is the volume fraction occupied by the spheres, are no longer valid. Indeed, for a moderate n, the Maxwell Garnett theory has been shown to describe the effective medium properties of cubic photonic crystals within a few percent of exact result till the first stop gap, and that even in the close-packed case (for dense dielectric spheres in air and for inverted opals see Refs. [4,5]). The latter also holds for core shell spherical particles [4]. (A useful parametrization of the Maxwell Garnett formula for the case of core shell spherical particles can be found in Ref. [4].)

Effective medium properties (effective dielectric permittivity and magnetic permeability tensors) of periodic arrays of arbitrarily shaped particles have been obtained in the quasi-static limit by L. Lewin [6] and N. A. Khizhnyak [7-9,11]. Their work also contains the idea of generating an effective magnetic response from a composite comprising inherently nonmagnetic materials.

First Lewin [6] provided a quasi-static extention of the Maxwell Garnett theory [1] for the case of a cubic array of spherical particles. (I wish I knew his work at the time of writing my publications [4,5]). Later on, Khizhnyak [7-9,11] generalized the results of Lewin for cubic arrays of spherical particles to the case of arbitrary periodic arrays of arbitrarily shaped particles. In Ref. [7], general formulas were obtained for the respective tensors of dielectric permittivity and magnetic permeability. Since Khizhnyak no longer required that either the lattice be cubic or the particles be spherical, such an artificial dielectric, or effective medium, could in general have anisotropic properties, that Khizhnyak analyzed in detail. In Ref. [8], the respective tensors of dielectric permittivity and magnetic permeability were investigated more closely for an effective medium composed of periodic arrays of spherical particles. Three-dimensional tetragonal, orthogonal, hexagonal, and monoclinic arrays were studied. On p. 2025, last paragraph the following observation was made: ``From Eqs. (2) and (5) an interesting property of artificial dielectrics follows. Even though an artificial dielectric is made from inherently nonmagnetic materials, at higher frequency it will exhibit both dielectric and magnetic anisotropy."

In Ref. [9], the respective tensors of dielectric permittivity and magnetic permeability were investigated in detail for an effective medium composed of periodic arrays of ellipsoidal particles. The role of multipole interactions was assessed. A brief comparison with experimental results by Kharadly and Jackson [10] was made. Having in mind an application for the construction of centimeter-wave lens antennae, Kharadly and Jackson [10] studied (both theoretically and experimentally) the effective permittivity of periodic arrays of perfectly conducting elements of simple geometric shapes (cylindrical rods, thin strips, spheres, and disks). However, Kharadly and Jackson [10] did not notice anything particular for centimeter waves and their effective permittivity remained to be equal or larger than one.

In 1959 Khizhnyak [11] obtained the effective medium parameters (effective permittivity and permeability) for artificial anisotropic dielectrics formed from two-dimensional square, rectangular, and rhombical lattices of infinite circular and elliptical bars and rods [11]. Sec. 5 of Ref. [11] discussed dispersive properties of the artificial dielectrics and the following observation was made: ``Upon substituting Eqs. (29)-(30) back into Eq. (22) one may find that resonance frequencies are possible at which the permittivity and permeability increase without bound." At the very end, the case of a regular array of cylinders with an axis inclined with respect to the two-dimensional periodicty plane was considered.

Relatively recent work by Waterman and Pedersen [12] provided again very detailed and explicit analytical and numerical results for the effective complex dielectric constant and permeability in the quasistatic and infinitesimal lattice (static) limits for several lattice geometries. In this regard, Eq. (35) in Sec. IV of P. C. Waterman and N. E. Pedersen [12] yields corrections to the Maxwell-Garnett formula (given by Eq. (33) of [12]), in the form of an expansion up to terms of order f**6, where f is the volume fraction occupied by the spheres. The results were shown to agree with existing static computations under appropriate conditions. According to the formula (35) of P. C. Waterman and N. E. Pedersen [12], the results depend only weakly on the actual periodic arrangment (e.g. whether simple cubic, body-centered cubic, or face-centered cubic lattice). The formula (35) yields the effective medium properties within a few percent up to volume fractions of about 90% of the maximum allowable values (spheres touching). A comparison with the Maxwell-Garnett formula confirmed that the Maxwell-Garnett formula describes the effective medium properties up to surprisingly large filling fractions. See in this regard also Fig. 2 in the article by W. Lamb et al [13]. Random arrays were also considered briefly, and the role of single-particle resonance effects was examined.

Although Waterman and Pedersen [12] quoted the work by Kharadly and Jackson [10], they were apparently unaware of the earlier results by Khizhnyak [7-9,11]. Yet even the Waterman and Pedersen work [12] appears to be largely unknown in the photonic crystals community that was formed merely a few years after the Waterman and Pedersen work [12] had been published.

Some recent results obtained using a quasi-static extention of the Maxwell Garnett theory for binary composites (= two different types of spheres embedded in a matrix) you can find in Ref. [14].

My recent work [15] then demonstrates that any composite which justifies an effective-medium Maxwell-Garnett description is in fact an artificial polaritonic like medium.

References:

  1. J. C. Maxwell Garnett, Colours in metal glasses and in metallic films, Phil. Trans. R. Soc. London 203, 385-420 (1904).

  2. Z. Hashin and S. Shtrikman, A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials, J. Appl. Phys. 33, 3125-3131 (1962).

  3. D. J. Bergman and K.-J. Dunn, Bulk effective dielectric constant of a composite with a periodic microgeometry, Phys. Rev. B 45, 13262-13271 (1992).

  4. A. Moroz and C. Sommers, Photonic band gaps of three-dimensional face-centered cubic lattices,
    J. Phys.: Condens. Matter 11, 997-1008 (1999). [physics/9807057] [story behind this article]

  5. A. Moroz, A simple formula for the L-gap width of a face-centered-cubic photonic crystal,
    J. Opt. A: Pure Appl. Opt. 1, 471-475 (1999). [physics/9903022]

  6. L. Lewin, The electrical constants of a material loaded with spherical particles, Proc. Inst. Elec. Eng. 94, 65-68 (1947).

  7. N. A. Khizhnyak, Artificial anisotropic dielectrics: I., Sov. Phys. Tech. Phys. 27, 2006-2013 (1957).

  8. N. A. Khizhnyak, Artificial anisotropic dielectrics: II., Sov. Phys. Tech. Phys. 27, 2014-2026 (1957).

  9. N. A. Khizhnyak, Artificial anisotropic dielectrics: III., Sov. Phys. Tech. Phys. 27, 2027-2037 (1957).

  10. M. M. Z. Kharadly and W. Jackson, The properties of artificial dielectrics comprising arrays of conducting elements, Proceedings IEE 100, 199-212 (1953).

  11. N. A. Khizhnyak, Artificial anisotropic dielectrics formed from two-dimensional lattices of infinite bars and rods, Sov. Phys. Tech. Phys. 29, 604-614 (1959).

  12. P. C. Waterman and N. E. Pedersen, Electromagnetic scattering by periodic arrays of particles, J. Appl. Phys. 59, 2609-2618 (1986).

  13. W. Lamb, D. M. Wood, and N. W. Ashcroft, Long-wavelength electromagnetic propagation in heterogeneous media, Phys. Rev. B 21, 2248-2266 (1980).

  14. V. Yannopapas and A. Moroz, Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges, J. Phys.: Condens. Matter. 17, 3717-3734 (2005) (minor erratum) (see also my accompanying F77 code EFFE2P).

  15. A. Moroz, Localized resonances of composite particles,
    J. Phys. Chem. C 113(52), 21604-21610 (2009).

Regular void lattices in ion- or neutron bombarded metals

J. H. Evans et al has demonstrated experimentally that, under ion- or neutron bombardement, regular body-centered-cubic void lattices form in metals, with lattice constant being usually several tens of nanometers - a truly metallic photonic nanocrystal [1,2]! Later on, this irradiation effect has been observed in various other metals. A review on this subject has been compiled by Krishan [3].

References:

  1. J. H. Evans, Observations of a regular void array in high purity molybdenum irradiated with 2 MeV nitrogen ions, Nature 229, 403-404 (1971).

  2. B. L. Eyre and J. H. Evans, Void Formed by Irradiation of Reactor Materials, eds. S. F. Pugh, M. H. Loretto, and D. I. R. Norris (British Nuclear Energy Society, London, 1971), p. 323.

  3. K. Krishan, Ordering of voids and gas bubles in radiation enviroments, Radiat. Eff. 66, 121-155 (1982).

Photonic crystal fibre enables lossless confined propagation in a core with lower refractive index than that of the cladding medium

The possibility of using Bragg reflection in a cylindrical fiber to obtain lossless confined propagation in a core with lower refractive index than that of the cladding medium (an omniguide) has been proposed and analysed by P. Yeh and collaborators [1,2].

References:

  1. P. Yeh and A. Yariv, Bragg Reflection Waveguides, Opt. Commun. 19, 427-430 (1976).

  2. P. Yeh, A. Yariv, and E. Marom, Theory of Bragg fiber, J. Opt. Soc. 68, 1196-1201 (1978).
[ An Early History of Acoustic and Photonic Crystals | Selected Links on Photonics, Photonic Crystals, Numerical Codes, Free Software ]
[ Negative refractive index material headlines long before Veselago work and going back as far as to 1904... ]
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