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.
- V. P. Bykov, Spontaneous emission in a periodic structure,
Sov. Phys. JETP 35, 269-273 (1972).
- 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 . (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.
how to download fulltext of any patent publication.
- S. Wang and S. K. Sheem, Two dimensional distributed feedback
devices and lasers, US patent publication US-3,884,549 A (May 20, 1975).
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 .
The conference contribution 
discusses interference, transitions to nonperiodic guides, beam steering,
negative refraction, and focusing as possible applications.
More accessible is later publication , 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. ) 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. ).
graphical representation of the observable propagation effects was
given in wave-vector diagrams, showing the directional dispersion of
the elementary waves in periodic structures.
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.)
- R. Ulrich and R. Zengerle, Optical Bloch waves in periodic
planar waveguides, (IEEE, New York, 1980), pp. TuB1/1-4
(conference technical digest).
- 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  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.)
- Winston E. Kock, Metal-Lens Antennas,
Proc. IRE 34, 828-836 (1946).
- J. Brown, Artificial dielectrics having refractive
indices less than unity,
Proc. IEEE, Monograph no. 62R, vol. 100, Pt. 4, pp. 51-62 (1953).
J. Brown and W. Jackson, The properties of
artificial dielectrics at centimeter wavelengths,
Proc. IEEE, paper no. 1699R, vol. 102B, pp. 11-21 (1955).
- 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).
- J. S. Seeley, The quarter-wave matching of
Proc. IEEE, paper no. 2736R, vol. 105C, pp. 103-106 (1958).
- A. Carne and J. Brown, Theory of reflections from the rodded-type
Proc. IEEE, paper no. 2742R, vol. 105C, pp. 107-115 (1958).
- 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).
- W. Rotman, Plasma simulation by artificial dielectrics and
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
In the case of a cubic array of spherical particles,
the static case is satisfactorily covered by the
Maxwell Garnett theory  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
The static limit has also been discussed in depth by
D. J. Bergman and K.-J. Dunn .
The usual assumptions of the
validity of the Maxwell Garnett type of composite geometry have been that:
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
(A useful parametrization
of the Maxwell Garnett formula for the case of core shell spherical particles
can be found in Ref. .)
the inclusions are assumed to be spheres
or ellipsoids of a size much smaller than the optical wavelength;
the distance between them is much larger than their characteristic size and
- much smaller than the optical wavelength.
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  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  provided a quasi-static
extention of the Maxwell Garnett theory 
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. , 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. , 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. , 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  was made.
Having in mind an application for the construction of
centimeter-wave lens antennae,
Kharadly and Jackson  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 
did not notice anything particular for centimeter waves
and their effective permittivity remained
to be equal or larger than one.
In 1959 Khizhnyak  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 . Sec. 5 of Ref. 
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
Relatively recent work by
Waterman and Pedersen  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 
yields corrections to the Maxwell-Garnett formula (given by
Eq. (33) of ), 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
According to the formula (35) of P. C. Waterman and N. E. Pedersen
, 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
Random arrays were also considered briefly, and the role of
single-particle resonance effects was examined.
Although Waterman and Pedersen 
quoted the work by Kharadly and Jackson ,
they were apparently unaware of the earlier results by Khizhnyak [7-9,11].
Yet even the Waterman and Pedersen work  appears to
be largely unknown in the photonic crystals community that
was formed merely a few years after the Waterman and Pedersen work 
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. .
My recent work 
then demonstrates that any composite which
justifies an effective-medium Maxwell-Garnett description
is in fact an artificial polaritonic like medium.
- J. C. Maxwell Garnett, Colours in metal glasses and in metallic films,
Phil. Trans. R. Soc. London 203, 385-420 (1904).
Z. Hashin and S. Shtrikman,
A Variational Approach to the Theory of the Effective
Magnetic Permeability of Multiphase Materials,
Appl. Phys. 33, 3125-3131 (1962).
D. J. Bergman and K.-J. Dunn,
Bulk effective dielectric constant of a composite with a periodic
Rev. B 45, 13262-13271 (1992).
- A. Moroz and C. Sommers,
Photonic band gaps of
three-dimensional face-centered cubic lattices,
Phys.: Condens. Matter 11,
[story behind this article]
- A. Moroz, A simple formula for the L-gap width of a
face-centered-cubic photonic crystal,
Opt. A: Pure Appl. Opt. 1,
- L. Lewin,
The electrical constants of a material loaded with spherical particles,
Proc. Inst. Elec. Eng. 94, 65-68 (1947).
- N. A. Khizhnyak, Artificial anisotropic dielectrics: I.,
Sov. Phys. Tech. Phys. 27, 2006-2013 (1957).
- N. A. Khizhnyak, Artificial anisotropic dielectrics: II.,
Sov. Phys. Tech. Phys. 27, 2014-2026 (1957).
- N. A. Khizhnyak,
Artificial anisotropic dielectrics: III.,
Sov. Phys. Tech. Phys. 27, 2027-2037 (1957).
- M. M. Z. Kharadly and W. Jackson,
The properties of artificial dielectrics comprising arrays
of conducting elements,
Proceedings IEE 100, 199-212 (1953).
N. A. Khizhnyak,
Artificial anisotropic dielectrics formed from
two-dimensional lattices of infinite bars and rods,
Sov. Phys. Tech. Phys. 29, 604-614 (1959).
P. C. Waterman and N. E. Pedersen,
Electromagnetic scattering by periodic arrays of particles,
J. Appl. Phys. 59, 2609-2618 (1986).
W. Lamb, D. M. Wood, and N. W. Ashcroft,
Long-wavelength electromagnetic propagation in heterogeneous media,
Phys. Rev. B 21, 2248-2266 (1980).
- 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)
(see also my accompanying F77 code EFFE2P).
- A. Moroz, Localized resonances of composite particles,
J. Phys. Chem. C 113(52), 21604-21610 (2009).
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 .
- J. H. Evans, Observations of a regular void array in high purity
molybdenum irradiated with 2 MeV nitrogen ions,
Nature 229, 403-404 (1971).
- 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.
- K. Krishan, Ordering of voids and gas bubles in
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].
- P. Yeh and A. Yariv, Bragg Reflection Waveguides,
Opt. Commun. 19, 427-430 (1976).
- 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...
Refreshing some of plasmonic headlines
Elementary properties of localized
surface plasmon resonances
My personal webpage
Full list of my scientific publications
List of my publications
My most frequently cited articles
© Alexander Moroz,
June 1, 2001 (last amended on June 6, 2010)