19.1 Photonic Band Gap Materials
495
(a)
(b)
(c)
Fig. 19.11 a Tetragonal square-spiral photonic crystal. The crystal shown here has a solid filling fraction of 30%. For
clarity, spirals at the corners of the crystal are highlighted with a different shade and height. The tetragonal lattice is
characterized by lattice constants a and b. The geometry of the square spiral is illustrated in the insets and is characterized
by its width, L, cylinder radius, r , and pitch, c. The top left inset shows a single spiral coiling around four unit cells.
b Oblique and edge views of a tetragonal square spiral structure grown using the GLAD (glancing-angle deposition)
process. Both markers are 1 µm. c Band structure for the direct structure crystal characterized by [L,C,r ]=[1.6,1.2,0.14]
and a spiral filling factor of 30%. The lengths are given in units of a, the lattice constant. The width of the PBG is 15.2%
relative to the center frequency for background dielectric constant of 1 and spiral material dielectric constant of 11.9.
The positions of high-symmetry points of the BZ are illustrated in the inset. Panel (a) reprinted and panel (c) adapted
and reprinted from [1599], with permission, ©2001 AAAS. Panel (b) reprinted from [1600] with permission, ©2002
ACS
from semiconductor physics, much of the interesting physics and numerous applications lie in defect
modes, i.e. localized electronic states due to doping and recombination at such centers. The equivalent
in PBG structures are point defects (one unit missing) or line defects (a line of units, straight, bend or
with sharp angles, missing). Such defects create localized states, i.e. regions for light localization. In
the case of line defects we deal with waveguides that can be conveniently designed and could help to
reduce the size of photonic and optoelectronic integrated circuits.
1D Model
We revisit our 1D scattering theory and create now a ‘defect’. A simple defect is the change of the
width of the center n 2 -region in a quarter-wave stack. For the numerical example, we choose N = 11,
n 1 = 1, n 2 = 2.
495
(a)
(b)
(c)
Fig. 19.11 a Tetragonal square-spiral photonic crystal. The crystal shown here has a solid filling fraction of 30%. For
clarity, spirals at the corners of the crystal are highlighted with a different shade and height. The tetragonal lattice is
characterized by lattice constants a and b. The geometry of the square spiral is illustrated in the insets and is characterized
by its width, L, cylinder radius, r , and pitch, c. The top left inset shows a single spiral coiling around four unit cells.
b Oblique and edge views of a tetragonal square spiral structure grown using the GLAD (glancing-angle deposition)
process. Both markers are 1 µm. c Band structure for the direct structure crystal characterized by [L,C,r ]=[1.6,1.2,0.14]
and a spiral filling factor of 30%. The lengths are given in units of a, the lattice constant. The width of the PBG is 15.2%
relative to the center frequency for background dielectric constant of 1 and spiral material dielectric constant of 11.9.
The positions of high-symmetry points of the BZ are illustrated in the inset. Panel (a) reprinted and panel (c) adapted
and reprinted from [1599], with permission, ©2001 AAAS. Panel (b) reprinted from [1600] with permission, ©2002
ACS
from semiconductor physics, much of the interesting physics and numerous applications lie in defect
modes, i.e. localized electronic states due to doping and recombination at such centers. The equivalent
in PBG structures are point defects (one unit missing) or line defects (a line of units, straight, bend or
with sharp angles, missing). Such defects create localized states, i.e. regions for light localization. In
the case of line defects we deal with waveguides that can be conveniently designed and could help to
reduce the size of photonic and optoelectronic integrated circuits.
1D Model
We revisit our 1D scattering theory and create now a ‘defect’. A simple defect is the change of the
width of the center n 2 -region in a quarter-wave stack. For the numerical example, we choose N = 11,
n 1 = 1, n 2 = 2.