3.3 Variable mesh
4.1.3 Woodpile structure
Due to the fact that the woodpile structure [65] can operate as a photonic crystal with a PBG at infrared wavelength [66], it has gained much interest.Using high resolution lithography, small samples of the woodpile structure consisting of a few unit cells have been made and measured [74].The main problem is that the current production technique does not allow to make usable large-scale structures, comprising hundreds of unit cells in each direction.
The woodpile is defined by three parameters: the width (w), height (h) and spacing (d).The layer by layer structure repeats itself each four layers, so the periodicity in the stacking direction is a = 4h, as shown in figure 4-4.In figure 4-5 we show the DOS of
Figure 4-3: TM-mode density of states of the sample of the inverted cylinder structure of figure 4-2.Size of the sample: 3× 4 units cells with a = 4.Computational parameters are (method,τ,∆t,M,S )=(T4S2,0.05,0.1,16384,1). (The DOS is not scaled in units 2/ac, since the frequency is not scaled in unitsωa.)
Figure 4-4: Woodpile structure.Characteristic parameters: width (w), height (h) and spac-ing (d).The periodicity is a= 4h and extends in the z-direction.
2× 2 × 4 units cells of the woodpile structure with parameters (w, h, d) = (0.55, 0.70, 2.00).
The filling is f = w/d = 0.275 and ε = 12.96.The bandgap extends from ωa/2πc = 0.490 toωa/2πc = 0.572, a width of 15.5%, comparable, but not equal to the bandgap width of 19.6%, found in reference [66], using the plane-wave expansion method.The fact that the woodpile structure contains a PBG is –naturally– due to its underlying symmetry, that is face-centered-tetragonal (fct), since the woodpile structure can also be obtained by replacing the110 chains of atoms in the diamond structure with rods.And for the diamond structure, it is known that it has a large PBG, for a sufficient dielectric constant and filling.
4.1 Photonic Crystals 65
Figure 4-5: Density of states as a function of frequency for the woodpile structure.Com-putational parameters are (method,τ,∆t,M,S )=(T4S2,0.075,0.075,8192,1), using periodic boundary conditions.
4.1.4 Minimal surfaces
Photonic crystals based on minimal surfaces can be described by the mathematical equation f (x, y, z) = t, which separates the high (εh) and low dielectric (εl) volumes within one unit cell:
ε(r) =
εh, if f (r) < t,
εl, if f (r) ≥ t , (4.6)
here, t is a structure parameter, with which one can adjust the volume fractions.In table 4-1, we list the mathematical expressions and generating symmetry (sub)groups from the International Tables of Crystallography for all the surfaces studied (see e.g.Ref.[75] for a more complete list).
The surface is minimal, i.e. the mean curvature is zero, if and only if t = 0.If the minimal surface divides space into equal volume fractions, the surface is called balanced.If the surface is not balanced, the inverted structure (replacingεh byεl) is different from the normal structure.Furthermore, a topology study of these surfaces in terms of Minkowski functionals can be found in reference [76].
The control parameter t can be used to tune the volume fraction of the high dielectric constant, to achieve fillings of approximately 25% for which one has the best chance to find a crystal with a PBG.However, for each minimal surface there are limiting values for t at which point the surface becomes disconnect and reduces to a simpler symmetry and the characteristic surface family disappears.For the D and P surfaces this happens at resp.
|t| ≤ 3 and |t| ≤ √
2, see Ref.[75]; they respectively reduce to simple cubic closed packed spheres and diamond closed packed spheres structures.For the gyroid minimal surface the limit is set at|t| ≤ 1.5, after which the structure reduces to gyroid closed packed spheres [77].
In figure 4-6 we show the bandgap maps of the G, D and C(±Y) minimal surfaces, and in
Surface Group Subgroup f (x, y, z) = 0 P Im¯3m Pm ¯3m cos(x)+ cos(y) + cos(z) = 0 D Pn ¯3m Fd ¯3m sin(x) sin(y) sin(z)+ sin(x) cos(y) cos(z)
+ cos(x) sin(y) cos(z) + cos(x) cos(y) sin(z) = 0 G Ia ¯3d I4132 sin(x) cos(y)+ sin(x) cos(z) + cos(x) sin(z) = 0 S Ia ¯3d I43d cos(2x) sin(y) cos(z)+ cos(x) cos(2y) sin(z)
+ sin(x) cos(y) cos(2z) = 0 C(P) Im¯3m Pm ¯3m cos(x)+ cos(y) + cos(z)
+4 cos(x) cos(y) cos(z) = 0 C(Y) I4132 P4332 − sin(x) sin(y) sin(z) + sin(2x) sin(y)
+ sin(2y) sin(z) + sin(x) sin(2z)
− cos(x) cos(y) cos(z) + sin(2x) cos(z) + cos(x) sin(2y) + cos(y) sin(2z) = 0 F-RD Fm¯3m Fm ¯3m 4 cos(x) cos(y) cos(z)− cos(2x) cos(2y)
− cos(2y) cos(2z) − cos(2x) cos(2z) = 0 C(Y**) Ia ¯3 I4132 3((sin(x) cos(y)+ sin(y) cos(z)
+ cos(x) sin(z))
+2((sin(3x) cos(y) + cos(x) sin(3z) + sin(3y) cos(z) − sin(x) cos(3y)
− cos(3x) sin(z) − sin(y) cos(3z)) = 0
Table 4-1: Minimal surfaces: name, group, subgroup and parametric surface equation.
figure 4-7 those of the C(Y), P and F-RD surfaces, together with an impression of the unit cell at t= 0.For the C(Y**) and the inverted F-RD surfaces, we present the bandgap maps in figure 4-8.All calculations were done using (method,τ,∆t,M,S )=(T4S2,0.075,0.075,8192,1).
The systems consisted of 4× 4 × 4 unit cells with a = 3, periodic boundary conditions and dielectric permittivityεh = 11.9.
The existence of full PBGs in the crystals defined by the P,G and D surfaces is already known [62].However, this is not the case for the other surfaces, C(Y), C(Y**), C(±Y) and (inverted) F-RD, and these structures therefore define new photonic crystals containing a full PBG.The remaining surfaces S,±Y, C(P) and I-WP showed no bandgaps at any filling fraction.The bandgap map for the C(Y**) displays a distortion in the center; this is due to a (low) mode inside the PBG, if we choose the threshold for the existence of a PBG higher, it disappears.Finally, we list the width1 of the largest bandgap for each minimal surface having a PBG, see table 4-2.
We conclude the analysis with a study of the size of the PBG as funtion of dielectric
1the bandgap width is defined as the width of the frequency range of the bandgap divided by the midgap frequency, and is usually expressed in percentage.
4.1 Photonic Crystals 67
Figure 4-6: Bandgap maps for Gyroid, Double Diamond and C(±Y) surfaces.
Figure 4-7: Bandgap maps for C(Y), Primitive and F-RD surfaces.
4.1 Photonic Crystals 69
Figure 4-8: Bandgap maps for C(Y**) and inverted F-RD surfaces.
Surface Fillingεh t Bandgap width (%)
G 0.205 -0.90 21.7
D 0.188 -0.75 20.3
C(±Y) 0.166 -1.30 8.1
F-RD 0.310 -0.60 6.6
P 0.241 -0.90 6.6
C(Y**) 0.151 -3.50 6.5
C(Y) 0.219 -1.25 6.2
F-RD inv 0.386 0.80 4.3
Table 4-2: Maximal bandgap widths of minimal surface structures
permittivity, for the Gyroid surface.See figure 4-9.At approximatelyε = 4 the bandgap emerges, and steadily grows with increasing permittivity.This low value is very promising, since there are more materials available with low permittivity than with high permittivity.
4.1.5 DOS and LDOS in finite sized clusters
With the availability of large scale fcc-structured closed-packed spheres of air in a dielectric background of Titania [60, 78], this system has become increasingly important.Since the PBG is relatively small (about 4%), the sensitivity to distortions, such as the dislocation of spheres and change in their radii, is important, and has proven to be non negligible [79, 80].
We will return to this issue in the next section, in the context of two dimensional systems.
Another important property, which has not received much attention in the literature2, is to which extent the local properties, for example the LDOS, inside a photonic crystal is affected by the surrounding material.In other words, if an infinite crystal has certain properties, then considering a certain fixed point in space, how large should the material
2probably due to the inevitable assumption of periodic boundary conditions that must be used for most current algorithms to compute a (L)DOS.
Figure 4-9: Bandgap map of the Gyroid at constant t= −1, as a function of ε.
surrounding that point be for the photonic crystal to have the same properties locally at that point? This we study in this section in relation to the width and depth of a PBG in the fcc closed packed spheres crystal.
The system size is defined as the total number of spheres of air that are located within the system, and are generated by using a virtual sphere with radius r, concentric with the central sphere.Spheres, the centers of which are located inside the virtual sphere, are entirely filled with air.A system generated according to this procedure is then placed within a larger dielectric cavity, see figure 4-10(right).To minimize the influence of the boundaries on the DOS in finite sized systems, we measure the local density of states (LDOS) averaged over the central air-sphere.If necessary, the LDOS can be refined by averaging the results of multiple measurements.However, if only the proof of the existence of a PBG is required, one measurement usually suffices.If there is no PBG in the LDOS, then additional measurements are not necessary.On the other hand, if there is a PBG in the DOS, more measurements should be performed for confirmation.
For the 3D system, we first obtain a reference DOS, by letting r→ ∞.In practice, this is achieved by taking a finite sized system filled entirely with unit cells (no virtual sphere is used), adopting periodic boundary conditions and placing a random wavepacket in all the air-spheres.See figure 4-10(left) for the DOS of the reference system containing 8× 8 × 8 FCC unit cells.The lattice constant is set at a= 3, and combined with a mesh size δ = 0.1, this ensures that the spheres are discretized with sufficient detail.The dielectric permittivity of the background was takenε = 11.9.In the figure, the PBG at (ω = 0.80 is clearly visible.
We study the depth of the PBG in the LDOS as a function of cluster size, i.e. the number of air-spheres n.Numerical results for a clusters built by n= 19, 249, 1409 spheres, together with the reference system, illustrate the decrease of the LDOS inside the PBG, in figure 4-11.
It is clear that we need a quantitative measure for the depth of the bandgap and for this purpose we choose the DOS per unit volume at the PBG frequency (i.e. (ω = 0.80) of a homogeneous system with an effective refraction index equal to the reference system, which is in this case ˜n= 1.79, denoted by ρ0((ω = 0.8), divided by the LDOS inside the PBG of a cluster of n spheres, denoted byρn((ω = 0.8).The relative bandgap depth obtained in
4.1 Photonic Crystals 71
Figure 4-10: Left: DOS as a function of frequency, for the reference system, a closed packed system of spheres of air in a dielectric background.The cen-ters of the spheres form a FCC structure, with lattice vectors given by a1 = a(1, 1, 0)/2, a2 = a(1, 0, 1)/2, and a3 = a(0, 1, 1)/2.Computational parameters are (method,τ,∆t,M,S )=(T4S2,0.075,0.075,16384,1), subject to periodic boundary conditions.
Right: system consisting of n= 43 spheres of air (grey) in a dielectric background (white).
Figure 4-11: (L)DOS as a function of frequency for fcc clusters consisting of n = 19, 249, 1409 spheres and the reference system.
such a way is measured as a function of cluster-size, but shown as function of the virtual sphere size r in figure 4-12.The choice to show the depth as a function of virtual sphere size is clear: on log scale the dependence is linear.The size of the virtual sphere can be
Figure 4-12: Relative bandgap depth of the closed-packed FCC system as a function of the virtual sphere size r in units a, the lattice constant.Dashed line: exp(2.25r/a).The relative bandgap depth of the PBG in the reference system is approximately 1· 1010.
interpreted as a horizon for particles inside the central sphere: in each direction the system resembles a normal fcc structure, up to distance r.If the exponential scaling of the PBG depth holds for larger systems than considered here, and in the next section we show that in two dimensions this is indeed the case, then we can extrapolate the value for which the LDOS of the central cluster would resemble3 the DOS of an infinitely sized system; this happens at approximately r = 10a.This value seems rather low, and could be a positive sign for possible applications.
Contrary to the quite smooth behavior of bandgap depth, the bandgap width does not scale so regularly, as a function of cluster-size, as can be seen in figure 4-13.The bandgap width is zero for the smaller sized clusters (up to n= 87), since the LDOS is higher than the zero level.For increasing cluster size, it then fluctuates and levels out to the reference system value.
Up to now, the LDOS for finite sized clusters was averaged over the entire central sphere.
However, it can be interesting to see how the LDOS depends on specific locations within the photonic crystal, because physically, the LDOS is directly related to the spontaneous emission of an atom [56, 81].At frequencyω0, an atomic excited state exponentially decays with a rate Γ = 2πρ(ω0).So, if the decay energy corresponds to a frequency inside the bandgap, we haveΓ = 0 and there will be no decay, but an atom will form a bound state with the photon to be emitted which can lead to interesting physical phenomena.
We now measure the LDOS inside the central sphere for a number of different locations inside the central sphere, for fixed cluster size n = 683.The results are shown in figure 4-14.As could be expected, the LDOS inside the PBG is lowest exactly in the center of the central sphere (dotted line).If the LDOS is averaged over the central sphere, it increases
3here we interpret two physical systems with the same bandgap depth as having the same physical properties, in other words, we neglect the possible differences in the (L)DOS for frequencies outside the PBG.
4.1 Photonic Crystals 73
Figure 4-13: Bandgap width as a function of clustersize (solid line).For the reference system we measured a width of 3.8% (dotted line).
Figure 4-14: LDOS as a function of frequency for a cluster of n= 683 spheres.Solid line:
LDOS at location r= (0.7, 0.1, 0.1)r0, where r0 = a√
2/4 is the radius of the sphere.Dashed line: LDOS averaged over central sphere.Dotted line: LDOS in center of sphere.Note that for the computation of the LDOS at a specific site, the initial wave-vector has non-zero (random) amplitudes for all the components of one Yee grid cell nearest to the location.
(dashed line), and becomes still higher for a location off the center (solid line).This is reflected in the widths of the bandgap for these three cases, as we measure widths of resp.
4.9%, 3.7% and 2.7% for the LDOS in the center, averaged over the sphere and off-center.
We conclude that LDOS is quite sensitive to the location inside a photonic crystal, and that
Figure 4-15: Left: Density of states per unit volume as function of frequency for the 2D ref-erence system.Computational parameters are (method,τ,∆t,M,S )=(T4S2,0.1,0.1,16384,1), subject to periodic boundary conditions.Right: the reference system, a closed-packed struc-ture of cylinders of air (white) in a dielectric background (grey).
it is worthwhile to exploit existing symmetries.
4.1.6 2D Photonic crystals revisited: imperfections
In the previous section, we briefly touched upon the subject of the robustness of a PBG in a photonic crystal suffering from random imperfections.Defects are important, because they will inevitably occur in the production of photonic crystals and may affect their properties significantly.This issue has received much attention for both two- and three dimensional systems [79, 80, 82–84].Some studies [79, 84] showed that considering the global properties of a disordered crystal, the size randomness of spheres/cylinders has a larger effect on the DOS than the site randomness.Here, we study the influence of defects on the local properties in a finite sized two dimensional photonic crystal consisting of simple cubic closed-packed cylinders of air in a dielectric background.Some work on finite size effects regarding this crystal has been done [85], but not in the context of imperfections.But, before we turn to the influence of defects, we study the effect on the system size on the depth of the PBGs.
We first compute the DOS of the reference system, consisting of 11× 11 unit cells of size a = 4 and ε = 11.4, using periodic boundary conditions.In figure 4-15 we see that there are four large PBGs in the DOS, in concert with figure 4-1.
The finite sized system is constructed by including only those cylinders whose centers lie within a virtual circle with radius r.We now study the amplitude of the PBGs in the LDOS of the central cylinder as a function of the radius of the virtual circle.The procedure to obtain the LDOS of the central cylinder is completely analogous to the 3D case: we place a random wavepacket in the area for which we want to measure the spatially averaged LDOS,
4.1 Photonic Crystals 75
Figure 4-16: Left: LDOS of the central cylinder as a function of frequency, for a system consisting of n= 1129 cylinders (solid line), and a system consisting of n = 441 cylinders (dotted line).Right: LDOS of the central cylinder as a function of frequency, for a system consisting of n = 1129 cylinders (solid line), and for the same system, but with a 1%
randomness in the size of the radii of the cylinders (dotted line).Computational parameters are (method,τ,∆t,M,S )=(T4S2,0.1,0.1,16384,1), subject to periodic boundary conditions.
the central cylinder, and from the time evolution of this wavepacket we extract the LDOS, by the method described in appendix B.Two typical results for the measurement of the LDOS of finite sized systems are shown in figure 4-16(left).
The depth of the ith bandgap for a system consisting of n cylinders is determined by measuring the ratio ρ0((ωi)/ρn((ωi), analogously to the 3D case.For the reference system, the effective refraction index, necessary to obtainρ0((ωi), is now ˜n= 1.664.In figure 4-17 we plot the depth of the bandgap for each of the four PBGs.From figure 4-17, we conclude that, just as in the 3D case, the depths of the PBGs increase exponentially as a function of the virtual circle size, until the depth of the reference system is reached.For every PBG, the convergence rate is slower than the 3D system considered in the previous section, but of the same order of magnitude.The first and third PBG converge faster than the second and fourth.Hence, the width of the PBGs, resp.20%, 2.5%, 2.5% and 2.5% for the first to fourth bandgap of the reference system, does not seem to play a role in the convergence rate.The horizon at which the physical properties of the the finite system become identical to that of a periodic system, for the spatial extension of the central cylinder, is located between r= 12a and r = 18a, depending on the PBG number.
Finally, we study the influence of lattice imperfections on the existence and amplitude of PBGs for our 2D system.Two types of imperfections are introduced in the crystal:
• Size randomness.The radius of the ith cylinder is given by ri= r0+ γra, where r0 is the normal cylinder radius andγris a random variable uniformly distributed over the interval [−∆r, ∆r].
• Site randomness.The (x, y) position of the ith cylinder differs from those in the periodical case by (γxa, γya), where γx andγy are both random variables uniformly distributed over the interval [−∆xy, ∆xy].
Figure 4-17: Relative bandgap depth of the each of the four PBGs in the 2D system as function of virtual circle size r in units a.Solid lines with markers represent the results of finite cluster measurements.Dashed lines are the functions exp(1.6r/a), exp(1.16r/a), exp(1.6r/a) and exp(1.4r/a) for the 1st, 2nd, 3rd and 4th PBG.The depths of the PBGs in the reference system (see figure 4-15) are shown as dotted lines at the values 3.8 · 108, 1.1 · 109, 3.3 · 109 and 4.3 · 108.
Clearly, the degree of randomness affects the existence and depth of the PBGs.A typical example of the influence of an imperfection on the LDOS is shown in figure 4-16(right).Here we see that the higher order PBGs (3 and 4) already begin to disappear at a 1% size randomness.For two different finite sized systems, consisting of n = 441 and n= 1129 cylinders, we systematically study the bandgap depth as a function of size and site randomness, see figure 4-18.These graphs show clearly that PBGs with a lower midgap frequency are more robust with respect to imperfections.The higher order PBGs (three and four) are reduced to noise level at about 2% size or site randomness; the second PBG disappears at 6% and the first PBG is present up to 10% randomness.Secondly, we note that the difference between size and site imperfections is small, though it seems that size defects destroy the PBGs more easily than site defects.This would corroborate the conclusions in references [79, 84], although we must emphasize that we study local effects here and not global effects.Thirdly, we conclude that the difference in system size (n= 441
4.1 Photonic Crystals 77
Figure 4-18: Relative bandgap depth of the each of the four bandgaps in finite sized systems (n= 441, 1192) as a function of the degree of size randomness ∆r and site randomness∆xy. All values should be considered with an error margin of a factor 10, a fluctuation estimated by performing multiple runs with the same parameters but different random configurations
Figure 4-18: Relative bandgap depth of the each of the four bandgaps in finite sized systems (n= 441, 1192) as a function of the degree of size randomness ∆r and site randomness∆xy. All values should be considered with an error margin of a factor 10, a fluctuation estimated by performing multiple runs with the same parameters but different random configurations