The plane-wave expansion method (PWEM), the multiple-scattering method (MSM) and the 3D finite-difference time-domain method (FDTD) are applied for simulations of propagation of electromagnetic waves through 3D colloidal photonic crystals. The system investigated is not a “usual” artificial opal with close-packed fcc lattice but a dilute bcc structure which occurs due to long-range repulsive interaction between electrically charged colloidal particles during the growth process. The basic optical properties of non-close-packed colloidal PhCs are explored by examining the band structure and reflection spectra for a bcc lattice of silica spheres in an aqueous medium. Finite size effects and correspondence between the Bragg model, band structure and reflection spectra are discussed. The effects of size, positional and missing-spheres disorder are investigated. In addition, by analyzing the results of experimental work we show that the fabricated structures have reduced plane-to-plane distance probably due to the effect of gravity during growth.
© 2010 OSA
Currently, one of the most wide-spread methods for the fabrication of 3D photonic crystals (PhC) for near-infrared and visible light is the self-organization of spherical colloidal microparticles [1–14]. Due to the process of sedimentation of submicron-sized spheres from a colloidal solution on a flat surface it is possible to fabricate ordered arrays of these spheres with 3D periodicity. Normally, this self-organized sedimentation process leads to the formation of close-packed layers of spheres on top of each other that corresponds to the growth of a fcc structure in  direction [1–7]. Such arrangements are commonly called artificial opals. However, there is another, a bit lesser-known, possibility to create 3D periodical colloidal structures. If the colloidal particles are statically charged then due to the long-range repulsive interaction between each other they can form a dilute structure with bcc or fcc lattice, depending on the concentration of particles [8–12].
The main aim of this paper is to investigate theoretically the wave propagation and the influence of disorder on the optical properties of non-close-packed colloidal PhCs with bcc lattice. We restrict ourselves to the case of the  propagation direction. Disorder can range from weak deviations from the ideal structure to completely random arrangements. An example for the latter case is a photonic glass . In this paper we consider deviations up to 40%.
In the second chapter we will illuminate the basic optical properties of non-close-packed colloidal PhCs by examining the band structure and reflection spectra for a bcc lattice of silica spheres in an aqueous medium. Finite size effects and correspondence between the Bragg model, band structure and reflection spectra will be discussed. The calculations of the reflectance presented in this chapter are made using the multiple-scattering method since this method allows examining spatially large structures under low consumption of computational resources. The band structure calculations are performed by the PWEM.
In the third chapter we present FDTD calculations of reflection spectra of PhCs with disorder. Three types of disorder are examined: disorder in radii, positional disorder and missing-sphere disorder. Detailed argumentation of utilizing FDTD method is given in the beginning of chapter three.
By comparison of simulations with the experimental reflection spectra [9,10] we will show in the fourth chapter that the experimental results can be well reproduced by the simulations and that the fabricated PhCs do not have a perfect bcc lattice but are slightly compressed in  direction. This compression is most probably due to the effect of gravity during growth.
2. The basic properties of diluted colloidal photonic crystals with bcc lattice
Here and below (except chapter 4) we use the following parameters in the simulations: nb = 1.33, nsph = 1.46, r = 0.18a, lattice type is bcc, where r is the radius of the spheres and a is the lattice constant of the bcc lattice, nb and nsph are the refractive indices of the background (aqueous medium) and of the silica spheres, respectively.
In Fig. 1 the photonic band structure calculated by the plane-wave expansion method (PWEM) for the 8 lowest bands is shown. Due to the very low refractive index contrast the structure does not exhibit a full photonic band gap. However, there is a narrow gap at the N point ( direction), shown enlarged in the inset. This unidirectional gap corresponds to the first-order Bragg reflection from the system of (110) planes.
Actually, in the case of  incidence, the 3D structure considered can be treated as an array of parallel partially reflecting planes or, in photonic crystal terms, as a 1D structure. Let us check the correspondence between full 3D consideration and simplified Bragg reflection model. The wavelength of the first-order Bragg reflection peak for normal incidence is given byEq. (1) is then a/λB = 0.5293. The center frequency of the band gap calculated by the PWEM and shown in the inset of Fig. 1 is a/λB = 0.5291. Thus, the agreement concerning the center frequency of the reflection peak can be treated as very good.
The reflection calculated by using the MULTEM2 program , applying the multiple scattering method (MSM), is presented in Fig. 2 . Different line colors correspond to different thicknesses of the PhC in  direction. N is the number of the (110) planes; vertical dotted lines show the edges of the pseudogap calculated by PWEM (Fig. 1). One can see a pronounced effect of the finite size of the PhC. This is easily understandable: since the contrast in refractive index between spheres and background is very weak, a large number of periods are required to suppress the incident wave in the frequency range of the reflection peak, thus the wave penetrates deeply into the PhC before all of it is reflected (see Fig. 4 in the next section). Figure 2 also shows that the reflection peak coincides with the gap at the N-point of the photonic band structure. Since the band structure calculations made by PWEM assume a PhC that is infinite in all directions we can state that for the current parameters in the MSM calculation 500 layers of spheres represent the reflectance of an infinite structure. Of course, PhCs with higher optical contrast and close-packed structure require fewer layers to fully develop the reflected wave [13,14]. An interesting point is that the simplest Eq. (1) gives already a sufficiently exact value of the center of the reflection peak; band structure calculations allow to derive also the width of the reflection peak of an infinite structure; however the shape of the reflection peak and the actual value of its maximum can be obtained only by real-space calculations.
3. FDTD calculations, influence of disorder
The calculations of the reflectance were performed also by using the finite-difference time-domain method (FDTD). The advantage of the FDTD is the ability to solve problems with complicated dielectric constant profiles. So, we applied the method to study the influence of disorder on the reflection peak mentioned above. The FDTD method solves the Maxwell equations directly in real space by replacing spatial and time derivatives of the field by finite differences, in other words dx, dy, dz, and dt are replaced by Δx, Δy, Δz, and Δt, respectively. Thus, the spatial computational area is divided into small cells of size Δx*Δy*Δz. The main requirement to this cell is that it should be at least 10 times smaller than the characteristic features size (in our case the spheres) and the wavelength. This requirement is the main limitation of the FDTD since spatially large problems (especially in 3D) consume a very large amount of computational resources: physical memory and computational time. Due to this limitation we cannot consider 500 planes of spheres. In the calculations presented below the PhC structures consist of 70 (110) planes.
In Fig. 3 the general view of the problem (without disorder) is depicted for xz cross-section (a) and full 3D view (b). The horizontal green bars in Fig. 3(a) are the monitors which record the flux of electromagnetic (EM) energy through them. The monitor at the top records the transmitted power, the one at the bottom the reflected one. The horizontal orange bar is the source radiating only towards the PhC. The electric field in incident wave is polarized along the y-direction ( crystallographic direction). The effect of different polarization (x-direction, [-110]) on the results presented below is insignificant. The propagation direction (z) corresponds to the  direction in the bcc lattice. The purple frame shows the edge of the computational domain with perfectly matched layer (PML) absorbing boundary conditions (ABC). There are two main points which force us to choose PML ABC instead of periodic boundary conditions: (i) since we will introduce random disorder below, the translational symmetry of the structure is broken that can result in some non-physical effects on the edges of the computational domain if using the periodic boundary conditions; (ii) PML ABC allow to account for scattering processes since the scattered (in x- and y- directions) electromagnetic energy will be absorbed and thus will not appear either in transmission or in reflection.
As we said in the previous section, the reflected wave develops over a large number of periods in propagation direction. This is illustrated in Fig. 4(a) showing the xz cross-section of the computational domain with the distribution of the Ey component of the electric field for the wavelength corresponding to the maximum of the reflection peak. The plane wave source is located at z = 0.5a. Please note, that the scales of z- and x-axes are different. In Fig. 4(b) the time-dependent transmission and reflection calculated at the wavelength of the reflection maximum are shown. As one can see, the system reaches steady state after cT = 250a. Since there are no absorptive materials and we are using PML ABC in our calculations we can estimate the scattering processes by the simple formula S = 1-T-R, where T is the transmittance and R is the reflectance and S is scattered intensity. The latter is defined as the intensity impinging the absorptive side walls (in x- and y-directions) of the computational domain. For the case of the photonic crystal without disorder [Fig. 4(b)] it is 0.0007 relative to the incident intensity.
Three types of disorder are considered: disorder in radii, positional disorder and missing-spheres disorder. Disorder in radii is introduced by a random change of the radius of each sphere. In Fig. 5(a) the solid black curve corresponds to the reflection of the perfect structure (without disorder), blue and red dotted curves are for disorder with maximum change in radii of 20% and 40% with respect to initial radii, respectively. We can state that for the given parameters of the PhC the disorder in radii has no effect on the reflection peak for 20% disorder and causes a peak reduction of only about 10% for 40% disorder.
Positional disorder is simulated by introducing random variations in x, y and z coordinates of the center of each sphere. In Fig. 5(b) the black dotted line is for the case without disorder, blue dotted line for disorder with amplitude 0.2a but only in x and y directions, and red dotted line is for disorder with amplitude 0.2a in all three directions. As one can see, the amplitude and FWHM of the peak are practically not affected if the spheres are displaced in the xy plane only. This is easy to understand considering that in this case all the spheres still stay within (110) planes. The situation changes when the spheres are displaced out of (110) planes (red dotted line). Then, the peak maximum value is more than twice lower now.
In order to simulate possible vacancies (missing spheres) we have randomly removed 20% of the spheres. In Fig. 5(c) the dotted curve shows the reflection peak for this type of disorder. A clearly visible decrease of the reflection peak maximum is observed in this case. It is necessary to note, that the model for the missing-spheres disorder is idealized. In a real structure, the spheres adjacent to a vacancy would be shifted due to missing repulsive force.
The scattering (S, as defined above) is essential only for red dotted curves in Figs. 5(a) and 5(b): 0.063 and 0.061, respectively. In all other cases with disorder it was less than 0.01. However, one should take into account that the transmittance and reflectance are calculated in near-field. In far-field the amount of scattered energy should be higher.
Interestingly, in all the cases we did not observe any broadening of the reflection peak due to the disorder but only the maximal value was affected.
4. Comparison with experimental data
In the experimental works [9,10] PhCs with the following nominal parameters were fabricated: nb = 1.35, nsph = 1.42, f = 0.035, r = 55 nm. Taking into account that there are two atoms per unit cell in the bcc lattice we can extract the lattice constant as
For f = 0.035 we obtain a = 341.5 nm (d = 241.5 nm), so r = 0.161a. The reflectance spectra for these new parameters are shown in Fig. 6 . We should note that in comparison to Fig. 2 the reflection peak is sufficiently narrower now.
By substituting the values of dielectric constants and spheres filling fraction into the Eq. (2) we obtain that the effective refractive index is neff = 1.352. Thus, from the Eq. (1) the wavelength of the reflection peak is λB = 652.9 nm. The wavelength of the reflectance peak shown in Fig. 6 is 653.0 nm which is again in very good agreement with the Bragg model, Eq. (1). However, this value does not coincide with that one obtained experimentally. In the experiments [9,10] the reflection peak was detected at the wavelength λB = 616 nm. We assume that the reason of this discrepancy is that the lattice in the charged colloidal PhC is not perfectly bcc, but slightly compressed in z-direction, so the distance between (110) planes is smaller than should be in a perfect bcc. It is easy to calculate that the reflection peak at λ = 616 nm corresponds to a plane-to-plane distance of 227.7 nm (instead of the nominal value of 241.5 nm). This reduction of the plane-to-plane distance can be attributed to the effect of gravity during the growth.
In conclusion, the effect of disorder on the reflection peak of dilute colloidal photonic crystals which consist of aqueous solutions of silica spheres is investigated theoretically. In the case of disorder in radii only a high amount (40% and more) has a visible effect on the reflection peak. Positional disorder decreases the maximal value of the peak only if the spheres are randomly shifted out of (110) planes (in the direction of the propagation of the electromagnetic wave). Random removal of 20% of silica spheres from the structure results in a clearly visible decrease of the maximum of the reflection peak. We did not observe any broadening of the reflection peak as effect of disorder but only the maximal value is reduced. The experimentally obtained spectral position of the reflection peak deviates from the calculated one by 6%. We interpret this discrepancy by a deviation from a perfect bcc lattice, viz. that a compression in  direction occurs during the growth of the colloidal PhC.
This work was supported by the Austrian Science Fund (FWF): project N 1104-NAN. The authors would like to thank Prof. Yo. Takiguchi for valuable discussions.
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