1. Introduction

The aesthetics and mathematical beauty of quasicrystals has fascinated mankind for centuries already [1, 2]. In 1984, Shechtman discovered three-dimensional quasicrystals [3]. X-ray diffraction from these structures showed unusual Laue diagrams with ten-fold symmetry [3]. In 2005, artificial three-dimensional icosahedral quasicrystals have been realized at microwave frequencies [4], in 2006 at optical frequencies [5]. For the latter, the behavior of the Laue diagrams taken with green light appeared similar to Shechtman’s result at first sight.

Three-dimensional (3D) quasicrystals can be thought of as the projection of a six-dimensional crystal onto three dimensions. Thus, 3D quasicrystals possess a pre-described long-range order but generally no periodicity. As a result, the corresponding Laue diffraction diagrams exhibit an infinite number of diffraction peaks in any finite solid angle – in sharp contrast to crystals. Often, the X-ray diffraction of 3D quasicrystals is well described by Bragg diffraction, i.e., by single scattering of photons. In this single-scattering limit, one neither expects a significant dependence of the various diffracted orders of the Laue diagram on the sample thickness nor on the photon wavelength (or frequency). Clearly, a frequency-independent behavior corresponds to an instantaneous response in the time domain.

The optical data from 3D photonic quasicrystals showed a number of mind boggling aspects that were not understood and that have subsequently been ascribed to imperfections of the fabricated samples [5]: First, time-resolving the response for incident femtosecond laser pulses revealed a large shift of the transmitted pulse maximum as well as a trailing exponential tail. Such tails are well known from disordered (“glassy”) dielectric systems [6] where multiple scattering of light in the disordered dielectric can lead to a delayed response in time. In contrast, such shifts and tails have not been reported for ideal crystals. Second, the measured transmittance and reflectance spectra of the fabricated 3D photonic quasicrystal samples are spectrally highly structured (not depicted in Ref. 5) and show an average level as low as just a few percent for both, transmittance and reflectance. Absorption of the dielectric plays no role at the frequencies of interest. Thus, a sum of transmittance and reflectance much smaller than unity means that the remaining fraction of light is scattered into the residual solid angle – as also known from disordered dielectric systems but not usually expected for ideal photonic crystals. From these conclusions it appeared as if progress in this field required significant improvements of the material quality of the 3D photonic quasicrystals before any experiments on their fundamental physics/optics would even come into reach.

In this context, it is clearly desirable to directly compare the experiments with results from a systematic microscopic theory of the ideal 3D photonic quasicrystal optical properties – accounting for multiple photon-scattering effects. In this letter, we present such results for what we believe is the first time. The surprise is that theory qualitatively describes all of the mentioned unexpected experimental observations previously made and even leads to an excellent qualitative match to experiment without attempting any fitting. This clearly shows that ideal (perfect) 3D photonic quasicrystals share some of their properties with disordered (“glassy”) systems. In particular, 3D photonic quasicrystals can mimic a “diffusive” behavior of light that is usually associated with disordered systems. Yet, we also find significant differences with respect to disordered systems.

2. Periodic approximants of three-dimensional photonic quasicrystals

Solving the 3D vector Maxwell equations for the dielectric 3D photonic quasicrystals of interest, poses a major computational challenge. Recently introduced approaches based on the 6D version of Maxwell’s equations [7], the solutions of which can be projected to 3D, are intellectually very appealing but have not delivered explicit results applicable to our experiments. Direct solutions via, e.g., finite-difference time-domain approaches for such large non-periodic 3D structures are presently out of reach. Efficient approaches taking advantage of the Bloch theorem can obviously not be used directly because of the lack of periodicity in the quasicrystal. Thus, we here combine the known approach of periodic approximants [8] of quasicrystals with scattering-matrix solutions [9] for these approximants.

Quasicrystal approximants are strictly identical to the actual quasicrystal within a (large) unit cell. Outside of that unit cell, they qualitatively resemble the quasicrystal appearance, but in a periodic manner. This aspect is graphically illustrated in Fig. 1 for the icosahedral 3D photonic quasicrystal of interest (top row) and for a local two-fold symmetry axis. From top to bottom row, Fig. 1 also shows the associated 3/2, the 2/1, and the 1/1 approximants. The unit cell is highlighted in red in each case. The construction principle of 3D quasicrystals has been described previously [5]. In essence, a 6D simple-cubic crystal is rotated in space and projected onto 3D (cut-and-project method [8]) in a manner such that each fictitious lattice point is connected to an adjacent one by a real dielectric rod if the original two lattice points were already connected. This step guarantees the mechanical stability of the resulting structure, i.e., it avoids floating parts of the structure. For the approximants, the same procedure is applied, but the rotation in 6D is by an angle with a rational tangent (rather than an irrational tangent for the actual quasicrystal), i.e., the m/n approximants are constructed by rotating a simple-cubic 6D lattice via the unitary matrix

The resulting lattice is projected onto 3D via the so-called “cut-and-project method” [8], as further outlined in Ref. 5. The edge length of the (cubic) unit cell of the m/n approximant is given by 2·(m·τ + n)·√2/(2τ ² + 2) times the rod length, l, where τ = (1 + √5)/2 is the golden mean. The left-hand side column of Fig. 1 illustrates the photonic quasicrystal and the approximants used in our calculations, the right-hand side column depicts corresponding actually fabricated polymer samples.

All samples discussed and shown in this letter are fabricated by direct laser writing using the commercially available photoresist SU-8 (MicroChem). Fabrication details can be found in Ref. 5. In order to avoid confusion, we note that the quasicrystals shown in Fig. 2 to Fig. 6 have a rod length of l=1 μm, whereas those shown in Ref. 5 and in Fig. 1 are for l=2 μm. As a result, the spectral features experience a blue shift in frequency by a factor of two. This shift allows us for employing very sensitive silicon light detectors in the visible the equivalent of which is not available for 1.5-μm wavelength (see Ref. 5). We will come back to these data below.

 

Fig. 1. Three-dimensional icosahedral photonic quasicrystal and its approximants. The left-hand side column shows computer generated images, the right-hand side column electron micrographs of corresponding SU-8 structures fabricated via direct laser writing. (a) and (b) exhibit the 3D quasicrystal (twofold local axis), (c) and (d) the 3/2 approximant, (e) and (f) the 2/1 approximant, and (g) and (h) the 1/1 approximant. The red regions in the theory highlight the unit cell of the periodic approximant.

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2.1 Scattering-matrix calculations

Our scattering-matrix calculations are based on a home-made computer code following the concepts outlined in Ref. 9. To ensure convergence for a given quasicrystal approximant, we account for as many as 8 orders. The real space discretization is 20 nm, which has to be compared with the rod length of l=1 μm and the 3D unit cell of about 4.454 μm in extent for the 2/1 approximant (see Fig. 1(e)). The special spatial shape of the “voxels” with an aspect ratio between axial and lateral feature sizes of about two resulting from the fabrication via direct laser writing [5] is accounted for (see Fig. 1). The combination of these aspects and the quoted numbers does allow us for reliable calculations for the 2/1 approximant shown in Fig. 1 indeed. Memory space and CPU times are, however, already excessive for the 3/2 approximant depicted in Fig. 1(c). Also, we have restrained ourselves to calculations for the twofold axis of the icosahedral 3D quasicrystals for reasons of complexity. Recall that in Ref. 5 experimental results for the two-, three-, and five-fold local real-space axis have been discussed. The respective thickness, L, of the photonic quasicrystal or approximant is quoted in the figure captions. In all calculations shown in this letter, the polymer (SU-8) refractive index is taken as n=1.6 (i.e., zero imaginary part). The glass substrate (half-space) underneath is explicitly accounted for, its refractive index is n=1.5.

For additional reference calculations (to be shown in Fig. 3), a 3D photonic crystal, a regular woodpile sample (see, e.g., Ref. 10) with a rod spacing of a=1 μm and a lattice constant of c=1.414 μm, is used.

2.2 Test of the approximant approach

To test the applicability of the approximant approach for our conditions, we compute angle-resolved (intensity) transmittance spectra for linearly polarized incident light and compare with the transmittance measured on fabricated quasicrystals and approximants consisting of 1-μm rods in Fig. 2. Precisely, “transmittance” (“reflectance”) here and in what follows refers to the emission of light into the forward (backward) direction in a pre-described finite opening angle ϑ around the incident (reflected) beam axis. In the experiments, the incident light that is imaged onto the samples corresponds to a finite opening angle of about 5 degrees. To mimic this angle averaging in the theory, we perform calculations for ideal plane waves for different angles of incidence and average these spectra over an angle of 5 degrees. Figure 2 shows that the transmittance properties of the approximants rapidly converge to those of the 3D photonic quasicrystal with increasing order of the approximant (i.e., with increasing size of the approximant unit cell). Furthermore, the scattering-matrix calculations agree well with the experimental data. Note that the 2/1 approximant already nicely resembles the 3D quasicrystal properties.

 

Fig. 2. (a) Calculated and measured (intensity) transmittance versus wavelength of light and versus angle of incidence with respect to the surface normal for structures with rod length l=1 μm and thickness L=4.45μm. The indicated approximants are illustrated in Fig. 1. Note the good overall qualitative agreement between experiment and theory that supports our approximant approach. (b) Line cuts of the experimental data for normal incidence of light (see also (a)) illustrate the convergence of the optical properties towards those of the 3D quasicrystal with increasing order of the approximant.

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3. Theoretical and experimental results

In the experiments of Ref. 5, the transmission of femtosecond optical pulses impinging onto 3D photonic quasicrystals has been studied. To directly connect to these experiments [5], we consider linearly polarized incident Gaussian optical pulses of 150-fs in duration in the approximant calculations. For each diffraction order, the square modulus of the Fourier transform of the frequency-dependent transmitted electric field orthogonal to the incident polarization [5] provides us with the time-resolved intensity. Analysis of the orthogonal polarization configuration is advantageous because the directly transmitted beam is blocked, such that one can observe predominantly the light which has interacted with the sample. The sum over the various diffracted orders within an opening angle of ϑ=27 degrees (corresponding to typical experimental conditions [5]) is depicted in Fig. 3. Here, the rod length is l=1 μm and the pulse center wavelength is 735 nm. Due to the scalability of Maxwell’s equations, the strictly identical result is obtained for l=2 μm and 1.47-μm center wavelength (compare experiments reported in Ref. 5). Amazingly, we find a very closely similar qualitative behavior: The maximum of the transmitted pulse is shifted with respect to time zero and an exponential tail develops – that has previously [5] been ascribed exclusively to disorder, i.e., to presumable imperfections of the samples. In contrast, Fig. 3 makes clear that such behavior can also arise as a result of the intrinsic physics of ideal 3D photonic quasicrystals. However, we are currently not in a position to quantify the relative contributions of intrinsic and extrinsic effects in the experiments. Figure 4 shows results similar to those in Fig. 3, but for several different center wavelengths of the incident Gaussian pulses, indicating that the qualitative behavior is generic, whereas the quantitative exponential time constants and temporal shifts clearly do depend on the parameters.

 

Fig. 3. Calculated time-resolved transmittance (solid) through a 2/1 approximant (l=1 μm, L= 8.9 μ m) of a photonic quasicrystal (see Fig. 1(e)) for normal incidence of a 150-fs Gaussian optical pulse centered around 735-nm wavelength. The emerging light is collected in an opening angle of ϑ=27 degrees around the surface normal and in the linear polarization orthogonal to the incident one. The calculated behavior nicely agrees with the experimental one that has previously been reported in Fig. 4 of Ref. 5 (not depicted here). Especially note the temporal shift and the long-time tail (red straight line with 90-fs time constant) with respect to the autocorrelation of the incident 150-fs Gaussian pulses (dashed). Further corresponding data are shown in Fig. 4.

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Fig. 4. Calculations as in Fig. 3 (2/1 approximant, l=1 μm, L=8.9 μm), but for different center wavelengths of the incident femtosecond optical pulses – represented as a false-color plot. (a) detection orthogonal to the incident linear polarization, (b) detection parallel to the incident linear polarization. The white lines indicate the inferred positions of the respective transmitted pulse maxima.

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It is clear that these temporal trailing tails necessarily require the occurrence of correspondingly rather narrow spectral features in the frequency domain for the same orthogonal detection. However, such narrow spectral features can easily be obscured by finite angle-averaging in the experiment. Thus, for the following normal-incidence transmittance experiments, great care has been taken that the ideal of an incident plane wave is actually nearly accomplished. To achieve this goal, we have introduced a 200-μm diameter pinhole directly in front of the focusing microscope lens, corresponding to a full opening angle of the incident light of only 1.5 degrees. To ensure that we still exclusively collect light that emerges from the photonic quasicrystal sample with a diameter of merely 50 μm, the transmitted light (collected within a cone of ϑ=24 degrees) is re-imaged onto an intermediate plane, where it is monitored by an infrared camera. Four adjustable knife edges in this intermediate image plane allow for selecting the sample area. As a result of all these steps, and given the employed black-body emission of a 100-W power halogen lamp operating at around 3000-K surface temperature, the light levels at the spectrometer are fairly low. Hence, we employ a grating spectrometer (1-nm resolution) connected to a sensitive liquid-nitrogen cooled back-illuminated charge-coupled-device (CCD) camera. Under these conditions, typical exposure times are several seconds. Normalization is with respect to the light transmitted by the bare glass substrate. In our previous spectroscopic experiments [5] as well as in the overview data shown in Fig. 2, the angular spread of the incident light has been too large to reveal the fine structures observed in this dedicated normal-incidence setup.

The left-hand side column of Fig. 5 shows the anticipated narrow spectral features in the normal-incidence sample transmittance. The overall qualitative agreement between experiment and theory is rather good – clearly we just cannot expect a one-to-one correspondence of all the detailed spectral maxima and minima. Yet, importantly, the characteristic scales do match. In contrast, for a simple 3D periodic photonic crystal of the woodpile type with similar spatial feature sizes, the transmittance in the orthogonal polarization configuration is close to zero – as expected from symmetry. For the parallel detection configuration (right-hand side column of Fig. 5), the higher photonic bands lead to highly structured transmittance spectra for both, woodpile and 3D photonic quasicrystal.

 

Fig. 5. Normal-incidence (intensity) transmittance spectra for linearly polarized incident light. The left-hand side column refers to the orthogonal detection configuration, the right-hand side column to the parallel one. The top row corresponds to theory for light impinging along the two-fold local symmetry axis of 2/1 approximants (l=1 μm, L=4.45 μm). The second row shows the experimental result obtained from corresponding 3D photonic quasicrystals oriented along different axes (as indicated). These data, which are taken for a small spread of the incident wave vector of light and for ϑ=24 degrees, are the frequency-domain analogue of the temporal tails shown in Fig. 3. For reference, the third row exhibits calculations for a 3D woodpile photonic crystal with similar feature sizes (a=1 μm, L=5.66 μm), and made from the same polymer.

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Intuitively, in sharp contrast to X-ray diffraction from Shechtman’s quasicrystals – where the refractive-index contrast is extremely small – multiple scattering of light is very important for our conditions, even though the polymer refractive index of n=1.6 with respect to air/vacuum (n=1) is fairly small for the standards of optics. Thus, the various spectral maxima/minima in Fig. 5 arise from the superposition of several complicated spatial interference pathways of light inside the 3D quasicrystal that are many wavelengths of light in length. Sensitively depending on the wavelength, the result is either mostly constructive or destructive.

3.1. Thickness dependence

This intuitive interpretation does suggest similarities with light scattering in disordered (“glassy”) photonic systems. There, one obtains an Ohmic behavior of the light transmittance T through a slab of thickness L, i.e., a scaling according to T~1/L. This scaling is quite distinct from Beer’s law according to T~exp(- const. L). For a simple periodic photonic crystal, one rather expects Beer’s law: Within the photonic bands, one has propagating Bloch waves, the amplitude of which does not decay. Hence, the exponent in Beer’s law is ideally zero (i.e., const.=0). Within a photonic stop band, the waves become evanescent, hence they rapidly decay exponentially. To investigate this aspect, Fig. 6 shows the calculated spectrally-resolved normal-incidence 3D 2/1 approximant transmittance integrated over all diffraction orders in the forward direction versus sample thickness. Clearly, the overall behavior is neither purely of Ohm’s nor purely of Beer’s type. For many wavelengths, the quasicrystal transmittance versus thickness rather drops within some thickness range to reach a finite constant value for larger thicknesses. Likely, this aspect can be interpreted in terms of forward scattering of light that is also responsible for the occurrence of the Laue diffraction peaks in forward direction. This observation suggests that the transport of light through 3D photonic quasicrystals is generally different from both, transport in photonic glasses and in photonic crystals. Unfortunately, fabricating 3D photonic quasicrystals with the large sample thicknesses shown in Fig. 6 is currently out of reach.

 

Fig. 6. (a) False color representation of calculated transmittance spectra versus thickness L of the 2/1 approximant (l=1 μm, edge length of the unit cell is 4.454 μm) of the 3D icosahedral quasicrystal. (b) shows selected cuts through this data set.

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4. Conclusions

In conclusion, we have shown that multiple scattering of light in 3D icosahedral photonic quasicrystals reveals certain aspects that resemble “diffusive” scattering of light in disordered photonic systems – even for ideal quasicrystals without any disorder or imperfections. Yet, the quasicrystal behavior also comprises aspects that are quite distinct from what is known for ideal disordered photonic systems – as becomes immediately obvious from, e.g., the beautiful diffraction patterns of photonic quasicrystals. It is interesting to note that our results obtained from periodic approximants of 3D quasicrystals imply that similar “diffusive” scattering of light can be expected for periodic photonic crystals with sufficiently complex unit cells and for small ratios of wavelength to lattice constant.