Abstract

Recent experiments on three-dimensional icosahedral dielectric photonic quasicrystals have shown several unexpected features: transmitted femtosecond pulses developed a trailing “diffusive” exponential tail and the sum of (zeroth-order) transmittance and reflectance was well below unity. These experimental findings have previously been ascribed to sample imperfections. Here, we analyze these findings by using 3D periodic approximants of the ideal photonic quasicrystals. We show that the experimental observations can be explained in terms of multiple scattering of light within these structures, i.e., in terms of intrinsic rather than purely extrinsic quasicrystal properties.

© 2009 Optical Society of America

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References

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  1. P. J. Lu and P. J. Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106-1110 (2007).
    [CrossRef] [PubMed]
  2. B. Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman and Company, New York, 1986).
  3. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
    [CrossRef]
  4. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
    [CrossRef] [PubMed]
  5. A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
    [CrossRef]
  6. A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).
  7. A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).
  8. C. Janot, Quasicrystals - A Primer (Clarendon Press, Oxford, 1992).
  9. D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
    [CrossRef]
  10. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
    [CrossRef]

2008 (1)

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

2007 (1)

P. J. Lu and P. J. Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106-1110 (2007).
[CrossRef] [PubMed]

2006 (1)

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

2005 (2)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

1999 (1)

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

1984 (1)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Avniel, Y.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Busch, K.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Cademartiri, L.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Chaikin, P. M.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Deubel, M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Hermatschweiler, M.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

Johnson, S. G.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

Koenderink, A. F.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).

Lagendijk, A.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).

Ledermann, A.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

Lu, P. J.

P. J. Lu and P. J. Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106-1110 (2007).
[CrossRef] [PubMed]

Man, W.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

McCauley, A. P.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

Megens, M.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

Ozin, G. A.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

Pereira, S.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Rodriguez, A. W.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Soukoulis, C. M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Steinhardt, P. J.

P. J. Lu and P. J. Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106-1110 (2007).
[CrossRef] [PubMed]

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

Toninelli, C.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

von Freymann, G.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Vos, W. L.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).

Wegener, M.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Wiersma, D. S.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

Nat. Mater. (2)

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006).
[CrossRef]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic crystal templates for telecommunications," Nat. Mater. 3, 444-447 (2004).
[CrossRef]

Nature (1)

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005).
[CrossRef] [PubMed]

Phys. Rev. B (3)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102:1-4 (2005).

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, "Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem," Phys. Rev. B 77, 104201:1-10 (2008).

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Science (1)

P. J. Lu and P. J. Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106-1110 (2007).
[CrossRef] [PubMed]

Other (2)

B. Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman and Company, New York, 1986).

C. Janot, Quasicrystals - A Primer (Clarendon Press, Oxford, 1992).

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Figures (6)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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.

Fig. 6.
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.

Equations (1)

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M = 1 2 · ( m / n ) 2 + 2 m / n m / n 0 1 0 1 0 0 1 m / n 1 m / n 1 1 m / n 0 m / n 0 m / n m / n 1 0 1 0 1 1 0 m / n 0 m / n 0 0 m / n 1 m / n 1

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