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Unfolding the band structure of non-crystalline photonic band gap materials.

Tsitrin S, Williamson EP, Amoah T, Nahal G, Chan HL, Florescu M, Man W - Sci Rep (2015)

Bottom Line: Our results demonstrate the existence of sizeable PBGs in these disordered structures and provide detailed information of the effective band diagrams, dispersion relation, iso-frequency contours, and their angular dependence.Slow light phenomena are also observed in these structures near gap frequencies.This study introduces a powerful tool to investigate photonic properties of non-crystalline structures and provides important effective dispersion information, otherwise difficult to obtain.

View Article: PubMed Central - PubMed

Affiliation: San Francisco State University, San Francisco, CA, 94132 USA.

ABSTRACT
Non-crystalline photonic band gap (PBG) materials have received increasing attention, and sizeable PBGs have been reported in quasi-crystalline structures and, more recently, in disordered structures. Band structure calculations for periodic structures produce accurate dispersion relations, which determine group velocities, dispersion, density of states and iso-frequency surfaces, and are used to predict a wide-range of optical phenomena including light propagation, excited-state decay rates, temporal broadening or compression of ultrashort pulses and complex refraction phenomena. However, band calculations for non-periodic structures employ large super-cells of hundreds to thousands building blocks, and provide little useful information other than the PBG central frequency and width. Using stereolithography, we construct cm-scale disordered PBG materials and perform microwave transmission measurements, as well as finite-difference time-domain (FDTD) simulations. The photonic dispersion relations are reconstructed from the measured and simulated phase data. Our results demonstrate the existence of sizeable PBGs in these disordered structures and provide detailed information of the effective band diagrams, dispersion relation, iso-frequency contours, and their angular dependence. Slow light phenomena are also observed in these structures near gap frequencies. This study introduces a powerful tool to investigate photonic properties of non-crystalline structures and provides important effective dispersion information, otherwise difficult to obtain.

No MeSH data available.


Related in: MedlinePlus

Dispersion relation reconstructed from measured phase data.(a) A sample plot of frequency vs. measured phase delay through the square lattice along the X direction (with 0° incident angle). (b) A sample plot of frequency vs. calculated wavenumber, for the square lattice along the X direction. (c) Dispersion relation along the irreducible Brillouin zone boundary reconstructed from measured phase data for the square lattice, plotted on top of the theoretically calculated bands results (thin solid curves). (Blue dots—the X direction, folded from data in Fig. 4b; Green dots—the M direction; and red dots—angles in between X and M). (d) Dispersion relation along an arbitrary direction in the HUD sample reconstructed from measured phase data.
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f4: Dispersion relation reconstructed from measured phase data.(a) A sample plot of frequency vs. measured phase delay through the square lattice along the X direction (with 0° incident angle). (b) A sample plot of frequency vs. calculated wavenumber, for the square lattice along the X direction. (c) Dispersion relation along the irreducible Brillouin zone boundary reconstructed from measured phase data for the square lattice, plotted on top of the theoretically calculated bands results (thin solid curves). (Blue dots—the X direction, folded from data in Fig. 4b; Green dots—the M direction; and red dots—angles in between X and M). (d) Dispersion relation along an arbitrary direction in the HUD sample reconstructed from measured phase data.

Mentions: Next, we construct the band diagram from the experimentally measured phase data. As mentioned above, the measured phase, ϕm. is the phase difference between the detected phase delay with and without the sample in place. For example, Fig. 4a plots frequency vs. the measured phase at 0° incident angle for the square lattice. Adding the phase, 2πfL/c, accumulated in air through the known sample length L with measured ϕm in each direction gives the actual phase delay in the sample, which can be directly converted into wavenumber /k/ of the propagating mode inside the sample along each incident angle, as shown in Fig. 4b. Contour plots for the resulting wavenumber dependence on frequency and angle in color, for both the square lattice and the HUD sample, are presented in the Supplementary Fig. S1. Figure 4c shows the reconstructed band diagram (along the irreducible first Brillouin zone boundary) for the square lattice sample using the measured wavenumber /k/ as a function of frequency along special symmetry directions of Γ-X (0° incidence, blue dots) and Γ-M (45° incidence, green dots), as well as the measured wavenumber and frequency at the edge of the stop bands between 0° to 45° (red dots). The results agree very well with the calculated band diagram (solid curves in Fig. 4c) for the same square lattice, hence validating our method of constructing effective band diagrams using microwave phase delay measurements through finite cm-scale samples. Applying the same method to the HUD sample data yields the results shown in Fig. 4d, a plot of frequency vs. the effective measured wavenumber /k/. As expected, the effective dispersion relation for HUD sample is statistically isotropic, with little difference between the band structures along various directions. The group velocity (the slope of the frequency vs. wavenumber shown in Fig. 4c,d) is constant for modes far away from the bandgap, indicating radiation propagation with a well-defined energy transport velocity. In the spectral regions close to the photonic band gap, the slope of the measured dispersion become noticeably smaller, indicating a combination between slow light effects and diffusive modes near the gap edges. It is interesting to notice that despite the unavoidable scattering inside the disordered system, the effective dispersion curve is in general smooth outside of the bandgap, suggesting a meaningful definition of group velocity. Actually, in disordered systems, it is often difficult or even impossible to define a group velocity. However, the HUD system is not random, but disordered with some very strong correlations due to the high degree of hyperuniformity, hence the strongly-correlated scattering still allows a meaningful average effective group velocity to be found.


Unfolding the band structure of non-crystalline photonic band gap materials.

Tsitrin S, Williamson EP, Amoah T, Nahal G, Chan HL, Florescu M, Man W - Sci Rep (2015)

Dispersion relation reconstructed from measured phase data.(a) A sample plot of frequency vs. measured phase delay through the square lattice along the X direction (with 0° incident angle). (b) A sample plot of frequency vs. calculated wavenumber, for the square lattice along the X direction. (c) Dispersion relation along the irreducible Brillouin zone boundary reconstructed from measured phase data for the square lattice, plotted on top of the theoretically calculated bands results (thin solid curves). (Blue dots—the X direction, folded from data in Fig. 4b; Green dots—the M direction; and red dots—angles in between X and M). (d) Dispersion relation along an arbitrary direction in the HUD sample reconstructed from measured phase data.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4542607&req=5

f4: Dispersion relation reconstructed from measured phase data.(a) A sample plot of frequency vs. measured phase delay through the square lattice along the X direction (with 0° incident angle). (b) A sample plot of frequency vs. calculated wavenumber, for the square lattice along the X direction. (c) Dispersion relation along the irreducible Brillouin zone boundary reconstructed from measured phase data for the square lattice, plotted on top of the theoretically calculated bands results (thin solid curves). (Blue dots—the X direction, folded from data in Fig. 4b; Green dots—the M direction; and red dots—angles in between X and M). (d) Dispersion relation along an arbitrary direction in the HUD sample reconstructed from measured phase data.
Mentions: Next, we construct the band diagram from the experimentally measured phase data. As mentioned above, the measured phase, ϕm. is the phase difference between the detected phase delay with and without the sample in place. For example, Fig. 4a plots frequency vs. the measured phase at 0° incident angle for the square lattice. Adding the phase, 2πfL/c, accumulated in air through the known sample length L with measured ϕm in each direction gives the actual phase delay in the sample, which can be directly converted into wavenumber /k/ of the propagating mode inside the sample along each incident angle, as shown in Fig. 4b. Contour plots for the resulting wavenumber dependence on frequency and angle in color, for both the square lattice and the HUD sample, are presented in the Supplementary Fig. S1. Figure 4c shows the reconstructed band diagram (along the irreducible first Brillouin zone boundary) for the square lattice sample using the measured wavenumber /k/ as a function of frequency along special symmetry directions of Γ-X (0° incidence, blue dots) and Γ-M (45° incidence, green dots), as well as the measured wavenumber and frequency at the edge of the stop bands between 0° to 45° (red dots). The results agree very well with the calculated band diagram (solid curves in Fig. 4c) for the same square lattice, hence validating our method of constructing effective band diagrams using microwave phase delay measurements through finite cm-scale samples. Applying the same method to the HUD sample data yields the results shown in Fig. 4d, a plot of frequency vs. the effective measured wavenumber /k/. As expected, the effective dispersion relation for HUD sample is statistically isotropic, with little difference between the band structures along various directions. The group velocity (the slope of the frequency vs. wavenumber shown in Fig. 4c,d) is constant for modes far away from the bandgap, indicating radiation propagation with a well-defined energy transport velocity. In the spectral regions close to the photonic band gap, the slope of the measured dispersion become noticeably smaller, indicating a combination between slow light effects and diffusive modes near the gap edges. It is interesting to notice that despite the unavoidable scattering inside the disordered system, the effective dispersion curve is in general smooth outside of the bandgap, suggesting a meaningful definition of group velocity. Actually, in disordered systems, it is often difficult or even impossible to define a group velocity. However, the HUD system is not random, but disordered with some very strong correlations due to the high degree of hyperuniformity, hence the strongly-correlated scattering still allows a meaningful average effective group velocity to be found.

Bottom Line: Our results demonstrate the existence of sizeable PBGs in these disordered structures and provide detailed information of the effective band diagrams, dispersion relation, iso-frequency contours, and their angular dependence.Slow light phenomena are also observed in these structures near gap frequencies.This study introduces a powerful tool to investigate photonic properties of non-crystalline structures and provides important effective dispersion information, otherwise difficult to obtain.

View Article: PubMed Central - PubMed

Affiliation: San Francisco State University, San Francisco, CA, 94132 USA.

ABSTRACT
Non-crystalline photonic band gap (PBG) materials have received increasing attention, and sizeable PBGs have been reported in quasi-crystalline structures and, more recently, in disordered structures. Band structure calculations for periodic structures produce accurate dispersion relations, which determine group velocities, dispersion, density of states and iso-frequency surfaces, and are used to predict a wide-range of optical phenomena including light propagation, excited-state decay rates, temporal broadening or compression of ultrashort pulses and complex refraction phenomena. However, band calculations for non-periodic structures employ large super-cells of hundreds to thousands building blocks, and provide little useful information other than the PBG central frequency and width. Using stereolithography, we construct cm-scale disordered PBG materials and perform microwave transmission measurements, as well as finite-difference time-domain (FDTD) simulations. The photonic dispersion relations are reconstructed from the measured and simulated phase data. Our results demonstrate the existence of sizeable PBGs in these disordered structures and provide detailed information of the effective band diagrams, dispersion relation, iso-frequency contours, and their angular dependence. Slow light phenomena are also observed in these structures near gap frequencies. This study introduces a powerful tool to investigate photonic properties of non-crystalline structures and provides important effective dispersion information, otherwise difficult to obtain.

No MeSH data available.


Related in: MedlinePlus