This work was partially supported by the Research Corporation for Science Advancement (Grant 10626), National Science Foundation (DMR-1308084), and the San Francisco State University internal award to W. M. We thank our collaborator Paul M. Chaikin from NYU for helpful discussions in experimental design and for providing the VNA system for us to use on site at SFSU. We thank our theoretical collaborators, the inventor of the HD PBG materials, Marian Florescu, Paul M. Steinhardt, and Sal Torquato for various discussions and for providing us the design of the HD point pattern and continuous discussions.

1. Design a 2D Hyperuniform Disordered Dielectric Structure11
<ol>
	<li>Chose a subclass of 2D hyperuniform disorder point pattern (blue circles in Figure 2) and partition it (blue lines in Figure 2) using Delaunay tessellation. A 2D Delaunay tessellation is a triangulation that maximizes the minimum angle for each triangle formed and guarantees there are no other points inside the circumcircle of each triangle11.</li>
	<li>Locate the centroids of each triangle (s...

<ol>
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The authors have nothing to disclose.

Starting from a hyperuniform disordered point pattern, 2D HD structures consisting rods and/or wall network can be designed to obtain a complete PBG for all polarization11. Based on the design, we constructed a template with holes and slots for assembling 2D Alumina rods and walls structures at cm-scale which could be tested with microwaves. We chose to work with microwaves, because cm-scale building blocks, such as Alumina rods and walls, are inexpensive, and easily handled. We have experimentally demonstrate...

The existence of a bandgap for photons has been the focus of many scientific works, starting from the earlier studies done by Lord Rayleigh on the one-dimensional stop-band, a range of frequencies that are forbidden from propagation through a periodic medium1. Research into electromagnetic wave (EM) propagation in periodic structures has really flourished in the last two decades after the seminal publications of E. Yablonovitch2,3 and S. John4. The term &#8220;photonic crystal&#8221; was coined by Yablonovitch to describe the periodic dielectric structures that possessed a photonic bandgap (PBG). &#160;

Photonic crystals are periodic dielectric structures possessing discrete translational symmetries, rendering them invariant under translations in directions of periodicity. When this periodicity is matched with the wavelengths of incoming electromagnetic (EM) waves, a band of frequencies becomes highly attenuated and may stop propagating. If wide enough, the ranges of the forbidden frequencies, also called stop bands, may overlap in all directions to create a PBG, forbidding the existence of photons of certain frequencies.

Conceptually, EM wave propagation in photonic crystals is similar to electron wave propagation in semiconductor materials, which have a forbidden region of electron energies, also known as a bandgap. Similar to the way engineers have employed semiconductors to control and modify the flow of electrons through semiconductors, PBG materials can be used for various applications requiring optical control. For example, PBG materials can confine light of certain frequencies in wavelength size cavities, and guide or filter light along line defects in them5. PBG materials are suggested to be used for controlling the flow of light for applications in telecommunication6, lasers7, optical circuits and optical computing8, and solar energy harvesting9.

A two-dimensional (2D) square lattice photonic crystal has 4-fold rotational symmetry. EM waves entering the crystal at different angles of incidence (for example, 0&#176; and 45&#176; with respect to the lattice planes) will face different periodicities. Bragg scattering in different directions leads to stop bands of different wavelengths that may not overlap in all directions to form a PBG, without very high refractive-index contrast of the materials. In addition, in 2D structures, two different EM wave polarizations, Transverse Electric (TE) and Transverse Magnetic (TM), often form bandgaps at differing frequencies, making it even harder to form a complete PBG in all directions for all polarizations5. In periodic structures, the limited choices of rotational symmetry lead to intrinsic anisotropy (angular dependence), which not only makes it hard to form a complete PBG, but also greatly limits the design freedom of functional defects. For example, waveguide designs are proven to be restricted along very limited choices of major symmetry directions in photonic crystals10.

Inspired to surpass these limitations due to periodicity, much research has been done in the past 20 years on unconventional PBG materials. Recently a new class of disordered materials was proposed to possess an isotropic complete PBG in the absence of periodicity or quasiperiodicity: the hyperuniform Disorder (HD) PBG structure11. The photonic bands do not have exact analytical solution in disorder structures. Theoretical study of the photonic properties of the disordered structures is limited to time&#8211;consuming numerical simulations. To calculate the bands, the simulation needs to employ a super-cell approximation method and the available computational power may limit the finite size of the super-cell. To calculate transmission through these structures, computer simulations often assume ideal conditions and thus neglect real world problems like the coupling between the source and detector, the actual incident EM wave profile, and alignment imperfections12. Furthermore, any modification (defect design) of the simulated structure would require another round of simulation. Due to the large size of the minimum meaning for super-cell, it is very tedious and impractical to systematically explore various defect design architectures for these disordered materials.

We can avert these computational problems by studying the disordered photonic structures experimentally. Through our experiments we are able to verify the existence of the complete PBG in HD structures. Using microwave experiments, we can also obtain phase information and reveal the field distribution and dispersion properties of existing photonic states in them. Using an easily modifiable and modular sample at cm-scale, we can test various waveguide and cavity (defect) designs in the disordered systems and analyze the robustness of the PBGs. This kind of analysis of complex disordered photonic structures is either impractical or impossible to obtain through numerical or theoretical studies.

The design process begins by selecting a &#8220;stealthy&#8221; hyperuniform point pattern13. Hyperuniform point patterns are systems in which the number variance of the points within a &#8220;spherical&#8221; sampling window of radius R, grows more slowly than the window volume for large R, that is, more slowly than Rd in d-dimensions. For example, in a 2D Poisson random distribution of point pattern, the variance of the number of points in domain R is proportional to R2. However, in a hyperuniform disorder point pattern, the variance of the points in a window of radius R, is proportional to R. Figure 1&#160;shows a comparison between a hyperuniform disordered point pattern and a Poisson point pattern11. We use a subclass of hyperuniform disordered point patterns called &#8220;stealthy&#8221;11.

Using the design protocol described in Florescu&#160;et al11, we construct a network of dielectric walls and rods, creating a 2D hyperuniform dielectric structure similar to a crystal, but without the limitations inherent to periodicity and isotropy. The wall networks are favorable for TE-polarization bandgap, while the rods are preferable for forming band gaps with TM-polarization.&#160; A modular design was developed, so that the samples can be easily modified for use with different polarizations and for introducing freeform waveguides and cavity defects. Due to the scale invariance of Maxwell&#8217;s equations, the electromagnetic properties observed in the microwave regime are directly applicable to the infrared and optical regimes, where the samples would be scaled to micron and submicron sizes.

<table border="0" cellpadding="0" cellspacing="0" width="599">
	<tbody>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">stereolithography machine</td>
			<td style="width:97px;">3D Systems</td>
			<td style="width:119px;">SLA-7000</td>
			<td style="width:167px;"></td>
		</tr>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">resin for base</td>
			<td style="width:97px;">3D Systems</td>
			<td style="width:119px;">Accura 60</td>
			<td></td>
		</tr>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">Alumina rods</td>
			<td style="width:97px;"></td>
			<td style="width:119px;"></td>
			<td>r=2.5mm, cut to 10.0cm height</td>
		</tr>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">Alumina sheets</td>
			<td style="width:97px;"></td>
			<td style="width:119px;"></td>
			<td>thickness 0.38mm, various width: from 1.0mm to 5.3mm with 0.2mm incerments</td>
		</tr>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">Microwave Generator</td>
			<td style="width:97px;">Agilent/HP</td>
			<td style="width:119px;">83651B</td>
			<td></td>
		</tr>
		<tr height="21">
			<td height="21" style="height:21px;width:216px;">S-Parameter Test set</td>
			<td style="width:97px;">Agilent/HP</td>
			<td style="width:119px;">8517B</td>
			<td></td>
		</tr>
		<tr height="42">
			<td height="42" style="height:42px;width:216px;">Microwave Vector Network Analyzer</td>
			<td style="width:97px;">Agilent/HP</td>
			<td style="width:119px;">8510C</td>
			<td></td>
		</tr>
	</tbody>
</table>

using microwave and macroscopic samples of dielectric solids to study the photonic properties of disordered photonic bandgap materials

Recently, disordered photonic materials have been suggested as an alternative to periodic crystals for the formation of a complete photonic bandgap (PBG). In this article we will describe the methods for constructing and characterizing macroscopic disordered photonic structures using microwaves. The microwave regime offers the most convenient experimental sample size to build and test PBG media. Easily manipulated dielectric lattice components extend flexibility in building various 2D structures on top of pre-printed plastic templates. Once built, the structures could be quickly modified with point and line defects to make freeform waveguides and filters. Testing is done using a widely available Vector Network Analyzer and pairs of microwave horn antennas. Due to the scale invariance property of electromagnetic fields, the results we obtained in the microwave region can be directly applied to infrared and optical regions. Our approach is simple but delivers exciting new insight into the nature of light and disordered matter interaction.

Our representative results include the first experimental demonstration of the existence of a complete and isotropic PBG in a two-dimensional (2D) hyperuniform disordered dielectric structure. Additionally we demonstrate experimentally the ability of this novel photonic structure to guide electromagnetic waves (EM) through freeform waveguides of arbitrary shape.

We have achieved the first confirmation ever of an isotropic complete PBG present in hyperuniform disorder dielectric structures. Here, we present our HD structure results and compare them to that of a periodic square lattice photonic crystal.
Figure 5 shows a semi-log plot of TE polarization transmission (dB) vs. frequency (GHz) for a hyperuniform disorder structure at one incident angle. This plot shows that the stop band region is located approximately between 8.5 and 9.5 G...

Watch this Scientific Journal Video about Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials at JoVE.com

Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials

Disordered structures offer new mechanisms for forming photonic bandgaps and unprecedented freedom in functional-defect designs. To circumvent the computational challenges of disordered systems, we construct modular macroscopic samples of the new class of PBG materials and use microwaves to characterize their scale-invariant photonic properties, in an easy and inexpensive manner.

Recently, disordered photonic materials have been suggested as an alternative to periodic crystals for the formation of a complete photonic bandgap (PBG). ...

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12.2K Views.  San Francisco State University. Disordered structures offer new mechanisms for forming photonic bandgaps and unprecedented freedom in functional-defect designs. To circumvent the computational challenges of disordered systems, we construct modular macroscopic samples of the new class of PBG materials and use microwaves to characterize their scale-invariant photonic properties, in an easy and inexpensive manner.