Subwavelength Photonic Band Gaps from Planar Fractals

We show by both experiment and theory that a specific class of planar conducting fractals possesses a series of self-similar resonances, leading to multiple gaps and pass bands for electromagnetic waves over an ultrawide frequency range. A double stack of these fractal patterns exhibits polarization-independent absolute gaps over a wide range of incidence angles. These characteristics are retained even when the fractal patterns are significantly subwavelength in all dimensions. In addition, the transmittance can be modulated by an external current source.

Figure 1

(a) A computer-generated image of a part of the fractal structure. The white lines corresponds to copper in the real sample. (b) Measured (circles) and calculated (lines) transmittance; and (c) reflectance of a 15-level fractal pattern for electromagnetic waves polarized with the $E$ vector along the $y$ axis.

Figure 2

Signals as a function of frequency (horizontal axis) photographed from the scope of the Spectral Analyzer. Panel (a) shows the amplitude of the transmitted microwave through the fractal due to the primary source. When the secondary current source was turned on and adjusted to be the same frequency of 2.13 GHz, the measured transmission was amplified (b) when they were in phase and diminished (c) when out of phase. The inset on the left is a schematic of the experimental setup. The inset on the right shows FDTD simulations of the transmittance modulation of a $y$-polarized 6.8 GHz microwave through the 7-level fractal plate, by a central-fed ac current of the same frequency. The magnitude of the modulation is governed by the phase difference (time delay) between the incident microwave and the central-fed current.

Figure 3

Transmission through two stacked identical fractal plates with a 90° rotation with respect to each other. The diamonds and squares denote measured normal transmission through the double stack, with $\overrightarrow{E}$ polarized along the $x$ and $y$-axes, respectively, showing the gaps to be polarization independent. The lines are calculated results for normal incidence (identical for both polarizations). The circles and triangles are transmissions measured with the plates rotated ($\theta$) and tilted ($\varphi$) by $30°$, respectively, from the normal.

Figure 4

The calculated $\mathrm{\text{gap}}/\mathrm{\text{midgap}}$ ratio as a function of the number of stacking layers for three spectral gaps at 1.6 (square), 4.4 (circle), and 13 (triangle) GHz. The dielectric substrates are 1 mm thick, separated by an air gap of 1 mm.