Millimeter wave localization: Slow light and enhanced absorption in random dielectric media

We exploit millimeter wave technology to measure the reflection and transmission response of random dielectric media. Our samples are easily constructed from random stacks of identical subwavelength quartz and Teflon wafers. The measurement allows us to observe the characteristic transmission resonances associated with localization. We show that these resonances give rise to enhanced attenuation even though the attenuation of homogeneous quartz and Teflon is quite low. We provide experimental evidence of disorder-induced slow light and superluminal group velocities, which, in contrast to photonic crystals, are not associated with any periodicity in the system. Furthermore, we observe localization even though the sample is only about four times the localization length, interpreting our data in terms of an effective cavity model. An algorithm for the retrieval of the internal parameters of random samples (localization length and average absorption rate) from the external measurements of the reflection and transmission coefficients is presented and applied to a particular random sample. The retrieved value of the absorption is in agreement with the directly measured value within the accuracy of the experiment.

Figure 1

Experimental setup. The vector network analyzer (VNA) generates centimeter waves which pass through a harmonic multiplier (HM), converting them to millimeter waves. Other components include isolators (I), a circulator (C), polyethylene lenses (L), harmonic detectors (HD), scalar (corrugated) horns (SH), and two phase-locked local oscillators (LO). The thick lines refer to coaxial cables, which are only used for the cm waves. The internal frequency synthesizer is locked to a rubidium clock, allowing for long-term stability. Finally, the vector receiver in the VNA down-converts the time-varying $E$ fields (Tx for transmitted and Rx for reflected) to a low frequency for digitization and display. The sample (PC) is at the focal plane of the transmitting horn. The thick black lines connecting the VNA and the millimeter wave components represent high-frequency microwave coaxial cables. Between the harmonic mixer/detectors and the scalar horn antennae, the millimeter waves propagate via waveguide.

Figure 2

Measured (top) and calculated (bottom) transmission and reflection response for a randomly layered sequence of 100 quartz and Teflon disks. Plotted are the transmission coefficients ($T$, dotted curves) and one minus the reflection coefficient ($1-R$, solid curves).

Figure 3

Shown is the solution to the transcendental Eq. (26), where $\lambda =l_{\mathrm{cav}}∕2l_{\mathrm{loc}}$ and $a$ is given in Eq. (28).

Figure 4

(Color) Localized $E$ field within the sample. Shown is a plot of the calculated $E$-field intensity (in dB) within the scattering medium. By comparing the frequencies with those in Fig. 2 one can see the strong localization of the field at transmission resonances.

Figure 5

Localized modes at regions of enhanced absorption. $E$-field intensity as a function of space inside the sample at four resonant frequencies used in the analysis in Table . Localization results in a varying degrees of field enhancement inside the random effective cavity. The stronger the field enhancement, the greater the path-length-induced absorption.

Figure 6

In order to interpret the fast and slow-light features of our measurement, we zoom in on a typical transmission resonance. For this purpose we could choose any of the resonances but here we zoom in around $97\mathrm{GHz}$. We show the phase (dashed curve) and amplitude (solid curve) of the $E$-field transmitted through the same sample as in Figs. 2, 4, as a function of frequency $f$ for the transmission resonance at $97\mathrm{GHz}$. In band gaps the phase is nearly stationary while at resonance the inverse phase derivative, ${[d\varphi \left(f\right)∕df]}^{-1}$ decreases markedly.