Coherent absorption reveals colors hidden in a grey Fabry-Pérot cavity

TL;DR

This work introduces a conductor-dielectric-conductor (CDC) grey cavity that yields spectrally flat single-beam optical observables while preserving color information in a coherently absorbed, phase-sensitive double-beam signal. The key relation $A_{db} = A_{sb}\left(1 - \cos(kd)\cos(\phi)\right)$ shows how phase and thickness control absorption without altering ordinary spectra. The authors implement a practical version using silicon nitride membranes with a Cr/Au bilayer, achieving near-ideal matching and demonstrating coherent absorption at $633~\text{nm}$ with phase shifts $\Delta \approx 2kd$. They further show a cryptographic application by encoding a color image in thickness variations and reconstructing it via coherent absorption, highlighting potential for secure, planar photonic information encoding and PUFs.

Abstract

Thin dielectric films are known to show distinct colors, responsible for the iridescence of various natural and artificial objects such as insect wings and soap bubbles. In the present article we show that a specialized thin film Fabry-Pérot resonator, that we name conductor-dielectric-conductor (CDC) matched cavity, appears instead completely grey when observed under ordinary conditions (i.e. by analyzing the transmitted or reflected incoherent white light). Nonetheless, the matched CDC cavity still retains spectral information, that clearly appear when the cavity is analyzed by the coherent absorption technique. The CDC system is simply a dielectric thin film sandwiched by two conducting interfaces, whose conductivity shall be appropriately matched depending upon the dielectric refractive index. In practical applications, the ideal conducting interfaces can be safely replaced by thin metal films, making the system easily applicable in particular for cryptographic purposes. We indeed demonstrated experimentally that a visually recognizable thin-film color pattern can be concealed to an ordinary observer, and finally recovered through a dedicated coherent absorption decoding apparatus.

Figures (11)

• Figure 1: Concept of the conductor-dielectric-conductor (CDC) grey cavity - A lossless dielectric layer sandwiched between conducting interfaces having a matched conductivity (a) exhibits flat single-beam reflectance $R$, transmittance $T$ and absorptance $A_{sb}$ spectra. Here, with "flatness" we mean that the ordinary spectra are independent of either wavelength (entering the wavevector $k = 2\pi n/ \lambda_0$) and thickness $d$. When considering a double-beam experiment, where coherent light is impinging on both sides of the cavity (c), the double-beam absorptance $A_{db}$ is modulated by the input beams' relative phase $\phi$ (d). Importantly, such modulation can be zero (for $kd = (2m+1)\pi/2$, with $m$ integer), or maximum (for $kd = m\pi$). Such maximum can reach unity when $2 A_{sb} = 1$, i.e. when $n = 2$ (see also panel (b)).
• Figure 2: Single and double beam absorptance of matched and mismatched CDC cavities - Maximum, minimum, and average double-beam spectral absorptance, calculated with respect to the input dephasing $\phi$, for several combinations of dielectric refractive index $n$ and interface conductivity $\sigma$. The average double-beam absorptance always coincides with the single-beam absorptance. Top row: spectra for the matched cavity ($\sigma = \sigma_M$), for different values of the dielectric refractive index. In all cases $\max\, / \min\, A_{db}$ are sinusoids and $\left\langle A_{db} \right\rangle = A_{sb}$ is constant. The case $\sigma = \sigma_M$, $n = 2$ represents a special case, where $\max\, A_{db}$ can reach unity. Middle row: absorptance spectra for $n = 2$ and variable, real $\sigma$. In general, $\max\, / \min\, A_{db}$ and $\left\langle A_{db} \right\rangle = A_{sb}$ are non-sinusoidal, oscillating functions. Bottom row: absorptance spectra for $n = 2$ and variable $\sigma$ where a spurious imaginary component is included. In general, $\max\, / \min\, A_{db}$ and $\left\langle A_{db} \right\rangle = A_{sb}$ are non-sinusoidal, oscillating functions, with warped lineshapes.
• Figure 3: Single- and double-beam measurements on variable-thickness CDC cavities - (a) Schematic of the investigated samples, that are conductor-dielectric-conductor cavities where the ideal, zero-thickness conductor is implemented by a thin metallic bilayer. (b) Photograph of an array of CDC membrane samples having different dielectric thickness, backside-illuminated with white light. All membranes appear identical (the top-right membrane is missing, leaving a hole as a blank reference). (c) Photograph of the same CDC membrane array, illuminated from the frontside with white light. The CDC samples show weakly saturated colors, with variable hues. A black frame is artificially introduced to better recognize the intrinsic colors, see also Extended Data Fig. 1. (d)-(f) Reflectance, transmittance, and single-beam (i.e. ordinary) absorptance spectra collected from the CDC sample array shown in (b)-(c). The quantitative spectra confirm the qualitative perceptual colors observed in the photographs. (g) Output light intensity observed when samples of different thickness are illuminated from two sides with coherent, dephased beams (as in Fig. 1c). Traces $I$ and $II$ refer, respectively, to output light intensity $I_{\mathrm{out}}^{I}$ and $I_{\mathrm{out}}^{II}$, detected on either sides of the membranes (cfr. Fig. 1c). The total output intensity ($I+II$) can be weakly or strongly modulated, depending on the dielectric thickness. (h) Maximum, average, and minimum (with respect to the dephasing $\phi$) double-beam absorptance as a function of the dielectric thickness. (i) Dependence of the phase $\Delta$ (see (g) for the definition) upon the dielectric thickness. In (h)-(i), points are experimental data and lines are full-wave simulations.
• Figure 4: Cryptography based on the colors hidden in CDC grey cavities - (a) Transmission photograph of a silicon nitride membrane where different regions have different thicknesses, resulting in a "smiley" image caused by interference colors. (b) Transmission photograph of the same membrane, coated on both sides with the metallic bilayer of Fig. 3a, resulting in CDC grey cavities where the structural colors are strongly suppressed. The colorbars of (a) and (b) are obtained by converting the simulated transmittance spectra to rgb values using the CIE1931 model with D65 illuminant and gamma correction. (c) Image of the "smiley" reconstructed through coherent absorption at fixed wavelength ($633\ \mathrm{nm}$). Here, the colors represent the phase $\Delta$ measured locally with the approach of Fig. 1c and Fig. 3g-i. The synthetic colors of (c) resemble the hue of the natural colors of (a), despite the metallized CDC sample being grey in ordinary, single-beam transmission. The size of the imaging window is $5 \times 5\ \mathrm{mm}$.
• Figure 5: Schematic of the conductor-dielectric-conductor (CDC) cavity.