Full-field broadband invisibility through reversible wave frequency-spectrum control: supplementary material

This document provides supplementary information to “ Full-field broadband invisibility through reversible wave frequency-spectrum control ,” https://doi.org/10.1364/OPTICA.5.0 0077 9 . The contents of this document include a brief description of the time-frequency duality of the Talbot effect (which is the basis of the reported work), a description of the signal metrics and characterization methods used in the reported work, as well as five supplementary figures. The results suggest that the cloaking bandwidth could be improved through the use of phase modulation sequences closer in shape to the ideal Talbot pattern.

The temporal and spectral phases involved in the Talbot effect are the key pieces of the proposed energy-preserving wavespectrum control method and spectral invisibility cloak. The relationship between such temporal and spectral phases was recently formalized [1], and it is known as the time-frequency duality of the Talbot effect.
Temporal Talbot self-imaging of a pulse train with period t r (corresponding to a frequency comb with free spectral range, FSR, ν r = t −1 r ) is achieved by propagation through a groupvelocity dispersive medium with a predominantly second-order dispersion (linear group delay as a function of frequency) that satisfies the following condition [1, 2], where β 2 is the second-order dispersion coefficient of the medium, i.e., the amount of second-order dispersion per unit length of the medium [3], z is the propagation length through the medium, and p and m are two mutually prime natural numbers (p = 1 in the reported experiments). After propagation, the comb FSR remains unaltered, but the k-th comb line acquires a phase given by, where σ is the sign of the parameter β 2 (σ = 1 if β 2 > 0, and σ = −1 if β 2 < 0). Under these conditions, the pulses of the corresponding temporal waveform are recovered without distortion at the output of the dispersive medium, but at a repetition rate m times faster than the input train (output pulse period of m −1 t r ). Furthermore, n-th output pulse of the obtained sequence acquires a phase given by [1], where s is a natural number, mutually prime with m, so that [4], where m is the parity of the parameter m ( m = 0 if m is even, and m = 1 if m is odd). The proposed spectral invisibility cloaking method relies on calculating the temporal Talbot phase associated to a specific Talbot condition for a given value of m, and cancelling it through the use of a phase modulation mechanism (details in Fig. S1a). This results in an increase of the comb FSR by the pulse rate multiplication factor m, or, more generally, in the reported formation of frequency gaps.
As shown above, the sign of the required temporal phase modulation sequence is determined by the sign of the secondorder dispersion coefficient, β 2 . We note that in the case of illu-mination by a wave with a periodic frequency comb spectrum (see S1a), the sign of the dispersion can be chosen arbitrarily and independently in both the input and output sections of the cloaking device (as long as the sign of the required temporal phase sequence is selected accordingly to each dispersive section). This property does not apply, however, to the case of illumination by a purely continuous spectrum (see S1b), where the signs of the dispersive sections at the input and output of the cloaking device, and the corresponding temporal phase sequences, must be opposite. We recall that the expressions of the spectral and temporal phases, required to produce spectral cloaking, are the exact same in the cases of a comb-like spectrum (several pulses interfering in the cloak) and a continuous spectrum (different frequency components of a single pulse interfering in the cloak), with the only difference that in the second, more general case, ν r is a free parameter, completely independent of the illumination wave's characteristics.

MULTI-LEVEL PERIODIC TALBOT PHASE SE-QUENCES
Although the reported demonstration uses a single RF tone driving signal to implement the phase modulation operations, it is important to note that this corresponds to a first-order approximation to the ideal temporal Talbot phase pattern with m = 2 (see Fig. S2b). The ideal phase modulation profiles resulting from Talbot conditions with arbitrary m factors are periodic sequences of phase steps. From a practical viewpoint, these can be implemented by an electronic arbitrary waveform generator. Fig. S3 shows a set of numerical simulations illustrating the formation of frequency gaps for different values of m, and the associated temporal Talbot phase modulation sequences, satisfying Eq. S3.
Additionally, Fig. S2c shows a comparison between the expected power spectra of the frequency gaps obtained with the prescribed ideal Talbot phase for m = 2, and the single-tone approximation used in the reported experiments. The results suggest that the cloaking bandwidth could be improved through the use of phase modulation sequences closer in shape to the ideal Talbot patterns.

CROSS-CORRELATION COEFFICIENT
The cross-correlation coefficient is a widely-employed metric for quantitative comparison of real-valued signals (used here to quantify the similitude between the illumination wave and the waves observed at the output of the cloaking device). It corresponds to the zero-lag sample of the cross-correlation between two signals, normalized to the zero-lag sample of the autocorrelations of each signal. For two real-valued signals x(t) and y(t), the definition of the cross-correlation coefficient, r x,y , is, This coefficient takes values between −1 and 1. Two realvalued signals satisfying x(t) = y(t) yield a cross-correlation coefficient r x,y = 1, while the value of the coefficient becomes −1 when x(t) = −y(t). If the two signals are real-valued and positive, the cross-correlation coefficient is defined between 0 and 1. The closer this coefficient is to 0, the more dissimilar the signals x(t) and y(t) are. The similarity between the signals x(t) and y(t) is then higher the closer the value of r x,y is to 1.

CHARACTERIZATION OF THE COMPLEX TEMPORAL ENVELOPE
The indirect reconstruction of the complex envelope of the involved pulses is achieved through spectral measurements, using a frequency-domain version of the PROUD technique (phase reconstruction through optical ultrafast differentiation), referred to as SPROUD (spectral phase reconstruction through optical ultrafast differentiation) [5].
The method reconstructs the complex temporal envelope (including both amplitude and phase profiles) of a signal under test (SUT), x(t) = |x(t)|e i x(t) , where i is the imaginary unit and |x(t)| and x(t) are the temporal amplitude and phase profiles of the SUT, respectively. This reconstruction requires the measurement of two power spectra: |X(ω)| 2 , the power spectrum of the SUT, (where ω = 2πν is the radial frequency variable, measured in rad), and |Y(ω)| 2 , the power spectrum resulting from modulating the amplitude of x(t) with a linear monotonic function of time. Following the properties of the Fourier transform, the result of such a modulation translates into a differentiation of the Fourier spectrum of the SUT, X(ω). In the reported experiments, this spectral differentiation was achieved through amplitude temporal modulation of the incoming optical waveform (SUT) in a LiNbO 3 Mach-Zehnder intensity modulator (MZM). The electronic modulation driving signal was a sinusoidal function, slowly-varying as compared to the temporal envelope of the SUT, where the required linear monotonic function was approximated by the rising slope of the sinusoidal cycle. For this purpose, the MZM was biased at the quadrature point (i.e., in the linear operation region, half way between the maximum and minimum transmission points). The group-delay frequency distribution of X(ω) (the derivative of the phase of X(ω) with respect to radial frequency) can be numerically reconstructed by means of the following equation, where |X(ω)| 2 and |Y(ω)| 2 are the measured power spectra of the SUT and the optical signal at the output of the MZM modulator, and the derivative of |X(ω)| on the right-hand side of Eq. S6 is performed numerically on the measured input amplitude spectrum. The parameters A, Ω and T 0 are associated to the temporal amplitude modulation process: A is the half amplitude of the RF tone referred to the half-wave voltage of the MZM, Ω is the frequency of the RF tone, and T 0 is the maximum throughput of the MZM. The values of these parameters in the reported experiment are: AΩ ≈ 7.8 ns −1 and T 0 ≈ 1. The spectral phase profile of the SUT, X(ω), is then obtained (except for an additive constant term) by numerical integration of the calculated group delay profile. The time-domain waveform of the SUT is finally reconstructed by simply calculating the inverse Fourier transform of the measured spectral amplitude, |X(ω)|, with the spectral phase profile, X(ω), calculated numerically, such that X(ω) = |X(ω)|e i X(ω) .   The designed spectral transformations introduce frequency gaps in the complex envelope of the spectrum, regardless of the pulse repetition rate. The design parameters correspond to the same implementation described in the Main text, with ν r = 19 GHz and m = 2.