An all-optical technique enables instantaneous single-shot demodulation of images at high frequency

High-frequency demodulation of wide area optical signals in a snapshot manner remains a technological challenge. If solved, it could open tremendous perspectives in 3D imaging, vibrometry, free-space communications, automated vision, or ballistic photon imaging in scattering media with numerous applications in smart autonomous vehicles and medical diagnosis. We present here a snapshot quadrature demodulation imaging technique, capable of estimating the amplitude and phase from a single acquisition, without synchronization of emitter and receiver, and with the added capability of continuous frequency tuning. This all-optical optimized setup comprises an electro-optic crystal acting as a fast sinusoidal optical transmission gate, and allows four quadrature image channels to be recorded simultaneously with any conventional camera. We report the design, experimental validation and examples of applications of such wide-field quadrature demodulating system that allowed snapshot demodulation of images with good spatial resolution and continuous frequency selectivity up to a few 100s of kilohertz.


Supplementary
: Flowchart of the frames processing involved in FAST-QUAD. The correction procedure and the quadrature mismatch correction algorithm allow intensityÎ, amplitudeÂ and phaseφ maps to be retrieved in a snapshot way from a single frame acquisition.
As sketched in Supplementary Figure 1, the corrected quadrature signals I and Q eventually allow the estimatd amplitude mapÂ to be obtained after a final correction step, which is required to compensate for the possible spatial inhomogeneity of the demodulation efficiency across the FOV. Similarly, the estimated phase mapφ is retrieved from the I and Q signals, after a phase unwrapping step to compensate for the inhomogeneous phase distribution imprinted by the isogyre pattern (see Methods). The inhomogeneity map and the phase unwrapping pattern are also calibrated from initial measurements on a homogeneous scene. The calibration procedure is not an easy task. Nevertheless, it has in principle to be performed once for a given imager. All experimental results presented in the manuscript have been obtained after a single, common, calibration procedure.
Due to experimental imperfections, the raw quadrature images obtained from subtraction of two image channels, respectivelyĨ = I 1 − I 2 andQ = Q 1 − Q 2 , have small deviation from being in quadrature phase. Moreover, a residual amplitude mismatch was also observed between the two raw quadrature imagesĨ andQ. Such differences are clearly visible on the raw quadrature data displayed in Supplementary Figure 2.a below, where the temporal evolution of the quadrature signalsĨ andQ are plotted as a function of time for a reference pixel (denoted with symbol × in Figure 5 of the main article), while applying a slowly varying (0.1 Hz) voltage ramp on the EO crystal (with exposure time 350 ms, sampling period 400 ms on an homogeneous scene).
Supplementary Figure 2: Illustration of the quadrature mismatch correction algorithm. a Original quadrature signals (Ĩ in red,Q in blue) of the reference pixel (marked with symbol × in Figure 5.b of the main article) as acquired by the camera over 200 frames (350 ms exposure time, 400 ms sampling period) while EO crystal voltage is slowly varied (0.1 Hz). The I-Q representation (green dots) shows clear amplitude mismatch, and slight deviation from 90 • phase (quadrature) between I and Q signals. b I and Q quadrature signals after correction showing equal amplitudes and perfect quadrature angle.
The experimental data in this case can be modeled by writing the two quadrature transmission functions at each pixel (i, j) of the FOV as where α (i, j) accounts for the amplitude mismatch between the two quadratures at pixel (i, j), whereas δφ (i, j) stands for phase deviation from perfect quadrature between TĨ (i, j) and TQ (i, j) . It can be noted that a pixel-dependent common phase factor φ (i, j) has been included in the above equation, which models the inhomogeneous phase distribution across the FOV, due to the isogyre pattern. As mentioned in Supplementary Note 1, this phase distribution is compensated in the last step of the frames processing during the phase unwrapping step to provide a correct estimated phase map. As a result, for the sake of clarity in the following description of the quadrature mismatch calibration and correction procedure, we will set φ (i, j) = 0 without loss of generality.
The objective of the quadrature mismatch calibration described below is to estimate the amplitude mismatch map α, as well as the phase quadrature mismatch δφ across the FOV, i.e., at each pixel (i, j). Several calibration methods exist for correcting the intensity and phase of a quadrature demodulator, especially for their application in RADARs [2,3]. We estimated the α and δφ maps from calibration, using a time-series image acquisition of N = 200 raw frames, such as the one presented here in Supplementary Figure 2.a for the reference pixel marked with a red cross symbol in Figure 5.b of the main article. The N = 200 images temporally sample the amplitude response of the two quadrature imagesĨ andQ over approximately 8 periods, with a spatially homogeneous constant (unmodulated) illumination on the FAST-QUAD prototype, while the voltage applied on the EO crystal was slowly varied (f d = 0.1 Hz). Indeed, at a given pixel (i, j) of the image, the transmission can be written as a N × 2 matrix whose covariance matrix yields showing that α (i, j) and δφ (i, j) can be easily estimated from the calibration data. It can be noted that in the ideal case, the covariance matrix should be equal to the identity matrix, denoting perfect amplitude balancing and exact quadrature phase. Thus, one can estimate a scaling and a rotation matrix at each pixel (i, j) that can be applied to the observed data X (i, j) = [Ĩ (i, j)Q(i, j) ] to balance the amplitudes and obtain perfect quadrature signals. To calibrate these correction matrices at each pixel, we follow a well-known method based on singular values decomposition (SVD) and similar to [3]. Briefly, the SVD of matrix , and one can easily extract the 2 × 2 real unitary matrix B T (i, j) , as well as the two first singular values stored in the 2 × 2 diagonal matrix Σ (i, j) (which is the 2 × 2 sub-matrix of the N × 2 matrix Σ (i, j) ). As shown in [3], it suffices to apply a correction matrix to the observed quadrature data X (i, j) to obtain corrected data X (i, j) R (i, j) . The efficiency of this correction algorithm can be checked in Supplementary Figure 2.b where the corrected quadratures I and Q are displayed, showing equalized amplitude and perfect quadrature phase.

SUPPLEMENTARY NOTE 3: DESCRIPTION OF THE SCENES BEING IMAGED
The scenes that were being imaged were illuminated with a 532 nm (Coherent Verdi) green laser. Acousto-optic modulators (AOM) (AA Opto-Electronic, MT80-A1-VIS) were used to obtain intensity-modulated beams. Indeed, by supplying the AOMs with RF signals whose amplitude was modulated at frequency f , the light diffracted in the first diffraction order shows an intensity modulation at frequency f with modulation index m 100 %. A sketch of the illumination setup is given in Supplementary Figure 3 below. Direct modulation of the LEDs or laser light sources can be preferentially envisaged in future developments and applications of FAST-QUAD. To obtain an image of an homogeneously illuminated scene, the modulated beam was directed into an 8 inches diameter integrating sphere (Labsphere CSTM-US-800C-100R) which created a uniform illumination field across its main circular aperture of 5 cm diameter. Note that during the experiments the sphere was mechanically connected to an electrodynamic shaker (with frequency far from the demodulation frequency, typ. ∼ 100 Hz) which created small vibrations of the sphere, thereby limiting the detrimental effect of speckle on the acquired images. To produce the images displayed in Figure 2 of the main article, a 4 cm mask of the logo of the Institut Foton was printed on a transparency film and positioned in front of the aperture of the integrating sphere. For the last experiments presented in Figure 4 of the main article, the two collimated laser spots (modulated at distinct frequencies with two AOMs) directly illuminated a rotating white diffuser (paper) used in place of the integrating sphere to avoid speckle (Figure 4.a). In Figure 4.b, the collimated laser beams were used to illuminate white-dotted images of the lock (unmodulated) and key (modulation frequency 5 kHz), both printed on a transparency film. Two lenses were finally used to superimpose the final images of the lock and key on the rotating diffuser and create the scene of Figure 4.b. This optical projection arrangement was subject to some geometrical aberrations in the projection lenses, hence resulting in apparent lower resolution of the images in Figure 4.b of the main article.