There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Wave propagation and group velocity

Modern digital cameras employ silicon focal plane array (FPA) image sensors featuring millions of pixels. However, it is possible to make a camera that only needs one pixel. In these cameras a spatial light modulator, placed before or after the object to be imaged, applies a time-varying pattern and synchronized intensity measurements are made with a single-pixel detector. The principle of compressed sensing then allows an image to be generated. As the approach suits a wide a variety of detector technologies, images can be collected at wavelengths outside the reach of FPA technology or at high frame rates or in three dimensions. Promising applications include the visualization of hazardous gas leaks and 3D situation awareness for autonomous vehicles. Rather than requiring millions of pixels, it is possible to make a camera that only needs one pixel. This Review details the working principle, advantages, technical considerations and future potential of single-pixel imaging.

Principles and prospects for single-pixel imaging

ERS SAR amplitude images are utilized to map ground displacements from a sub-pixel correlation method. It yields a ground two-dimensional displacement field with independent measurements about every 128m in azimuth and 250m in range. The accuracy depends on the characteristics of the images. For the Landers test case, the 1-σ uncertainty is 0.8m in range and 0.4m in azimuth. We show that this measurement provides a map of major surface fault ruptures accurate to better than 1km and information on coseismic deformation comparable to the 92 GPS measurements available. Although less accurate, this technique is more robust than SAR interferometry and provides complementary information since interferograms are only sensitive to the displacement in range.

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Measuring ground displacements from SAR amplitude images: Application to the Landers Earthquake

Enhancement of light extraction from light emitting diodes

In this article we present a novel approach for determining the electromagnetic modes of photonic multilayer structures. We combine the plane wave expansion method with the method of lines resulting in a fast and accurate computational technique which we named the plane wave admittance method. In addition, we incorporate perfectly matched layers at the boundaries parallel to the multilayer surfaces which allow for easy determination of leaky modes. The convergence of the method is verified for the case of photonic crystal slab showing very good agreement with the results obtained with full three-dimensional plane wave expansion method while the numerical effort is largely reduced. The numerical implementation of the method will be soon available on the web.

PlaneWave Admittance Method — a novel approach for determining the electromagnetic modes in photonic structures

We report on experimental evidence of unidirectional transmission of terahertz waves through a pair of metallic gratings with different periods. The gratings are optimized for a broadband transmission in one direction, accompanied with a high extinction rate in the opposite direction. In contrast to previous studies, we show that the zero-order nonreciprocity cannot be achieved. Nonetheless, we confirm that the structure can be used successfully as an asymmetric filter.

Asymmetric transmission of terahertz radiation through a double grating.

We present a closed-form image reconstruction method for single-pixel imaging based on the generalized inverse of the measurement matrix. Its numerical cost scales proportionally with the number of measured samples. Regularization of the inverse problem is obtained by minimizing the norms of the convolution between the reconstructed image and a set of spatial filters. The final reconstruction formula can be expressed in terms of matrix pseudoinverse. At high compression, this approach is an interesting alternative to the methods of compressive sensing based on l1-norm optimization, which are too slow for real-time applications. For instance, we demonstrate experimental single-pixel detection with real-time reconstruction obtained in parallel with measurement at a frame rate of 11 Hz for highly compressive measurements with a resolution of 256 × 256. To this end, we preselect the sampling functions to match the average spectrum obtained with an image database. The sampling functions are selected from the Walsh-Hadamard basis, from the discrete cosine basis, or from a subset of Morlet wavelets convolved with white noise. We show that by incorporating the quadratic criterion into the closed-form reconstruction formula, we can use binary rather than continuous sampling and reach similar reconstruction quality as is obtained by minimizing the total variation. This makes it possible to use cosine- or Morlet-based sampling with digital micromirror devices without advanced binarization methods.

Real-time single-pixel video imaging with Fourier domain regularization.

We demonstrate numerically the diffraction-free propagation of sub-wavelength sized optical beams through simple elements built of metal-dielectric multilayers. The proposed metamaterial consists of silver and a high refractive index dielectric, and is designed using the effective medium theory as strongly anisotropic and impedance matched to air. Further it is characterised with the transfer matrix method, and investigated with FDTD. The diffraction-free behaviour is verified by the analysis of FWHM of PSF in the function of the number of periods. Small reflections, small attenuation, and reduced Fabry–Perot resonances make it a flexible diffraction-free material for arbitrarily shaped optical planar elements with sizes of the order of one wavelength.

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Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers

We compare the properties of subwavelength imaging in the visible wavelength range for metal-dielectric multilayers operating in the canalization and the resonant tunnelling regimes. The analysis is based on the transfer matrix method and time domain simulations. We show that Point Spread Functions for the first two resonances in the canalization regime are approximately Gaussian in shape. Material losses suppress transmission for higher resonances, regularise the PSF but do not compromise the resolution. In the resonant tunnelling regime, the MTF may dramatically vary in their phase dependence. Resulting PSF may have a sub-wavelength thickness as well as may be broad with multiple maxima and a rapid phase modulation. We show that the width of PSF may be reduced by further propagation in free space, and we provide arguments to explain this surprising observation.

Rafal Kotynski

Papers

PlaneWave Admittance Method — a novel approach for determining the electromagnetic modes in photonic structures

Asymmetric transmission of terahertz radiation through a double grating.

Real-time single-pixel video imaging with Fourier domain regularization.

Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers

Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes