Geoffroy LEROSEY

Abstract: Reconfigurable intelligent surfaces (RISs) are gaining huge momentum in the field of wireless communications due to the paradigm shift that they bring. Indeed, they allow making any environment electromagnetically smart and dynamically reconfigurable for more efficient and greener wireless communications. As physicists, we proposed to use electronically tunable metasurfaces to shape the electromagnetic waves carrying our wireless communications in reflection almost ten years ago, inspired by some works that we and colleagues did in the field of wave control in complex media. In this article, we review the seminal works that led us to propose this concept, starting from the original one that is time reversal. Then, we propose a physicist's point of view of RISs using a comparison with phase conjugation. Finally, we highlight what we think are their limitations, relying on both our knowledge of wave control and our study of them over a decade.

Abstract: Reverberation chambers are currently used to test electromagnetic compatibility as well as to characterize antenna efficiency, wireless devices, and MIMO systems. The related measurements are based on statistical averages and their fluctuations. We introduce a very efficient mode stirring process based on electronically reconfigurable metasurfaces (ERMs). By locally changing the field boundary conditions, the ERMs allow us to generate a humongous number of uncorrelated field realizations even within small reverberation chambers. We fully experimentally characterize this stirring process by determining these uncorrelated realizations via the autocorrelation function of the transmissions. The uniformity criterion parameter σ dB, as defined in the IEC 61000-4-21 standard, is also investigated and reveals the performance of this stirring. The effect of short paths on the two presented quantities is identified. We compare the experimental results on the uniformity criterion parameter with a corresponding model based on random matrix theory and find good agreement, where the only parameter, the modal overlap, is extracted by the quality factor.

Abstract: © 2020 American Physical Society Wave-chaotic systems underpin a wide range of research activities from fundamental studies of quantum chaos via electromagnetic compatibility up to more recently emerging applications, such as microwave imaging for security screening, antenna characterization, or wave-based analog computation. To implement a wave-chaotic system experimentally, traditionally cavities of elaborate geometries (bow tie shapes, truncated circles, or parallelepipeds with hemispheres) are employed because the geometry dictates the wave field's characteristics. Here, we propose and experimentally verify a conceptually different approach: a cavity of regular geometry but with tunable boundary conditions, experimentally implemented by leveraging a reconfigurable metasurface reflect array. This approach offers an alternative stirring mechanism and enables a fuller study of random matrix theory in connection with wave chaos.

Abstract: © 2020 American Physical Society. Wave propagation control is of fundamental interest in many areas of physics. It can be achieved with wavelength-scaled photonic crystals, hence avoiding low-frequency applications. By contrast, metamaterials are structured on a deep-subwavelength scale, and therefore usually described through homogenization, neglecting the unit-cell structuration. Here, we show with microwaves that, by considering their inherent crystallinity, we can induce wave propagation carrying angular momenta within a subwavelength-scaled collection of wires. Then, inspired by the quantum valley Hall effect in condensed-matter physics, we exploit this bulk circular polarization to create modes propagating along particular interfaces. The latter also carry an edge angular momentum whose conservation during the propagation allows wave routing by design in specific directions. This experimental study not only evidences that crystalline metamaterials are a straightforward tabletop platform to emulate exciting solid-state physics phenomena at the macroscopic scale, but it also opens the door to crystalline polarized subwavelength waveguides.

Abstract: © 2019 IEEE. Metamaterials are artificial media which provide exotic propagation propertiesthanks to their structuration on scales that are much smaller than the free-space wavelength of operation. Consequently, the propagation in these systems is usually described by the use of effective parameters. However, although this description allows to envision metamaterial engineering at a mesoscopic scale, it neglects the effects owing to the spatial patterning at the scale of the unit cell. Yet, it has been shown that the wave physics within a specific class of metamaterial, namely the locally resonant ones, can be accurately explained with the use of band structures [1].

Abstract: © 2019 IEEE. We propose using complex media as platforms to build any linear operator using wavefront shaping. We demonstrate that system composed of multimode fiber and spatial light modulator can act like any linear operator and can be reconfigured at will. We experimentally performed several 16×16-single-shot operations.

Abstract: © 2019, The Author(s). Future wireless networks are expected to constitute a distributed intelligent wireless communications, sensing, and computing platform, which will have the challenging requirement of interconnecting the physical and digital worlds in a seamless and sustainable manner. Currently, two main factors prevent wireless network operators from building such networks: (1) the lack of control of the wireless environment, whose impact on the radio waves cannot be customized, and (2) the current operation of wireless radios, which consume a lot of power because new signals are generated whenever data has to be transmitted. In this paper, we challenge the usual “more data needs more power and emission of radio waves” status quo, and motivate that future wireless networks necessitate a smart radio environment: a transformative wireless concept, where the environmental objects are coated with artificial thin films of electromagnetic and reconfigurable material (that are referred to as reconfigurable intelligent meta-surfaces), which are capable of sensing the environment and of applying customized transformations to the radio waves. Smart radio environments have the potential to provide future wireless networks with uninterrupted wireless connectivity, and with the capability of transmitting data without generating new signals but recycling existing radio waves. We will discuss, in particular, two major types of reconfigurable intelligent meta-surfaces applied to wireless networks. The first type of meta-surfaces will be embedded into, e.g., walls, and will be directly controlled by the wireless network operators via a software controller in order to shape the radio waves for, e.g., improving the network coverage. The second type of meta-surfaces will be embedded into objects, e.g., smart t-shirts with sensors for health monitoring, and will backscatter the radio waves generated by cellular base stations in order to report their sensed data to mobile phones. These functionalities will enable wireless network operators to offer new services without the emission of additional radio waves, but by recycling those already existing for other purposes. This paper overviews the current research efforts on smart radio environments, the enabling technologies to realize them in practice, the need of new communication-theoretic models for their analysis and design, and the long-term and open research issues to be solved towards their massive deployment. In a nutshell, this paper is focused on discussing how the availability of reconfigurable intelligent meta-surfaces will allow wireless network operators to redesign common and well-known network communication paradigms.

Abstract: © 2019, The Author(s), under exclusive licence to Springer Nature Limited. Multichannel wireless systems have become a standard solution to address our information society’s ever-increasing demand for information transfer. The capacity that such systems can achieve is ultimately limited by the channel diversity in a given propagation medium, and numerous approaches to reduce channel cross-talk by engineering software or hardware details of the signals and antenna arrays have been proposed. Here we show that optimal channel diversity can be achieved by physically shaping the propagation medium itself. Using a reconfigurable metasurface placed inside a random environment, we tune the disorder and impose perfect orthogonality of wireless channels. We report experiments in the microwave domain in which we impose equal weights of the channel matrix eigenvalues for up to 4 × 4 systems, and almost equal weights in larger systems. We also demonstrate enhanced wireless image transmission in an office room in which we augmented the 3 × 3 system’s number of effectively independent channels from two to the optimum of three.

Abstract: © 2018 IEEE. Thanks to their particularly efficient, low frequency Minnaert resonance, air bubbles are known to be excellent candidates for the realization of acoustic metamaterials. Here, we demonstrate that the introduction of pair-wise spatial correlations between the bubbles can result in double negativity. This can occur when the bubble pairs are arranged either in random or periodic configurations. Predictions for both types of structure will be presented and the influence of dissipation on the doubly negative behaviour discussed.

Abstract: © 2018, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional. Our goal is to propose an efficient approach to characterize the eigenvalues and eigenfunctions of the Helmholtz equation with mixed (Dirichlet and Neumann) boundary conditions. Our approach is based on layer potentials. We extend the eigenvalue characterization known for Neumann boundary conditions to the case of mixed boundary conditions. The problem is motivated by the need of such methods for real-time wave-field shaping by electronically tunable surfaces.

Abstract: © 2018 The Author(s). Detecting and analysing motion is a key feature of Smart Homes and the connected sensor vision they embrace. At present, most motion sen sors operate in line-of-sight Doppler shift schemes. Here, we propose an alternative approach suitable for indoor environments, which effectively constitute disordered cavities for radio frequency (RF) waves; we exploit the fundamental sensitivity of modes of such cavities to perturbations, caused here by moving objects. We establish experimentally three key features of our proposed system: (i) ability to capture the temporal variations of motion and discern information such as periodicity ("smart"), (ii) non line-of-sight motion detection, and (iii) single-frequency operation. Moreover, we explain theoretically and demonstrate experimentally that the use of dynamic metasurface apertures can substantially enhance the performance of RF motion detection. Potential applications include accurately detecting human presence and monitoring inhabitants' vital signs.

Abstract: © 2017 American Physical Society. Wave-front shaping has emerged over the past decade as a powerful tool to control wave propagation through complex media, initially in optics and more recently also in the microwave domain with important applications in telecommunication, imaging, and energy transfer. The crux of implementing wave-front shaping concepts in real life is often its need for (direct) feedback, requiring access to the target to focus on. Here, we present the shaping of a microwave field based on indirect, unsolicited, and blind feedback which may be the pivotal step towards practical implementations. With the example of a radio-frequency harvester in a metallic cavity, we demonstrate tenfold enhancement of the harvested power by wave-front shaping based on nonlinear signals detected at an arbitrary position away from the harvesting device.

Abstract: © 2017 Optical Society of America. Optical microscopy offers a unique insight of biological structures with a sub-micrometer resolution and a minimum invasiveness. However, the inhomogeneities of the specimen itself can induce multiple scattering of light and optical aberrations which limit the observation to depths close to the surface. To predict quantitatively the penetration depth in microscopy, we theoretically derive the single-to-multiple scattering ratio in reflection. From this key quantity, the multiple scattering limit is deduced for various microscopic imaging techniques such as confocal microscopy, optical coherence tomography and related methods.

Abstract: © 2017 The Author(s). Many efforts have been devoted to wave slowing, as it is essential, for instance, in analog signal computing and is one prerequisite for increased wave/matter interactions. Despite the interest of many communities, researches have mostly been conducted in optics, where wavelength-scaled structured composite media are promising candidates for compact slow light components. Yet their structural scale prevents them from being transposed to lower frequencies. Here, we propose to overcome this limitation using the deep sub-wavelength scale of locally resonant metamaterials. We experimentally show, in the microwave regime, that introducing coupled resonant defects in such metamaterials creates sub-wavelength waveguides in which wave propagation exhibit reduced group velocities. We qualitatively explain the mechanism underlying this slow wave propagation and demonstrate how it can be used to tune the velocity, achieving group indices as high as 227. We conclude by highlighting the three beneficial consequences of our line defect slow wave waveguides: (1) the sub-wavelength scale making it a compact platform for low frequencies (2) the large group indices that together with the extreme field confinement enables efficient wave/matter interactions and (3) the fact that, contrarily to other approaches, slow wave propagation does not occur at the expense of drastic bandwidth reductions.

Abstract: © 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Topological insulators, a hallmark of condensed matter physics, have recently reached the classical realm of acoustic waves. A remarkable property of time-reversal invariant topological insulators is the presence of unidirectional spin-polarized propagation along their edges, a property that could lead to a wealth of new opportunities in the ability to guide and manipulate sound. Here, we demonstrate and study the possibility to induce topologically non-trivial acoustic states at the deep subwavelength scale, in a structured two-dimensional metamaterial composed of Helmholtz resonators. Radically different from previous designs based on non-resonant sonic crystals, our proposal enables robust sound manipulation on a surface along predefined, subwavelength pathways of arbitrary shapes.

Abstract: © 2016 IEEE.We report on the passive measurement of time-dependent Green's functions in the optical frequency domain with low-coherence interferometry. Inspired by previous studies in acoustics and seismology, we show how the mutual coherence function of a broadband and incoherent wave-field can directly yield the Green's functions between scatterers of a complex medium. Both the ballistic and multiple scattering components of the Green's function are retrieved. This simple and powerful approach directly yields a wealth of information about the medium under investigation. In particular, it allows to investigate locally the growth of the diffusive halo within the scattering medium. Local measurements of transport parameters can thus be performed and allow to image a strongly scattering layer with a unprecedented resolution of a few transport mean free paths. This constitutes a major breakthrough compared to state-of-the-art techniques such as optical diffuse tomography.

Abstract: Weintroduce an open microwave cavity that has a wall replaced by a sub-wavelength grating. Usually, sub-wavelength gratings show very low transmission. In our experiment, this phenomenon is compensated by the microwave cavity that finally allows all the energy to be transmitted. We study the far field emission of this system and show that coupling the cavity with a sub-wavelength grating gives rise to a zero order emission only at discrete angles and frequencies. We study the relations between angles of emissions and frequencies, the influence of geometric parameters such as the grating fill factor and the behavior of a chaotic cavity. We show that it allows us to make a configurable system that may have many applications in the fields of communications, detection and imaging, and may allow the study of open microwave cavities on a fundamental point of view. © 2013 IEEE.

Abstract: © 2016 American Physical Society. Materials which possess a high local density of states varying at a subwavelength scale theoretically permit the focusing of waves onto focal spots much smaller than the free space wavelength. To do so, metamaterials - manmade composite media exhibiting properties not available in nature - are usually considered. However, this approach is limited to narrow bandwidths due to their resonant nature. Here, we prove that it is possible to use a fractal resonator alongside time reversal to focus microwaves onto λ/15 subwavelength focal spots from the far field, on extremely wide bandwidths. We first numerically prove that this approach can be realized using a multiple-channel time reversal mirror that utilizes all the degrees of freedom offered by the fractal resonator. Then, we experimentally demonstrate that this approach can be drastically simplified by coupling the fractal resonator to a complex medium, here a cavity, that efficiently converts its spatial degrees of freedom into temporal ones. This makes it possible to achieve deep subwavelength focusing of microwave radiation by time reversing a single channel. Our method can be generalized to other systems coupling complex media and fractal resonators.

Abstract: Locally resonant metamaterials derive their effective properties from hybridization between their resonant unit cells and the incoming wave. This phenomenon is well understood in the case of plane waves that propagate in media where the unit cell respects the symmetry of the incident field. However, in many systems, several modes with orthogonal symmetries can coexist at a given frequency, while the resonant unit cells themselves can have asymmetric scattering cross-sections. In this paper we are interested in the influence of symmetry breaking on the hybridization of a wave field that includes multiple propagative modes. The A 0 and S 0 Lamb waves that propagate in a thin plate are good candidates for this study, as they are either anti-symmetric or symmetric. First we designed an experimental setup with an asymmetric metamaterial made of long rods glued to one side of a metallic plate. We show that the flexural resonances of the rods induce a break of the orthogonality between the A 0/S 0 modes of the free-plate. Finally, based on numerical simulations we show that the orthogonality is preserved in the case of a symmetric metamaterial leading to the presence of two independent polariton curves in the dispersion relation.

Abstract: We report on imaging in random scattering media. Our approach is based on the measurement of a reflection matrix between a spatial light modulator and a camera. We take advantage of the memory effect to filter the multiple scattering noise and improve the detection and imaging of objects embedded in scattering media. © 2014 Optical Society of America.

Abstract: We report on the passive measurement of time-dependent Green's functions in optics with low-coherence interferometry. Inspired by previous studies in acoustics and seismology, we show how the correlations of a broadband and incoherent wave-field can directly yield the Green's functions between scatterers of a complex medium. © 2014 Optical Society of America.

Abstract: © 2015 American Physical Society. We report on the passive measurement of time-dependent Green's functions in the optical frequency domain with low-coherence interferometry. Inspired by previous studies in acoustics and seismology, we show how the correlations of a broadband and incoherent wave field can directly yield the Green's functions between scatterers of a complex medium. Both the ballistic and multiple scattering components of the Green's function are retrieved. This approach opens important perspectives for optical imaging and characterization in complex scattering media.

Abstract: In this article, we investigate composite media which present both a local resonance and a periodic structure. We numerically and experimentally consider the case of a very academic and simplified system that is a quasi-one dimensional split ring resonator medium. We modify its periodicity to shift the position of the Bragg bandgap relative to the local resonance one. We observe that for a well-chosen lattice constant, the local resonance frequency matches the Bragg frequency thus opening a single bandgap which is at the same time very wide and strongly attenuating. We explain this interesting phenomenon by the dispersive nature of the unit cell of the medium, using an analogy with the concept of white light cavities. Our results provide new ways to design wide and efficient bandgap materials.

Abstract: We report on the optical measurement of the backscattering matrix in a weakly scattering medium. A decomposition of the time reversal operator allows selective and efficient focusing on individual scatterers, even through an aberrating layer. © 2012 OSA.

Abstract: In this talk, we show how the concept of hybridization band gap in metamaterials can be utilized to create antennas for MIMO applications. Those strongly decoupled antennas present at the same time a very small form factor and a very low correlation. To that aim, we first explain briefly the concept of hybridization between a resonator and the free space waves continuum. Then we expose the methodology we use to design multi-ports antennas based on that concept. We present results of several antennas designed using this idea, especially in the wifi bands, and give potential solutions for multi-band compact MIMO antennas for LTE applications. © 2012 IEICE.

Abstract: Breaking the diffraction barrier in the visible part of the electromagnetic spectrum is of fundamental importance. Far-field subwavelength focusing of light could, for instance, drastically broaden the possibilities available in nanolithography, light-matter interactions and sensing at the nanoscale. Similarly, imaging with a nanometric resolution could result in incredible breakthroughs in soft matter and biology. There have been numerous proposals in this regard based on metamaterials, structured illumination methods or diffractive optical components. The common denominator of all these approaches resides in their monochromatic nature. Here we show that using polychromatic light in dispersive metamaterials allows us to circumvent many limitations associated with previous monochromatic approaches. We design a plasmonic metalens based on metallic nanorods that, when used with broadband light fields, can beat the diffraction limit for imaging and focusing from the far field. © 2012 Macmillan Publishers Limited. All rights reserved.

Abstract: In complex media such as white paint and biological tissue, light encounters nanoscale refractive-index inhomogeneities that cause multiple scattering. Such scattering is usually seen as an impediment to focusing and imaging. However, scientists have recently used strongly scattering materials to focus, shape and compress waves by controlling the many degrees of freedom in the incident waves. This was first demonstrated in the acoustic and microwave domains using time reversal, and is now being performed in the optical realm using spatial light modulators to address the many thousands of spatial degrees of freedom of light. This approach is being used to investigate phenomena such as optical super-resolution and the time reversal of light, thus opening many new avenues for imaging and focusing in turbid media. © 2012 Macmillan Publishers Limited. All rights reserved.

Abstract: We experimentally measure the monochromatic transmission matrix (TM) of an optical multiple scattering medium using a spatial light modulator together with a phase-shifting interferometry measurement method. The TM contains all the information needed to shape the scattered output field at will or to detect an image through the medium. We confront theory and experiment for these applications and study the effect of noise on the reconstruction method. We also extracted from the TM information about the statistical properties of the medium and the light transport within it. In particular, we are able to isolate the contributions of the memory effect and measure its attenuation length. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

Abstract: This article is the first one in a series of two dealing with the concept of a 'resonant metalens' we introduced recently. Here, we focus on the physics of a medium with finite dimensions consisting of a square lattice of parallel conducting wires arranged on a sub-wavelength scale. This medium supports electromagnetic fields that vary much faster than the operating wavelength. We show that such modes are dispersive due to the finiteness of the medium. Their dispersion relation is established in a simple way, a link with designer plasmons is made, and the canalization phenomenon is reinterpreted in the light of our model. We explain how to take advantage of this dispersion in order to code sub-wavelength wavefields in time. Finally, we show that the resonant nature of the medium ensures an efficient coupling of these modes with free space propagating waves and, thanks to the Purcell effect, with a source placed in the near field of the medium. © 2011 Taylor & Francis.

Abstract: Recently, a method has been proposed by I. Vellekoop et al. [1] to focus light through a multiple scattering material, using a spatial light modulator as a tool to shape the incoming beam to obtain a maximal interference on a speckle spot of the output speckle pattern. The result is a bright, diffraction limited, spot which can be several hundred times brighter than the rest of the speckle. © 2011 IEEE.

Abstract: Optical imaging relies on the ability to illuminate an object, collect and analyse the light it scatters or transmits. Propagation through complex media such as biological tissues was so far believed to degrade the attainable depth, as well as the resolution for imaging, because of multiple scattering. This is why such media are usually considered opaque. Recently, we demonstrated that it is possible to measure the complex mesoscopic optical transmission channels that allow light to traverse through such an opaque medium. Here, we show that we can optimally exploit those channels to coherently transmit and recover an arbitrary image with a high fidelity, independently of the complexity of the propagation. © 2010 Macmillan Publishers Limited. All rights reserved.

Abstract: The theory of monochromatic time-reversal mirrors (TRM) or equivalently phase conjugate mirrors is developed for electromagnetic waves. We start from the fundamental time-symmetry of the Maxwell's equations. From this symmetry, a differential expression similar to the Lorentz reciprocity theorem is deduced. The radiating conditions on TRM are expressed in terms of 6-dimension Green's functions. To predict the time reversal focusing on antenna arrays, a formalism that involves impedance matrix is developed. We show that antenna coupling can dramatically modify the focal spot. Especially, we observe, that in some circumstances, sub-wavelength focusing on a bi-dimensional array may arise. © 2010 IEEE.

Abstract: An experimental validation of high data rate communication for a time reversal (TR) ultra wide-band (UWB) communication system is performed using binary pulse amplitude modulation (BPAM) in two different dense multipath propagation channels for different data rates (15.62-Mbps≤Rb≤1-Gbps). From the measured received signals, signal, interference and noise contributions are separated. At very high data rates, interference has the most dominant contribution of all. Furthermore, without any processing and equalisation at the receiver, bit error rate (BER) performance is compared for different Rb It is shown that for Rb≤125-Mbps, TR system gives a good BER performance. Finally, the authors introduce a modified TR scheme in which total bandwidth of the TR system is divided into Nsub-bands contributing equal power in the power spectral density (PSD). This technique enables a flat PSD of the TR transmitted signal, reduces inter symbol interference (ISI) and therefore improves the BER performance of the system. © 2010 © The Institution of Engineering and Technology.

Abstract: We show that all the spatiotemporal degrees of freedom available in a complex medium can be harnessed and converted into spatial ones. This is demonstrated experimentally through an instantaneous spatial inversion, using broadband ultrasonic waves in a multiple scattering sample. We show theoretically that the inversion convergence is governed by the total number of degrees of freedom available in the medium for a fixed bandwidth and demonstrate experimentally its use for complex media investigation. We believe our approach has potential in sensing, imagery, focusing, and telecommunication. © 2009 The American Physical Society.

Abstract: We employ time reversal for deep subwavelength focusing in plasmonic periodic nanostructures. The strong anisotropy enables propagating modes with very large transverse wave vector and moderate propagation constant, facilitating transformation of diffraction-limited plane waves to high- K Bloch waves in the plasmonic nanostructure. Time reversal is used to excite the waves in the nanostructure at the exact amplitude and phase to focus the incident light to dimensions well below the diffraction limit at any point in the structure, exemplifying a true subdiffractional confinement and resolution. © 2009 The American Physical Society.

Abstract: We present an approach for subwavelength focusing of microwaves using both a time-reversal mirror placed in the far field and a random distribution of scatterers placed in the near field of the focusing point. The far-field time-reversal mirror is used to build the time-reversed wave field, which interacts with the random medium to regenerate not only the propagating waves but also the evanescent waves required to refocus below the diffraction limit. Focal spots as small as one-thirtieth of a wavelength are described. We present one example of an application to telecommunications, which shows enhancement of the information transmission rate by a factor of 3.

Abstract: In this letter, time reversal is applied to wideband electromagnetic waves in a reverberant room. To that end a multiantenna time reversal mirror (TRM) has been built. A 150 MHz bandwidth pulse at a central frequency of 2.45 GHz is radiated by a monopolar antenna, spread in time due to reverberation, recorded at the TRM, time reversed, and retransmitted. The time-reversed wave converges back to its source and focus in both time and space. The time compression is studied versus the number of antennas in the TRM and its bandwidth. The focal spot is also measured thanks to an eight-channel receiving array. © 2006 American Institute of Physics.

Abstract: We present a method to transmit digital information through a highly scattering medium in a MIMO-MU (multiple input multiple output multiple users) context. It is based on iterations of a time-reversal process, and permits us to focus short pulses, both spatially and temporally, from a base antenna to different users. This iterative technique is shown to be more efficient (lower inter-symbol interference and lower error rate) than classical time-reversal communication, while being computationally light and stable. Experiments are presented: digital information is conveyed from 15 transmitters to 15 receivers by ultrasonic waves propagating through a highly scattering slab. From a theoretical point of view, the iterative technique achieves the inverse filter of propagation in the subspace of non-null singular values of the time-reversal operator. We also investigate the influence of external additive noise, and show that the number of iterations can be optimized to give the lowest error rate. © 2004 IOP Publishing Ltd.