The enigma of imaging in the Maxwell fisheye medium
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The resolution of optical instruments is normally limited by the wave nature of light. Circumventing this limit, known as the diffraction limit of imaging, is of tremendous practical importance for modern science and technology. One method, super-resolved fluorescence microscopy was distinguished with the Nobel Prize in Chemistry in 2014, but there is plenty of room for alternatives and complementary methods such as the pioneering work of Prof. J. Pendry on the perfect lens based on negative refraction that started the entire research area of metamaterials. In this thesis, we have used analytical techniques to solve several important challenges that have risen in the discussion of the microwave experimental demonstration of absolute optical instruments and the controversy surrounding perfect imaging. Attempts to overcome or circumvent Abbe’s diffraction limit of optical imaging, have traditionally been greeted with controversy. In this thesis, we have investigated the role of interacting sources and detectors in perfect imaging. We have established limitations and prospects that arise from interactions and resonances inside the lens. The crucial role of detection becomes clear in Feynman’s argument against the diffraction limit: “as Maxwell’s electromagnetism is invariant upon time reversal, the electromagnetic wave emitted from a point source may be reversed and focused into a point with point-like precision, not limited by diffraction.” However, for this, the entire emission process must be reversed, including the source: A point drain must sit at the focal position, in place of the point source, otherwise, without getting absorbed at the detector, the focused wave will rebound and the superposition of the focusing and the rebounding wave will produce a diffraction-limited spot. The time-reversed source, the drain, is the detector which taking the image of the source. In 2011-2012, experiments with microwaves have confirmed the role of detection in perfect focusing. The emitted radiation was actively time-reversed and focused back at the point of emission, where, the time-reversed of the source sits. Absorption in the drain localizes the radiation with a precision much better than the diffraction limit. Absolute optical instruments may perform the time reversal of the field with perfectly passive materials and send the reversed wave to a different spatial position than the source. Perfect imaging with absolute optical instruments is defected by a restriction: so far it has only worked for a single–source single–drain configuration and near the resonance frequencies of the device. In chapters 6 and 7 of the thesis, we have investigated the imaging properties of mutually interacting detectors. We found that an array of detectors can image a point source with arbitrary precision. However, for this, the radiation has to be at resonance. Our analysis has become possible thanks to a theoretical model for mutually interacting sources and drains we developed after considerable work and several failed attempts. Modelling such sources and drains analytically had been a major unsolved problem, full numerical simulations have been difficult due to the large difference in the scales involved (the field localization near the sources and drains versus the wave propagation in the device). In our opinion, nobody was able to reproduce reliably the experiments, because of the numerical complexity involved. Our analytic theory draws from a simple, 1–dimensional model we developed in collaboration with Tomas Tyc (Masaryk University) and Alex Kogan (Weizmann Institute). This model was the first to explain the data of experiment, characteristic dips of the transmission of displaced drains, which establishes the grounds for the realistic super-resolution of absolute optical instruments. As the next step in Chapter 7 we developed a Lagrangian theory that agrees with the simple and successful model in 1–dimension. Inspired by the Lagrangian of the electromagnetic field interacting with a current, we have constructed a Lagrangian that has the advantage of being extendable to higher dimensions in our case two where imaging takes place. Our Lagrangian theory represents a device-independent, idealized model independent of numerical simulations. To conclude, Feynman objected to Abbe’s diffraction limit, arguing that as Maxwell’s electromagnetism is time-reversal invariant, the radiation from a point source may very well become focused in a point drain. Absolute optical instruments such as the Maxwell Fisheye can perform the time reversal and may image with a perfect resolution. However, the sources and drains in previous experiments were interacting with each other as if Feynman’s drain would act back to the source in the past. Different ways of detection might circumvent this feature. The mutual interaction of sources and drains does ruin some of the promising features of perfect imaging. Arrays of sources are not necessarily resolved with arrays of detectors, but it also opens interesting new prospects in scanning near-fields from far–field distances. To summarise the novel idea of the thesis: • We have discovered and understood the problems with the initial experimental demonstration of the Maxwell Fisheye. • We have solved a long-standing challenge of modelling the theory for mutually interacting sources and drains. • We understand the imaging properties of the Maxwell Fisheye in the wave regime. Let us add one final thought. It has taken the scientific community a long time of investigation and discussion to understand the different ingredients of the diffraction limit. Abbe’s limit was initially attributed to the optical device only. But, rather all three processes of imaging, namely illumination, transfer and detection, make an equal contribution to the total diffraction limit. Therefore, we think that for violating the diffraction limit one needs to consider all three factors together. Of course, one might circumvent the limit and achieve a better resolution by focusing on one factor, but that does not necessary imply the violation of a fundamental limit. One example is STED microscopy that focuses on the illumination, another near–field scanning microscopy that circumvents the diffraction limit by focusing on detection. Other methods and strategies in sub-wavelength imaging –negative refraction, time reversal imaging and on the case and absolute optical instruments –are concentrating on the faithful transfer of the optical information. In our opinion, the most significant, and naturally the most controversial, part of our findings in the course of this study was elucidating the role of detection. Maxwell’s Fisheye transmits the optical information faithfully, but this is not enough. To have a faithful image, it is also necessary to extract the information at the destination. In our last two papers, we report our new findings of the contribution of detection. We find out in the absolute optical instruments, such as the Maxwell Fisheye, embedded sources and detectors are not independent. They are mutually interacting, and this interaction influences the imaging property of the system.
Thesis, PhD Doctor of Philosophy
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