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We describe a method for learning a group of continuous transformation operators to traverse smooth nonlinear manifolds. The method is applied to model how natural images change over time and scale. The group of continuous transform operators is represented by a basis that is adapted to the statistics of the data so that the innitesimal generator for a measurement orbit can be produced by a linear combination of a few basis elements. We illustrate how the method can be used to efciently code time-varying images by describing changes across time and scale in terms of the learned operators.

The one-dimensional model in Eq. (1) is an approximation to realistic organic semiconductors, which typically adopt two-dimensional transport network8,13,29. In the Supplementary Fig. 4 we demonstrate that this approximation is valid for anisotropic materials by comparing the simulated one-particle spectral function with experimental angle resolved ultraviolet photoemission spectra (ARUPS)30, and at the end of the section we go beyond the one-dimensional model to discuss the isotropy effect on the different transport regimes.

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The PA regime and the TL regime can be further confirmed by semi-analytical results. In Fig. 2c we compare C(t)H-P of our numerical simulation with C(t)PA of phonon-assisted charge transport theory24:

Thanks Cory! I figured it out. The software required to view the Chandra fits images is SAOImage DS9. It allows the user to zoom into the x-ray image with sub-arcsecond precision, and even export in jpg, png, or tiff. Non-image fits files are accessed from within the software by accessing header data.

Transverse Anderson localization of light allows localized optical-beam-transport through a transversely disordered and longitudinally invariant medium. Its successful implementation in disordered optical fibres recently resulted in the propagation of localized beams of radii comparable to that of conventional optical fibres. Here we demonstrate optical image transport using transverse Anderson localization of light. The image transport quality obtained in the polymer disordered optical fibre is comparable to or better than some of the best commercially available multicore image fibres with less pixelation and higher contrast. It is argued that considerable improvement in image transport quality can be obtained in a disordered fibre made from a glass matrix with near wavelength-size randomly distributed air-holes with an air-hole fill-fraction of 50%. Our results open the way to device-level implementation of the transverse Anderson localization of light with potential applications in biological and medical imaging.

Here we demonstrate optical image transport using transverse Anderson localization of light, specifically in a disordered optical fibre. The possibility of using disordered optical fibre for some form of image transport was expected, given the earlier demonstration of spatial beam multiplexing in p-ALOF (ref. 8). The novelty of the presented work is in demonstrating that the image transport quality can be of a comparable or higher quality than the commercially available multicore imaging optical fibres. It is remarkable that the high quality image transport is achieved because of, not in spite of, the high level of disorder and randomness in the imaging system.

Multicore optical fibres have been used extensively in high-resolution optical imaging11; however, the transmitted images are inherently pixelated due to the discrete nature of the light-guiding array of cores, and the inter-core coupling can reduce the image contrast and result in blurring12,13. Certain structural non-uniformities such as variations in the size of the cores were shown by ref. 14 to improve the image transport quality. A weakly disordered fibre array was also studied in ref. 15 and was shown to induce diffusive spreading or localization at a few sites across the fibre. High numerical aperture guiding cores were also suggested by ref. 13 to reduce core-to-core coupling and blurring in imaging applications.

A highly disordered optical fibre with large refractive index fluctuations can transport high quality images, as it provides a high degree of structural non-uniformity as well as a sufficiently large local numerical aperture. More rigorously, the image transport quality is due to the transverse Anderson localization phenomenon that creates localized transport channels with finite radii (localized optical modes) through the disordered imaging waveguide16,17. A higher amount of disorder and a larger level of fluctuation in the refractive index provides stronger beam localization, hence improving the image resolution. It is also responsible for the reduction in the value of the s.d. in the localized beam radius as a consequence of the self-averaging behaviour7,18,19, ensuring uniform image transport quality across the fibre facet. The coherent transverse coupling and blurring is considerably reduced, because the transverse disorder results in strong spatial incoherence across the beam, akin to using incoherent light to eliminate speckles in an imaging system. Therefore, even a laser can be readily used for illumination in this image fibre to obtain a higher signal-to-noise ratio, without worrying about its undesirably high spatial coherence.

The fabrication of the p-ALOF used in these imaging experiments is described in detail in (refs 7, 20); In brief, it is composed of 40,000 strands of poly methyl methacrylate (PMMA) and 40,000 strands of poly styrene (PS) drawn to a square profile with a side width of 250-m and site sizes of about 0.9-m (see Methods). A magnified scanning electron microscope (SEM) image of a portion of the tip of the p-ALOF is shown in Fig. 1a, where the PMMA sites (refractive index of 1.49) are darker compared with the lighter PS sites (refractive index of 1.59). Also shown in Fig. 1b, for comparison, is an SEM image of a portion of the tip of a glass disordered fibre earlier reported in ref. 21, where the darker sites are the airholes. In the following, we demonstrate high-quality optical image transport through the p-ALOF related to Fig. 1a mediated by transverse Anderson localization of light. We also argue that a higher disorder density is required for quality image transport through the glass disordered fibre of Fig. 1b and a higher air-hole fill-fraction of nearly 50% should result in a very high-quality image fibre.

Similar to the experimental results of Fig. 5, the numerical simulation shown in Fig. 6 indicates that the visual image quality of the transported image through the p-ALOF is better or comparable to the Fujikura image fibres. The experiment and numerics are in reasonable agreement, but there are differences, as well. Possible sources of difference can be traced back to uncertainties in relating the experiment to numerics. The MSSIM values are 0.637 for Fig. 6a, 0.615 for Fig. 6b and 0.6257 for Fig. 6c.

In the experiment, a good precision in the butt-coupling of the test target to the input fibre is needed to obtain high-quality output images. An important source of uncertainty is the surface quality of the fibres, determining the precision in coupling the input profile from the test target and coupling the output to the CCD camera. The Fujikura image fibres are cleaved and polished using commercial-grade equipment, while the p-ALOF is hand-cleaved and polished using the lapping paper from Thorlabs. These variations and uncertainties cannot be easily accounted for in the numerical simulation without extensive surface quality characterizations of p-ALOF and Fujikura image fibres and are likely not very illuminating, considering that the p-ALOF images can be improved if specialized equipment for cleaving and polishing polymer fibres are used to improve its surface quality. Another possible source of uncertainty is the degree of spatial coherence of the laser used to illuminate the test target. The spectral bandwidth of the laser also contributes the fuzziness of the experiment, while the single-frequency numerical simulation looks more pixelated.

The simulated images of multicore image fibres in Fig. 6 look quite different from the experiment in Fig. 5, even when one takes into account the uncertainties mentioned above. This issue was pointed out in ref. 14, where it was suggested that the discrepancy between theory and experiment is because the image fibres are not composed of identical cores. The published core size specification is likely the average value with potentially a large s.d. While it is possible to build a concrete model to investigate this issue in detail, it is beyond the scope and interests of this manuscript.

It should be noted that the images in Fig. 7 appear to be of higher quality compared with Figs 5 and 6, simply because they are of different sizes and resolutions. The MSSIM values are 0.8923 for Fig 7a and 0.6263 for Fig. 7b.

For image transport through multicore imaging optical fibres, a longer optical wavelength increases the inter-core coupling strength and therefore lowers the quality of image transport. A similar effect can be observed in Anderson localized optical fibres as the mean localized beam radius has been shown to be larger for a longer wavelength9.

We note that not only is the average localized beam radius lower in a properly designed glass-air disordered optical fibre compared with a p-ALOF, but there also are fewer sample-to-sample variations in the localized beam radius. Therefore, a higher image transport quality is expected with a more uniform performance across the fibre.

The image transport quality is comparable to or better than some of the best commercially available multicore image fibres with less pixelation and higher contrast. In practice, the imaging resolution in p-ALOF is limited by the quality of the cleaving and polishing of the p-ALOF surface, and the maximum transport distance is limited by the optical attenuation as well as the variations in the thickness of p-ALOF in the draw process. The ultimate disordered image fibre will be made from a glass matrix with wavelength-size randomly distributed air-holes with an air-hole fill-fraction of 50%. The low optical attenuation in glass-air material is essential for transporting images over longer distances than reported here. The large index difference between the glass matrix and the air-holes and 50% air-hole fill-fraction provides maximum scattering in the transverse plane to reduce the localization beam radius and to minimize the width of the imaging point spread function. The large transverse scattering is also responsible for reducing the beam-to-beam variation of the localization radius. 2351a5e196