The "lensless endoscope" represents the ultimate limit in miniaturization of imaging tools: an image can be transmitted through a (multi-mode or multi-core) fiber by numerical or physical inversion of the fiber's pre-measured transmission matrix. However, the transmission matrix changes completely with only minute conformational changes of the fiber, which has so far limited lensless endoscopes to fibers that must be kept static which limits its practical applications. In our work, we show how the properties of specialty multi-core fibers can help to overcome this challenge.
Cellular-level microscopic imaging has long been a vital tool in biomedical research. Recent years have seen numerous efforts to miniaturize imaging instruments with the aim of opening the door to cellular-level imaging in behaving animals.
A recent example is the miniaturized head mounted microscope for fluorescence imaging on freely behaving animals reported in Ref.  (weight 2 g) . This approach has light source, filters, imaging optics, and CMOS camera integrated into a head mounted device.
A different approach bases the light delivery and collection on optical fiber which brings the advantage that light source and detectors can be remote rather than integrated in the head mounted device which then only houses miniaturized actuators and/or imaging optics. As a consequence and added benefit, optical fiber-based miniaturized microscopes/endoscopes are able to use pulsed laser sources and perform non-linear imaging---to date the only demonstrations have been with optical fiber-based systems, see for example the demonstrations of two-photon fluorescence imaging in Ref.  (3 mm x 40 mm) which employs a piezo-electric actuator ; and Ref.  (2.15 g, 1 cm^3) which employs a MEMS mirror to perform point-scanning imaging . For an overview of the most "conventional" optical fiber-based approaches, see Ref. .
A new approach to fiber-based endoscopes came about in 2011 [5-10] when spatial-light modulator technology made it possible to measure and invert the transmission matrix (TM) of a multi-mode fiber (MMF) or multi-core fiber (MCF). This approach does away with the need to have imaging optics between fiber and sample and consequently, this approach is often termed "Lensless endoscopes" and represents the ultimate limit in miniaturization, since the head-mounted device can be as small as an optical fiber itself (diameter typically less than few hundred micrometers).
Cellular-level imaging in live mice by lensless endoscopes has recently been demonstrated for the first time in Refs. [11,12].
In our earlier work we have shown how lensless endoscopes based on MCF simplify many of the considerations pertaining to the TM . In particular, in Ref. , we showed that the extrinsic contribution to the TM of a MCF is simply a diagonal matrix with complex elements of unit norm and argument which is linear in the transverse coordinate, i.e. a matrix with only two free parameters.
In both the MMF or MCF case, however, the great challenge remains that following each conformational change of the fiber either the additional extrinsic contribution or the new TM must be experimentally quantified whether directly or indirectly in order for aberation-free imaging to continue. This is the main obstacle standing between us and a flexible lensless endoscope which would open the possibility for minimally-invasive imaging in behaving animals, and one of the principal drivers of our current work.
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