A filament of an incompressible highly viscous fluid that is supported at its ends sags under the influence of gravity. Its instantaneous shape resembles that of a catenary, but evolves with time. At short times, the shape is dominated by bending deformations. At intermediate times, the effects of stretching become dominant everywhere except near the clamping boundaries where bending boundary layers persist. Finally, the filament breaks off in finite time via strain localization and pinch-off.

We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate and (ii) the full Stokes equations. In either case, the plate is stabilised by a combination of viscous resistance, surface tension and buoyancy relative to an underlying denser fluid. In the limit of vanishing thickness, plates buckle at a shear rate γ/(µd) independent of buoyancy, where 2d is the plate thickness, γ is the average surface tension between the upper and lower surfaces and µ is the fluid viscosity. For thicker plates stabilised by an equal surface tension at the upper and lower surfaces, at and above onset, the most unstable mode has moderate wavelength, is stationary in the frame of the centre-line, spans the width of the plate with crests and troughs aligned at approximately 45 • to the walls and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) approximately 0.6 and are stabilised only by internal viscous resistance. We show that the viscous plate model can only accurately describe the onset of buckling for vanishingly thin plates but provides an excellent description of the most unstable mode above onset. Finally, we show that by modifying the plate model to incorporate advection and make the model material frame-invariant, it is possible to extend its predictive power to describe relatively short, travelling waves.

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Executive SummaryThe Defense Advanced Research Projects Agency (DARPA) asked the Institute for Defense Analyses (IDA) to assemble a summary of work on gradient index optics completed under funding from the Agency. The purpose was to archive the efforts, motivation, and accomplishments that the Agency supported and to provide a reference for any future programs that explore this new area of optics.Although optical instruments, such as lenses and mirrors, dating back thousands of years ago have been unearthed, lenses were not put into practical use until the invention of eyeglasses in the thirteenth century. The microscope and the telescope were invented in the seventeenth century, but no revolutionary changes occurred in these complex lens systems for the next 200 years. Most lens systems remained homogeneous multi-element systems with spherical glass surfaces and fixed optical properties. Recently, however, new materials and lens designs have been developed, inspired by properties of biological eyes. Materials with a gradient index (GRIN) allow the development of compact systems that have high focusing power while correcting for aberration.In 2002, DARPA initiated the Bio-Optic Synthetic Systems (BOSS) program, which aimed to synthesize the components of a biologically inspired vision system and demonstrate a level of performance beyond that of standard optical imaging systems (i.e., with reduced size and complexity). Out of more than a dozen exploratory efforts, four efforts were selected for further development and demonstration: fluidic adaptive zoom lenses, foveated imaging, photon sieves, and nanolayer lenses.As an example, Case Western Reserve University (CWRU) and the Naval Research Laboratory (NRL) collaborated on the nanolayer lens effort to create a synthetic lens that would mimic the structure and capabilities of an octopus eye. The CWRU/NRL team used a forced assembly nanolayer coextrusion process to form films that had a tailored refractive index consisting of thousands of nanolayers of two different polymers that had different refractive indices. The films of various n were stacked to create a refractive index range (Δn) and formed into hemispheres, which were combined to form the synthetic bi-convex octopus lens. A zoom lens system constructed from three of these GRIN lenses was demonstrated on a small unmanned aerial vehicle (UAV).Limited manufacturing capabilities have hindered practical applications of GRIN lenses. In 2008, DARPA initiated the Manufacturable Gradient Index Optics (M-GRIN) program to address GRIN lens manufacturing issues in the development of low-cost, iii customizable GRIN-based optics for use in a variety of military systems. This effort included a significant focus on metrology, where new tools were required for monitoring the polymer nanolayer process for GRIN optics. Performers in the M-GRIN program included the following teams:

Abstract. For media with heterogeneous index of refraction, rays follow curved paths. Methods exploiting knowledge of the relationship between refractive index distribution and ray paths to estimate the refractive index distribution based on laser beam deflection, displacement, and mode conversion on passage through the media are described. The work covers axial, radial (cylindrical), and spherical refractive index distributions. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.