Transport Ill Electron flow with folds and caustics, resulting from injection at the lower right fol lowed y branching due to the rough landscape over which the electrons are travelling. Loosely speaking, caustics are edges, lines along which one object or space ends and another begins. But edges If the object being rendered is a smooth, three dimensional form, light will usually collect or diminish rapidly at an o are usually much more. In a drawing, caustics determine where a line should fall, and where it should begin and end edge, and detail will accumulate there. This is because the caustic of a curved surface is where we look tangent to (i.e. along) the surface. If we imagine the surface as a thin shell of smoky plastic in front of a uniform gray sky, then the edges(caustics)will be very dark, because there light must pass through much more material to get through than at a typical place. This points to the more general concept of caustics: places where accumulation occurs Whether by training or by instinct, we associate a line in a simple drawing with a caustic in the real world. But caustics are not always found at the obvious places. For example, look at the " proper"way to draw a torus(donut) from a certain perspective. This simple drawing is much more convincing than the way that might seem more obvious The reason is that the top edge persists up to a certain point and then just vanishes; where it vanishes we stop the line. There is no reason for the top edge to stop where the bottom one meets it, as in the incorrect drawing. Caustics are found everywhere, and nowhere are they as beautiful as when looking through a thin folded translucent sheet, such as translucent kelp. One of the caustics we are bound to see is called a cusp It happens when a flat part of the kelp develops a fold as we follow along a blade. At a definite point, we start to see two new edges or caustics arise where before there were none: Light coming through folded kelp is an example of projection, in this case from three dimensions into two( think of taking a picture of the kelp: it is three dimensional, but its image will h ave to live on the two dimensional surface of the film
Transport III Electron flow with folds and caustics, resulting from injection at the lower right fol lowed by branching due to the rough landscape over which the electrons are travelling. Loosely speaking, caustics are edges, lines along which one object or space ends and another begins. But edges are usually much more. In a drawing, caustics determine where a line should fall, and where it should begin and end. If the object being rendered is a smooth, three dimensional form, light will usually collect or diminish rapidly at an edge, and detail will accumulate there. This is because the caustic of a curved surface is where we look tangent to (i.e. along) the surface. If we imagine the surface as a thin shell of smoky plastic in front of a uniform gray sky, then the edges (caustics) will be very dark, because there light must pass through much more material to get through than at a typical place. This points to the more general concept of caustics: places where accumulation occurs. Whether by training or by instinct, we associate a line in a simple drawing with a caustic in the real world. But caustics are not always found at the obvious places. For example, look at the "proper" way to draw a torus (donut) from a certain perspective. This simple drawing is much more convincing than the way that might seem more obvious: The reason is that the top edge persists up to a certain point and then just vanishes; where it vanishes we stop the line. There is no reason for the top edge to stop where the bottom one meets it, as in the incorrect drawing. Caustics are found everywhere, and nowhere are they as beautiful as when looking through a thin folded translucent sheet, such as translucent kelp. One of the caustics we are bound to see is called a cusp It happens when a flat part of the kelp develops a fold as we follow along a blade. At a definite point, we start to see two new edges or caustics arise where before there were none: Light coming through folded kelp is an example of projection, in this case from three dimensions into two (think of taking a picture of the kelp: it is three dimensional, but its image will h ave to live on the two dimensional surface of the film)
Transport IV Electrons launched from the bottom fan out and then form branch, as indirect effects of travelling over bumps These images render electron flow paths in a"two dimensional electron gas". Inspired by the experiments of Mark Topinka, Brian Leroy, and Prof. Robert Westervelt at Harvard. Theory performed by Scot Shaw of my group, and me These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape. The electrons have more than enough energy to ride over any bump in the landscape, and the concentrations of electron flow( white/pink in Dendrite, and darkest streaks in Transport Ill) are newly discovered indirect effects of that bumpy ride. The channeling or branching was unexpected and has serious implications for small electronic devices of the future. These two images are excellent examples of 1)the degree to which I manipulate the raw data to get an image, and 2 )more important still, the wonderful way nature emulates herself in different contexts. Thus, the folding of the electron trajectories looks like looking through translucent kelp! Or, like ridges on a mountain
Transport IV Electrons launched from the bottom fan out and then form branch, as indirect effects of travelling over bumps. These images render electron flow paths in a "two dimensional electron gas". Inspired by the experiments of Mark Topinka, Brian Leroy, and Prof. Robert Westervelt at Harvard. Theory performed by Scot Shaw of my group, and me. These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape. The electrons have more than enough energy to ride over any bump in the landscape, and the concentrations of electron flow (white/pink in Dendrite, and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride. The channeling or branching was unexpected and has serious implications for small electronic devices of the future. These two images are excellent examples of 1) the degree to which I manipulate the raw data to get an image, and 2) more important still, the wonderful way nature emulates herself in different contexts. Thus, the folding of the electron trajectories looks like looking through translucent kelp! Or, like ridges on a mountain
Transport IX Paths of electrons followed for a short time, representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live In Transport IX we see the paths of electrons followed for a short time, representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live. The distinct and overlapping patterns resulted from the particular hills and valleys encountered from a new location It is seen that the branches emanating from different initial launch points cross and seem independent, confirming that they are not due to any fixe features in the potential landscape but rather are due to the history of encounters with hills and valleys"upstream
Transport IX Paths of electrons followed for a short time, representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live. In Transport IX we see the paths of electrons followed for a short time, representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live. The distinct and overlapping patterns resulted from the particular hills and valleys encountered from a new location. It is seen that the branches emanating from different initial launch points cross and seem independent, confirming that they are not due to any fixe features in the potential landscape but rather are due to the history of encounters with hills and valleys "upstream
Transport VI Electrons launched from the upper left fan out and then form branches, as indirect effects of travelling over bumps Transport Vll Electron flow over a two dimensional hilly terrain. Electron flow over a two dimensional bumpy surface; the electrons were injected from the top in a uniform sheet, all initially heading straight down. The focussing and defocussing effects of the hilly
Transport VI Electrons launched from the upper left fan out and then form branches, as indirect effects of travelling over bumps. Transport VII Electron flow over a two dimensional hilly terrain. Electron flow over a two dimensional bumpy surface; the electrons were injected from the top in a uniform sheet, all initially heading straight down. The focussing and defocussing effects of the hilly
terrain are clearly seen Transport XI Electron tracks in nano device launched at various places. The bunching or branching of electron tracks depends on where the electrons are launched. Electron flow started at several points on a single random potential surface, some moved and copied Small-scale electronic devices, the size of a bacterium or even a hundred times smaller, inevitably have minute imperfections in them which cause electrons to scatter and spread out as they progress through the device. We recently discovered that the electrons tend to bunch up and form branches, as is seen in many of the Transport images. This image sh ows that the branching is not a matter only of the landscape over which electrons are traveling, but also depends on where the electrons begin their joumey. In this image, the electrons are launched at different places, over a very small range of initial angles, represented by the narrow"Stems". Smaller initial differences in angle grow quickly, as evidenced by the fanning out of electron paths. This is the beginning of the eventual chaotic motion of these electrons. However, the branching is also evident. Note that the branches cross. This means that the branches are not following specific valleys in the landscape, but are rather indirect effects caused by focusing as electrons travel over bumps and hill
terrain are clearly seen. Transport XI Electron tracks in nano device launched at various places. The bunching or branching of electron tracks depends on where the electrons are launched. Electron flow started at several points on a single random potential surface, some moved and copied Small-scale electronic devices, the size of a bacterium or even a hundred times smaller, inevitably have minute imperfections in them which cause electrons to scatter and spread out as they progress through the device. We recently discovered that the electrons tend to bunch up and form branches, as is seen in many of the Transport images. This image shows that the branching is not a matter only of the landscape over which electrons are traveling, but also depends on where the electrons begin their journey. In this image, the electrons are launched at different places, over a very small range of initial angles, represented by the narrow "Stems". Smaller initial differences in angle grow quickly, as evidenced by the fanning out of electron paths. This is the beginning of the eventual chaotic motion of these electrons. However, the branching is also evident. Note that the branches cross. This means that the branches are not following specific valleys in the landscape, but are rather indirect effects caused by focusing as electrons travel over bumps and hill