Due to a busy day yesterday and some amount of illness occurring today, I decided to use this time to go over a few papers I have read and include some important parts and my opinions on them within this blog post. This will be a useful reminder and will hopefully help cement my understanding of the papers. I may continue with this tomorrow and then do the refraction without a refraction map tomorrow as well.

The first paper I re-read was U. Leonhardt and T. G. Philbin, “Chapter 2 Transformation Optics and the Geometry of Light,” Prog. Opt., vol. 53, pp. 69–152, May 2009. This paper deals with the history of the transformation optics and the recent research into the connection between this and metamaterials. It goes into detail about metamaterials, stating that they are materials with electromagnetic properties that originate from man-made sub-wavelength structures. These have the ability to be tailored according to practical needs and desires and while they are not the solution to all engineering problems, they have inspired a great deal of research into what they are capable of. Some of the research done with metamaterials refers to the realm of optics and what possibilities were available there. Some of the research was into invisibility cloaks, electromagnetic cloaking devices and perfect lenses based on the theory of negative refraction.

How it relates to my project is within the discussion of Transformation Optics, something that Professor J.B. Pendrey talks about in his lecture on the science of invisibility. The idea is that using Fermat’s principle, a grid of light rays illuminating a non-uniform medium would all be curved. This leads to the question of whether you’d be able to transform away all curvature of the grid so that each curved ray would appear straight in the transformed space. This would make the bending of light simply an illusion of choosing the wrong co-ordinate system and a byproduct of cartesian linear thinking. In this case, you could create invisibility when the transformable grid contains a hole, making anything in the hole invisible. Thus making the transformation medium act as a cloaking device.

This could be used by my project. By changing the co-ordinate system in a select area, you could affect how the light bends in that area and make the inner circle invisible, or at least unlit. This could make too perfect an invisibility cloak however, which is not one of the aims of my project. Though this kind of model could raise ideas about how the person/object would see the outside world. Could you work the system so that light that enter is deflected but light that escapes from within the shell passes through so that you can see? I also think that creating a simulation of this theory would definitely be interesting based on how possible it would be.

The paper talks about Maxwell’s equations and being able to use them in systems that are not right-handed Cartesian, though the way it does this is still a little unclear to me. As far as I can tell it interprets the equations as being in a right-handed Cartesian space but with a medium. It describes this on page 34 with a diagram and stating that the way to think about it is with two spaces. In the first space, the system remains the same with no medium. The second space has a medium is created through a transformation and these co-ordinate are then interpreted as being right-handed Cartesian in that space. It describe the transformation requried to create a hole in space as r = R1 + r’(R2 -R1)/R2. Where R1 is the radius of the invisible region and R2 is the radius of the outer region. The R1-R2 describes the thickness of the cloaking layer.

The section of perfect invisibility confirms the idea that perfect invisibility will be very difficult, if not impossible, to achieve because of the need for the ray travelling forward to travel around the sphere in the same amount of time as it should travel straight. This means it needs to travel at a rate of infinity for phase velocity! However imperfect devices can be done, and this does coincide with the idea my supervisor had to have rays that travel through the center dead on to pass through a whole of some sort, providing light in a small space to the occupant, perhaps providing a small point of vision and it would allow the light to travel as quickly as normal. This may produce the effect of a dot where a dot shouldn’t be however.

This is the first paper I have re-read, I will add my opinions on another paper in another post.