Optical Cloaking of a Spherical Shape

Creating an Illusion of Invisibility using MetamaterialsApril 1, 2010

Optical CloakingAuthors:

Amanda Booth David DrorDaniel FrasquilloJaron Gubernick

With thanks to Dr. Gabitov for his help.

Our goal:To explore mathematical methods that have been previously constructed that successfully model the cloaking of objects, otherwise known as invisibility.

To do this the use of metamaterials is necessary; therefore an understanding of metamaterials is required to convert our mathematical model back into English for practical applications.

We also wish to explore other applications for metamaterials, such as optical black holes.

Why should we care?Invisibility has been the realm of myth for so long that it would be a clear triumph of science to be able to make something macroscopic invisible.

Military applications would include avoiding detection through radar or from regular visible light.

The better we understand how to control light, the better telescopes, detectors, and other optical instruments we can create.

Some materials and records decay in the presence of light. Making these objects ‘invisible’ would shield them from the light and preserve them.

What has been done?Dr. John Pendry, a physicist, has developed mathematical methods using metamaterials to produce cloaking devices.

Graduate students at the University of Arizona have also worked on the theory presented by Pendry.

Previous Work

Metamaterials

Metamaterials are artificially engineered materials that produce unusual optical properties, namely negative refractive indices.

At the moment, most of the success at making these materials are for radio waves or individual wavelengths of visible light.

It is interesting to note that metamaterials do not occur anywhere in nature!!

Scanning electron microscope image of a metamaterial

developed for microwaves

Where did the idea for metamaterials

originate?Scientists first observed

materials reflecting light in a unique way in nature. The Blue

Morpho Butterfly utilizes iridescence in its wings to reflect blue light from its

surroundings. For years, people were trying to extract dye from

these butterflies until they realized the stunning color

exists from microscopic scales in the wings that reflect incident

light in successive layers. Observations like this may have

lead to an increased study in light wave refraction.

Interpretation of Metamaterials

Mathematicians observed that for a material with an index of refraction, n may be represented by the term n2 = εμ, where ε is the dielectric constant, and μis the magnetic permeability.

Solving for n yields two solutions, one positive and one negative. Scientists wanted to create a material that has a negative n solution.

It was only by knowing such a solution was possible and then attempting to recreate it through experimenting with various ε and μ’s that metamaterials were actually discovered.

Snell’s Law

η1sinθI= η2sinθT

This is the formula for the angle of refraction of light (using the symbols in the image to the right).

η1, η2 are the refractive indices of the two materials and θI, θTare the angles between the ray of light and the normal .

Negative Refractive Index

The ‘rays’ in the 4th quadrant give examples of how common materials affect light. The 3rd

quadrant provides an illustration of how metamaterials refract the light.

The deflection of the incoming light ray is given by Snell’s law : η1sinθI= η2sinθT .

It is fairly straightforward to see that if η2 is negative, then the angle of refraction will be the opposite of conventional refraction.

Negative Refractive Index

Another effect of negative refractive index is that is makes the waves inside of a pulse flow in the opposite direction from the pulse itself.

The math for this can be seen from the formula for the speed of light in a medium:

v = c / n .

V is speed of light in the medium, c is speed of light in vacuum, and n is index of refraction.

It is clear that if n is negative, v is negative too.

http://people.ee.duke.edu/~drsmith/negative_index_about.htm

We are using metamaterials to make rays of light refract around the object and then convene to their

original course once on the other side.

Maxwell’s Equations

∇×E=-μ(r)μ0∂H/∂t

∇×H=+ε(r)ε0∂E/∂t

where μ(r) represents the magnetic permeability and ε(r) is the electrical permittivity.

It is important to understand the significance of Maxwell’s

equations relating to the problem at hand. Since the

equations are applicable under coordinate transformations, any shape can be modeled as long as the proper coordinate

transformation can be determined:

How did we model invisibility?

One way to look at the effects of refraction is by using coordinate transformations instead of refraction explicitly.

As you can see on the left, as the light passes through a different set of coordinates it continues to travel in a straight line, but what looks straight inside the local coordinates looks bent from the outside.

Why do we need metamaterials for invisibility?

We need an index of refraction that changes in a continuous controlled manner, and our only understanding of how to do this is through the use of metamaterials.

What we did:

The paper we used (Pendry et al), actually explicitly states a straightforward coordinate transformation to cloak a sphere. If the new coordinates are r’, θ’ and φ’ , and the sphere we are trying to cloak has a radius R1, and the cloak surrounding the sphere has a radius R2 the coordinate transformation to cloak a sphere is:

r’ = (R2-R1)*r/R2 + R1

θ’ = θ

φ’ = φ

We applied this transformation to simulated rays of light (3-D ‘straight’ lines) in a specific situation where R1 = .5 and R2 =1, and found that it worked as claimed; the sphere almost never got touched by the light.

Pictures!

Transforming these equations under spherical coordinates yields:

ε′r′= μ′r′= [R2/(R2-R1)][(r′-R1)/r′]2, (9)

for R1 > r′> R2

Or, for our specific case:

ε′r′= μ′r′= 2(.5r/(.5r+.5))^2

ε′θ′= μ′θ′= ε′ϕ′= μ′ϕ′= R2/(R2-R1) = 2 (10)

where R1 is the inner radius and contains the object and R2 is the outer radius. The waves must only distort the coordinate grid where R1 < r < R2, so this is the only region affected by the different μ and ε values described in equations 9 and 10.

By determining the values for μ(r) and ε(r), the properties for designing a metamaterial may be specified.

What we learned:The rays actually did not get bent if the were coming in straight through the center of the sphere. The reason for this is that the angle of the light (both θ and φ) would not change over the course of this transformation, and so all the transformation would do is effect the rate at which the light gets to the center.

The above effect does not actually matter in practice because the percentage of light that is coming in perfectly through the center in real life is infinitely small, and therefore would not be visible.

Difficulties for Invisibility in Real Applications

Manipulating the previous set of equations, in order to achieve invisibility we need a index of refraction varying continuously between 2/9 and 1/2.

The issue is, even though metamaterials are produced in the lab, those that effect visible light only effect a narrow portion of the wavelengths in the correct way. This means the object might become invisible to green light but not to red or blue. Work is still being done from the practical end to make what we modeled possible.

A lot of shapes in real life are not spheres. The coordinate transformations and therefore the configuration of the materials for non-spherical shapes in much more difficult.

Where do we go from here?

Modeling non-spherical shapes, particularly something that might be able to cover a human being: say a cylinder with a half sphere on top.

Figuring out how to focus light at a specific point (used in lensing, various instruments, and potentially even a way to make solar power more efficient.)

Determining, from observing model behavior, where some difficulties might arise putting this into practice.

Where do we go from here?

We plan on expanding the idea we have of cloaking a sphere and making it into a shape similar to that of a cylinder/hemisphere combination

Optical Black HolesThis is an example of redirection of light rays due to a negative index of refraction may present possibilities for optical black holes

Optical black holes redirect all incoming light to a single point, causing potential for highly effective generation of light energy into solar energy, for example

Work in progress

References:Pendry,John.Taking the wraps off cloaking. The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, UK. Physics 2, 95 (2009).

http://www.dailymotion.com/video/x6jvnm_lecture-john-pendry-invisibility-cl_tech

Crosskey M., Nixon A., & Schick L. Invisibility in Metamaterials: Transformations and Ray Tracing. June 26, 2009.