1. Metamaterials, Cloaking, and Acoustics
Steven A. Cummer
Electrical and Computer Engineering Department
Duke University
Other Team Members:
Prof. David Smith (Duke)
Prof. Sir John Pendry (Imperial College London)
Prof. David Schurig (NC State)
Dr. Anthony Starr (SensorMetrix, Inc.)
Prof. Steve Cummer
20 February 2008

2. Presentation Overview
Metamaterials and cloaking theory development are independent but practical realization tightly connected.
Acoustics ideas are entirely built on comparable ideas from electromagnetics.
Easiest to describe in essentially chronological order:
Electromagnetic metamaterials
Electromagnetic cloaking
Acoustic cloaking and metamaterials
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Prof. Steve Cummer
20 February 2008

3. How to Control the Electromagnetic Properties of a Material?
Mechanical and other properties of materials engineered all the time. Why not EM?
Electromagnetic properties of natural materials are fairly limited:
Few magnetic materials
Few strongly anisotropic materials
Available dielectric constants not continuous
How can you design and fabricate a “material” with the properties you need?
Two approaches.
IMAGE: need some picture of campaign hardware
Prof. Steve Cummer
20 February 2008

4. One Approach: Photonic Bandgap Materials
Idea dates to Yablonovitch [PRL, 1987.]
Resonant (Bragg) scattering from defects or structure spaced every half wavelength.
Occurs in nature and now in engineered devices such as optical fiber.
Properties: almost always anisotropic, depends critically on half-wavelength structure, can’t be smaller.
Prof. Steve Cummer
20 February 2008

5. Another Approach: Metamaterials
Common definition: artificial subwavelength structure that generates net magnetic and/or electric dipole moment in response to applied fields.
Mimics the physics of conventional materials (Si shown here).
Properties: isotropic or anisotropic, in principle doesn’t have to be periodic, structure must be subwavelength (how small is an interesting question).
Shelby et al., Science, 2001
Prof. Steve Cummer
20 February 2008

6. Metamaterials History
Like many good ideas, history goes back a long time.
Brown [1953], Rotman [1961]: array of wires aligned with electric field create a large electric susceptibility.
Schelkunoff and Friis [1952]: capacitively loaded loop creates a resonant magnetic susceptibility.
Last 7 years have seen lots of MM building on the independent rediscovery and extension of these ideas by Pendry [1996, 1999].
Prof. Steve Cummer
20 February 2008

7. Magnetic Metamaterials
Need a big electric or dipole moment per volume to create non-free space.
Split ring resonator [Pendry et al., 1999] resonantly amplifies the induced voltage.
Results in a large magnetic susceptibility (+ or –) near resonant frequency.
Isotropy can be controlled.
In theory arrangement doesn’t have to be regular, but in practice it is easier.
Bext x MB
Prof. Steve Cummer
20 February 2008

8. Electric Metamaterials
For permittivity, wire array produces cutoff (Drude) response, but electrical continuity is a challenge.
Or can make self-resonant elements that create an electric dipole moment in response to an applied electric field [Schurig et al., APL, 2006].
Again, isotropy can be controlled, most positive and negative values possible.
But bandwidth limited.
Eext ME
Prof. Steve Cummer
20 February 2008

9. Metamaterial Resurgence: Negative Refractive Index
Much of metamaterial research in past 10 years originally motivated by one idea.
By combining resonant electric and magnetic elements, could make a material with negative  and  at the same frequency, i.e. a negative refractive index?
Idea explored theoretically by Veselago [1968], who derived many unusual reversals (Doppler, etc.) in negative index material (NIM).
But idea didn’t go anywhere because no one knew how to make such a material.
But in 1999 all the pieces were in place to actually do it.
Prof. Steve Cummer
20 February 2008

10. Negative Refraction
Negative refraction first experimentally demonstrated by Shelby et al. [Science, 2001].
Some controversy erupted over some theoretical issues, but these were quickly resolved.
Cummer, APL, 2003
Shelby et al., Science, 2001
Prof. Steve Cummer
20 February 2008

11. Electromagnetic Metamaterials Summary
Electromagnetic properties can be engineered with precise control using metamaterial ideas: negative, large positive, smoothly inhomogeneous, anisotropic, etc.
Some limitations related to bandwidth and losses.
Many possible applications: antennas, lenses, surfaces, radomes, etc.
Electromagnetic material design space dramatically broadened, but not always easy to make an already optimized device work better with metamaterials.
Prof. Steve Cummer
20 February 2008

12. Cloaking and Transformation Optics
Is it possible to smoothly bend light around an object?
No backscatter, no shadow = effectively invisible.
Can there really be such an interesting solution still lurking in classical electromagnetics? Pendry et al. [Science, 2006] showed how it can be done.
Key realization: coordinate transformations on electromagnetic fields are completely equivalent to a nonuniform permittivity and permeability.
Curve space by opening a hole (mapping 0 to R2 to R1 to R2): everything, including electromagnetic fields, are curved around the hole.
Or, surround the “hole” with a shell from R1 to R2 containing very specific permittivity and permeability: electromagnetic fields are curved around the hole (but nothing else).
Prof. Steve Cummer
20 February 2008

13. Cloaking Theory Example
Example: cloaking a 2D cylinder.
Required  and specified by theory.
Strongly anisotropic, values from 0 to very large (not negative).
10 years ago this would have been completely unrealizable, especially anisotropy.
With metamaterials, however, there is hope of actually creating such a material.
Prof. Steve Cummer
20 February 2008

14. Cloaking Theory Simulations
Theory undoubtedly wonderful, but it gives no clues as to how sensitive the solution is to small parameter perturbations.
Is it like perfect focusing in that it completely falls apart if the material parameters aren’t realized with unachievable precision?
Numerical simulations are a very good tool for answering this question.
COMSOL Multiphysics enables full tensor description of  and , even off-diagonal components (needed for cartesian coords).
Plane wave or Gaussian beam incident on cloaked PEC scatter.
BCs either absorbing or equivalent to periodic.
Prof. Steve Cummer
20 February 2008

15. Baseline Simulations: No Scatterer and No Cloak
No scatterer: plane wave is undisturbed.
No cloak: strong scattering especially in forward (shadow) direction.
Prof. Steve Cummer
20 February 2008

16. Ideal Cloak Simulations
Ideal cloak smoothly bends electromagnetic power around scattering object.
Validates original prediction in no-approximations form.
Scattering is small, even in forward direction (but not zero).
Simulating cloaking physics not especially challenging, bodes well for experiment.
Parameter sensitivity not extreme.
Prof. Steve Cummer
20 February 2008

17. Non-Ideal Cloak Simulations
Concept is robust.
Loss: absorbs but does not scatter.
Staircase approximation not too bad.
Reduced parameter set: worse but basic ray and phase front bending still visible.
Prof. Steve Cummer
20 February 2008

18. Cloak Design (1) Goal to demonstrate basic physics of field bending.
2D TE polarization (Ez, Hr, H), reduced parameter set gives easiest path to realization. Only radially varying radial component of permeability.
Approximate continuous permeability variation with 10 discrete layers.
Step 1: Design 10 different magnetic resonators to give 10 different values (from 0 to about 1) for radial permeability at a single frequency.
This is done with simulations of single metamaterial particles.
Prof. Steve Cummer
20 February 2008

19. Cloak Design (2)
Step 2: Pattern each of 10 cells onto sheets of flexible printed circuit board material.
Step 3: Bend into circles per original design.
Result: A good approximation of a material with a continuously variable radial permeability.
Cheap to fab, design requires only modest simulations.
Prof. Steve Cummer
20 February 2008

20. Cloaking Experiment
Fields measured in field mapping chamber [Justice et al., Opt. Exp., 2006].
Prof. Steve Cummer
20 February 2008

21. Cloaking Measurements
Prof. Steve Cummer
20 February 2008

22. What Next for Electromagnetic Cloaking?
Make a better one: challenging for metamaterial design.
Other wave systems?
Other applications of “transformation optics”?
Invisibility at visible wavelengths? Losses are much too big at this point to be useful.
Transformation optics offers a new way of manipulating electromagnetic fields with engineered materials.
Prof. Steve Cummer
20 February 2008

23. Acoustic Cloaking
Natural to wonder whether the ideas behind transformation optics [Pendry et al., Science, 2006] can be applied to other kinds of waves.
Coordinate transformation invariance linked to relativity, maybe does not work for non EM waves?
Milton et al. [New J. Phys., 2006] applied coordinate transform approach to general elastodynamics with a specific assumption about how vectors have to transform.
Found that equation form is not preserved, even for acoustics.
Concluded that ideal elastic or acoustic cloaking was not theoretically possible.
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Prof. Steve Cummer
20 February 2008

24. 2D Acoustic Cloaking Some aspects of wave behavior are very general.
Maybe non-ideal but still useful acoustic cloaking possible?
We showed that 2D acoustics and 2D electromagnetics have exactly the same equation form [Cummer and Schurig, New J. Phys., 2007].
Thus 2D acoustic cloaking (i.e., a cylinder), and general sound field manipulation in 2D, is feasible.
Requires a fluid with inhomogeneous bulk modulus and anisotropic effective mass density.
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Prof. Steve Cummer
20 February 2008

25. 3D Acoustic Cloaking
slightly smaller than background
near background to very large
near background to very large
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No clear EM/acoustic analogy holds for three dimensions (i.e., a sphere).
But scattering theory can be used to derive the acoustic parameters of a theoretically perfect 3D spherical cloaking shell [Cummer et al., PRL, 2008].
Requires similar fluid properties, details slightly different than 2D.
Almost certain it can be shown that arbitrary sound field manipulation can be done with specific material properties, analogous to electromagnetics.
Prof. Steve Cummer
20 February 2008

26. Effective Mass Anisotropy
Both 2D and 3D acoustic cloaking require anisotropic effective mass density.
Strange sounding idea, but not difficult to imagine how to realize.
Milton et al. [NJP, 2006] describe a conceptual model of a composite with anisotropic effective mass density.
Springs mean that when force is applied, the magnitude of the net motion in different directions is not the same.
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Prof. Steve Cummer
20 February 2008

27. A Simpler Method for Realizing Anisotropic Effective Mass
Simple rigid scatterers are also resonant.
Torrent and Sanchez-Dehesa [NJP, 2008]: array of rigid scatterers in a fluid controls the anisotropy of the effective mass density of the array.
Nonspherical scatterers almost certainly give greater control over that key parameter.
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Design approach same as EM: simulate single material cells, assemble into a functional material.
Prof. Steve Cummer
20 February 2008

28. General Cloaking Limitations
Electromagnetic metamaterial losses are difficult to control and are large enough that it would be difficult to build an X band cloaking shell larger than ~10–20 wavelengths.
Losses are low in many rigid materials and so a higher quality, lower loss acoustic metamaterial is a realistic possibility.
Electromagnetic cloaking is inherently bandlimited because of speed of light issues.
No fundamental speed limit on acoustic waves, hence broadband acoustic cloak is in principle possible.
Thinner cloaking shells are more challenging to realize.
Cloaking theory + metamaterials give a completely new way to manipulate and reduce scattering of large objects, even forward scattering.
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Prof. Steve Cummer
20 February 2008

29. Conclusions
Metamaterial ideas are proven in their ability to yield engineered electromagnetic materials with desired effective, bulk properties such as strong anisotropy.
There is every reason to expect that these same properties can be engineered into acoustic metamaterials.
These engineered properties are exactly what is required to realize the newly discovered electromagnetic and acoustic cloaking shells.
There are undoubtedly practical limitations to how well these shells can perform in practice, i.e. thickness, scatter reduction, losses.
But the field has made a LOT of progress very quickly, and I would not be surprised to see things move equally quickly in acoustics and further in electromagnetics.
Prof. Steve Cummer
20 February 2008

30. Perfect Focusing with NIMs
Pendry [PRL, 2000] showed that the amplitude of evanescent waves is restored by a negative index slab in the same way as phase restored for propagating waves.
Causal simulations [Cummer, APL, 2003] showed that occurs exactly as predicted by Pendry [PRL, 2000].
Substantial limitations include exponential material sensitivity, rendering it a largely near field effect.
Prof. Steve Cummer
20 February 2008

31. Active Metamaterials
Metamaterials approach lends itself to embedding active devices into structure to expand capabilities at both RF and optical.
Lots of work presently on switching and tunable elements to switch between two states or continuously tune material properties.
Plenty to be done here: challenges are low loss elements and similarity from element to element.
But what about powered active devices, such as amplifiers?
In principle, active devices can eliminate losses and control dispersion.
Prof. Steve Cummer
20 February 2008

32. Gain Metamaterials
Resonant particles can do a wonderful job, but properties like loss and dispersion are difficult to control.
Resonant particles work by resonant gain:
What if we let an amplifier do the work in generating gain?
Certainly more complicated, but there are advantages. For example, gain is not nearly as frequency dependent.
Prof. Steve Cummer
20 February 2008

33. Non-Reciprocal Metamaterials
Have done a bunch of initial experiments, but I will jump to some very exciting (to me) results for a full metamaterial.
How to make a one-way material at RF?
Non-reciprocal 1D dispersion relation:
Non-reciprocal magnetoelectric coupling breaks symmetry and results in a single polarization non-reciprocal metamaterial.
Prof. Steve Cummer
20 February 2008

34. Non-Reciprocal Metamaterial Measurements
Built a 5 cell-wide slab of this “material”.
Measured 2-way TEM wave transmission through the material: Highly non-reciprocal, just as we’d hoped.
Prof. Steve Cummer
20 February 2008

35. Cloak Scattering
Interesting sidebar: How does the near-ideal cloak scatter?
It scatters like a 1D line at the center of the cloaked region.
Pretty unusual: not many electrically large objects that scatter isotropically.
Especially surprising because these computations are done on a unstructured grid.
Prof. Steve Cummer
20 February 2008