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# Metamaterials: negative mass density and anisotropic mass

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**Update: February 2012**

1) The lecture notes are on Wikiversity at http://en.wikiversity.org/wiki/Waves_in_composites_and_metamaterials

2) The book on the topic can be bought from Amazon at http://www.amazon.com/Introduction-Metamaterials-Waves-Composites/dp/1439841578

3) Solutions and errata can be found at node/9727

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Is it possible for materials to have negative mass densities, anisotropic masses, and other such unintuitive characteristics? Recent work on certain composite material has shown that that indeed such things are possible (see for example Liu, Chan, Sheng, (2005), Physical Review B, 71, 014103).

The study of metamaterials is just beginning and will probably lead to interesting new discoveries. Mechanicians have somewhat of a head start in the matter because researchers in the field of classical electrodynamics have already made some significant strides with metamaterials.

Prof. Graeme Milton has been teaching a class on metamaterials this semester (Spring, 2007). I have typeset his lecture notes and added in some details for the beginner. You can find the lecture notes here. Please let me know how the notes can be improved and if you find any mistakes.

**Update: 27 June, 2007**:

Here's a link to David Schurig 's talk at the Fermilab Colloquium Lectures. The talk is titled "*The Transformation Design Method and Metamaterials: Tools to realize Invibility and Other Interesting Effects*". The talk discusses the contents of Lecture 5 and Lectures 22/23 without most of the mathematical details.

**Links to lecture notes from Prof. Milton's course:**

**Lecture 1**: Rainbows [pdf]**Lecture 2**: Airy Theory [pdf]**Lecture 3**: Maxwell's Equations in Media [pdf]**Lecture 4**: Fresnel's Equations [pdf]**Lecture 5**: Perfect Lenses and Negative Density Materials [pdf]**Lecture 6**: Anisotropic Mass and Generalization [pdf]**Lecture 7**: Elastodynamics and Electrodynamics [pdf]**Lecture 8**: Acoustic Metamaterials and Negative Moduli [pdf]**Lecture 9**: Fading Memory/Waves in Layered Media [pdf]**Lecture 10**: Airy solution and WKB solution [pdf]**Lecture 11**: TE waves in multilayered media [pdf]**Lecture 12**: Continuum limit and propagator matrix [pdf]**Lecture 13**: Waves in layered media and point sources [pdf]**Lecture 14**: Point sources and EM vector potentials [pdf]**Lecture 15**: Mie Theory and Bloch's Theorem [pdf]**Lecture 16**: Bloch Waves and the Quasistatic Limit [pdf]**Lecture 17**: Bloch Waves in Elastodynamics and Bubbly Fluids [pdf]**Lecture 18**: Duality Relations/Phase Interchange Identity/Laminates [pdf]**Lecture 19**: Backus Formula for Laminates/Rank-1 Laminates [pdf]**Lecture 20**: Hierarchical Laminates/Hilbert space formalism [pdf]**Lecture 21**: Effective tensors using Hilbert space formalism [pdf]**Lecture 22**: Transformation-based Cloaking in Electromagnetism [pdf]**Lecture 23**: Transformation-based Cloaking continued [pdf]**Lecture 24**: Willis equations for Elastodynamics [pdf]

- Biswajit Banerjee's blog
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## Comments

## waves in composite materials

Dear Biswajit,

Thank you for posting those precious lecture notes on waves in composite materials. I have a question, how to derive the equation (5) in lecture 8, for low frequency processes?

Thank you very much.

Henry.

## Momentum balance in frequency domain

Henry,

You can look at Section 2 in Lecture 7. For a material that has a frequency dependent mass, causality requires an equation of the form given in equation (22). A Fourier transform of that equation (and the use of the convolution theorem) gives equation (23) which is the same as equation (5) in Lecture 8. I did check the signs when I worked out the algebra. However, you may want to check that to see whether everything is correct.

## Update: Waves in composites and metamaterials

I have updated my notes on the "Waves in Composites and Metamaterials" course. The last four lectures in the course (Lectures 21 through 24) will be posted in a couple of weeks. I have given links to the lecture notes below for your convenience. These notes will also be of interest to those who work in the field of homogenization of composites in the quasistatic limit.

Lecture 1: Rainbows [pdf]Lecture 2: Airy Theory [pdf]Lecture 3: Maxwell's Equations in Media [pdf]Lecture 4: Fresnel's Equations [pdf]Lecture 5: Perfect Lenses and Negative Density Materials [pdf]Lecture 6: Anisotropic Mass and Generalization [pdf]Lecture 7: Elastodynamics and Electrodynamics [pdf]Lecture 8: Acoustic Metamaterials and Negative Moduli [pdf]Lecture 9: Fading Memory/Waves in Layered Media [pdf]Lecture 10: Airy solution and WKB solution [pdf]Lecture 11: TE waves in multilayered media [pdf]Lecture 12: Continuum limit and propagator matrix [pdf]Lecture 13: Waves in layered media and point sources [pdf]Lecture 14: Point sources and EM vector potentials [pdf]Lecture 15: Mie Theory and Bloch's Theorem [pdf]Lecture 16: Bloch Waves and the Quasistatic Limit [pdf]Lecture 17: Bloch Waves in Elastodynamics and Bubbly Fluids [pdf]Lecture 18: Duality Relations/Phase Interchange Identity/Laminates [pdf]Lecture 19: Backus Formula for Laminates/Rank-1 Laminates [pdf]Lecture 20: Hierarchical Laminates/Hilbert space formalism [pdf]## thanks for your notes

Thank you, Biswajit,

I learned a lot from your notes， especially from those detailed reductions。

Who are the students attending this course?

## Re: Students

There were five graduate students taking the course for credit. I believe three were from Applied Math and two were from Geophysics. There were five others who were sitting in. I'm not sure about their backgrounds.

## linked to my blog:

linked to my blog: explosion science and engineering

## Update 2: Waves in composites and metamaterials

The final four lectures have been posted on my webpage. I have provided the links below for your convenience.

Lecture 21: Effective tensors using Hilbert space formalism [pdf]Lecture 22: Transformation-based Cloaking in Electromagnetism [pdf]Lecture 23: Transformation-based Cloaking continued [pdf]Lecture 24: Willis equations for Elastodynamics [pdf]## New paper on elastodynamics of metamaterials

Graeme Milton has posted a copy of his latest paper on elastic metamaterials on arXiv. The paper is called

"New metamaterials with macroscopic behavior outside that of continuum elastodynamics"and can be downloaded from http://arxiv.org/abs/0706.2202.For background on what Graeme is talking about you can take a look at Lecture 24 above (on the Willis equations for elastodynamics).

Abstract:

Metamaterials are constructed such that, for a narrow range of frequencies,

the momentum density depends on the local displacement gradient, and the stress

depends on the local velocity. In these models the momentum density generally

depends not only on the strain, but also on the local rotation, and the stress

is generally not symmetric. A variant is constructed for which, at a fixed

frequency, the momentum density is independent of the local rotation (but still

depends on the strain) and the stress is symmetric (but still depends on the

velocity). Generalizations of these metamaterials may be useful in the design

of elastic cloaking devices.

## Online and editable version of Metamaterial Notes

Biswajit graciously provided the LaTeX source for these excellent notes and I am using them as an example of open source content. I have converted the raw LaTeX to a very similar markup that is more readily handled by the Mediawiki parser. You can access the content at http://dssl.mne.psu.edu/nsfevo . So far I have ten lectures up. The first few are complete while the rest need figures, references, etc. added. All of these pages are editable. Clicking on the 'edit' tab at the top of the page will allow you to view the source code and make changes. I've refrained from correcting the few typos I saw in the notes so please feel free to correct them if you'd like. More substantial changes are also welcome as the content can always be reverted should the authors not like the changes.

Eric

## Editable notes

Thanks Eric for the converted notes. How long did it take you to do the conversion? Was most/all of the process automatic? There are a couple of things that could be improved/added:

## Editable Notes

I've gotten pretty good at writing regular expressions and it takes only a few minutes to convert each set of notes. The equations are all generated on the fly so if you change the LaTeX, a new image is generated.

I have not decided what markup I want to use for lists. Wiki markup is simpler so I am leaning towards it. Plus, I think it will be slightly easier to convert wiki lists to LaTeX than the other way around. (I plan to allow any page to be output as pure LaTeX and/or compiled and downloaded as PDF. I am planning to implement footnotes as popup windows like when you hover over an equation reference. This should be trivial to do but I have not had a chance.

The appearance of the equations has been a problem (but only on Windows and Linux). I use Macs and everything is served from a Mac. The equations look wonderful in all Mac browsers and I was flabbergast when I checked the pages on Windows (with IE and FF). Basically, I have learned that Windows (the OS) does a horrible job scaling graphics. I render the equations at 300dpi and then set the height in 'em' units so that if you increase the page font size, the equations will also scale. Unfortunately, this requires Windows and Linux to scale the images, making them look horrible. I can render the equations at a lower resolution so that they do not need scaled but then will look funny if the font size is changed in the browser. Right click on an equation and load it in a separate browser window/tab. It will look fine. I'm still trying to figure out if there is a kludge to get around this problem.

Thus, your suggesting is actually what is causing the problem. There are two ways to see the equations as I do. One is to download the Safari for Windows beta since it uses its own image scaling algorithms or to view the screenshots I will post.

## Re: Equations in editable notes

I zoomed into a few equations and it looks like Firefox is not antialiasing the scaled images. Is there any way of building antialiasing into the images themselves? I think that will fix the problem.

## Anti-aliasing equations

I'm using Imagemagik's convert to render the ps to png with the following php code...

// imagemagick convert ps to image and trim picture

$command = $this->_convert_path.' -verbose -density '.$this->_formula_density.' -trim -transparent "#FFFFFF" '.$this->_tmp_filename.'.ps '.$this->_tmp_filename.'.'.$this->_image_format;

// try no transparent for crappy Windoze scaling...// $command = $this->_convert_path.' -verbose -density '.$this->_formula_density.' -trim '.$this->_tmp_filename.'.ps '.$this->_tmp_filename.'.'.$this->_image_format;

I've tried all kinds of things to fix this. convert should default to anti-alias the images. I've tried explicitly turning it on (and off) with no effect. If you open an equation in an image editor and zoom in, you will see that they are anti-aliased (and I thought this actually might be the problem at one point).

## Re: Anti-aliasing equations

You're right - the images are antialiased. In that case, what is Safari doing to make the images look OK? You could try to ask Dr. Nelson Beebe for some pointers. He's the author of pstopngtops and is a TeX guru among other things.

## Re: Anti-aliasing equations

It's not just Safari but every browser on Mac OS X. I think the browsers are calling an OS routine to do the scaling and Mac OS X just has better scaling routines than Windows/Linux (although I have not tried Vista). I think Apple ported their image scaling routine with Safari for Windows so that it looks better. I.e. Safari for Windows is not calling the OS for scaling.