Seeing through tissue and Kramers-Kronig

There is a paper in Science this week that is just a great piece of work.  The authors find that by dyeing living tissue with a particular biocompatible dye molecule, they can make that tissue effectively transparent, so you can see through it.  The paper includes images (and videos) that are impressive. 

Seeing into a living mouse, adapted from here.

How does this work?  There are a couple of layers to the answer.  

Light scatters at the interface between materials with dissimilar optical properties (summarized mathematically as the frequency-dependent index of refraction, \(n\), related to the complex dielectric function \(\tilde{\epsilon}\).   Light within a material travels with a phase velocity of \(c/n\).).  Water and fatty molecules have different indices, for example, so little droplets of fat in suspension scatter light strongly, which is why milk is, well, milky.  This kind of scattering is mostly why visible light doesn't make it through your skin very far.  Lower the mismatch between indices, and you turn down scattering at the interfaces.  Here is a cute demo of this that I pointed out about 15 (!) years ago:

Frosted glass scatters visible light well because it has surface bumpiness on the scale of the wavelength of visible light, and the index of refraction of glass is about 1.5 for visible light, while air has an index close to 1.  Fill in those bumps with something closer to the index of glass, like clear plastic packing tape, and suddenly you can see through frosted glass.  

In the dyed tissue, the index of refraction of the water-with-dye becomes closer to that of the fatty molecules that make up cell membranes, making that layer of tissue have much-reduced scattering, and voilà, you can see a mouse's internal organs.  Amazingly, this index matching idea is the plot device in HG Wells' The Invisible Man!

The physics question is then, how and why does the dye, which looks yellow and absorbs strongly in the blue/purple, change the index of refraction of the water in the visible?  The answer lies with a concept that very often seems completely abstract to students, the Kramers-Kronig relations.  

We describe how an electric field (from the light) polarizes a material using the frequency-dependent complex permittivity \(\tilde{\epsilon}(\omega) = \epsilon'(\omega) + i \epsilon''(\omega)\), where \(\omega\) is the frequency.  What this means is that there is a polarization that happens in-phase with the driving electric field (proportional to the real part of \(\tilde{\epsilon}(\omega)\)) and a polarization that lags or leads the phase of the driving electric field (the imaginary part, which leads to dissipation and absorption).   

The functions \(\epsilon'(\omega)\) and \(\epsilon''(\omega)\) can't be anything you want, though. Thanks to causality, the response of a material now can only depend on what the electric field has done in the past.  That restriction means that, when we decide to work in the frequency domain by Fourier transforming, there are relationships, the K-K relations, that must be obeyed between integrals of \(\epsilon'(\omega)\) and \(\epsilon''(\omega)\).  The wikipedia page has both a traditional (and to many students, obscure) derivation, as well as a time-domain picture.  

So, the dye molecules, with their very strong absorption in the blue/purple, make \(\epsilon''(\omega)\) really large in that frequency range.  The K-K relations require some compensating changes in \(\epsilon'(\omega)\) at lower frequencies to make up for this, and the result is the index matching described above.  

This work seems like it should have important applications in medical imaging, and it's striking to me that this had not been done before.  The K-K relations have been known in their present form for about 100 years.  It's inspiring that new, creative insights can still come out of basic waves and optics.