Research

• A list of my publications is here.
• Here is my Ph.D dissertation.

Introduction to the concept of negative refractive index materials

The idea of negative refractive index acoustic materials has its roots in negative refractive index optical materials. Since my background is not optics, I hope  possible minor inconsistencies in the description of the optical case will be tolerated.

Light travels as a wave and is governed by what’s called the Maxwell’s equation. This equation, together with Lorentz force law describes the wave nature of the electromagnetic field. The speed at which light travels is a universal constant denoted by the letter c. On a side note, the whole of special relativity can be derived from the one assumption (which is not really an assumption but an experimental truth) that the speed of light in vacuum is constant for all observers moving with constant velocities. Anyhoo, the speed at which light of a given frequency travels in a certain material depends upon ‘c’ and ‘n’ where ‘n’ is called the refractive index of the material. ‘n’ depends upon the square root of the product of two constants, the permittivity (e) and the permeability (m) of the material. Now as we all know, square root is a funny thing, especially when applied to the product of two quantities. As long as both e and m are either positive or negative, the root of their product would be a real number. This means a real refractive index and it means that light can travel in such materials.

Naturally occurring materials, for any given frequency, have either both e and m positive or one of them negative. But there is no physical law which prohibits e and m to both become negative at the same frequency. Veselago, in his 1968 seminal paper, predicted what would happen if ever we were to find a material with simultaneously negative e and m (called negative refractive index materials) and his findings were very interesting. He predicted that in such a hypothetical material several interesting phenomenon would occur, the most understandable of them being the reversal of Doppler shift. Doppler shift, in sound, is the apparent increase in the pitch of the horn when the train is approaching you. If you were to stand in a material with negative refractive index, you would hear the pitch lowering as the train approaches you. There were other striking properties that Veselago predicted and they had sci-fi practical uses – if only we could ever find such materials. About the turn of the century, they managed to create such materials, and since then research in the field has been vibrant.

A similar concept applied to sound waves. In the simplest case, the constants e and m are replaced by density and stiffness of the material but the essential idea remains the same. But even then, no practical examples of negative refractive index acoustic materials have come till date. One possible reason for this is that acoustic waves are more complicated than light waves and our theoretical understanding of such hypothetical materials is still limited. My current research is towards better such understanding.

Introduction to ultrasonic guided waves:

The broad area of my research during my Ph.D. was ‘ultrasonic guided waves’. I used to work on simulation and modeling but during the last 1.5 years I  concentrated on the theoretical aspects of guided waves. In the following paragraphs I have made an attempt at explaining my research to someone who might only have a basic understanding of the physics involved.

‘Sound’ is the interaction of material disturbance, travelling in the air, with the eardrum. This material disturbance is known as a ’stress wave’. Same stress wave  also travels in any other material when it is struck with a hammer or insonified with a transducer. In an infinite 3 dimensional medium these waves travel unhindered and both the solution and analysis of such waves are easy. But when these waves are constrained by boundaries , like in the case of a thin plate or a circular pipe, their behavior becomes very complex. Such constrained waves are called ‘guided waves’. They continue to generate  a lot of research interest because they can travel long distances with little attenuation. Hence they are ideal for long range inspection of defects in structures such as rails and aircraft wings. There is a rule of thumb that the higher the frequency of the wave (or smaller the wavelength), the smaller the defects it is sensitive to.  Therefore, high frequency (ultrasonic) guided waves are the craze.

The caveat here is that guided waves are very complex to analyze. Even in the simplest of structures, they exhibit what is called multimodal-dispersive behavior. Multimodal means that at a given frequency, more than 1, typically thousands of modes potentially exist in the structure and dispersive means that most of those modes, at least all which are of any interest usually, travel at velocities which are in turn dependent upon frequency. So any attempt at using guided waves requires that this behavior be well understood.

Simulation and modeling:

My initial research (first 2-2.5 years) consisted of computer modeling of such guided waves in complex structures. One way to do this is by using the Finite Element method but it is highly computationally intensive and there is virtually no physical insight in the end. So we used what’s called a Semi Analytical Finite Element (SAFE) method. My contribution till this point was small. Later though, I was able to join SAFE and FE in a tool that had the advantages of both methods. This hybrid method (unimaginatively called Global-Local) had the computational efficiency and physical insight of SAFE and the versatility of FE and still remains a  useful tool.

Nonlinear guided waves:

At this point I became interested in the nonlinear aspects of guided waves. I also wanted to do some theoretical work but frankly speaking, the kind of physics I work in has been around for so long that it appears that most of the fundamental problems have already been solved and what all remain, if any, are excruciatingly difficult. So I was surprised to find that there was a very fundamental physical phenomenon in nonlinear plate waves (Lamb waves) that was neither known nor had a mathematical explanation. Part of the problem lay in the unbearably cumbersome mathematics of the solution. It was possible to solve the problem, but owing to the mathematical complexity,  it wasn’t possible to find meaningful physical implications of the solution. Fortunately though, I was able to find a workaround wherein the underlying physics could be understood without going into the complete solution.  This reasoning was subsequently extended to rod waves for other results.