# Accomplishments

Professor Soukoulis has played and continues to play a dominant role in the fields of electron, light localization, random lasers, photonic band gap materials, and left-handed materials. He is also a fellow of the American Physical Society, the Optical Society of America and of the American Association for the Advancement of Science. His research is well recognized throughout the world, since he is one of 1000 physicists who are most cited in the world for the period of 1981-1987.

Please Select a Topic:

- Relation of conductance with the transmission coefficient (g=(e
^{2}/h) T) - Connection of localization with the problem of the bound state in a potential well
- Fractal character of the wavefunction at the mobility edge
- Scaling properties in highly anisostropic systems
- Energy density CPA: A new effective medium
- Photonic Band Gap Materials
- Left-Handed Materials and Metamaterials

### 1. Relation of conductance with the transmission coefficient (g=(e^{2}/h) T)

In collaboration with Professor E. Economou, Professor Soukoulis was the first prove, by the use of the Kubo-Greenwood formula, that the conductance g of a 1d disordered system is equal to g=(e^{2}/h) T, where T is the transmission coefficient (see PRL 46. 618 (1981) and PRL 47, 973 (1981)). Landauer and the rest of the community were arguing that g=(e^{2}/h) T/(1-T) instead. This formula gives that g approaches infinity for a perfect system, which has T =1. While the Economou-Soukoulis formula of g=(e^{2}/h) T gives a finite g, even for a perfect system. It is by now well known that the Economou-Soukoulis formula applies to the two probe measurements, while the Landauer formula applies to the four probe measurements. Unfortunately, not all the community gives to Economou and Soukoulis their deserved credit. Both formulas are called Landauer's formula.

### 2. Connection of localization with the problem of the bound state in a potential well

In collaboration with Professor Economou, they have connected the self-consistent theory of localization with the problem of a quantum-mechanical bound established by observing that the self-consistent equation of the diffusion coefficient D in the localized regime is reminiscent of the Schrodinger equation for a particle of mass m in a potential well of depth Vo. This formal equivalence suggests that the problem of electron localization is a random potential is equivalent to the problem of finding a bound state in a shallow potential well. Note that a quantum-mechanical particle is always bound to a potential well in dimensions d=1,2 which explains the absence of extended states for d<=2. In 3d the potential well needs to be sufficiently attractive for a particle to bound, which explains that a localization transition takes place at a critical value of disorder. Economou, Soukoulis and their groups have used this theory, which then called potential well analogy, (PWA) to calculate the mobility edge trajectory, the localization length and the conductivity which are in excellent agreement with independent numerical data. Economou and Soukoulis have demonstrated that the PWA is capable in producing in results agreement with independent, very reliable numerical approaches developed by their groups. The advantage of the PWA approach over numerical approaches is analytical and gives results, even for complicated quantities, almost immediately. The PWA approach applied by our group to electronic localization, phonon localization and recently to light localization with tremendous success. (See PRB 28, 1093 (1983); PRB 30, 1686 (1984); PRB 34, 2253 (1986); PRB 36, 8649 (1987); PRB 44, 4685 (1991); PRB 45, 7724 (1992)).

### 3. Fractal character of the wavefunction at the mobility edge

Soukoulis and Economou back in 1984, was the first group to numerically demonstrate that the wavefunction of a 3d disordered system has fractal characteristics. They also obtained the fractal dimensionality for d=3, which still agrees with the very recent numerical results of very large systems. Their work open the road to a lot of work about fractal and multifractal behavior of disordered systems (see PRL 52, 6 (1984); PRL 53, 616 (1984); PRB 33, 4936 (1986)).

### 4. Scaling properties in highly anisostropic systems

Recently the problem of Anderson localization in anisotropic systems has attracted considerable attention, largely due to the fact that a large variety of materials, including the high-Tc materials are highly anisotropic. Economou and Soukoulis recently showed that in highly anisotropic systems of weakly coupled planes, states are localized in the direction parallel and perpendicular to the plane at exactly at the same amount of critical disorder, in support of the one-parameter scaling theory which excludes the possibility of having a wave function localized in one direction and extended in the other two. In addition they have demonstrated that the scaling functions of the isotropic system are recovered once the dimension of the system in each direction is chosen to be proportional to the localization length. (see PRL 76, 3614 (1996); PRB 56, 4297 (1997); PRB 56, 12221 (1997)).

### 5. Energy density CPA: A new effective medium

Professor Soukoulis, in collaboration with K. Busch have recently developed a new method for efficient accurate calculations of the transport properties of random media. It is based on the principle that the energy density should be uniform when averaged over length scales larger than the size of the scatterers. It has been successfully applied to both "scalar" and "vector" classical wave calculations, with favorable agreement with experiments. This new approach is of general use and can be easily extended to treat different types of wave propagation in random media, and therefore, has led to a better understanding of their properties. (see PRL 75, 3442 (1995); PRB 54, 893 (1996); PRB 57, 277 (1998).

### 6. Photonic Band Gap Materials

Over the past ten years, the Ames Laboratory (AL) group of Professors Soukoulis and Ho at Iowa State University, has achieved international recognition for their work on the theoretical understanding and the experimental realization of photonic band gap (PBG) materials. Besides initial theoretical breakthroughs which energized the field of PBG research, this group has produced mature computer programs (based upon band structure, transfer matrix, and equation of motion techniques) which allow for the rapid design of PBG structures and are invaluable for the analysis and optimization of the experimentally realized materials. They have designed, fabricated, and characterized PBG structures based upon their patented layer-by-layer dielectric structure, and obtained the highest frequency 3D PBG to be manufactured today. They have also studied the role of defects, disorder, and absorption in different classes of PBGs. In addition the AL group developed metallic PBGs and utilized PBGs in building high-directivity antennas and directional filters. These studies have significantly added to our present understanding of PBGs and accelerated the development of novel applications. The AL group is internationally recognized as one of the creators of the new field, that of PBG materials. The development of 3D PBG structures started in 1990 with the accurate calculation by Ho, Chan and Soukoulis, showing that the diamond structure has a full gap. The woodpile structure was first introduced by the AL group in 1994, and gives the largest photonic band gap, even today (2010). More than 100 publications and more than 70 invited talks on the subject have been given by the AL group at many prestigious international meetings, including the APS and EPS March meetings, IEEE, NATO, OSA, and MRS meetings. Six review articles and one in the 1997 McGraw-Hill Yearbook of Science and Technology have been published on PBGs by the AL group. Professor Soukoulis organized three NATO conferences on PBGs in 1992, 1995, and 2000. He was the director of PECS-VI that took place in Crete in June 2005. Their work was highlighted editorially in Science News, Science, Nature, Optics and Photonic News, Optical Engineering Reports, and Scientific American. Two U.S. patents were granted to the PBG effort at Ames Laboratory (5,335,240 and 5,406,573).

### 7. Left-Handed Materials and Metamaterials

Prof. Soukoulis’ group is among the prime movers and initiators of the development of left-handed materials (LHMs). This revolutionary concept of negative refraction prompted objections, based on perceived violations of causality, momentum conservation, and Fermat’s principle. Soukoulis’s group refuted all of these objections and showed that properly designed LHMs give negative refraction and sub-wavelength imaging resolution. They combined fabrications, experiments, sophisticated simulations, and analytical reasoning to infer the sign of the refractive index. Soukoulis’ group developed easier to fabricate designs, which are more compact, matchable to free space, and amenable to many possible applications. To extend LHMs to telecommunications and optical frequencies, Prof. Soukoulis was instrumental in providing designs fabricated by the Univ. of Karlrhuhe group, which exhibited negative refraction at 1.5 microns, and at 780 nm, the highest frequency achieved to date. This is the first time ever a material has shown a negative refraction index for visible light! Prof. Soukoulis’ group found fundamental limits on the highest attainable frequency for negative refraction, and provided designs that give 3d isotropic negative index refraction. In addition, they used chiral structures that provide an alternative root to negative n, strong optical activity, as well as a broadband optical polarizer. The materials that have been ‘fabricated’ recently at the nano-scale, have spectacular properties, and Prof. Soukoulis’ group will drive this field forward into a wonderful landscape of new possibilities through his innovative designs and experimental achievements.