Abstract:
We report experiments that demonstrate the quantization of the conductance of photonic crystal waveguides. To obtain a diffusive wave, we have added all the transmitted channels for all the incident angles. The conductance steps have equal height and a width of one half the wavelength used. Detailed numerical results agree very well with the novel experimental results.

Abstract:
In pseudo integrable systems diffractive scattering caused by wedges and impurities can be described within the framework of Geometric Theory of Diffraction (GDT) in a way similar to the one used in the Periodic Orbit Theory of Diffraction (POTD). We derive formulas expressing the reflection and transition matrix elements for one and many diffractive points and apply it for impurity and wedge diffraction. Diffraction can cause backscattering in situations, where usual semiclassical backscattering is absent causing an erodation of ideal conductance steps. The length of diffractive periodic orbits and diffractive loops can be detected in the power spectrum of the reflection matrix elements. The tail of the power spectrum shows $\sim 1/l^{1/2}$ decay due to impurity scattering and $\sim 1/l^{3/2}$ decay due to wedge scattering. We think this is a universal sign of the presence of diffractive scattering in pseudo integrable waveguides.

Abstract:
We study the propagation of waves in quasi-one-dimensional finite periodic systems whose classical (ray) dynamics is diffusive. By considering a random matrix model for a chain of $L$ identical chaotic cavities, we show that its average conductance as a function of $L$ displays an ohmic behavior even though the system has no disorder. This behavior, with an average conductance decay $N/L$, where $N$ is the number of propagating modes in the leads that connect the cavities, holds for $1\ll L \lesssim \sqrt{N}.$ After this regime, the average conductance saturates at a value of ${\mathcal O}(\sqrt{N})$ given by the average number of propagating Bloch modes $$ of the infinite chain. We also study the weak localization correction and conductance distribution, and characterize its behavior as the system undergoes the transition from diffusive to Bloch-ballistic. These predictions are tested in a periodic cosine waveguide.

Abstract:
We derive the semiclassical WKB quantization condition for obtaining the energy band edges of periodic potentials. The derivation is based on an approach which is much simpler than the usual method of interpolating with linear potentials in the regions of the classical turning points. The band structure of several periodic potentials is computed using our semiclassical quantization condition.

Abstract:
The Kubo formula for the conductance of classically chaotic systems is analyzed semiclassically, yielding simple expressions for the mean and the variance of the quantum interference terms. In contrast to earlier work, here times longer than $O( \log \hbar^{-1} )$ give the dominant contributions, i.e. the limit $\hbar \rightarrow 0$ is not implied. For example, the result for the weak localization correction to the dimensionless conductance of a chain of $k$ classically ergodic scatterers connected in series is $-{1 \over 3} [ 1 - (k+1)^{-2} ]$, interpolating between the ergodic ($k = 1$) and the diffusive ($k \rightarrow \infty$) limits.

Abstract:
The Wigner time delay of a classically chaotic quantum system can be expressed semiclassically either in terms of pairs of scattering trajectories that enter and leave the system or in terms of the periodic orbits trapped inside the system. We show how these two pictures are related on the semiclassical level. We start from the semiclassical formula with the scattering trajectories and derive from it all terms in the periodic orbit formula for the time delay. The main ingredient in this calculation is a new type of correlation between scattering trajectories which is due to trajectories that approach the trapped periodic orbits closely. The equivalence between the two pictures is also demonstrated by considering correlation functions of the time delay. A corresponding calculation for the conductance gives no periodic orbit contributions in leading order.

Abstract:
The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on waveguides" by Linton and McIvor.

Abstract:
This paper presents an analytical formula to evaluate even- and odd-mode characteristics of infinitely parallel coplanar waveguides (CPW) with the same dimensions in each CPW, given name as periodic coplanar waveguides (PCPW). The analysis yields a closed-form expression based on the quasi-TEM assumption and conformal mapping transformation. Calculated results show that both the even- and odd-mode characteristic impedances are in good agreements with the results generated by numerical solvers and available experimental data. The results are important especially for highly demand on miniaturization of circuit design to place multiple CPWs in parallel.

Abstract:
We perform a detailed numerical study of the distribution of conductances $P(T)$ for quasi-one-dimensional corrugated waveguides as a function of the corrugation complexity (from rough to smooth). We verify the universality of $P(T)$ in both, the diffusive ($\bra T \ket> 1$) and the localized ($\bra T \ket\ll 1$) transport regimes. However, at the crossover regime ($\bra T \ket \sim 1$), we observe that $P(T)$ evolves from the surface-disorder to the bulk-disorder theoretical predictions for decreasing complexity in the waveguide boundaries. We explain this behavior as a transition from disorder to deterministic chaos; since, in the limit of smooth boundaries the corrugated waveguides are, effectively, linear chains of chaotic cavities.

Abstract:
We consider electron waveguides (quantum wires) in graphene created by suitable inhomogeneous magnetic fields. The properties of uni-directional snake states are discussed. For a certain magnetic field profile, two spatially separated counter-propagating snake states are formed, leading to conductance quantization insensitive to backscattering by impurities or irregularities of the magnetic field.