Semianalytic treatment for propagation in finite photonic crystal waveguides

Summation of a Schlömilch type series

The ability to accurately and efficiently characterize multiple scattering of waves of different nature attracts substantial interest in physics. The advent of photonic crystals has created additional impetus in this direction. An efficient approach in the study of multiple scattering originates from the Rayleigh method, which often requires the summation of conditionally converging series. Here summation formulae have been derived for conditionally convergent Schlömilch type series ∑ s = − ∞ ∞ Z n ( | s D − x | ) × e − i n arg ⁡ ( s D − x )   e i s D sin ⁡ θ 0 , where Z n ( z ) stands for any of the following cylindrical functions of integer order: Bessel functions J n ( z ), Neumann functions Y n ( z ) or Hankel functions of the first kind H n ( 1 ) ( z ) = J n ( z ) + i Y n ( z ) . These series arise in two-dimensional scattering problems on diffraction gratings with multiple inclusions per unit cell. It is shown that the Schlömilch series involving Hankel functions or Neumann functions can be expressed as an absolutely converging series of elementary functions and a finite sum of Lerch transcendent functions, while the Schlömilch series of Bessel functions can be transformed into a finite sum of elementary functions. The closed-form expressions for the Coates's integrals of integer order have also been found. The derived equations have been verified numerically and their accuracy and efficiency has been demonstrated.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Analysis of arbitrary defects in photonic crystals by use of the source-model technique

A novel method derived from the source-model technique is presented to solve the problem of scattering of an electromagnetic plane wave by a two-dimensional photonic crystal slab that contains an arbitrary defect (perturbation). In this method, the electromagnetic fields in the perturbed problem are expressed in terms of the field due to the periodic currents obtained from a solution of the corresponding unperturbed problem plus the field due to yet-to-be-determined correction current sources placed in the vicinity of the perturbation. Appropriate error measures are suggested, and a few representative structures are presented and analyzed to demonstrate the versatility of the proposed method and to provide physical insight into waveguiding and defect coupling mechanisms typical of finite-thickness photonic crystal slabs.

Ultracompact resonant filters in photonic crystals

A novel design for an ultracompact, high-Q notch-rejection filter is presented, and an analytic expression for the transmission properties is derived. This folded directional coupler shares the properties of a Fabry-Perot resonator and a directional coupler. We compare and contrast the device to high-Q Fabry-Perot cavities in photonic crystal waveguides.