We present a novel type of a waveguide, which consists of several rows of periodically placed dielectric cylinders. In such a nanopillars photonic crystal waveguide, light confinement is due to the total internal reflection, while guided modes dispersion is strongly affected by waveguide periodicity. Nanopillars waveguide is multimode, where a number of modes is equal to the number of rows building the waveguide. We present a detailed study of guided modes properties, focusing on possibilities to tune their frequencies and spectral separation. An approach towards the specific mode excitation is proposed and prospects of nanopillars waveguides application as a laser resonator are discussed.
©2004 Optical Society of America
Photonic crystals (PhCs) are known for offering a set of unique options to control the flow of light by acting as waveguides, cavities, dispersive elements, etc [1–4]. Photonic crystal waveguides (PCW) are one of the promising examples of PhCs applications at micron and submicron length-scales. PCW can be formed by removing one or several lines of scatterers from the PhC lattice. Light confinement is obtained due to a complete photonic bandgap (PBG). PCW based on PhCs formed by different two-dimensional (2D) lattices of both air holes in a dielectric background and dielectric rods in air were reported [1–3]. The former platform has been chosen as a dominated fabrication principle and a lot of work has been done to find optimum designs of straight PCWs, bends, splitters, coupling systems, etc .
The principle questions addressed in PCW theories are (i) whether a complete PBG exists for a given lattice symmetry, (ii) how wide is it and (iii) whether it is possible to obtain defect (guided) modes inside a PBG? There exist many known systems possessing a complete PBG. For example, a triangular lattice of air holes in a dielectric background has a complete PBG for TE (transverse electric) polarization, while a square lattice of dielectric rods in air has a PBG for TM (transverse magnetic) polarization. Recently, it was reported, that a complete PBG can be obtained in rectangular  and quasiperiodic  lattices of dielectric rods in air.
In the same time, the total reflection due to a complete PBG is not the only waveguiding mechanism in a PhC. Unique anisotropy of PhC can cancel out a natural diffraction of the light, leading to the self-guiding of a beam in a non-channel PCW [7–9]. Another waveguiding mechanism is the total internal reflection (TIR) in one-dimensional (1D) periodic array of dielectric rods .
In this paper, we address a periodic system of dielectric rods, to show that such a system can effectively guide light due to TIR. Guided modes of a 1D periodic array of dielectric rods were studied in . The cases of an infinite lattice of rods treated as 2D PhC or 2D PhC slab were previously presented in [10–13]. Our aim here is to study a very intriguing intermediate design comprising several rows of dielectric rods. Recent progress in fabrication of different nanopillars structures proves the relevance of such a study. For example, fabrication of 2D lattices of Si nanopillars [14,15], chains of coupled microresonators with quantum wells  and CdTe-based photonic dots  have been reported.
In Section 2 of this paper, we show that a limited number of rows can effectively confine light and therefore can be regarded as a PCW. The mode structure of such PCW is discussed afterwards. In Section 3, an influence of dielectric constant and radius of rods, as well as, lattice geometry upon guided modes properties are analyzed. Finally, in Section 4, we propose a promising concept towards an application of a nanopillars PCW as a laser resonator with ab-initio implemented distributed feedback. Section 5 concludes the paper.
2. Guided modes of a nanopillars PCW
In paper  it was found that a single row of periodically placed dielectric rods has two localized guided TM modes: the first one, fundamental mode, has an even symmetry and the second one — an odd symmetry. The second mode splits from the upper continuum at the very end of the irreducible Brillouin zone (IBZ) leaving a wide gap between two modes. The waveguide is effectively single-mode in a wide frequency range.
Attaching one, two or more identical 1D periodic waveguides in parallel with the original one produces a coupled-waveguide structure. It is well known in optoelectronics that in the strong coupling regime, this leads to the splitting of the original modes into n modes, where n is a number of coupled waveguides . In Fig. 1, dispersion diagrams for 2-, 3-, 4- and 5-rows of 1D periodic array of dielectric rods are shown. All rods are placed in the square lattice vertices. Further, we will refer to such structures as nanopillar PCWs and designate them as W2, W3, W4 and W5, correspondingly. Here it is assumed, that nanopillars PWCs are extended in z direction. Dielectric constant of rods is ε=13.0 and their radius is r=0.3a, where a is a lattice constant of a square lattice. Dispersion diagrams were calculated using the plane-wave expansion method in 2D . To model nanopillars PCW the supercell method was employed. In all calculations, the supercell consists of one period in the z direction and 20 periods in the x direction, where n periods are occupied by dielectric rods. Calculations were performed for TM polarization.
Wn waveguide is a multimode waveguide with n modes grouped near the lowest original mode of a 1D waveguide. The group of n modes is separated from the higher frequency modes by a sizable bandgap. All n lowest guided modes are flat near the end of BZ. This fact leads to almost equidistant distribution of flat tails of the modes for limiting values of wave vectors close to kz=0.5 In Fig. 2, field patterns for the four lowest modes of W4 waveguide are shown for this frequency range. Field patterns were calculated by 2D FDTD method . Monochromatic excitation with the corresponding mode symmetry was used. We applied periodic boundary conditions in the z direction and absorbing [perfect matched layers (PML)] boundary conditions in the x direction. The field patterns were plotted after the steady state regime has been reached. The field is effectively localized within the waveguides region.
Analyzing the dispersion diagrams in Fig. 1 one realizes, that the n lowest guided modes of Wn waveguide are localized in the certain regions of ω, kz space. This can be understood using the following arguments. A surface truncation of an infinite 2D PhC imposes that modes of a semi-infinite crystal are modes of the infinite crystal, projected onto the direction of the crystal surface (in our case it is Γ-X direction) [1, 21–23]. In other words it means that for certain kz component of the wave vector all modes of the infinite crystal with allowed combinations of (ω, kx) are supported by the truncated crystal. A continuous band of modes exists in a semi-infinite PhC and it is bounded by Γ-X (the solid blue line in Fig. 1) and X-M (the dashed blue line in Fig. 1) dispersion curves of an infinite PhC. Truncation of the semi-infinite PhC from the second side, leaving only n integer rows of pillars, brings a new condition for the modes of the remaining system. Now, not all (ω, kx) pairs for the particular kz are allowed in the system, but only those, which support the resonant conditions in transverse direction. The spectrum becomes discrete instead of continuous. Consequently, all dispersion curves of nanopillars PCW are localized between Γ-X and X-M bands of an infinite crystal and their number is equal to n (Fig. 1).
3. Modes dispersion engineering
It is well known that by varying the filling factor and dielectric constant of rods one can tailor the frequency range of the 2D PhC bands. In addition, the use a rectangular Bravais lattice  opens another possibility to affect the crystal band structure. Taking into account, that nanopillars PCW modes are bounded by Γ-X and X-M bands of an infinite crystal, these can be used for a proper adjustment of nanopillars waveguide modes, especially their flat tails near the edge of IBZ, on the spectrum.
First, decreasing the dielectric constant of the rods simply shrinks the gap uprising in frequency, so the flat n-modes bundle travels up and intersects with higher frequency modes. In Fig. 3 dispersion diagrams of the W4 waveguide are shown for the dielectric constant of rods equals to ε=4.0, 9.0, 13.0.
Second, by changing the crystal filling factor (the radius of the rods) the position of a bandgap is disturbed following the rule of a thumb : the bigger the effective index, the lower the frequencies of the gap.
Finally, increasing the horizontal side of a square lattice by coefficient m leads to the expansion or shrinkage of the frequency range of projected bands depending on whether coefficient m>1 or m<1. Examples are shown in Fig. 4 for the case of three Bravais lattices:
square lattice (m=1.0) and rectangular lattices with m=0.5 and m=2.0. Changing the coefficient m can control both the frequency separation and positions of waveguide modes.
4. Given mode excitation
To analyze whether it is possible to excite the single mode of a nanopillars PWC, we calculated the transmission spectra of 20 periods long W4 waveguide. 2D FDTD method  with PML boundary conditions at all sides and the resolution of 16 grid points per lattice constant was used. Modes were excited by a gaussian-shaped temporal impulse with initial spatial amplitude distribution reflecting the symmetry of a chosen mode (Fig. 2). Fields were monitored by input and output detectors and transmitted waves intensities were normalized by the ones of incident waves. In Fig. 5, transmission for different modes is compared with the dispersion diagram. The positions of the cut off frequencies for different modes are resembled by spectra. Intensive ripples on transmission curves are attributed to the Fabry-Perot resonance at the ends of the waveguide.
Transmission spectra proved that the waveguiding effect in nanopillar PCW is promising for controlling the propagation of light. The final conclusion about transmission rates can be made only after full 3D FDTD calculation, which will be presented elsewhere. Nevertheless we can speculate that even 2D calculation for PCWs presented by other groups showed a reasonable agreement in the behavior of theoretical and experimental curves . In addition in , it was proposed that nanopillars in Si, SIO or SiGe can be fabricated tall enough to localize fields effectively in vertical direction and to have optimistic 3D results.
Flat tails of the nanopillars waveguide modes near the IBZ edge reflects a very low group velocity of these modes. Modes near flat tails should have a smallest energy decay rate in a finite waveguide. Then, if a given single mode can be excited, an intriguing possibility of lasing in a nanopillars PCW arises. To find modes with a smallest energy decay rate, we used FDTD method [20, 24]. After initial pulsed excitation with different spatial patterns electromagnetic energy decays with different rates for different frequencies. The modes with the highest quality factor remain excited during the longest time leading to the strong peaks in the energy spectra. The spatial patterns of the excitation were chosen to follow the mode symmetry (Fig. 2, 6). In insets of Fig. 6, open dots mark the rods where the dipoles were placed for a particular mode. In fact, energy spectrum for a given excitation shows the strong peak in the spectral position of the flat tail of the corresponding mode accompanied by a weak satellite peak at the frequency of the mode with the same parity (Fig. 6).
If rods with an active material in it can be individually addressed, a promising design of a laser resonator with distributed feedback based on the nanopillars PWC can be proposed. By simultaneously exciting only rods reflecting the symmetry of the requested mode the lasing frequency can be switched among n modes of the Wn nanopillars waveguide. By choosing a proper material, lattice geometry and rods diameter lasing frequencies can be adjusted for a given spectral range and a given spectral resolution.
In conclusion we have proposed a novel type of 2D PWC, comprising several rows of dielectric rods. Nanopillars waveguide possesses the system of localized modes separated nearly equidistantly at the boundary of the BZ. The factors having the influence upon modes dispersion have been analyzed. The most important among them might be the ratio of the rectangular Bravais lattice sides. In spite of a multimode nature of the waveguide, transmission spectra prove the possibility of a single mode excitation by imposing specific symmetry conditions onto a field source. We believe that nanopillars PCW bear a great potential for application as an effective laser resonator with a distributed feedback.
This work was partially supported by the EU-IST project APPTech IST-2000-29321 and the German BMBF project PCOC 01 BK 253.
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