## Abstract

A hybrid one-dimensional photonic crystal waveguide—a slab waveguide surrounded by one-dimensional photonic crystal with omnidirectional reflection bands, is presented; the transmission/reflection characteristics of the hybrid waveguides with a semi-cylinder defect are investigated by numerical simulation. Calculated results indicate that, as long as the semi-cylinder radius is appropriately adjusted, the high reflectance of 96.7%, nearly zero transmittance and low loss of 3% can be observed. The hybrid waveguide with a semi-cylinder defect can be used as a waveguide reflector. The simple and compact waveguide reflector is expected to be applied to highly dense photonic integrated circuits after further research.

© 2010 OSA

## 1. Introduction

In a bulk two-dimensional photonic crystal (2D PC), if the radii of a row of rods/holes are adjusted, a line-defect two-dimensional photonic crystal waveguide (2D PCW) can be formed, and light is confined within and transmitted along the defect channel by photonic band gaps (PBGs). However, there is the flat band edge of defect modes due to periodicity along the guided-wave direction, so the guided-wave band available is usually narrow. To obtain a larger guided-wave bandwidth, through a row of rods/holes in defect channel replaced by a slab waveguide (SW), a hybrid 2D PCW is proposed [1]. The hybrid 2D PCW has a large enough bandwidth of guided-wave mode. The dispersion curves of hybrid 2D PCW are close to those of SW, so the 2D PC surrounding SW is only regarded as a perturbation [2]. However, the existence of PBGs can effectively inhibit scattering loss from surface roughness caused by manufacture in SW [3].

The omnidirectional reflection bands (OFBs) of one-dimensional photonic crystals (1D PCs) is presented [4]; both dielectric mirrors [5] and Bragg fibers [6,7] based on OFBs have already been manufactured. Recently, a one-dimensional photonic crystal waveguide (1D PCW) is proposed [8,9], and light can be confined within and transmitted along defect channel full of air by OFBs.

Inspired by these points, a hybrid 1D PCW is proposed in this paper, and its dispersion characteristics are studied. The transmission/reflection characteristics of hybrid 1D PCW with a semi-cylinder defect are investigated, and a simple and compact waveguide reflector is constructed. These numerically computed results are discussed and some important conclusions are drawn.

## 2. Waveguide structures and their dispersion characteristics

Fig. 1(a)
is a schematic drawing of symmetrical SW; the refractive index of its clad and core are n_{0} and n_{2}, respectively; its core width is h. If the SW is surrounded by 1D PC with OFBs, a hybrid 1D PCW is formed, as shown in Fig. 1(b), where h_{0} is the total width of defect channel. 1D PC consists of alternating layers of material with different index/width: n_{1}/h_{1} and n_{2}/h_{2}, where is h_{1} + h_{2} = *a*, and *a* is lattice constant. Here, take n_{0} = 1 (air), n_{1} = 1.6 (polystyrene), n_{2} = 4.6 (tellurium), h_{0} = 1.75*a*, h_{1} = 0.75*a* and h_{2} = 0.25*a*.

When a light beam is incident into a periodical multilayer stack with alternative indices of n_{2}-n_{1}-n_{2}-n_{1}-n_{2}-n_{1}……from a medium with a lower index n_{0} (n_{0}<n_{1}<n_{2}), the omnidirectional reflection will occur [4,5]. For the parameters used in this paper, the frequency range of OFB for TE modes, 0.151-0.282 [*ωa/2πc*], is marked in Fig. 2
. In the following studies on dispersion and transmission characteristics of the hybrid 1D PCW, the frequency range is focused on, and the influence of OFB on SW is considered.

As an example for TE modes, the dispersion curves of SW and hybrid 1D PCW are calculated by using plane wave expansion method based on a super-cell [10]. The results for h = 0.15*a* and 0.35*a* are shown in Fig. 2, respectively. It can be seen that, in frequency range less than 0.282 [*ωa/2πc*], there is only zero-order TE mode (TE_{0}) in the two waveguides; TE0 mode difference between the two waveguides with identical h is decreasing with the increase of frequency, because the light confinement in SW is stronger and stronger, and the influence of 1D PC on isolated SW is smaller and smaller. Figs. 3(a)
, 3(b), 3(c) and 3(d) are E_{y} field distributions at A, B, C and D points of the dispersion curves in Fig. 2, respectively. It can be observed that Ey difference between A (PCW) and B (SW) points at the propagation constant 0.4 [*2π/a*] is obvious, while E_{y} difference between C (PCW) and D (SW) points at 0.9 [*2π/a*] is very little /negligible.

## 3. A semi-cylinder defect in hybrid one-dimensional photonic crystal waveguides

A semi-cylinder defect with radius *r* and index n_{2} is introduced into a sidewall of the SW with h = 0.35*a* and that of the hybrid 1D PCW with h = 0.35*a*, as shown in Figs. 4(a)
and 4(b), respectively. The influences of defect radius *r* on transmission characteristics of the two waveguides are studied by using definite-difference time-domain method [11].

For the SW with a semi-cylinder defect shown in Fig. 4(a), Figs. 5(a)
, 5(b), 5(c) and 5(d) are transmission(T), reflection (Г) and loss spectra, corresponding to *r* = 0.19*a*, 0.21*a*, 0.29*a* and 0.34*a*, respectively. It can be seen that, with the increase of *r*, the transmittance is decreasing and the radiation loss is increasing, while the reflectance is always low. Furthermore, the valley frequency of transmittance spectrum exactly correspond the peak frequency of loss spectrum, and this means larger radiation loss will occur when there is a larger disturbance in a sidewall of SW.

For the hybrid 1D PCW with a semi-cylinder defect shown in Fig. 4(b), Figs. 6(a)
, 6(b), 6(c) and 6(d) are transmission (T), reflection (Г) and loss spectra, corresponding to *r* = 0.19*a*, 0.21*a*, 0.29*a* and 0.34*a*, respectively. It can be observed that there are a sharp transmission valley and a sharp reflection peak, and the minimum transmittance is always nearly zeros regardless of *r*, while the maximum reflectance is changed with *r* due to radiation loss. With the comparison between Fig. 5 and Fig. 6, it can be seen that the radiation loss is well inhibited by introducing 1D PCW with OFBs.

The relation curve between maximum reflectance and defect radius *r* is given in Fig. 7
, where there is 0.967 of the maximum reflectance for *r* = 0.21*a*. According to Fig. 6(b), at reflection peak frequency 0.2734[*c*/*a*], radiation loss and transmittance are 0.03 and nearly zero, respectively.

## 4. A waveguide reflector

Base on the reflection/transmission characteristics of hybrid 1D PCW with a semi-cylinder defect, a waveguide reflector is constructed and shown in Fig. 8(a)
. The waveguide reflector has good performances such as high reflectance, nearly zero transmittance and low loss. Moreover, the waveguide reflector is simple and compact: a semi-cylinder with appropriate radius can be located in any necessary position, only if its circular center is located at either sidewall of SW; the width of reflection layer on each side of SW is only 4*a*.

The distributions of *E _{y}* field component in the waveguide reflector with defect radius r = 0.21

*a*at 0.2735 [

*c/a*], is given in Fig. 8(b). If the optical communication wavelength of 1.55 µm is located at the frequency 0.2735 [

*c/a*], the required lattice constant

*a*is 0.4239 µm, the radius of semi-cylinder

*r*is 0.0890 µm, the width of waveguide core

*h*is 0.1484 µm, the total width of device is 4.6629 µm.

## 5. Conclusions

A hybrid one-dimensional photonic crystal waveguide is presented, and this waveguide with a semi-cylindrical point defect can be used as a waveguide reflector, whose good performances such as high reflection, nearly zero transmission and low loss are demonstrated. The simple and compact waveguide reflector in flattened form will be studied further, and it is expected to be applied to highly dense photonic integrated circuits.

## Acknowledgement

The research has been partially supported by the National Nature Science Foundation of China (61007027) and the Fundamental Research Funds for the Central Universities, China.

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