Apodized photonic crystal waveguide gratings

Apodized photonic crystal (PC) waveguide gratings are proposed to suppress sidelobes which appear in reflection spectra of usual PC waveguide gratings. By using specific functions (Gauss and Gauss-cosine functions) for the longitudinal refractive index distribution, it is possible to suppress sidelobes in the reflection spectra of PC waveguide gratings efficiently. The apodization is realized by simply changing diameters of dielectric pillars adjacent to the PC waveguide core. It is shown that by using Gauss-cosine functions for the apodization, Bragg frequency of the waveguide grating becomes insensitive to the magnitude of perturbation leading to the possibility of designing waveguide gratings with arbitrarily Bragg frequency and bandwidth by modulating geometrical parameters only.


Introduction
Optical devices based on photonic crystals (PCs) have attracted considerable attention in recent years because of their unique features such as a photonic band gap (PBG) effect, high anisotropies, and so on. Because these features enable the size of PC devices to be in the order of wavelength scale, various kinds of PC devices such as optical directional couplers, microcavities, and Mach-Zehnder interferometers, have been studied so far [1]- [6]. More recently, waveguide gratings based on PCs were proposed as an alternative component for realizing ultrasmall optical functional devices [7]. Waveguide gratings play a very important role in integrated photonics, and can be used for various optical devices, for example, as wavelength filters, mode converters, reflectors, and so on. One problem arising in waveguide gratings is sidelobes which appear in their reflection spectrum [7], [8]. It was shown that by using apodization techniques for the longitudinal refractive index distribution of the waveguide grating, the sidelobes can be suppressed in the reflection spectrum of fiber Bragg gratings [8].
In this paper, apodized PC waveguide gratings are proposed to suppress sidelobes which appear in reflection spectra of usual PC waveguide gratings. By using specific functions (Gauss and Gauss-cosine functions) for the longitudinal refractive index distribution, it is possible to suppress sidelobes in the reflection spectra of PC waveguide gratings efficiently. The apodization is realized by simply changing diameters of dielectric pillars adjacent to the PC waveguide core. It is shown that by using Gauss-cosine functions for the apodization, the Bragg frequency of the waveguide grating becomes insensitive to the magnitude of perturbation thus leading to the possibility of designing waveguide gratings with arbitrarily Bragg frequency and bandwidth by modulating geometrical parameters only. All the simulations are done by using rigorous finite-element time-domain beam propagation method (FETD-BPM) [9], [10] to ensure accurate analysis. The parameters of the calculations are selected as follows W = 10, R = 4 10 − , ΔT = 1fs, and N max = 4 10 2× , where W, R, ΔT, and N max stand for the number of perfectly matched layers, the theoretical reflection coefficient, the time interval of calculations, and the number of the calculation steps, respectively.

Photonic crystal waveguide gratings with Gaussian apodization
Here, we consider a PC waveguide grating as shown in Fig. 1 composed of dielectric pillars (rods) on square array with lattice constant a, where the diameter of rods are taken as d = 0.5a. By assuming silicon-on-insulator configuration (SOI), refractive indexes of rods and background material are, respectively, taken as 3.4 (Si) and 1.45 (SiO 2 ). Experiments of the system of pillars have been already demonstrated in Refs.
[11]- [13]. This PC has a PBG for TE modes which extends from a/λ = 0.242 to 0.289 with λ being the free-space wavelength. The propagating mode exists entirely in the PGB region. The waveguide core is formed by eliminating a row of rods, and the rods adjacent to both sides of the waveguide are alternately replaced by rods with different diameters to construct PC waveguide grating. The period is N and the diameters of modulated rods are d n (n = 1 ~ N) as shown in Fig. 2. Usual PC waveguide gratings without apodization [7] are formed for d 1 = d 2 = ··· = d n . Here, Gaussian apodization is used for the longitudinal refractive index distribution of PC waveguide gratings. In this case, d n is given by where d c is a maximum variation of the rod diameter, ω 0 is a spot size, and Λ = 2a is a one period of grating. Figure 2 shows the longitudinal d n variation of PC waveguide gratings with Gaussian apodization for d c = 0.05a, ω 0 = 0.25NΛ, and N = 52. Dashed line in Fig. 3 shows the reflection spectrum of the PC waveguide grating with Gaussian apodization for d c = 0.05a, ω 0 = 0.25NΛ, and N = 52, while the dash-dot line in Fig.   3 shows the reflection spectrum of the PC waveguide grating without apodization for d n = 0.45a and N = 52. We can see that the sidelobes in the reflection spectra are effectively suppressed by using apodization. However, reflection at the Bragg frequency is reduced. Solid line in Fig. 3 shows the reflection spectrum of PC waveguide gratings with Gaussian apodization for d c = 0.05a, ω 0 = 0.25NΛ, and N = 104. It is possible to obtain larger reflection for larger grating periods, however, small sidelobes appear in the lower frequency region. Figure 4 shows the transmission spectra of PC waveguide gratings with Gaussian apodization (d c = 0.05a and N = 104) for different values of ω 0 . Although larger reflections can be obtained by increasing the value of ω 0 , sidelobes in lower frequency region also become larger. This is due to the fact that by increasing the value of ω 0 , the longitudinal refractive index distribution comes closer to that of PC waveguide gratings without apodization. Figure 5 shows the dispersion curves of the PC waveguide grating with d n = 0.45a. We can see the anti-crossing of the propagating mode around the Bragg frequency.

Photonic crystal waveguide gratings with Gauss-cosine apodization
Although the sidelobes in the reflection spectrum of PC waveguide gratings can be suppressed by using Gaussian apodization as shown in the previous section, the suppression is not enough in the low frequency region [8]. Here, PC waveguide grating with Gauss-cosine apodization is proposed to suppress sidelobes further. Figure 6 shows a PC waveguide grating with Gausscosine apodization. All the structural parameters with regard to PC are the same as in the previous section. To form Gauss-cosine PC waveguide gratings, diameters of rods adjacent to the core are changed to d i (i = 1 ~ 2N) as shown in Fig. 6 which is given by  Fig. 10. Though PC waveguide gratings can be constructed with the system of holes as well as that of pillars, the system of holes is particularly discussed in Ref. [

Bragg frequency and bandwidth of photonic crystal waveguide gratings
The performance of waveguide gratings is characterized by their Bragg frequency, bandwidth, and reflection. These characteristics have to be carefully designed to meet the required performance. Reflection can be easily tuned by changing the number of periods of the grating. Bragg frequency and bandwidth can be tuned by changing the magnitude of refractive index in the grating. Because Bragg frequency and bandwidth of the grating are closely related to the magnitude of perturbation, both quantities are simultaneously changed for different perturbation. Therefore, it is difficult to make the Bragg frequency of the waveguide grating constant for different bandwidths. This is illustrated in Figs. 11 and 12. In particular, Figs. 11 and 12 show the transmission spectra of PC waveguide gratings for different values of d c without apodization (N = 52) and with Gaussian apodization (ω 0 = 0.25NΛ, and N = 104), respectively. For lager perturbation (larger values of d c ), larger bandwidth can be obtained and Bragg frequency is shitted to higher frequency region. Figure 13 shows the transmission spectra of PC waveguide gratings with Gauss-cosine apodization for different values of d c (ω 0 = 0.5NΛ, and N = 104). Although larger bandwidth can be obtained for larger perturbation, Bragg frequencies are almost constant. This is quite unusual and can be explained as follows. Figure 14(a) shows one period of PC waveguide grating with Gauss-cosine apodization. This is composed of two PC waveguides, that is, a PC waveguide having rods with diameters d i−1 (d i−1 > d) adjacent to the core (we call it "waveguide A" hereafter) and a PC waveguide having rods with diameters d i (d i < d) adjacent to the core (we call it "waveguide B" hereafter). Solid, dashed, and dash-dot lines in Fig.  14(b) show dispersion curves of input PC waveguide, waveguide A with d i−1 = 0.525a, and waveguide B with d i = 0.475a, respectively. Suppose β, β A , and β B are the propagation constants of input PC waveguide, waveguides A and B, and Δβ A = β A − β and Δβ B = β − β B , Bragg condition is given by According to Eq. (2), |d − d i−1 | ≈ |d − d i |, and therefore, Δβ A ≈ Δβ B . In this case, Eq. (4) is reduced to βa/2π = 0.25. This means that Bragg frequency is constant regardless of the magnitude of perturbation and Bragg reflection occurs at the frequency satisfying above condition. From Fig. 14(b), the frequency at which βa/2π = 0.25 is a/λ = 0.27 and this is consistent with the results of Fig. 13. Δω is defined as |ω + − ω − | where ω ± are frequencies at which transmission is 0.5. While Bragg frequency and bandwidth are simultaneously changed with d c for PC waveguide gratings without apodization and with Gaussian apodization, Bragg frequency is almost constant for Gauss-cosine apodization. Usually, to obtain arbitrary Bragg frequency and bandwidth, both geometrical parameters and refractive indices of materials have to be modulated. By using Gauss-cosine apodization, PC waveguide gratings having arbitrary Bragg frequency and bandwidth can be simply designed by modulating the geometrical parameters only.

Conclusions
We have proposed apodized PC waveguide gratings for suppressing the sidelobes appearing in the reflection spectra of usual PC waveguide gratings. It was shown that by using Gaussian and Gauss-cosine apodization, the sidelobes were effectively suppressed. The apodization can be realized by simply changing the geometrical parameters of PC waveguide gratings. Furthermore, it was shown that Bragg frequency of PC waveguide gratings with Gauss-cosine apodization is almost insensitive to the magnitude of perturbation leading to the possibility of designing PC waveguide gratings having arbitrarily Bragg frequency and bandwidth without modulating material properties. Although discussions about the application of PC waveguide gratings presented in this paper are limited to wavelength selective filter, other interesting applications of PC waveguide gratings may include slow light [14], because the group velocity of light can be slowed down near the stop band edge as in the coupled resonator optical waveguides [14].