Bragg gratings are used in many integrated optical components for a large variety of applications including optical signal processing, high-speed optical communications, sensing systems and networking [1-4]. Examples of prevalent grating-based components are add/drop filters for wavelength-division-multiplexing (WDM) communication systems, grating-assisted couplers, dispersion compensators, distributed-feedback and distributed-Bragg-reflector lasers [5-8].
Great compatibility with CMOS structures is made by silicon-on-insulator (SOI) technology, a promising candidate for large-scale integration of optics and electronics on a single silicon platform [5, 7, 9]. Therefore, in the past decades, SOI waveguides have been attractive choices for integration with Bragg grating structures [8, 10, 11]. A main group of SOI structures are strip waveguides which are used in many applications, including WDM add/drop filters and grating couplers [4, 12, 13]. In strip waveguides, or photonic wires, usually submicron cross sections are used [9, 10]. Moreover, the strong confinement of light in the core due to the high index contrast between Si and SiO_{2} is a beneficial feature of strip waveguides. So, as a result of small waveguide dimensions and high confinement of light, even small perturbations on the sidewalls can result in high coupling strength [10].
Various methods are suggested for implementing Bragg gratings on the SOI waveguides. Two main categories are: (1) using ion implantation for refractive index modulation [14], and (2): physically corrugating the top surface or the sidewalls of the waveguides [9, 11, 15].
light rays reflects back around the Bragg wavelength when it propagates through the Bragg gratings. The Bragg condition is satisfied at the Bragg wavelength,
where
In this paper, four apodization functions are introduced for SOI strip waveguides with sidewall corrugated gratings. Here, the analysis is done using the coupled-mode theory (CMT) and the transfer-matrix method (TMM). The effects of apodization functions on the side-lobe level, the full width at half maximum (FWHM), and the reflectivity of the grating reflection spectrum are investigated. Improvement of the apodization functions, in order to achieve better side-lobe reduction and filtering characteristics, is accomplished by changing the parameters of the apodization functions.
Physical perturbation of the waveguide causes the refractive index modulation and brings the coupling between the forward and backward propagation modes. It occurs when the phase-matching condition of Eq. (1) is satisfied or simply when
Consider Ψ_{1}(
where
where the relative permittivity,
Since no gain or loss is assumed,
where
Here Δ
where
Equations (6) and (7) can be rewritten in the following form [19]:
where S is a 2× 2 matrix. The relation between the fields at
where C is the transfer matrix and
Matrix exponentials in Eq. (11) can be solved by the Pade approximation [19] and the matrices of S_{1} and S_{2} are represented by [7]:
Thus, the total transfer matrix can be described as follows:
By using Eq. (14), the reflectivity and the transmittivity of the grating can be represented by:
The relation between the reflectivity
As it was mentioned, by using the CMT the relation between the forward and the backward propagating modes can be described through a set of equations. On the other hand, by utilizing the TMM, the grating structure is divided into N segments with uniform gratings. For each segment, applying the CMT results in a transfer matrix which can relate the fields at two ends of the segment. Therefore, overall changes in the grating (along the structure as a result of the apodization functions) can be modeled by multiplying the transfer matrices of segments.
Typical structures of sidewall corrugated SOI strip waveguide (SWC-SOI-SW) with uniform gratings and an exemplary structure with apodized gratings are shown in Fig. 1.
The waveguide consists of a silicon layer over the surface of a buried oxide layer with the thicknesses of 220 nm and 2 μm, respectively. For all waveguides reported in this paper, the strip width, W, the grating period, Λ, and the duty cycle are considered to be 500 nm, 310 nm and 50%, respectively. The corrugation width, ΔW, and the grating length L are changed for different waveguides. The phase-matching condition, given in Eq. (1), and the effective refractive index of the fundamental mode of the SWC-SOI-SW are depicted in Fig. 2(a). It was assumed that the corrugation width and the grating length are 5 nm and 620 μm, respectively, The crossing point of the two diagrams in Fig. 2(a) shows the resonant wavelength of the grating structure. The transmission and the reflection spectra of the waveguide are shown in Fig. 2(b).
As is clear, the resonant peak of the transmission spectrum is compatible with the Bragg wavelength of Eq. (1). The response exhibits the FWHM of 3.31 nm, the high side-lobe level of -1.8 dB, and the reflectivity of 100%. Suppression of these high side-lobes can be performed by using the apodized gratings. To do this, many types of grating apodization can be utilized. The apodization functions introduced in this paper are:
Exponential 1:
Exponential 2:
where
The schematic of the apodized waveguide with the apodization function of
Spectral features of the apodized waveguide with the apodization function of Exponential 1
As it is evident, by varying the apodization parameters, the value of the FWHM, the side-lobe levels and the reflectivity would be changed dramatically. An inspection in the results reveals that the FWHM is lower than the un-apodized or uniform waveguide. Therefore, this function can be used for applications with the requirement of the low bandwidth grating structures such as WDM filters [3, 10]. By comparing the results, it is found that, increasing the parameters
On the other hand, among the reported results in Table 1, the lowest values of the FWHM and the side-lobe are 1.09 nm and -15.7 dB, respectively, which can be obtained for
Spectral features of the apodized waveguide with the apodization function of Exponential 1 for b=c= 1, a=0.8 and different lengths
As shown in Table 2, grating length increment leads to an increase in the reflectivity and the side-lobe level while the FWHM decreases and reaches below 1nm. Here, the minimum value of the FWHM is 0.92 nm which is obtained for L=1000 μm.
In Fig. 6 the schematic and the spectra of the apodized waveguide with the apodization function of
Spectral features of the apodized waveguide with the apodization function of Exponential 2
The simulations of apodized waveguides are performed using the structures with the grating length of 620 μm and the corrugation width of 2 nm. As it is clear, compared to the un-apodized waveguide with the corrugation width of 5 nm, in some cases, the FWHM of the apodized waveguide is increased considerably. From Table 3, it can be seen that by enhancing the apodization f-parameter, the FWHM decreases dramatically. Moreover, the side-lobe level decreases when this parameter increases from
The schematic of the apodized waveguide with the apodization function of
[TABLE 4.] Spectral features of the apodized waveguide with the apodization function of Polynomial
Spectral features of the apodized waveguide with the apodization function of Polynomial
As shown in Fig. 7, the apodized waveguide presents extremely good filtering behavior and the side-lobe oscillations of the reflection spectra are entirely removed. The calculated smooth spectra can be utilized in applications such as WDM communication systems where the presence of side-lobes is considered to be a drawback. Fig. 7(b) demonstrates the impact of changing the apodization k-parameter on the amplitude of the side-lobes, for apodization function of
The impacts of enhancing the grating length on the reflection spectra of the apodized structure with the apodization function of
Spectral features of the apodized waveguide with the apodization function of Polynomial for j=0.6, k=2 and different lengths
Comparing the simulation results of un-apodized structure, it is clear that the completely-flat apodized spectra are much narrower than that of the un-apodized one. For
The schematic of the apodized waveguide with the last apodization function of
[TABLE 6.] Spectral features of the apodized waveguide with the apodization function of z-power
Spectral features of the apodized waveguide with the apodization function of z-power
As shown in Fig. 9(b), by increasing the apodization
A comparison between the un-apodized and apodized waveguides for
On the other hand, by decreasing the apodization
Spectral features of the apodized waveguide with the apodization function of z-power for m=1, n=？1, p=0, q=-1 and different lengths
As can be deduced from Fig. 10 and Table 7, higher reflectivity and lower FWHM are obtained by increasing the grating length. By comparing the spectra of un-apodized and the apodized waveguides, it is found that much narrower spectra with lower side-lobes can be achieved by using the apodization. Moreover, the FWHM of 0.74 nm for the grating length of 1300 μm is the minimum value among the apodized waveguides proposed in this paper.
In this paper, four apodization functions, applied to SOI strip waveguide with sidewall corrugated gratings are proposed. The effects of apodization functions on the FWHM, the side-lobe level, and the reflectivity are studied. Compared to the reported results for un-apodized waveguide by Wang