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Abstract
We propose a new kind of planar waveguide Bragg grating structure, i.e., planar waveguide moiré grating (MG), which is formed by two transverse adjacent gratings with slightly different Bragg wavelengths. It is found that this kind of structure shows the same light properties as the conventional MG that is realized by superimposing two Bragg gratings. Because the proposed MG structure is a planar pattern, the fabrication becomes much easier if applying a semiconductor microfabrication process, which is very beneficial for its applications in photonic integrated devices. Similar to the well-known Vernier effect, the coupling coefficient distribution can be easily adjusted by the alignment of the two adjacent gratings. Consequently, some special grating profiles can be achieved, such as perfect apodization with two sides of the coupling coefficient approaching zero. One important potential application of these specific features is the distributed feedback (DFB) semiconductor laser for improved properties, such as reduced spatial-hole burning and more power extraction. Some design examples are also given in this paper.
© 2017 Optical Society of America
1. Introduction
Bragg grating has a wide range of applications in optical communications [1–3]. In order to realize specific light responses, some grating structures have been proposed, such as phase shift [4], apodization [5], chirp [6], and sample [7]. Among these structures, moiré grating (MG) is one of the important structures, which was firstly proposed by Reid et al in 1990. It is formed by superimposing two Bragg gratings with slightly different periods [8]. It exhibits some special features such as perfect apodization with cosine profile and π phase shift in the middle. Therefore, based on MG, photonic devices on fiber platform, like multipassband resonators [9], dispersion compensators [10] and encoder/decoder [11] have been demonstrated. Besides, MG has also been applied in volume Bragg grating (VBG) for advanced light devices, for example, a volume monolithic device based on moiré pattern was demonstrated in photo-thermo-refractive (PTR) glass [12]. But up to now, the MGs are mainly fabricated in the photorefractive materials as mentioned above. Moreover, the fabrication method is mainly limited to double holographic exposure with slightly and high precisely controlled grating period difference (only ~0.1 nm) [8], which gives rise to very poor uniformity in practical fabrications. As a result, the applications are highly limited. Though some other methods have been proposed, such as, the scanning electron microscope (SEM) moiré method [13], atomic force microscope (AFM) moiré method [14], nanoimprint lithography and a hot embossing system [15]. But all these methods are expensive and time-consuming, so they are not suitable for large-scale manufacturing.
On the other hand, with the fast development of the photonic integrated circuits, more and more photonic devices are required to be monolithically integrated on the single chip. As a result, it requires that the fabrication of the devices must be compatible with semiconductor microfabrication processes, such as photolithography and etching. Unfortunately, the fabrication of the conventional MG structure is not compatible with these processes, since it is not a planar structure. In order to overcome this issue, we propose a new kind of planar MG structure, in which, two gratings with slightly different periods are located on two transverse sides of the waveguide. The light properties can be controlled by the two gratings, which is similar as the conventional MGs. For example, coupling coefficient distribution can be controlled by the alignment of the two gratings that can be understood in analogy with the well-known Vernier effect [16]. Particularly, the fabrication of this structure is compatible with the current microfabrication process. In addition, we also theoretically proved that, this structure can be equivalently realized by sampled grating, where the basic grating is uniform while only the sampling pattern are specifically designed. Because of large scale of the sampling pattern (several micrometers), the whole cost can be achieved by common holographic exposure and photolithography, resulting in a highly reduced cost, which is very beneficial to the practical applications of the MG based photonic integrated devices.
An important application of the proposed MG structure is the distributed feedback (DFB) semiconductor lasers, which are the key elements in the optical communications [17]. Because of the MG induced grating apodizations and phase shift, some improved performances can be easily achieved, such as to suppress spatial hole burning (SHB), to increase single longitudinal mode (SLM) properties or to increase light power extraction [18, 19]. Especially a perfect apodizaion with half period cosine profile can be realized in the proposed MG structure. By contrast, it is nearly impossible to realize such an apodization by directly change the index modulation under the current fabrication processes [18]. Some DFB semiconductor laser structures are also suggested and simulated in this paper to exhibit the effect of the MG gratings for improved light properties. Therefore, the proposed structure shows good design flexibility and opens a new way for the practical applications in the grating based photonic integrated devices.
2. Principle
The schematic of the proposed planar MG is shown in Fig. 1. It consists of two adjacent uniform gratings located in two transverse sides of the waveguide with equal index modulation amplitudes and slightly different grating periods. The morphology of the two gratings are assumed to be rectangular ridge-shaped gratings with the duty cycles of 0.5 as illustrated in Fig. 1(a). The periods of two gratings are Λ1 and Λ2 respectively and the width of the MG is W. Figures 1(b)-1(d) give the detailed views of the grating structure at different positions of the MG.
Fig. 1 The schematic of the proposed MG with (a) the full view of the grating structure and detailed views of grating structures with (b) the low coupling coefficient region, (c) high coupling coefficient region, and (d) the low coupling coefficient region again.
The relative permittivity perturbation of the proposed MG structure can be expanded as a Fourier series. Here, the 1st order component is taken into consideration and expressed as [23],
where, and Δn denote the average refractive index and refractive index modulation of the gratings, respectively. φ1 and φ2 are the initial phases of the two gratings. Equation (1) can be rewritten as,where, and . In Eq. (2), the phase term, i.e., , denotes a uniform grating with the period of ΛS. Therefore, according to the coupled mode theory [20], the coupling coefficient of the proposed planar MG can be written as,where, and are the normalized electric field component of fundamental mode and its conjugate component. We complete the integration of Eq. (3) and obtain that,where, γ00 is the half power confinement factor, which is expressed as .Through the above analysis, an equivalent refractive index modulation can be extracted for the proposed MG as,
It can be seen that there are a rapidly varying component and a slowly varying envelope in Eq. (5), which is as the same as the conventional MG [8]. This result shows that the proposed structure should have the same light property as the conventional MG. The principle can also be understood as the well-known Vernier effect, where the structure property is manipulated by the alignment of the two adjacent gratings. Consequently, the special grating structures such as perfect apodization with cosine profile and π phase shift can be realized. These special features show potential applications for the photonic integrated devices that based on grating [5, 21, 22].
3. Simulated results
Finite-difference time domain (FDTD) simulations are performed to investigate the proposed structure. Here, the two dimensional (2D) model is used to reduce the simulation time and memory. The fundamental TE pulsed mode is launched as light source. Fast Fourier Transform (FFT) is applied to calculate both the reflection and transmission spectra. The designed structure parameters are described in the following. The grating length (L) is 1126.0 μm, which is equal to one period of the slowly varying envelope (ΛC) in Eq. (2). The waveguide width (W) is 1.0 μm. The average normalized coupling coefficient () is about 2.0. The periods of the two adjacent gratings are 237.5 nm and 237.6 nm, respectively, with the initial phase difference (Δφ = φ2 – φ1) of π. Figure 2(a) shows the refractive index profile of the equivalent MG structure of Eq. (5), where one slowly varying period is plotted. Here, the π phase shift in the middle is caused by the sign abrupt change of the slowly varying cosine-function. In order to confirm that the planar MG and the conventional MG have the same properties, the conventional MG is also calculated for comparison. Figure 2(a) is the MG structure used for simulation and Fig. 2(b) is the corresponding simulated light response. The transmission spectrum, which was simulated by FDTD, agrees well with that of the conventional MG calculated by transfer matrix method (TMM) [8]. By contrast, there is an additional radiation loss around 1.537 μm in the proposed structure, which should be caused by the leaky mode generated in the antisymmetric grating region [see Figs. 1(b) and 1(d)] where TE0 and TE1 mode order conversion occurs, which will be further explained in the “Discussions” part.
Fig. 2 (a) The equivalent refractive index modulation profile of the MG structure with the length of ΛC and the initial phase difference of π. The blue solid line and the red dash line mean the rapidly varying component and the slowly varying envelope, respectively. (b) The corresponding simulated transmission and reflection spectra of the planar MG and the reflection spectrum of the conventional MG.
Side lobe is one of the main drawbacks of the Bragg gratings which may cause serious crosstalk in the related photonic devices and systems. In order to suppress side lobes, apodization is usually used [23]. But how to realize perfect apodization by directly changing the index modulation profile is still a huge challenge [5, 18, 26]. Here, if MG effect is used, various apodizations can be easily achieved. For example, if the half-period cosine envelope (HPC) with the length of ΛC/2 and the initial phase difference of π are adopted, a perfect apodization can be realized as shown in Fig. 3(a). Figure 3(b) shows the corresponding simulation results. The side lobes are fully suppressed. In addition, other types of apodization profiles can also be realized, if different grating parameters are chosen. Figure 3(c) shows the asymmetrical apodization, where the quarter-period cosine envelope (QPC) with length of ΛC/4 and the initial phase difference of π are applied. The corresponding transmission and reflection spectra are given in Fig. 3(d). This structure can be used in the DFB semiconductor laser for more output power [18]. Besides, we can also choose the grating region with length of ΛC/4 and the initial phase difference of π/2 to achieve a grating profile with middle apodization and π phase shift as shown in Fig. 3(e). The corresponding simulated transmission and reflection spectra are shown in Fig. 3(f). Because of the π phase shift, there is a transmission peak at the center of the stop-band. This structure is very beneficial for the DFB lasers to obtain good SLM resonance and suppressed SHB [17].
Fig. 3 The equivalent refractive index modulation profiles of (a) the HPC with the initial phase difference of π, (c) the QPC with the initial phase difference of π, and (e) the QPC with the initial phase difference of π/2, where the blue solid line and the red dash line mean the rapidly varying component and the slowly varying envelope, respectively. (b), (d) and (f) show the corresponding transmission and reflection spectra of (a), (c) and (e).
Here, the surface grating has been taken for study. Meanwhile, the proposed MG structure can also be achieved by other kinds of gratings such as lateral grating [24–26], which can be fabricated along with the waveguide by one-step photolithography (or E-beam lithography) and etching. Moreover, the devices with MG structures are usually polarization-dependence because of the different effective refractive indexes between TM and TE modes. However, the light properties of TM modes are similar as TE modes.
4. One realization method: two-dimensional sampled grating
The proposed planar MG can be fabricated by the current fabrication processes, such as electron-beam lithography (EBL) and ultraviolet exposure [13]. But the cost of these methods are still high. Here, we propose a new method for easy fabrication with low cost, i.e., sampled grating method [7, 27]. The schematic of the proposed sampled MG (SMG) structure is shown in Fig. 4. The basic grating is uniform with the period of Λ0 [see Fig. 4(b)]. The sampling pattern with two sampling periods, i.e., P1 and P2, are superimposed on the basic grating [see Figs. 4(a) and 4(b)]. Because the basic grating is uniform and the sampling pattern is several micrometers, the fabrication can be highly simplified to one step holographic exposure and another step of photolithography with micrometer scale. Then, the whole processes become much easier.
Fig. 4 The schematic of the SMG structure with (a) the sampling pattern, (b) the basic grating and (c) the corresponding SMG.
Here, we assume the initial phase of the basic grating is φ0 and the initial phases of the sampling pattern are φs1 and φs2, respectively. According to the Fourier analysis [27], the relative permittivity perturbation of the proposed SMG structure can be expanded as,
where, Fm is the Fourier coefficient of mth order sub-grating. The −1st order sub-grating is selected here. Then, we can see that, the structure of the −1st order sub-grating can be easily adjusted by changing the two adjacent sampling patterns. Similarly, we can also extract an equivalent refractive index modulation for the −1st order sub-grating in proposed SMG as,Fortunately, there is no initial phase φ0 of the basic grating in Eq. (7). This means the initial phase change of the basic grating does not affect the properties of the proposed SMG, which benefits the fabrication of the proposed SMG very well because the initial phase of the basic grating is hard to control. As an example, a grating structure is designed and simulated. The basic grating period is 230.0 nm. The sampling periods are 7.28 μm and 7.19 μm, respectively. As the same as the above MG, the average normalized coupling coefficient () of −1st order sub-grating of the SMG is about 2.0. The width and the length of SMG are 1.0 μm and 1126.0 μm, respectively. The simulation spectra of the SMG are shown in Fig. 5(a). Figure 5(b) shows the detailed spectra of the −1st order response which agrees well with the spectra in Fig. 2(b). Here, it is noted that, if the SMG is applied in the DFB semiconductor lasers, in order to avoid the influence of the 0th order sub-grating, the sampling period should be designed small enough to make the 0th wavelength far away from the –1st wavelength which is located in the gain region of the semiconductor material [18].
Fig. 5 (a) The simulated transmission and reflection spectra of the SMG. (b) The detailed light response of the −1st order sub-grating.
5. Applications in DFB semiconductor lasers
Because of the special feathers of the MG as analyzed above, it can be applied to design the advanced DFB semiconductor lasers. Several types of DFB laser are designed with InP material system and calculated by TMM algorithm [28]. The parameters of the Epi-wafer are the same as those in [29]. Table 1 shows the structure parameters of different DFB lasers. Here, the grating length is equal to the length of the laser cavity. All the average normalized coupling coefficients () are about 2.0. In order to obtain the SLM property, the initial phase difference and the middle π phase shift are carefully designed.
Table 1. Parameters of different DFB laser types
The calculated performances of different DFB lasers are shown in Fig. 6. The reflectivities of the two facets of the DFB lasers are assumed to be zero. It shows that the laser performances can be controlled by the different grating apodizaion profiles. As shown in Fig. 6(a) and the calculated flatness (F) in Table 1 [30], the SHB can be suppressed when middle apodizaion structures [see Fig. 3(e)] are used [30]. The normalized threshold gain margins (ΔαL) are all larger than 0.4 as shown in Fig. 6(b), which indicates the SLM operation can be ensured because ΔαL is larger than 0.25 [30]. We also designed the DFB lasers with asymmetric apodization profiles. The normalized light intensities are shown in Fig. 6(c). More light can be extracted from one laser facet with the SLM operation at the same time (ΔαLs > 0.25) [30]. Therefore, the proposed MG should be benefit for improving the performances of the DFB laser. In addition, besides the grating, all the other fabrication processes are the same as the normal DFB lasers.
Fig. 6 (a), (b) The normalized intensity distributions of different laser types, which have been shown in Table 1. (b), (d) The corresponding normalized threshold gain margins.
6. Discussions
6.1 Radiation mode
According to the coupling mode theory, the antisymmetric structure in some regions of the proposed MG [see Figs. 1(b) and 1(d)] can lead to mode conversion between forward TE0 mode and backward TE1 mode [27]. Then, the TE1 leaky mode can be formed because the designed waveguide only supports the TE0 mode. The radiation wavelength can be calculated by [31],
where neff0, neff1 represent the effective refractive index of TE0 and TE1 modes, respectively. λr is the wavelength of the radiation mode. As can be seen in Fig. 2(b), strong radiation occurs around 1.537 μm, which agrees well with the results that obtained from Eq. (8). Similar as the calculation of the coupling coefficient between forward and backward TE0 guided modes, we can get the coupling coefficient (κ01) for the conversion between TE0 guided mode and TE1 leaky mode as,where, γ01 is the overlap integration in the half waveguide region between the TE0 guided mode and the TE1 leaky mode, which is expressed as . Because much more power of the TE1 leaky mode is out of the waveguide, γ01 is much smaller than the aforementioned γ00. Consequently, an equivalent refractive index modulation can also be obtained as,Because the monitor was set just behind the grating, most of the light power of the TE1 leaky mode can still be collected. Here, two apodization structures are selected for simulation as shown in the inserted figures in Fig. 7. Different light spectra are obtained due to the different grating structures. Specifically, a transmission peak can be seen in Fig. 7(a), which agrees well with the property of grating structure with two π phase shifts [12].
Fig. 7 The simulated transmission and reflection spectra of the leaky mode in the MG with the initial phase difference of π and the grating length of (a) ΛC, (b) ΛC/4. The inserted figures are the profiles of the corresponding Δne01, where the blue solid line and the red dash line mean the rapidly varying component and the slowly varying envelope, respectively.
6.2 Bandwidth limitation
Generally, arbitrary structures of the two adjacent gratings can be used for more light properties, except for the above uniform grating [8–10]. The arbitrary phase functions of two gratings can be expressed as Φ1(z) and Φ2(z), respectively. Then, the equivalent refractive index of the grating can be written as,
If we properly design the Φ1(z) and Φ2(z), some advanced photonic devices, such as, multichannel filter [8,32], can be realized by MG structures.
But because of the radiation loss in nature, the working bandwidth of the proposed structure should be carefully designed to avoid the crosstalk coming from the radiation. Here, as an example, we assume that two adjacent gratings are linear chirped with the same chirp coefficients (c). The two initial grating periods of the chirped MG are Λc1 and Λc2, respectively. In order to simplify the analysis, the two edges of the stop band can be approximately expressed as 2neff0Λc1 and 2neff0(1 + c)Λc2. Similarly, the two edges of the radiation mode are also approximately expressed as (neff0 + neff1)(Λc1 + Λc2)/2 and (neff0 + neff1) (1 + c)(Λc1 + Λc2)/2. In order to avoid the crosstalk of the radiation, the start of the stop band should be larger than the end of the radiation wavelength, which leads to,
Therefore, the limiting condition for the chirp coefficient can be approximately expressed as,
6.3 Fabrication analysis
During the practical fabrication, two main fabrication errors must be taken into consideration, namely, the error of the grating position along the waveguide (grating position error) and the error of the initial phase difference between the two transverse adjacent gratings (initial phase error).
The grating position along the waveguide may be randomly changed, which is called random grating position error [see Figs. 8(a) and 8(b)]. Here, g denotes the random grating position error. For the current fabrication technology, it is easy to control the grating position error within 50.0 nm. Moreover, we consider the grating position errors of each grating line are independent and follow the Gauss distribution [33]. Accordingly, the coupling coefficient of MG with random grating position error can be expressed as,
where, and , which indicate the two overlap integrations between the forward and backward modes. In addition, the schematic of the ideal, error, and real coupling coefficient distributions have been shown in Fig. 8(c). It can be seen that there is just a little difference between the κi(z) and κr(z), which means the properties of the MG with random grating position error are slightly influenced [see Fig. 8(d)].Fig. 8 (a) The schematic of the random grating position error. (b) The random grating position error along the waveguide. (c)The schematic of the ideal, error, and real coupling coefficient distributions. (d) The corresponding simulated transmission spectra of the ideal and the MG with random grating position error.
While in some cases, the grating position error along the waveguide could be uniform, which is called constant grating position error. It is also analyzed in the following. Here, a constant grating position error index had been defined as δ = 2d/W × 100%, where d denotes the offset value between the MG and the waveguide, as shown in Fig. 9(a), which comes from the fact that the interface of the two transverse gratings is usually not in the middle of the waveguide. Accordingly, the coupling coefficient of MG with constant grating position error can be written as,
where, γm and γu can be expressed as, andFig. 9 (a) The schematic of the constant grating position error. (b) The coupling coefficient distributions curves versus different constant grating position errors. (c) The corresponding simulated transmission spectra with different constant grating position errors and the inserted figure is the transmission peak versus the constant grating position error.
It is found that there is a direct current (DC) component in the κ1(z), that is proportional to |d|. The value increases with the increase of |d|, as illustrated in Fig. 9(b), resulting in the decrease of the transmission peak as shown in Fig. 9(c). The inserted figure shows the curve of the transmission peak versus the constant grating position error δ. Here the waveguide structure parameters are the same as those used in Fig. 2. It can be seen that when the error δ is 10% (|d| = 50 nm), the light is slightly influenced. But when the error δ is 30% (|d| = 150 nm), nearly no transmission peak can be seen.
Besides, the initial phase difference (Δφ) between two transverse adjacent gratings can lead to the variation of the MG structure according to Eq. (5). This variation introduced in the fabrication is the above mentioned initial phase error [see Fig. 10(a)]. In order to analysis this error, a phase index φe is defined to denote the error as φe = Δφ – Δφ’. Here, Δφ and Δφ’ are the ideal and the fabricated initial phase difference, respectively. Then, the profile of the MG is changed by the φe, as illustrated in Fig. 10(b), leading to the varied light response as shown in Fig. 10(c). The inserted figure in Fig. 10(c) is the calculated curves of 3dB bandwidth versus the error φe. While for the SMG structure, the equivalent SMG profile is only related to the initial phase difference between the two sampling patterns (φs2 – φs1) according to Eq. (7). Hence the controlling of the fabrication error of SMG is much easier than the conventional MG, since the initial phase error of the sampling pattern is as the same scale as the critical dimension of sampling structure.
Fig. 10 (a) The schematic of the initial phase error. (b) The effective grating profiles with the different initial phase errors. (c) The corresponding transmission spectra and the inserted figure is the 3dB bandwidth of the transmission peak versus the initial phase error.
7. Conclusion
We first proposed a planar MG, which consists of two adjacent gratings with slightly different grating periods. According to the theoretical analysis, the planar MG shows the same properties as the conventional MG which is realized by superimposing two Bragg gratings. Because the proposed MG is planar structure, the fabrication is compatible with the current microfabrication processes. Besides, SMG structure is also proposed to further reduce the fabrication cost. Since the coupling coefficient distribution can be adjusted by changing the initial phase difference of two adjacent gratings and the length of MG, various apodization profiles can be realized, which is very beneficial for the applications such as DFB semiconductor lasers, as well as the other optical integrated devices based on gratings.
Funding
National Natural Science Foundation of China (61435014 11574141, 61504170, 61504058, and 61640419); Jiangsu Province Natural Science Foundation of China (BK20141168, and BK20140414); National “863” project of China (2015AA016902); The Fundamental Research Funds for the Central Universities (021314380092).
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