1. Introduction

Anti-symmetric periodic structure (APS) has attracted wide interests for functional photonic devices, such as the APS based metallic-silicon waveguide to realize the parity-time symmetry [1], and the APS based waveguide grating to realize mode conversion in add and drop multiplexer [2]. Meanwhile, as a typical periodic structure, waveguide Bragg grating is an important element in optical communication systems for its wide applications [3–6 ]. To realize some special performances, Bragg grating should be carefully designed with specific nano-scale structures, such as phase shift or chirp. However, it would cause high-cost fabrication. Reconstruction-equivalent-chirp (REC) technique, which is based on predesigned sampling pattern to equivalently realize complex nano-scale structures, has been widely studied to design and fabricate DFB semiconductor lasers and other grating based devices [7–9 ], The fabrication cost is reduced with only common holography exposure and one-step micrometer-scale photolithography. No mechanical errors such as stitching error are introduced and the process time is greatly reduced compared with Electron Beam Lithography (EBL) [10]. Besides, the fabrication precision is also significantly improved, which has been confirmed by experiments [7].

Nevertheless, sampled grating can produce multi-order reflections, which can result in the interaction between different order resonances in some cases, especially for DFB laser based on REC technique [11]. Therefore, the 0th order resonance must be avoided to maintain stable single longitude mode (SLM) operation. Usually the sampling period should be small enough to make 0th order wavelength far away from gain region. But it will reduce the linewidth of the corresponding photomask, thus increasing the fabrication cost of photomask and reducing wavelength precision [9]. For this reason, some methods have been proposed to suppress 0th order resonance [12–14 ], such as, the periodically injection blocking technique, the multi-exposure sampled grating structure and the chirped grating in a bent waveguide.

In this paper, an anti-symmetrically sampled Bragg grating (ASBG) with single mode waveguide is proposed and studied for the first time. There is an initial π phase difference between transversely upper and lower sections of the waveguide to form the basic (seed) grating with APS. No light is reflected because the coupling coefficient between the forward and backward guided modes becomes zero. Besides, the APS grating is analogous to the tilted grating with radiation mode coupling. When anti-symmetric sampling structure is imposed on the APS seed grating to form ASBG, the resonance of 0th sub-grating can be fully suppressed while the ± 1st sub-gratings are equivalent to uniform gratings. As a result, the interference of the 0th order resonance can be well eliminated. In addition, some error analyses of the proposed structure are also given for practical applications.

2. Principle

The schematic of the proposed ASBG structure is shown in Fig. 1 . The seed grating has an initial π phase difference between the transversely upper and lower sections as shown in Fig. 1(b) and the grating period isΛ₀. Then, another anti-symmetric sampling structure [see Fig. 1(a)] is patterned on the seed grating to form the ASBG as shown in Fig. 1(c). In order to realize the equivalent π phase shift (π-EPS), there is an abrupt shift with half sampling period (P/2) in the middle of the sampling structure [11].

 

Fig. 1 Schematic of the proposed ASBG structure with (a) anti-symmetric sampling pattern with π-EPS, (b) APS seed grating and (c) corresponding sampled grating.

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The index modulation of the proposed ASBG can be Fourier expanded as [15]

Here Δn is the refractive index modulation and F_m is the Fourier coefficient of m^th sub-grating. Here, ΔP is P/2 and ΔΛ is Λ/2 for initial π phase difference in sampling structure and seed grating respectively. Since the ± 1st sub-gratings are usually used as working gratings and the 0th sub-gating has the largest index modulation, only these three sub-gratings are analyzed in detail as follows.

For the ± 1st sub-gratings, since the phase term of 2π[(ΔΛ/Λ₀) ± (ΔP/P)] is zero or 2π, Eq. (1) becomes the pattern of uniform grating. To achieve the required coupling coefficient in the ± 1st sub-gratings, the coupling coefficient of seed grating should be three times of that the normal uniform grating [11,13 ].Therefore, they perform as normal gratings without anti-symmetric pattern. Besides, π-EPS can be introduced in the middle of the gratings with the help of abrupt half-period shift in sampling pattern as shown in Figs. 2(a) and 2(c) . But for the 0th sub-grating, grating structure is anti-symmetric according to Eq. (1).

 

Fig. 2 Schematic of the refractive index of the (a) + 1st sub-grating (b) 0th sub-grating and (c) −1st sub-grating.

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Coupled-mode theory is used to further investigate the characteristics of the 0th sub-grating [2, 16 ]. At the Bragg condition, the maximum reflectivity from mode a to mode b (a and b represent different modes) is given by

Here λ is the operating wavelength, e_a (x, y) and e_b (x, y) are the electric field distribution profiles of mode a and mode b. In addition, ζ(x, y) describes the refractive index perturbation. In the APS waveguide Bragg grating, the expression can be given by

That is to say, Kac_ab is zero and no light will be reflected. Consequently, no resonance occurs in the 0th sub-grating.

Finally, the Bragg reflection of the 0th sub-grating in ASBG can be totally avoided while the ± 1st sub-gratings perform as that of normal gratings. It should be mentioned that, because the APS grating also can be considered as part of the centered lattice, there are some reciprocal lattice vectors which are tilted to waveguide. As a result, the APS performs similar to the tilted waveguide grating where radiation mode coupling occurs [17]. But the radiation is weak and nearly no reflective light or resonant light in the waveguide.

3. Simulation results

2D finite-difference time-domain (FDTD) simulations are performed to investigate the proposed structure. The fundamental TE pulsed mode is launched as light source. One monitor is set to collect the reflective light and another to collect the transmission light. Fast Fourier Transform (FFT) is used to calculate both the reflection and transmission spectra. Detailed parameters used in the simulation are listed in Table 1 .

Table 1. Parameters Used in the Simulation

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The transmission and reflection spectra of the ASBG structure are calculated and compared with the normal π-EPS grating structure. From Fig. 3(a) and 3(b) , the notch depth and the extinction rate of ± 1st sub-gratings are nearly the same. Therefore, the ASBG and with normal sampled grating based on REC technique should have the same performances for DFB lasers or other passive filters. The simulated model is a simplified structure and the waveguide structure can also be optimized for improved performance in practical applications.Meanwhile, the reflections of 0th sub-grating between these two structures are completely different. There is nearly no light reflected in the ASBG but intense reflection appearing in normal π-EPS grating. Thus the simulation results are in good agreement with the above theoretical analysis. The transmission peaks in ± 1st sub-gratings can be found in both ASBG and normal structure, which are caused by π–EPS. Hence, the intensive resonance can be built in the transmission point for single mode DFB laser or passive filters.

 

Fig. 3 The calculated transmission and reflection spectra of (a) the designed ASBG and (b) the normally sampled grating with π-EPS. The inserted figures are the corresponding reflection spectra.

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In addition, the wavelength spacing between + 1st and −1st sub-grating can be adjusted by sampling period. Smaller sampling period can lead to wider wavelength spacing [9]. But if the 0th resonance is fully suppressed, sampling period can be significantly enlarged. Here around 129 nm wavelength spacing is achieved. Then, the interaction between ± 1st sub-gratings can be fully avoided for such large wavelength spacing. For example, the 3dB bandwidth of gain spectrum of usually used DFB semiconductor laser is around 60 nm much smaller than 129 nm. Therefore, the sampling period can be enlarged to obtain relatively small wavelength spacing but low interaction is still satisfied. In addition, according to Eq. (2) and (3) , refractive index-change or waveguide length can be increased, or waveguide structure can be optimized to increase the overlap between grating region and optical modes. Thus, the coupling coefficient can be optimized. Meanwhile, it can be found that ripples happen on the shorter wavelength of the 0th order resonance mode [see Fig. 3(a) and Fig. 4(b) ], which should be caused by radiations from equivalently tilted gratings corresponding to the reciprocal lattice vectors in seed grating. In order to further understand the effect, detailed analysis is given in the following.

 

Fig. 4 (a) The schematic of APS seed grating related to centered lattice with basic vectors of ${\stackrel{\rightharpoonup }{a}}_1 , {\stackrel{\rightharpoonup }{a}}_2$ and corresponding reciprocal lattice vectors where ${\stackrel{\rightharpoonup }{e}}_x,{\stackrel{\rightharpoonup }{e}}_z$ are two axises and (b) wavelength of radiation mode coupling versus different waveguide widths for APS grating and tilted grating respectively. The inserted figure is their transmission spectra with waveguide width of 1.0 μm.

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APS grating can be considered as part of centered lattice, of which the reciprocal lattice vectors are expressed as ${\stackrel{\rightharpoonup }{K}}_h=h_1{\stackrel{\rightharpoonup }{b}}_1+h_2{\stackrel{\rightharpoonup }{b}}_2$, with ${\stackrel{\rightharpoonup }{b}}_1$, ${\stackrel{\rightharpoonup }{b}}_2$ are the basic vectors and h₁, h₂ donate all the position of reciprocal lattice vectors. The basic vectors are expressed as ${\stackrel{\rightharpoonup }{b}}_1=-2\pi /w{\stackrel{\rightharpoonup }{e}}_x+2\pi /{\Lambda }_0{\stackrel{\rightharpoonup }{e}}_z$ and ${\stackrel{\rightharpoonup }{b}}_2=-4\pi /w{\stackrel{\rightharpoonup }{e}}_x$ respectively. Since magnitude of high order reciprocal lattice vectors decrease rapidly, only several basic cases are taken into consideration. For both the conditions of (h₁, h₂) equal to (0, 1) and (1, 1), the reciprocal lattice vectors are shown in Fig. 4(a), which can be considered as two symmetrically tilted gratings. Radiation mode coupling occurs when phase-matching condition between guided mode and tilted grating is satisfied. Thus, the equivalent tilted grating is realized in the APS waveguide and equivalent tilted angle is θ = arctan(Λ₀/w).

In order to verify the theoretical analysis, an APS grating and a true tilted grating are simulated with the same waveguide parameters. Because the equivalent tilted angle in APS changes with waveguide width according to the expression of tilted angle, the same tilted angle is also designed along with variation of waveguide width in true tilted grating. Figure 4(b) shows the curve of radiation wavelength peak versus waveguide width. The variation of radiation wavelength of APS grating fits well with that of tilted grating. That is to say, the equivalent tiled grating can be realized in the APS waveguide grating, which may has some potential applications such as sensors [18]. In addition, the characteristics of TM mode are the same as those of the TE mode.

4. Discussion

Piezo actuator technique, with the accuracy of ± 10 nm, combined with double-holographic exposure may be a potential solution [19]. Nanoimprint lithography (NIL) could be another way to fabricate the anti-symmetric seed grating with precise nano-patterns [20]. Because the different grating structures can be controlled by large-scale sampling pattern with unchanged seed grating, only low-cost photomask with μm-scale sampling pattern is changed. Therefore, only one mask with high precision for seed grating is used, resulting in low-cost fabrication.

During practical fabrication, the potential errors are caused by imperfect fabrication of seed grating such as overlapped, separated and mixed up interface between two lateral sections and misalignment of the sampling pattern and the ridge layer.

In order to analyze seed grating pattern errors, we defined normalized reflection as NR = R_e/R_ideal. Here, R_ideal denotes the reflection peak without error and R_e denotes the reflection peak with error. The interface between two lateral sections may be overlapped (d<0) or separated (d>0) as shown in Fig. 5(a) (inserted figures). Here, d denotes the magnitude of the error. It is found that normalized reflection of + 1st sub-grating (NR ₊₁) are slightly decreased. Even when d is ± 75 nm, the decrease is only 15% while the 0th sub-grating is nearly unchanged.

 

Fig. 5 The calculated curves of normalized reflection with (a) ideal anti-symmetric sampling pattern imposed on seed grating with different lateral errors and (b) ideal anti-symmetric sampling pattern imposed on seed grating with mixed up interface errors, where d represents the magnitude of the two errors. (c) The suppression factor (S) versus the initial phase difference in the seed grating along longitudinal direction.

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The interface is usually mixed up due to imperfect fabrication, which is also taken into consideration as shown in Fig. 5(b). Fortunately, this error nearly has no influence on the performance. NRs of 0th and ± 1st are nearly unchanged with different magnitude of errors.

Besides, the initial phase difference between two sections may be not exactly equal to π shown in the inserted figure in Fig. 5(c). This error can also be considered that the value of ΔΛ is not exactly equal to Λ/2 in Eq. (1). Thus both the phase of 0th sub-grating and ± 1st sub-gratings are changed. Here, we define a factor to evaluate the suppression effectiveness (S) of Bragg reflection, which is expressed as S = 1-NR = 1-R_e/R_ideal.

As shown in Fig. 5(c) that near the region of π, 0 and 2π, the curve S changes very slowly. While it changes sharply from around 1/2π to 3/4π and from around 5/4π to 3/2π. The phase of 0th sub-grating is around π. Since the ± 1st sub-gratings are around 0 or 2π, they are within the slowly changing region. Therefore, by properly controlling the longitudinal error, the performance of ASBG in practical applications can be guaranteed.

Another potential error is the misalignment between the sampling pattern and anti-symmetric seed grating at the lateral direction. It can be seen in Fig. 6(a) that the NR ₊₁ changes with the sampling alignment error while that of the 0th sub-grating is nearly unchanged. In addition, error in alignment between ridge waveguide and sampled grating is also calculated as shown in Fig. 6(b). This error has little effect on the + 1st sub-grating. However, the Bragg reflection of the 0th sub-grating increases with magnitude of ridge alignment error. Therefore, both the two kinds of alignments need to be controlled within an appropriate region to guarantee the required performances of the ASBG.

 

Fig. 6 The normalized reflection NR with (a) different overlay alignment errors for sampled grating (Waveguide width are 1.0 μm and 2.0 μm respectively) and (b) different ridge alignment error with waveguide width of 1.0 μm and 2.0 μm respectively, where both d represent the size of the error.

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For the common exposure machines, the accuracy can reach 100 nm with well-designed alignment key such as key pattern with vernier effect. In fact, practical waveguide width forDFB lasers is usually up to around 2.0 μm. As a result, the error tolerance can be further improved. To further reduce the error, stepper can be applied, the alignment accuracy of which can be around 50 nm and even up to around 30 nm. As shown in Fig. 6, it is found that the 50 nm error has slight influence on the performance. If waveguide is up to 2.0 μm, the error tolerance is much improved.

5. Conclusion

The ASBG structure is proposed and studied for the first time. It is found that the 0th sub-grating can be totally eliminated while the ± 1st sub-gratings work the same as that of π-EPS grating. In addition, equivalent tilted grating effect is also found in the APS grating. Because the interaction between the 0th and ± 1st sub-gratings is avoided, it can be used for some potential applications such as DFB lasers with improved SLM operation. In addition, the error analysis for ASBG is also given for practical applications.