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Abstract
Due to the ability of changing light propagation path direction, curved waveguide Bragg grating (CWG) plays an important role in photonic integrated circuits. In this paper, we proposed a cascaded sampled Bragg grating on tilted waveguide (CSBG-TW) structure to equivalently realize CWG. As an example, by designing two-dimensional (2D) sampled gratings, the direction of +1st sub-grating vector in CSBG-TW can be changed. Then if a curved waveguide is divided into several sections of tilted waveguide, we can keep the grating direction being always parallel to the longitudinal direction of each section of tilted waveguide, while the basic grating is uniform. Hence, the required CWG can be equivalently realized, and the light responses such as reflection Bragg wavelength shift and backward mode convert caused by the tilted grating in curved waveguide can be compensated for. The results show that the sampling structures of CSBG-TW is micro-scale and the difference between reflection intensity between the CSBG-TW with four section tilted waveguide and CWG as design target is less than 0.1 dB. Compared with CWG, the CSBG-TW allows convenient holographic exposure and the wavelength can be accurately controlled. Therefore, the CSBG-TW can be used in various photonic integrated devices that require changing propagation paths.
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1. Introduction
The curved Bragg grating (CWG) has wide applications as it can change light propagation path direction and achieve required light response simultaneously [1–3]. Compared with traditional straight Bragg grating (SWG), the CWG is favored as the superiority of the more compact size [4]. Over the past decades, various photonic devices have been proposed based on CWGs, including distributed feedback (DFB) lasers, mode converter, and laser array [5–7].
However, the fabrication of CWG requires lithography with high accuracy due to the grating direction usually changes continuously along the curved waveguide [8–10], especially when fine nano-scale grating structure is inserted for special responses [11], such as π-phase-shift grating for light resonance [12] and chirped grating for recompression of dispersed light pulse [13]. Although high-precision CWG can be fabricated by Electron Beam Lithography (EBL), it still suffers from problems such as stitching errors, proximity effects, and long-time writing [14]. To overcome these drawbacks, Xiangfei Chen et al. proposed the reconstruction-equivalent-chirp (REC) technique to realize the light response of waveguide gratings by micro-scale fabrication [15,16]. In the REC technique, the sampling structures are microscale. Therefore, the sampling structures can be obtained by micro-lithography. The basic grating can be fabricated by standard holographic exposure, which significantly reduces the fabrication accuracy requirement, cost, and period [17,18]. However, it should be noted that grating based on REC has an obvious drawback that its coupling coefficient is only 1/π of common grating [19,20].
In this paper, we proposed and studied a cascaded sampled Bragg grating on tilted waveguide (CSBG-TW) using the rigorous finite-difference time-domain (FDTD) method using REC technique. The decrease of coupling coefficient of +1st sub-grating is compensated by increasing the etching depth of basic grating. The simulation results show that CSBG-TW can equivalently realize the same light response as a CWG at the same curved waveguide by designing the two-dimensional (2D) sampled gratings and dividing curved waveguide into multi-section tilted waveguide. As a design example, the direction of +1st sub-grating vector in CSBG-TW can change along multi-section tilted waveguide. Then we can keep the grating direction being always parallel to the longitudinal direction of each section of tilted waveguide, while the basic grating is uniform. Therefore, the light responses such as reflection Bragg wavelength shift and backward mode convert caused by the tilted grating can be compensated. The CSBG-TW realizes the phase matching of the multiple light responses of multi-cascaded by designing sampling structures. The multiple light responses can be equivalent to one light response transmitted in the waveguide when the phase matching condition is satisfied. In addition, the CSBG-TW simplifies the analysis process of curved waveguide by dividing curved waveguide into multi-section tilted waveguide and designs the grating pattern of the interface between the two grating sections to avoid the grating phase shift. Owing to the micro-scale of 2D sampling structures of CSBG-TW, the light response can be accurately controlled compared with the traditional CWG. The proposed CSBG-TW extends the REC technique into a 2D Bragg grating in curved waveguide for the first time, which can reduce fabrication accuracy requirements and pave a new path for changing the direction of light propagation path with complex light responses for photonic integrated devices.
2. Principle
2.1 Principle of C-TSBG based on sampling structures
Figure 1(a) shows a 2D schematic of three-cascaded tilted Bragg grating (3C-TBG). Here, φdu is the tilted angle of uth cascaded diffracted light wave vector, ΛTu and φu are the period and tilted angle of the uth cascaded tilted grating respectively. Figure 1(b) shows the corresponding 2D phase-matching conditions, which can be written as [21]:
where ${{\vec{K}}_\textrm{I}}$ is the incident light wave vectors, ${{\vec{K}}_{\textrm{du}}}$ is the uth cascaded diffracted light wave vector, and ${{\vec{K}}_{\textrm{Tu}}}$ is the uth cascaded grating vectors. The ${{\vec{K}}_{\textrm{du}}}$ is expressed as: where n is the refractive index of material, λ is the Bragg wavelength of light in vacuum, ${{\vec{e}}_{\textrm{du}}}$ denotes the unit direction vectors of ${{\vec{K}}_{\textrm{du}}}$. The λ is determined by [22]:According to (3), in order to accurately control the Bragg wavelength, the fabrication of the cascaded tilted Bragg grating (C-TBG) requires high accuracy due to the ΛTu is nano-scale and the tilted angle of uth cascaded tilted grating φu also has great influence on the Bragg wavelength. Therefore, we proposed the cascaded tilted sampled Bragg grating (C-TSBG) to equate the C-TBG. Figure 2(a) shows the 2D schematics of the three-cascaded tilted sampled Bragg grating (3C-TSBG), which can equivalent to 3C-TBG. Here, αu is the tilted angle of the uth cascaded sampling structure relative to the y-axis, Λsu and Λ0 are the periods of the uth cascaded sampling structure and basic grating along the perpendicular direction of grating plane, respectively.
According to the Fourier analysis of the sampled grating [15,21], rating section refractive index modulation of each grating section of +1st order sub-grating (Δn + 1u) can be expressed as:
2.2 Principle of CSBG-TW to equivalently realize CWG
Curved waveguide can be considered as a multi-section tilted waveguide with tilted angle of each section gradually changing. Grating in each waveguide section can be individually designed. Hence, we can design the sampled grating in each tilted waveguide section to equivalently realize a required CWG. In detail, the +1st order sub-grating angles of the CSBG-TW gradually change based on C-TSBG. As an example, Fig. 3 shows the 2D schematics of the proposed three-cascaded sampled Bragg grating on tilted waveguide (3CSBG-TW) [see Fig. 3(a)], which is formed by sampling structures with three different tilted angles and sampling periods [see Fig. 3(b)] superimpose on the uniform basic grating [see Fig. 3(c)]. Here, the tilted angle of uth section waveguide is θu, the basic grating period is Λ0, and the periods along the uth section waveguide direction of the sampling structures are Λsm1, Λsm2 and Λsm3, respectively. The tilted angles relative to the y-axis of the sampling structures are αm1, αm2 and αm3, respectively.
Figure 4(a) shows the 2D schematics of CSBG-TW and Figs. 4(b) shows the corresponding sum of uth cascaded +1st order sub-grating vector. Here, the tilted angle of the last section waveguide equals to the tilted angle (θ) of the end of CWG. We can find that light responses of reflection Bragg wavelength shift and backward mode convert caused by the tilted grating on tilted waveguide can be compensated for when the angle of ${{\vec{K}}_{\textrm{+1u}}}$ (φmu) is equivalent to θu. Therefore, the same light response of CWG is realized by setting the sampling structures [see Fig. 4(c)]. It should be noted that the large difference of tilted waveguide angles between adjacent waveguide sections may result in the change of grating period, which causes the grating phase shift. According to Eq. (4), if the sampling period at position L0 has s shift denoted by ΔΛsm, the sampled grating can be expressed by s(L-ΔΛsm) instead of s(L) when L > L0. Then, the index modulation of the +1st order sub-grating can be expressed as [23]:
Fig. 4. 2D schematics of (a) CSBG-TW, (b) grating vector of +1st order sub-grating and (c) the corresponding CWG.
According to Eqs. (8) and (9), the phase shift is basically compensated when the value of ΔΛsm/Λsm is in the tolerance range. Hence, we properly design the grating pattern of the interface between the two grating sections to obtained the smaller value of ΔΛsm/Λsm, which can avoid the grating phase shift.
According to Eqs. (1) and (4), the CSBG-TW can realize the CWG light response with the required curved waveguide structure when the angle of ${{\vec{K}}_{\textrm{+1u}}}$ (φu) is equal to the tilted angle of CSBG-TW uth cascaded waveguide θu. Hence, the corresponding period and tilted angle of sampling structure can be expressed as:
According to Eqs. (10) and (11), the equivalent C-TBG can be obtained, and CSBG-TW can realize the CWG light response with the required curved waveguide structure.
3. Simulation
3.1 Light response of C-TSBG
Figure 5 shows the three-dimensional (3D) schematics of sampling structures and uniform basic grating that make up the C-TSBG. The same light response as the required C-TBG is obtained from the +1st Fourier sub-grating of C-TSBG [see Fig. 5(a)] by designing the micro-scale sampling structures [see Fig. 5(b)] and keeping the basic grating uniform [see Fig. 5(c)].
Fig. 5. 3D schematics of (a) the C-TSBG, which consists of (b) sampling structure and (c) uniform basic grating.
As an example, we employ the FDTD method to numerically study the light response of the 3C-TBG and 3C-TSBG structures when the two kind of grating tilted angles φ both are 5°, 6° and 7°. The material of the core layer and substrate are both Si3N4 and the refractive index are 1.44 and 2.0, respectively. The height and width of the core layer are 250.0 nm and 2.2 µm respectively, which can support the TE0 and TE1 mode at same time. These characteristic parameters of 3C-TBG and 3C-TSBG refer to a previous article [20]. The reflection spectra are shown in Fig. 6. It can clearly see that the light response equivalence of the C-TSBG to the C-TBG is realized by designing the sampled gratings of the C-TSBG. In addition, the Bragg wavelength of C-TSBG can be designed by only changing the tilted angles and periods of sampling structure while the basic grating is uniformed [24].
Fig. 6. Reflection spectra of the 3C-TBG and 3C-TSBG when the tilted angles φu are 5°, 6° and 7° of two kind of gratings.
The uth cascaded tilted angle φu of C-TBG and C-TSBG, the uth cascaded tilted angle αu and uth cascaded sampling structures period Λsu of the C-TSBG can be calculated according to (8) and (9). The relationships between φu, αu and Λsu are shown in Fig. 7 and the characteristic parameters are listed in Table 1. We find that the Λsu decreases with the φu increases. Inversely, the αu increases with φu. Besides, when the light response of the C-TBG is the same as that of the C-TSBG, according to (2), the reflection Bragg wavelength of C-TBG changes 5 nm when the tilted angles φu of C-TBG changes by 1°, while the reflection Bragg wavelength of C-TSBG changes 0.5 nm when the tilted angle αu of the C-TSBG sampling structure changes 1°. This implies that the influence of the angle error on the wavelength can be reduced by sampling structures.
Fig. 7. Relationship between tilted angle αu and period Λsu of sampling structure in C-TSBG for different tilted angle φu.
Table 1. Characteristic parameters of C-TSBG and C-TBG for different φu
3.2 Equivalent realization of CWG based on CSBG-TW
The CSBG-TW can equivalently realize the light response of the CWG with the same curved waveguide structure by designing the sampling structures.
As an example, Fig. 8 shows the 3D schematics of three-cascaded sampled Bragg gratings on tilted waveguide (3CSBG-TW) and the equivalent CWG. The material of waveguide core with the CSBG-TW and CWG is assumed as Si3N4 with a refractive index of 2. The sampling structure and uniform basic grating [see Fig. 8(a)] that make up the 3CSBG-TW [see Fig. 8(b)]. In addition, the 3CSBG-TW consists of a straight waveguide and three section waveguides with different tilted angles [see Fig. 8(c)]. Here, the tilted angle of each waveguide section increases continuously, i.e. θ3>θ2>θ1. The periods along the waveguide direction of each cascaded sampling structures are Λms1, Λms2 and Λms3, respectively. The tilted angles relative to the y-axis of each section sampling structures are αm1, αm2 and αm3, respectively. Figure 8(d) shows the cross-section of the 3CSBG-TW. Here, the periods of the basic gratings of all the waveguide section are uniform and denoted as Λm0, which can be fabricated by standard holographic exposure. Figures 8(e)–(f) show the equivalent CWG and the top view of the CWG. The period and tilted angle of the CWG is Λc and θ, respectively. By designing the sampling structures of the CSBG-TW, the light response equivalent to that of the CWG is obtained.
Fig. 8. 3D schematics of the (a) sampling structures and uniform basic grating that comprise (b) the 3CSBG-TW; (c) the top view and (d) cross-section of the 3CSBG-TW. (e) 3D schematics of the CWG, and (f) top view of the CWG.
Figure 9 shows the reflection spectra of the four-cascaded sampled Bragg gratings on tilted waveguide (4CSBG-TW) and CWG with an incident wavelength of 1550 nm when the tilted angles of the tilted waveguide section angle θ are 1°, 2°, 3°, 4°, 5°, and 6°. The results show that the direction of +1st sub-grating vector in CSBG-TW can change along multi-section tilted waveguide by designing two-dimensional sampling structures of the 4CSBG-TW. Then we can keep the grating direction being always parallel to the longitudinal direction of the multi-section tilted waveguide to compensate for the light responses as reflection Bragg wavelength shift and backward mode convert caused by the tilted grating. Hence, the +1st sub-grating of 4CSBG-TW can equivalently realize the light response of CWG with the same tilted angle of curved waveguide θ. The intensity of the reflection of 4CSBG-TW and CWG for different θ are listed in Table 2. It can find that the difference in reflection intensity between the 4CSBG-TW and CWG (ΔR) is less than 0.1 dB. More importantly, due to the sampling structures of 4CSBG-TW are micro-scale, the fabrication of 4CSBG-TW is more conveniently and the CSBG-TW wavelength can be accurately controlled compared with the CWG.
Fig. 9. Reflection spectra of the 4CSBG-TW and CWG with wavelength of 1550 nm for different tilted angles (θ) of curved waveguide: (a) 1°, (b) 2°, (c) 3°, (d) 4°, (e) 5°, and (f) 6°.
Table 2. Reflection intensity of the 4CSBG-TW and CWG for different θ
Furthermore, for a more complex spatial-variant CMG, we can also design a complex spatial-variant sampling structure to equivalently realize the same light response. Figure 10(a) shows the reflection spectra of π-phase-shift CWG and the corresponding π-phase-shift CSBG-TW. Figure 10(b) shows the 2D schematic of π-phase-shift CSBG-TW. It can clearly see that the simulated reflection spectra of the equivalent π-phase-shift CSBG-TW nearly presents the same performance of π-phase-shift CWG.
Fig. 10. (a) Reflection spectra of the π-phase-shift CWG and π-phase-shift CSBG-TW, (b) 2D schematic of equivalent π-phase-shift CSBG-TW.
Noteworthily, the number of cascaded waveguides of the CSBG-TW (Nc) has great impact on its performance. We analyzed the CSBG-TW for different Nc when the light propagation path θ is 4°. Figures 11(a)–11(e) and Table 3 show the reflection spectra and the reflection intensity of the CSBG-TW and CWG for different Nc values, respectively. We can find that the reflection intensity of CSBG-TW increases, and the wavelength shift is compensated with the increase of cascade number from Nc = 2 to Nc = 6. Nevertheless, we can find that the length of each cascade is so small that the sampling structures cannot compensate for the light responses as reflection Bragg wavelength shift and backward mode convert when Nc = 7 from Fig. 11(f). Table 3 shows that the different in reflection intensity between the 7CSBG-TW and CWG are 1.03 dB, which larger than 0.1 dB [see Fig. 11(f)]. It means that the best CSBG-TW performance can be obtained by choosing an appropriate Nc.
Fig. 11. Reflection spectra of the CSBG-TW and CWG for different tilted waveguide section numbers Nc when the titled angle of curved waveguide is 4°: (a) two-cascaded (2CSBG-TW), (b) three-cascaded (3CSBG-TW), (c) four-cascaded (4CSBG-TW), (d) five-cascaded (5CSBG-TW), (e) six-cascaded (6CSBG-TW), (f) seven-cascaded (7CSBG-TW).
Table 3. Reflection intensity of the CSBG-TW and CWG for different Nc
Noteworthily, the values of different tilted angles between the two consecutive gratings of the CSBG-TW decreases with the increase of cascade number from Nc = 2 to Nc = 6. Therefore, the number of cascaded waveguides of the CSBG-TW (Nc) has great impact on its performance. We analyzed the CSBG-TW for different Nc when the light propagation path θ is 4°. Figures 11(a)–11(e) and Table 3 show the reflection spectra and the reflection intensity of the CSBG-TW and CWG for different Nc values, respectively. We can find that the reflection intensity of CSBG-TW increases, and the wavelength shift is compensated with the increase of cascade number from Nc = 2 to Nc = 6. Besides, according to Eqs. (8) and (9), the reflection spectra and the reflection intensity of 3CSBG-TW, 4CSBG-TW and 5CSBG-TW almost similar when the change value of ΔΛsm/Λsm caused by different Nc is in the small range. Nevertheless, we can find that the length of each cascade is so small that the sampling structures cannot compensate for the light responses as reflection Bragg wavelength shift and backward mode convert when Nc = 7 from Fig. 11(f). Table 3 shows that the different in reflection intensity between the 7CSBG-TW and CWG are 1.03 dB, which larger than 0.1 dB [see Fig. 11(f)]. It means that the best CSBG-TW performance can be obtained by choosing an appropriate Nc.
4. Discussion
Through the above analysis, the performance of the CSBG-TW is greatly impacted by αm and Λms. According to Eqs. (10) and (11), we theoretically analyzed the influence of Λ0 on αm and Λms due to the value of the Λ0 has an effect on αm and Λms. The wavelength of +1st sub-grating is constant by setting Λ+1 invariable. Figure 12 shows the calculated αm and θ of CSBG-TW when the value of Λ+1 is 397.24 nm and θ is 4° for different Λ0. It can be seen that αm and Λms decrease as Λ0 increases. For example, when Λ0 is 425.43 nm, αm is 48° and Λms is 6.23 µm, while when Λ0 is 412.42 nm, αm is 64° and Λms is 11.59 µm. The value of αm increases 16°and the value of Λms decreases 5.36 µm when the value of Λ0 from 425.43 nm to 412.42 nm. Therefore, the Λ0 has the great influence on αm and Λms and we can choose the appropriate Λ0 according to different applications of CSBG-TW.
The smaller value of Λms can completely compensate for reflection Bragg wavelength shift and backward mode convert caused by the tilted grating. As an example, we simulated and analyzed the reflection spectra of the 7CSBG-TW at θ=4° when Λ0 is 422.87 nm and 425.43 nm. As shown in Fig. 13, when Λ0 is 425.43 nm, the intensity of the reflection of the 7CSBG-TW is 0.97 dB, which is larger than -2.04 dB, which is the value when Λ0 is 422.87 nm. Therefore, a better CSBG-TW performance can be obtained by using the smaller value of Λms.
Fig. 13. Reflection spectra of the 7CSBG-TW at θ=4° when Λ0 is 422.87 nm and 425.43 nm, respectively.
5. Conclusion
In summary, we proposed CSBG-TW structure to equivalently realize light response of the CWG on same curved waveguide by designing 2D sampling structures. Compared with CWG, CSBG-TW allows conveniently holographic exposure and the wavelength can be accurately controlled. The proposed CSBG-TW extends the REC technique into a 2D Bragg grating for curved waveguide for the first time and may provide a new method for design light responses on curved waveguide for photonic devices.
Funding
Natural Science Foundation of Fujian Province (2020J01777, 2021J01972, 2021J05180, 2022H0048, 2022J011102); Chinese National Key Basic Research Special Fund (2018YFE0201200); National Natural Science Foundation of China (11704223, 61975075).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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