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SOI波导基本结构一般为Si衬底上生长一定厚度的SiO2绝缘缓冲层,中间为波导芯层Si (ncore=3.47),上包层为SiO2 (ncladding=1.444),产生超高折射率差。设计的超高消光比偏振器结构如图1所示,主要由SOI输入波导、倾斜Bragg光栅波导及输出波导三部分构成。倾斜Bragg光栅由Si与SiO2两种不同材料介质层周期排列构成。该结构仅需一次曝光刻蚀,制作工艺十分简单。
图 1 倾斜Bragg光栅SOI波导偏振器原理结构
Figure 1. Schematic layout of titled Bragg grating-based polarizer on SOI waveguide
其中Bragg光栅周期为a,两种介质层的宽度分别为a1与a2,对应有效折射率分别为n1与n2。θ为Bragg光栅与光传输方向的倾斜夹角,ϕ为关于光栅界面法线方向的入射角,d为相应的周期间距。满足以下几何关系:
$$ \begin{split} &\theta + \phi = {90^ \circ } \\ &{a_1} + {a_2} = a,\;\;a \cdot \sin \theta = d \end{split} $$ (1) 满足Bragg相位条件的光经过光栅发生高效反射,而其他部分光会透射传输,利用此原理可实现TE与TM模式的分离。
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Bragg光栅由宽度分别为d1与d2的两种介质层交替组成,结构如图2所示。利用一维光子晶体能带理论可计算相应Bragg光栅的带隙分布,准确地进行光栅参数设计。
图 2 光在一维光子晶体结构中的传输特性
Figure 2. Properties of light propagated in one-dimensional photonic crystal structure
光在两种介质层中的场E1、H1与E2、H2关系可由传输矩阵 [11]表示:
$$ \left[ {\begin{array}{*{20}{l}} {{E_{\text{1}}}} \\ {{H_{\text{1}}}} \end{array}} \right] = {M_2}\left[ {\begin{array}{*{20}{l}} {{E_{\text{2}}}} \\ {{H_{\text{2}}}} \end{array}} \right] $$ (2) 经过一个周期,传输矩阵为:
$$ \begin{split} &{M_d} = \left[ {\begin{array}{*{20}{c}} {{{{m}}_{11}}}&{{{{m}}_{12}}} \\ {{{{m}}_{21}}}&{{{{m}}_{22}}} \end{array}} \right] = {M_1} \cdot {M_2} =\\ &\left[ {\begin{array}{*{20}{c}} {\cos {\delta _1}}&{ - \dfrac{i}{{{p_1}}}\sin {\delta _1}} \\ { - i{p_1}\sin {\delta _1}}&{\cos {\delta _1}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\cos {\delta _2}}&{ - \dfrac{i}{{{p_2}}}\sin {\delta _2}} \\ { - i{p_2}\sin {\delta _2}}&{\cos {\delta _2}} \end{array}} \right] \end{split} $$ (3) 式中:
${\delta _i} = \dfrac{{2\pi }}{\lambda }{n_i}{d_i}\cos {\phi _i}{\text{ (}}i{\text{ = 1,2)}}$ ; p取值与偏振模式相关,对于TE模式:${p_i} = \sqrt {{{{\varepsilon _i}} \mathord{\left/ {\vphantom {{{\varepsilon _i}} {{\mu _i}}}} \right. } {{\mu _i}}}} \cos {\phi _i}$ ;而对于TM模式:$ {p_i} = \sqrt {{{{\mu _i}} \mathord{\left/ {\vphantom {{{\mu _i}} {{\varepsilon _i}}}} \right. } {{\varepsilon _i}}}} \cos {\phi _i} $ 。根据Bloch-Fouquet理论,Bloch波矢K满足关系:
$$ K = \dfrac{i}{d}\ln \left\{ {\frac{1}{2}({{{m}}_{11}} + {{{m}}_{22}}) \pm } {\sqrt {{{\left[ {\frac{1}{2}({{{m}}_{11}} + {{{m}}_{22}})} \right]}^2} - 1} } \right\} $$ (4) 利用Bloch波矢K的取值,可判断光在一维光子晶体结构中的传输状态。若K为实数,即
${{\left| {{{{m}}_{{\text{11}}}} + {{{m}}_{{\text{22}}}}} \right|} \mathord{\left/ {\vphantom {{\left| {{{{m}}_{{\text{11}}}} + {{{m}}_{{\text{22}}}}} \right|} 2}} \right. } 2} \lt 1$ ,对应光子晶体导带;若K为虚数,相应满足${{\left| {{{{m}}_{{\text{11}}}} + {{{m}}_{{\text{22}}}}} \right|} \mathord{\left/ {\vphantom {{\left| {{{{m}}_{{\text{11}}}} + {{{m}}_{{\text{22}}}}} \right|} 2}} \right. } 2} \gt 1$ ,则光处于消逝态(光子晶体禁带),从而得到一维光子晶体的能带分布[12]。对角频率
$\omega $ 关于$2\pi c/d$ (2π/d为一维光子晶体的倒易空间点阵基矢)进行归一化,得到归一化频率$\varpi $ 表达式为:$$ \varpi = \frac{\omega }{{2\pi c/d}} = \frac{{\omega d}}{{2\pi c}} = \frac{d}{\lambda } $$ (5) 式中:c为真空中光速;λ为真空中波长。取介质层宽度比例h=d1/d,则d2=d(1-h)。波矢的平行分量ky由基矢2π/d表示为
${k_y} = k' \cdot ({{2\pi } \mathord{\left/ {\vphantom {{2\pi } d}} \right. } d})$ 。给定h,结合公式(3)与(4),根据Bloch波矢K的取值,可以计算得到两种偏振模式下一维光子晶体在$\omega - {k_y}$ 平面上的能带分布,如图3所示(灰色区域为导带,白色区域为禁带)。当ky=0时,ϕ=0,对应垂直入射。图中横坐标数值
$k{'}={k}_{y}\cdot(d/2\pi)$ ,纵坐标的数值$\varpi =\omega \cdot \left(d/2\pi c\right)$ ,则$\varpi ,k{'}$ 之间的关系为:$$ \varpi = \frac{k}{{\sqrt {\mu {\varepsilon _1}} }} \cdot \frac{d}{{2\pi c}} = \frac{{{k_y}c}}{{{n_1}\sin \phi }} \cdot \frac{d}{{2\pi c}} = \frac{1}{{{n_1}\sin \phi }}k' $$ (6) 则
$\varpi - k'$ 直线的斜率即为$1/{n_1}\sin \phi $ ,由此可得到不同入射角ϕ下光子带隙的分布。选择ϕ角入射时,TE模式禁带的下边界(${\varpi _1}$ )与TM模式导带的上边界(${\varpi _2}$ ),即可得到重叠带隙。根据公式(5),$d$ 为常数,归一化角频率${\varpi _1}\sim{\varpi _2}$ 对应的波长范围为:$$ {\lambda }_{1}\text=d\text{/}{\varpi }_{1}\text{,}{\lambda }_{2}\text=d\text{/}{\varpi }_{2} $$ (7) 取偏振器工作的中心波长
${\lambda _0} = ({{{\lambda _1} + {\lambda _2})} \mathord{\left/ {\vphantom {{{\lambda _1} + {\lambda _2})} 2}} \right. } 2}$ ,从而Bragg光栅周期可由以下公式确定:$$ d = \frac{{2{\lambda _0}{\varpi _1}{\varpi _2}}}{{{\varpi _1} + {\varpi _2}}} $$ (8) 再根据公式(1),即可确定倾斜光栅周期a、倾斜角θ等结构参数,从而实现倾斜Bragg光栅型波导偏振器的设计。
上述方法利用能带理论来设计光栅参数,也可对光在波导中的传输特性进行分析。为实现TE模式高效反射,TM低损传输,应满足如下Bragg条件:
$$ \begin{split} &{L_B}^{\rm TE} = {n_1}^{\rm TE}{d_1}\cos \phi + {n_2}^{\rm TE}{d_2}\sqrt {1 - {{\left(\frac{{{n_1}^{\rm TE}}}{{{n_2}^{\rm TE}}}\right)}^2}{{\sin }^2}\phi } = \frac{{{\lambda _0}}}{2} \\ & {L_B}^{\rm TM} = {n_1}^{\rm TM}{d_1}\cos \phi + {n_2}^{\rm TM}{d_2}\sqrt {1 - {{\left(\frac{{{n_1}^{\rm TM}}}{{{n_2}^{\rm TM}}}\right)}^2}{{\sin }^2}\phi } \lt \frac{{{\lambda _0}}}{2} \end{split}$$ (9) 式中:
${n_1}^{\rm TE}、{n_1}^{\rm TM}、{n_2}^{\rm TE}、{n_2}^{\rm TM}$ 分别为TE与TM模式在相应介质层的有效折射率,可通过数值仿真计算得到。利用上式可以验证能带理论设计参数的可靠性。
Design of Bragg grating-based ultra-high extinction ratio polarizer on silicon-on-insulator waveguide
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摘要: 波导偏振器是片上集成相干光学系统中的关键器件之一,超高消光比、低损耗、紧凑型波导偏振器的设计一直是研究的热点。基于绝缘体上硅平台的倾斜Bragg光栅被用于实现超高消光比波导偏振器结构。利用一维光子晶体能带理论分别计算TE和TM模式光的能带结构分布,选择TE模式禁带与TM导带重叠带隙设计光栅,可实现TM模式低损传输,而TE模式被Bragg光栅高效反射,从而产生超高偏振消光比。3D FDTD仿真表明:16 μm倾斜Bragg光栅波导偏振器可在中心波长1550 nm附近70 nm的带宽内,实现大于37 dB的超高消光比,器件的损耗小于0.64 dB;进一步增加光栅周期数,当长度为25 μm时,消光比可提高至46 dB。Bragg光栅倾斜角与刻蚀宽度偏差仿真表明:设计的结构加工误差容限较大,同时该结构仅需一次曝光刻蚀,工艺流程简单。Abstract: The waveguide polarizer is a key component of an on-chip integrated coherent optical system, and it has attracted much interest to design a waveguide polarizer with an ultrahigh extinction ratio, low excess loss and compact size in photonic integrated circuits. A tilted Bragg grating-based polarizer on a silicon-on-insulator platform was proposed and designed by one-dimensional photonic crystal band gap theory. The energy band structure distribution for the TE and the TM modes were calculated using the energy band theory of one-dimensional photonic crystals, and the overlap gap between the forbidden band of theTE mode and the transmission band of the TM mode were used to determine the grating structure parameters. As a result, the TM mode light could pass through the Bragg grating waveguide with a low excess loss, while the TE mode shows almost complete reflection, introducing an extremely high polarization extinction ratio. The 3D FDTD simulation results suggest that a 16 μm titled Bragg grating-based waveguide polarizer could achieve an ultrahigh extinction ratio of more than 37 dB at the central wavelength of 1550 nm over a broad bandwidth of 70 nm, and the excess loss of the device is less than 0.64 dB. With an increasing length of waveguide to 25 μm, the extinction ratio could further reach up to 46 dB. The effects of the tilt angle and etching deviation on the performance of the polarizer were also studied, and the results show that the designed structure has a good fabrication tolerance. In addition, the polarizer only needs one-step etching with a simple fabrication process.
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Key words:
- polarizer /
- Bragg grating /
- one-dimensional photonic crystal /
- SOI waveguide
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