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平衡零拍探测技术可以有效抑制共模信号,放大差模信号,其探测原理图如图1所示[30-31]。
信号光场与LO光场在一个50/50分束器上干涉。干涉后输出两个光场分别为
$\hat c$ 和$\hat d$ :$$\tag{1a} \hat c\left( t \right) = \sqrt {{\eta _1}} \left[ {{{\hat a}_s}\left( t \right) - {\rm{exp}}\:\left( {i\theta } \right){{\hat a}_L}\left( t \right)} \right]/\sqrt 2 {\mkern 1mu} $$ $$\tag{1b} \hat d\left( t \right) = \sqrt {{\eta _2}} \left[ {{{\hat a}_s}\left( t \right) + {\rm{exp}}\:\left( {i\theta } \right){{\hat a}_L}\left( t \right)} \right]/\sqrt 2 $$ 式中:
$\hat a_s^ + $ 和${\hat a_s}$ 分别为信号光场的产生和湮灭算符;$\hat a_L^ + $ 和${\hat a_L}$ 为本振光场的产生和湮灭算符;${\eta _1}$ 和${\eta _2}$ 表示PD1和PD2的量子效率;$\theta $ 为信号光场${\hat a_s} = \alpha + \delta {\hat a_s}$ 与LO光场${\hat a_L} = \beta + \delta {\hat a_L}$ 间的相位差,在实验中,可以通过改变本振光光路上带压电陶瓷(PZT)的镜子来实现对相位差$\theta $ 的控制,然后通过光电探测器测量光场$\hat c$ 和$\hat d$ 。由于LO光场强度远大于信号光场的强度,即$\beta \gg \alpha $ 。并且利用$\delta {\hat X_a}(\theta )$ 和$\delta {\hat X_b}(\theta )$ 来表示场涨落项${\hat a_s}$ 和${\hat a_L}$ 在相对相位$\theta $ 处的正交涨落算符,那么两个光电管的光电流可表示为:$$\tag{2a} {\hat I_c}\left( t \right) = \frac{{{\eta _1}}}{2}\left[ {\beta \delta {{\hat X}_a}(\theta ) + \beta \delta {{\hat X}_b}(\theta )} \right] + \frac{{{\eta _1}}}{2}{\beta ^2} $$ $$\tag{2b} {\hat I_d}\left( t \right) = \frac{{{\eta _2}}}{2}\left[ { - \beta \delta {{\hat X}_a}(\theta ) + \beta \delta {{\hat X}_b}(\theta )} \right] + \frac{{{\eta _2}}}{2}{\beta ^2} $$ 在实际的平衡零拍探测中,除了量子效率存在差异外,两个光电探测器之间还存在着增益因子和电子元件上的差异,这些可以用探测器的共模抑制比表示。由除光电二极管之外的电子元件差异引起的不平衡项用
${\eta _{imb}}$ 表示。不平衡项${\eta _{imb}}$ 被添加到平衡零拍探测器的一个臂上,用于表示探测器其他电子元件引起的不平衡。因此,这两个光电探测器上的光电流的差值可表示为:$$ \begin{split} {{\hat I}_ - }\left( t \right) = &{{\hat c}^ + }(t)\hat c(t) - {{\hat d}^ + }(t)\hat d(t) =\\ &\frac{{\beta {\eta _1}}}{2}\left[ {(1 + G)\delta {{\hat X}_a}(\theta ) + (1 - G)\delta {{\hat X}_b} + \beta (1 + G)} \right] \end{split} $$ (3) 式中:
$G = {\eta _2}{\eta _{imb}}/{\eta _1}$ 为不平衡因子,其表征了BHD的不平衡元素。通过取光电流${\hat I_ - }$ 的平方,可以计算信号光噪声方差$V$ :$$ \begin{split} V\left( {{I_ - }} \right) = & \frac{{{\beta ^2}\eta _1^2}}{4} \times \\ & \left[ {V\left( {{{\hat X}_a}(\theta )} \right){{(1 + G)}^2} + V\left( {{{\hat X}_b}} \right){{(1 - G)}^2}} \right]{\mkern 1mu} {\mkern 1mu} \end{split} $$ (4) 当阻档Signal光场时,Signal光场对应于真空状态,此时方差
$V\left( {{{\hat X}_a}(\theta )} \right)$ 是光场SNL。将信号光场注入平衡零拍探测系统,测量微弱信号光场的噪声谱的测量值可表示为:$$ {S_m} = 10\lg \frac{{{{(1 + G)}^2} + V\left( {{{\hat X}_b}} \right){{(1 - G)}^2}}}{{V\left( {{{\hat X}_a}\left( \theta \right)} \right){{(1 + G)}^2} + V\left( {{{\hat X}_b}} \right){{(1 - G)}^2}}}{\mkern 1mu} {\mkern 1mu} $$ (5) 如果BHD达到平衡,即不平衡因子G等于1,并且LO光场的强度噪声达到SNL,则公式(5)可简化为:
$$ {S_m} = {S_{{\text{real}}}} = - 10\lg \left( {V\left( {{{\hat X}_a}(\theta )} \right)} \right) $$ (6) 此时信号光场的噪声测量值等于实际值。否则,测量值和实际值会存在一个偏差E,E取决于平衡零拍探测的共模抑制比和LO光场本振的强度噪声,由下式给出:
$$ \begin{split} E = & - 10\lg \left( {V\left( {{{\hat X}_a}(\theta )} \right)} \right) - \\ & 10\lg \frac{{{{(1 + G)}^2} + V\left( {{{\hat X}_b}} \right){{(1 - G)}^2}}}{{V\left( {{{\hat X}_a}(\theta )} \right){{(1 + G)}^2} + V\left( {{{\hat X}_b}} \right){{(1 - G)}^2}}}{\mkern 1mu} {\mkern 1mu} \end{split} $$ (7) 依据公式(7),计算可得探测偏差随本振光噪声及不平衡度的变化关系,其中图2(a)为信号光场正交振幅位相噪声方差为10时,测量偏差随LO光场的强度噪声和平衡因子G的变化关系。当不平衡度为0 dB时,既BHD的共模抑制比很好时,所引起的探测偏差0;如果不平衡度不为0 dB时,随本振光噪声的增加,引起逆向测量误差。
图 2 探测偏差随本振光噪声及不平衡度的变化关系图
Figure 2. Relation diagram of detection deviation with local vibration noise and unbalance degree
图2(b)为信号光场正交振幅噪声方差为0.1时,测量偏差随LO光场的强度噪声和平衡因子G的变化关系。当不平衡度为0 dB时,即BHD的共模抑制比很好时,所引起的探测偏差为0;如果不平衡度不为0 dB时,随本振光噪声的增加,引起正向测量误差。
只有当BHD具有较好的平衡特性以及消除LO光的经典噪声才能对信号光场的正交振幅和位相噪声进行精密测量,计算可得:
$$ \Delta {i_ - } \propto {\text{ }}{\Delta ^2}\hat x\left( \theta \right) = {\Delta ^2}\left( {{{\hat X}_s}{\rm{sin}}\:\theta - {{\hat Y}_s}\rm{cos}\:\theta } \right){\mkern 1mu} {\mkern 1mu} $$ (8) 式中:
${\hat X_s} = \left( {{{\hat a}_s} + \hat a_s^ + } \right)/2$ 和${\hat Y_s} = \left( {{{\hat a}_s} - \hat a_s^ + } \right)/2 i$ 为量子态的正交振幅算符和正交位相算符。分析可知,当相位差$\theta = 0$ 时,BHD测量Signal光场量子态的正交位相分量上的噪声方差$(\Delta \langle {\hat Y_s}\rangle ){{\text{ }}^2}$ ;当相位差$\theta = \pi /2$ 时,测量Signal光场量子态的正交振幅分量上的噪声方差${(\Delta \langle {\hat X_s}\rangle )^2}$ ,若挡住Signal光场,即输入端口为真空场,测量的是真空场的量子噪声起伏[32]。 -
该BHD噪声模型是基于跨阻电路的噪声分析建立的。图中两个虚线框内为两臂光电二极管的等效电路。BHD的电子学噪声[34]主要包括四种相互独立的噪声:(1)运算放大器反馈电阻的热噪声;(2)由PD暗电流引起的散粒噪声以及PD并联电阻的热噪声;(3)运算放大器的输入电流电压噪声。
(1)反馈电阻
${R_f}$ 的热噪声可以表示为:$$\tag{9a} {e_{{R_f},Thermal}} = \sqrt {4KT\Delta f/{R_f}} \cdot \left| {{Z_s}} \right| $$ $$\tag{9b} {Z_s} = {R_f}||(1/j2\pi f{C_f}) $$ 式中:
$K$ 为玻耳兹曼常数;$T$ 为热力学温度;$\Delta f$ 为单位测量带宽;${Z_s}$ 为跨阻抗放大电路的增益阻抗;$f$ 为分析频率;${C_f}$ 为跨阻电路的反馈电容;$||$ 表示两个元件之间的并联关系。在BHD中,采用${R_f}$ 为100 kΩ,温漂系数为1 ppm(1ppm=10−6),${C_f}$ 为82 μF,$f$ 为1 Hz,计算可得${e_{{R_f},Thermal}} = 39.9\; {\rm{nV}}/\sqrt {{\rm{Hz}}}$ 。(2)光电二极管所引起的BHD的电子学噪声,包括暗电流引起的散粒噪声以及并联电阻产生的热噪声,假设
${I_{dk1}}$ 和${I_{dk2}}$ 分别为PD1和PD2的暗电流,则它们引起的散粒噪声$\Delta {i_{{\rm{PD}}1,Dark}}$ 和$\Delta {i_{{\rm{PD}}2,Dark}}$ 为:$$\begin{split} \\ \Delta {i_{{\rm{PD}}1,Dark}} = \sqrt {2e{I_{dk1}}\Delta f} {\mkern 1mu} {\mkern 1mu} \end{split} $$ (10) $$ \Delta {i_{{\rm{PD}}2,Dark}} = \sqrt {2e{I_{dk2}}\Delta f} $$ (11) 式中:e为电荷量。采用C30642光电二极管,暗电流为10 nA,则
$\Delta {i_{PD1,Dark}}$ 和$\Delta {i_{PD2,Dark}}$ 为0.566 nA,所以与反馈电阻产生的热噪声相比可以不予考虑。PD1和PD2的并联电阻产生的热噪声可表示为:$$\tag{12a} \Delta {i_{{\rm{PD}}1,Thermal}} = \sqrt {4KT\Delta f/{R_{d1}}} {\mkern 1mu} {\mkern 1mu} $$ $$\tag{12b} \Delta {i_{{\rm{PD}}2,Thermal}} = \sqrt {4KT\Delta f/{R_{d2}}} {\mkern 1mu} {\mkern 1mu} $$ 式中:
${R_{d1}} = {V_{bias}}/{I_{dk1}}$ 和${R_{d2}} = {V_{bias}}/{I_{dk2}}$ 为PD1和PD2的并联电阻。根据上式可知,加载光电二极管上的偏置电压${V_{bias}}$ 的噪声将直接影响光电二极管并联电阻产生的热噪声,传统7805输出5 V电压的偏差为±3%以上,输出电压温漂为0.8 mV/℃,这将直接导致热噪声的增加,所以必须采用低温漂及输出电压稳定的偏置供电电压。(3)运算放大器所引起的BHD电子学噪声,包括运算放大器输入电流和电压噪声。当运算放大器的输入电流噪声为
${I_{noi}}$ 时,则输入电流噪声经过运算放大后器转化后的电压噪声和来自运算放大器输入电压噪声${e_{in}}$ 为:$$\tag{13a} {e_{TIA,Current}} = {I_{noi}} \cdot \left| {{Z_s}} \right| \cdot \sqrt {\Delta f} {\mkern 1mu} {\mkern 1mu} $$ $$\tag{13b} {e_{TIA,Voltage}} = {e_{in}} \cdot \left| {{Z_n}} \right| \cdot \sqrt {\Delta f} $$ $$\tag{13c} {Z_n} = ({Z_d} + {Z_s})/{Z_d} $$ 式中:
$\left| {{Z_n}} \right|$ 为运算放大器的电压噪声增益;这里$ {Z_d} $ 表示跨阻放大电路的输入阻抗。Zd可以写为:$$ {Z_d} = \frac{1}{{j2\pi f{C_{in}}}}||\left({R_{d1}}||{R_{d2}}||\frac{1}{{j2\pi f{C_{d1}}}}||\frac{1}{{j2\pi f{C_{d2}}}}\right) $$ (14) 式中:
${C_{d1}}$ ,${C_{d2}}$ 和${C_{in}}$ 分别为PD1和PD2的结电容以及放大电路的输入电容。由于构成BHD的电子学噪声的噪声都是相互独立,因此BHD的电子学噪声可以表示为:
$$ {e_{EL,Noise}} = \sqrt {e_{{R_f},Thermal}^2 + e_{TIA,Voltage}^2} $$ (15) 将公式(9a)和公式(13b)代入公式(15)中,则总的电子学噪声能够表达为:
$$ {e_{EL,Noise}} = \sqrt {\left( {4KT/{R_f} \cdot {{\left| {{Z_s}} \right|}^2} + e_{in}^2 \cdot {{\left| {{Z_n}} \right|}^2}} \right) \cdot {\text{ }}\Delta f} {\mkern 1mu} {\mkern 1mu} $$ (16) 所以,在较低频段,探测电路的电子学噪声的主要来源是反馈电阻
${R_f}$ 的热噪声和运放的输入电压噪声。
Research on low noise balanced homodyne detection system for space-based gravitational wave detection (Invited)
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摘要: 空间引力波探测频段位于0.1 mHz~1 Hz范围内,在该频段内包含了更大特征质量和尺度的引力波波源信息。目前,基于不同尺寸及空间轨道的大型激光干涉空间引力波探测计划已经逐步实施,其中在干涉仪的激光光源系统中,需要抑制激光强度噪声及频率噪声等,光电探测作为激光噪声表征及抑制的第一级器件,其性能将直接影响激光噪声抑制效果。通过选定低噪声芯片、高稳定偏压系统的基础上,采用自减电路及跨阻放大电路进行整体电路设计;在电磁屏蔽、低温漂系数元件、低噪声供电以及主动温控等技术手段实现了高增益低噪声平衡零拍探测系统的研制;结合快速傅里叶变换法以及对数轴功率谱密度算法对其增益、带宽等性能进行评估测试,并进一步对激光的强度噪声在0.05 mHz~1 Hz频段进行探测表征。实验结果表明:所研发平衡零拍探测电子学噪声谱密度在1 mHz~1 Hz的频率范围内在3.6×10−5 V/Hz1/2以下,小于空间引力波探测对激光光源噪声要求;进一步当入射光功率为400 μW时,测量得到平衡零拍探测系统在0.1 mHz~1 Hz的频率范围内增益在20 dB以上;激光强度噪声谱密度在1 mHz处为3.6×10−2 V/Hz1/2,实现低噪声光电探测及激光强度噪声表征,为空间引力波探测中激光强度噪声表征及抑制等方面提供关键器件支撑。Abstract: The space-based gravitational wave detection frequency band is located in the range of 0.1 mHz-1 Hz, because the gravitational wave source information with larger characteristic quality and scale is contained in the aforementioned frequency band. At present, large-scale laser interferometer space-based gravitational wave detection projects based on different sizes and space orbits have been gradually implemented. It should be emphasized that the laser intensity noise and frequency noise should be suppressed in the laser source system of the interferometer. Moreover, as the first level device of laser noise characterization and suppression, the performance of photoelectric detection will directly affect the effect of laser noise suppression. First of all, on the basis of selecting low noise chip and high stable bias system, the whole circuit was designed by self-reducing circuit and transimpedance-amplifying circuit. In addition, in electromagnetic shielding, low temperature drift factor element, low noise power supply and active temperature control and other technical means, realize the development of high gain and low noise balanced homodyne detection system. Finally, the gain and bandwidth of the photodetector were evaluated and tested by combining the fast Fourier transform method and the number line power spectral density algorithm, and the intensity noise of the laser was detected and characterized in the 0.05 mHz-1 Hz band by using the detector. The experimental results show that the electronic noise spectral density of the balanced homodyne detector is less than 3.6×10−5 V/ Hz1/2 in the frequency range of 1 mHz-1 Hz, which is less than the noise requirement of the laser source for space-based gravitational wave detection. When the incident light power is 400 μW, the gain of the balanced homodyne detection system is measured to be more than 40 dB in the frequency range of 0.1 mHz-1 Hz. What’s more, the spectral density of laser intensity noise is 3.6×10−2 V/ Hz1/2 at 1 mHz. Low noise photoelectric detection and laser intensity noise characterization are achieved, which provide key device support for laser intensity noise characterization and suppression in space-based gravitational wave detection.
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图 6 BHD性能测试结果。(a)研发的BHD在2~500 Hz频段性能结果;(b)研发的BHD在0.05 mHz~1 Hz频段的性能测试结果,高精度万用表采样率为2 S/s,采样时间为5 h,LPSD处理数据重叠率为0.3
Figure 6. BHD performance test results. (a) Performance results of the BHD developed in the 2-500 Hz frequency band; (b) Performance test results of the developed BHD in 0.05 mHz-1 Hz band. The sampling rate of the high-precision multimeter is 2 S/s, the sampling time is 5 h, and the data overlap rate of LPSD processing is 0.3
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