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Laser parameters requirement for third-generation ground-based gravitational wave detection

Li Qing-Hui Li Wei Sun Yu Wang Ya-Jun Tian Long Chen Li-Rong Zhang Peng-Fei Zheng Yao-Hui

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Laser parameters requirement for third-generation ground-based gravitational wave detection

Li Qing-Hui, Li Wei, Sun Yu, Wang Ya-Jun, Tian Long, Chen Li-Rong, Zhang Peng-Fei, Zheng Yao-Hui
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  • Gravitational waves (GWs), predicted by the general relativity of Albert Einstein, are ripples in space-time caused by massive accelerating objects. Since the first direct observation of GWs in 2015, more and more binary black hole mergers and neutron star merger were detected by the laser interferometer gravitational-wave observatory (LIGO) and the Virgo interferometric detector. The construction of the third-generation (3G) gravitational wave detector(GWD), whose sensitivity is ten times that of the second-generation (2G) GWD (Advanced LIGO and Virgo), can not only push the gravitational wave astronomy towards the edge of the observable universe, but also test the fundamental laws of physics and study the nature of matter. By utilizing the abandoned underground mines, Shanxi university proposes to construct a 3G ground-based gravitational wave detector with an arm length of 10 km and a strain sensitivity of 10–24 Hz–1/2, improving the location accuracy of wave source by participating in the global GWD network. The construction of 3G GWD has many technical challenges, including ultrahigh large-scale vacuum system, ultrastable seismic isolation system, high-precision control system, high-quality laser and quantum source. Theoretically, the sensitivity of GWD with equal arm length is not limited by the laser source noise. However, in the actual scenario, the sensitivity is limited by the differences in arm length, arm cavity linewidth, arm reflectivity, arm mass, arm power, and the laser parameters. In this work, based on the design sensitivity (10–24 Hz–1/2) of dual-recycled Fabry-Perot Michelson interferometer, we propose the requirements for an ultra low-noise laser, including wavelength, amplitude noise, frequency noise, beam pointing noise and fundamental mode purity. The results show that in order to achieve the design sensitivity at the Fourier frequency of 100 Hz, the wavelength of the laser source should be 1.5 μm, the output power should be higher than 200 W, the amplitude noise should be better than 10–8 Hz–1/2, and the frequency noise should be better than 100 Hz/Hz1/2. To achieve the sensitivity of 10–24 Hz–1/2 at 10 Hz analysis frequency, the requirements for the amplitude and frequency noise of the laser source are much more stringent. This study lays a solid foundation for the analysis of laser source noise and the decomposition of interferometer indexes .
      Corresponding author: Zheng Yao-Hui, yhzheng@sxu.edu.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2020YFC2200402), the National Natural Science Foundation of China (Grant Nos. 62027821, 11874250, 62035015, 12174234), the Key Research and Development Projects of Shanxi Province (Grant No. 201903D111001), and the Program for Sanjin Scholar of Shanxi Province.
    [1]

    Abbott B P, Abbott R, Abbott T D, et al. 2016 Phys. Rev. Lett. 116 061102Google Scholar

    [2]

    Abbott B P, Abbott R, Abbott T D, et al. 2016 Phys. Rev. Lett. 116 241103Google Scholar

    [3]

    Vermeulen S M, Relton P, Grote H, et al. 2021 Nature 600 424Google Scholar

    [4]

    Bailes M, Berger B K, Brady P R, et al. 2021 Nat. Rev. Phys. 3 344Google Scholar

    [5]

    Abbott R, Abbott T D, Abraham S, et al. 2021 Astrophys. J. Lett. 915 L5Google Scholar

    [6]

    Badaracco F, Rossi C D, Fiori I, Harms J, Miyo K, Paoletti F, Tanaka T, Washimi T, Yokozawa T 2021 Phys. Rev. D 104 042006Google Scholar

    [7]

    Hall E D, Kuns K, Smith J R, et al. 2021 Phys. Rev. D 103 122004Google Scholar

    [8]

    Buikema A, Cahillane C, Mansell G L, et al. 2020 Phys. Rev. D 102 062003Google Scholar

    [9]

    Adhikari R X 2014 Rev. Mod. Phys. 86 121Google Scholar

    [10]

    Bond C, Brown D, Freise A, Strain K A 2016 Living Rev. Relativ. 19 3Google Scholar

    [11]

    Matichard F, Lantz B, Mittleman R, et al. 2015 Classical Quant. Grav. 32 185003Google Scholar

    [12]

    P. Nguyen, Schofield R M S, Effler A, et al. 2021 Classical Quant. Grav. 38 145001Google Scholar

    [13]

    Adhikari R X, Arai K, Brooks A F, et al. 2020 Classical Quant. Grav. 37 165003Google Scholar

    [14]

    Hammond G, Hild S, Pitkin M 2014 J. Mod. Optic. 61 S10Google Scholar

    [15]

    Heurs M 2018 Philos. T. R. Soc. A 376 20170289Google Scholar

    [16]

    Danilishin S L, Khalili F Y, Miao H 2019 Living Rev. Relativ. 22 2Google Scholar

    [17]

    Sigg D 1997 LIGO Report No. LIGO- T970084-00 D

    [18]

    Rana A 2004 Ph. D. Dissertation (Cambridge: Massachusetts Institute of Technology)

    [19]

    Somiya K, Chen Y 2006 Phys. Rev. D 73 122005Google Scholar

    [20]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500325

    [21]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500461

    [22]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500559

    [23]

    Cahillane C 2021 Ph. D. Dissertation (Pasadena: California Institute of Technology)

    [24]

    Cahillane C, Mansell G L, Sigg D 2021 Opt. Express 29 42144Google Scholar

    [25]

    Buonanno A, Chen Y 2001 Phys. Rev. D 64 042006Google Scholar

    [26]

    Pitkin M, Reid S, Rowan S, Hough J 2011 Living Rev. Relativ. 14 5Google Scholar

    [27]

    Kwee P 2010 Ph. D. Dissertation (Hanover: Wilhelm Leibniz University)

    [28]

    Chen Z, Guo M, Zhang R, Zhou B, Wei Q 2018 Sensors 18 02603Google Scholar

    [29]

    Degallaix J, Komma J, Forest D, Hofmann G 2014 Classical Quant. Grav. 31 185010Google Scholar

    [30]

    Khalaidovski A, Steinlechner J, Schnabel R 2013 Classical Quant. Grav. 30 165001Google Scholar

    [31]

    Biscans S, Gras S, Blair C D, Driggers J, Evans M, Fritschel P, Hardwick T, Mansell G 2019 Phys. Rev. D 100 122003Google Scholar

  • 图 1  引力波探测器光路图

    Figure 1.  Diagram of the optical layout of Gravitational wave detection.

    图 2  探测灵敏度与激光功率的关系图

    Figure 2.  Detection sensitivity as a function of laser power.

    图 3  激光源振幅噪声耦合传递函数图  (a) DARM偏移、辐射压力差异和对比度缺陷引起的振幅噪声耦合; (b) 辐射压力差异引起的振幅噪声耦合

    Figure 3.  Coupling transfer function of laser amplitude noise: (a) Amplitude noise coupling due to DARM offset, radiation pressure difference and contrast defect; (b) amplitude noise coupling due to radiation pressure difference.

    图 4  探测灵敏度与激光源振幅噪声关系图

    Figure 4.  Detection sensitivity as a function of laser amplitude noise.

    图 5  激光频率噪声耦合传递函数图

    Figure 5.  Coupling transfer function of laser frequency noise.

    图 6  探测灵敏度与激光源频率噪声关系图

    Figure 6.  Detection sensitivity as a function of laser frequency noise.

    表 1  山西大学引力波探测干涉仪参数表

    Table 1.  Parameter of Shanxi University gravitation waves detection interferometer.

    参数符号表示数值
    臂长$ L $10 km
    激光波长$ \lambda $1550 nm
    激光频率$ {\nu _0} $1.94 × 1014 Hz
    ITM, ETM质量$ M $200 kg
    约化质量$ \mu = \dfrac{{{m_{\text{i}}}{m_{\text{e}}}}}{{{m_{\text{i}}} + {m_{\text{e}}}}} = \dfrac{M}{2} $100 kg
    $ \delta \mu = \dfrac{{{\mu _x} - {\mu _y}}}{2} $–0.001 kg
    ITM透射率$ t_{\text{i}}^{\text{2}} $1.4%
    ETM透射率$ t_{\text{e}}^2 $5 × 10–6
    PRM透射率$ t_{\text{p}}^2 $3%
    SRM透射率$ t_{\text{s}}^{\text{2}} $20%
    激光功率$ {P_{{\text{in}}}} $200 W
    功率循环腔增益$ g_{\text{p}}^2 = {\left( {\dfrac{{{t_{\text{p}}}}}{{1 - {r_{\text{p}}}{r_{\text{a}}}}}} \right)^2} $120
    信号循环腔增益$ g_{\text{s}}^{\text{2}} = {\left( {\dfrac{{{t_{\text{s}}}}}{{1 + {r_{\text{s}}}{r_{\text{a}}}}}} \right)^2} $0.06
    臂腔增益$ g_{{\text{arm}}}^{\text{2}} = {\left( {\dfrac{{{t_{\text{i}}}}}{{1 - {r_{\text{e}}}{r_{\text{i}}}}}} \right)^2} $284
    臂腔反射率$ {r_{\text{a}}} = \dfrac{{ - {r_{\text{i}}} + {r_{\text{e}}}}}{{1 - {r_{\text{i}}}{r_{\text{e}}}}} $0.99929
    $ \delta {r_{\text{a}}} = \dfrac{{{r_{{\text{a}}x}} - {r_{{\text{a}}y}}}}{2} $31 × 10–6
    反射率导数$r_{\text{a} }' = \dfrac{ {t_{\text{i} }^{\text{2} }{r_{\text{e} } } }}{ { { {\left( {1 - {r_{\text{i} } }{r_{\text{e} } } } \right)}^2} } }$283.5
    臂腔线宽$ {f_c} = \dfrac{c}{{4{\text{π }}L}}\lg \left( {\dfrac{1}{{r_{\text{i}}^{\text{2}}r_{\text{e}}^{\text{2}}}}} \right) $14.6 Hz
    $ \delta {f_c} = \dfrac{{{f_{cx}} - {f_{cy}}}}{2} $0.05 Hz
    臂腔精细度$ F = \dfrac{{{\text{π }}\sqrt {{r_{\text{i}}}{r_{\text{e}}}} }}{{1 - {r_{\text{i}}}{r_{\text{e}}}}} $445.5
    总损耗$ T $2.24 %
    臂腔功率$ {P_{\text{a}}} = \dfrac{1}{2}{P_{{\text{laser}}}}g_{\text{p}}^{\text{2}}g_{{\text{arm}}}^2 $3.4 MW
    $ \delta {P_{\text{a}}} = \dfrac{{{P_{ax}} - {P_{ay}}}}{2} $–6.5 kW
    CARM腔
    线宽
    ${f_{cc} } = \dfrac{c}{ {4{\text{π } }L} }\lg \left( {\dfrac{ {1 + {r_{\text{i} } }{r_{\text{p} } } }}{ { {r_{\text{i} } }{r_{\text{e} } } + {r_{\text{p} } }{r_{\text{e} } }\left( {t_{\text{i} }^{\text{2} } + r_{\text{i} }^{\text{2} } } \right)} } } \right)$0.06 Hz
    DARM腔
    线宽
    $ {f_{{\text{rse}}}} = \dfrac{c}{{4{\text{π }}L}}\lg \left( {\dfrac{{1 - {r_{\text{i}}}{r_{\text{s}}}}}{{{r_{\text{i}}}{r_{\text{e}}} - {r_{\text{s}}}{r_{\text{e}}}\left( {t_{\text{i}}^{\text{2}} + r_{\text{i}}^{\text{2}}} \right)}}} \right) $131 Hz
    Schnupp
    不对称
    $ {l_{{\text{sch}}}} = {l_x} - {l_y} $0.08 m
    DARM偏移$ \Delta {L_{{\text{DC}}}} $10–13 m
    高阶模耦合$ {q_{{\text{HOM}}}} $1 × 10–7 W/RAN
    $ {k_{{\text{HOM}}}} $8 × 10–17 m/Hz
    DownLoad: CSV

    表 2  熔融石英和硅材料的物理性质对比

    Table 2.  Comparison of physical properties of fused silica and silicon materials.

    比较参数硅(~123 K)熔融石英(~300 K)
    密度/(g·cm3)3.432.21
    折射率(@1.5 μm)~3.48411.445
    热膨胀系数/K–10.001 × 10–65.5 × 10–7
    热导率/
    (W·(m K)–1)
    598.31.38
    吸收系数/cm–11.11×10 @1064 nm4×10–6@1064 nm
    3.2×10–8 @1550 nm2×10–5@1550 nm
    机械损耗角/rad0.00139 × 10–61 × 10–4
    杨氏模量/GPa131.173
    泊松比0.2790.17
    DownLoad: CSV
  • [1]

    Abbott B P, Abbott R, Abbott T D, et al. 2016 Phys. Rev. Lett. 116 061102Google Scholar

    [2]

    Abbott B P, Abbott R, Abbott T D, et al. 2016 Phys. Rev. Lett. 116 241103Google Scholar

    [3]

    Vermeulen S M, Relton P, Grote H, et al. 2021 Nature 600 424Google Scholar

    [4]

    Bailes M, Berger B K, Brady P R, et al. 2021 Nat. Rev. Phys. 3 344Google Scholar

    [5]

    Abbott R, Abbott T D, Abraham S, et al. 2021 Astrophys. J. Lett. 915 L5Google Scholar

    [6]

    Badaracco F, Rossi C D, Fiori I, Harms J, Miyo K, Paoletti F, Tanaka T, Washimi T, Yokozawa T 2021 Phys. Rev. D 104 042006Google Scholar

    [7]

    Hall E D, Kuns K, Smith J R, et al. 2021 Phys. Rev. D 103 122004Google Scholar

    [8]

    Buikema A, Cahillane C, Mansell G L, et al. 2020 Phys. Rev. D 102 062003Google Scholar

    [9]

    Adhikari R X 2014 Rev. Mod. Phys. 86 121Google Scholar

    [10]

    Bond C, Brown D, Freise A, Strain K A 2016 Living Rev. Relativ. 19 3Google Scholar

    [11]

    Matichard F, Lantz B, Mittleman R, et al. 2015 Classical Quant. Grav. 32 185003Google Scholar

    [12]

    P. Nguyen, Schofield R M S, Effler A, et al. 2021 Classical Quant. Grav. 38 145001Google Scholar

    [13]

    Adhikari R X, Arai K, Brooks A F, et al. 2020 Classical Quant. Grav. 37 165003Google Scholar

    [14]

    Hammond G, Hild S, Pitkin M 2014 J. Mod. Optic. 61 S10Google Scholar

    [15]

    Heurs M 2018 Philos. T. R. Soc. A 376 20170289Google Scholar

    [16]

    Danilishin S L, Khalili F Y, Miao H 2019 Living Rev. Relativ. 22 2Google Scholar

    [17]

    Sigg D 1997 LIGO Report No. LIGO- T970084-00 D

    [18]

    Rana A 2004 Ph. D. Dissertation (Cambridge: Massachusetts Institute of Technology)

    [19]

    Somiya K, Chen Y 2006 Phys. Rev. D 73 122005Google Scholar

    [20]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500325

    [21]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500461

    [22]

    Izumi K, Sigg D, Kawabe K 2016 LIGO Report No. LIGO-T1500559

    [23]

    Cahillane C 2021 Ph. D. Dissertation (Pasadena: California Institute of Technology)

    [24]

    Cahillane C, Mansell G L, Sigg D 2021 Opt. Express 29 42144Google Scholar

    [25]

    Buonanno A, Chen Y 2001 Phys. Rev. D 64 042006Google Scholar

    [26]

    Pitkin M, Reid S, Rowan S, Hough J 2011 Living Rev. Relativ. 14 5Google Scholar

    [27]

    Kwee P 2010 Ph. D. Dissertation (Hanover: Wilhelm Leibniz University)

    [28]

    Chen Z, Guo M, Zhang R, Zhou B, Wei Q 2018 Sensors 18 02603Google Scholar

    [29]

    Degallaix J, Komma J, Forest D, Hofmann G 2014 Classical Quant. Grav. 31 185010Google Scholar

    [30]

    Khalaidovski A, Steinlechner J, Schnabel R 2013 Classical Quant. Grav. 30 165001Google Scholar

    [31]

    Biscans S, Gras S, Blair C D, Driggers J, Evans M, Fritschel P, Hardwick T, Mansell G 2019 Phys. Rev. D 100 122003Google Scholar

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Publishing process
  • Received Date:  26 March 2022
  • Accepted Date:  15 April 2022
  • Available Online:  09 August 2022
  • Published Online:  20 August 2022

Laser parameters requirement for third-generation ground-based gravitational wave detection

    Corresponding author: Zheng Yao-Hui, yhzheng@sxu.edu.cn
  • 1. State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
  • 2. Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
  • 3. School of Physical and Electronic Engineering, Shanxi University, Taiyuan 030006, China
Fund Project: Project supported by the National Key Research and Development Program of China (Grant No. 2020YFC2200402), the National Natural Science Foundation of China (Grant Nos. 62027821, 11874250, 62035015, 12174234), the Key Research and Development Projects of Shanxi Province (Grant No. 201903D111001), and the Program for Sanjin Scholar of Shanxi Province.

Abstract: Gravitational waves (GWs), predicted by the general relativity of Albert Einstein, are ripples in space-time caused by massive accelerating objects. Since the first direct observation of GWs in 2015, more and more binary black hole mergers and neutron star merger were detected by the laser interferometer gravitational-wave observatory (LIGO) and the Virgo interferometric detector. The construction of the third-generation (3G) gravitational wave detector(GWD), whose sensitivity is ten times that of the second-generation (2G) GWD (Advanced LIGO and Virgo), can not only push the gravitational wave astronomy towards the edge of the observable universe, but also test the fundamental laws of physics and study the nature of matter. By utilizing the abandoned underground mines, Shanxi university proposes to construct a 3G ground-based gravitational wave detector with an arm length of 10 km and a strain sensitivity of 10–24 Hz–1/2, improving the location accuracy of wave source by participating in the global GWD network. The construction of 3G GWD has many technical challenges, including ultrahigh large-scale vacuum system, ultrastable seismic isolation system, high-precision control system, high-quality laser and quantum source. Theoretically, the sensitivity of GWD with equal arm length is not limited by the laser source noise. However, in the actual scenario, the sensitivity is limited by the differences in arm length, arm cavity linewidth, arm reflectivity, arm mass, arm power, and the laser parameters. In this work, based on the design sensitivity (10–24 Hz–1/2) of dual-recycled Fabry-Perot Michelson interferometer, we propose the requirements for an ultra low-noise laser, including wavelength, amplitude noise, frequency noise, beam pointing noise and fundamental mode purity. The results show that in order to achieve the design sensitivity at the Fourier frequency of 100 Hz, the wavelength of the laser source should be 1.5 μm, the output power should be higher than 200 W, the amplitude noise should be better than 10–8 Hz–1/2, and the frequency noise should be better than 100 Hz/Hz1/2. To achieve the sensitivity of 10–24 Hz–1/2 at 10 Hz analysis frequency, the requirements for the amplitude and frequency noise of the laser source are much more stringent. This study lays a solid foundation for the analysis of laser source noise and the decomposition of interferometer indexes .

    • 2015年, 美国LIGO(laser interferometer gravitation waves observatory)第一次直接探测到两个黑洞并合的引力波事件, 标志着天文学研究进入引力波时代[1-2]. 天文事件的探测范围和速率依赖于激光干涉仪灵敏度, 且随灵敏度的提升而提升[3-5]. 对基本物理学和宏观宇宙学更多基本问题认识的需求驱动引力波探测器技术的不断进步和灵敏度的不断提升. 为此, 美国加州理工学院和欧洲欧空局(European Space Agency, ESA)计划建设第三代地基引力波探测装置(cosmic explorer, CE[6]和Einstein telescope, ET[7]). 山西大学在山西省政府的支持下提出在地下的废弃矿井中建设基于量子光源的引力波探测大型地基观测装置, 第一阶段预期灵敏度达10–24 Hz–1/2, 阶段目标实现以后, 通过继续优化升级各个分系统, 且引入频率依赖的压缩光源, 探测灵敏度有望达到10–25 Hz–1/2量级, 其探测频带、谱分辨能力、信号持续时间等均达到第三代地基引力波天文台的要求. 该装置的部署能够填补中国地基引力波探测装置的空白, 进一步提升现有引力波探测网络的波源定位精度.

      第二代探测装置受干涉仪臂长短、注入干涉仪激光功率受限、测试质量基底和镀膜热噪声大等因素限制, 灵敏度(10–23 Hz–1/2量级)提升面临挑战[8-9]. 与第二代地基引力波探测装置相比, 第三代探测装置, 拟解决低频震动噪声, 中频热噪声及高频散粒噪声等问题, 实现灵敏度提升一个数量级的目标[10]. 为了高效隔离低频震动噪声, 选址为地下200 m左右的废弃矿井, 并且采用准零刚度震动隔离系统设计[11-12]. 为了解决热噪声导致的注入功率受限问题, 第三代地基引力波探测装置预计选用硅作为测试质量基底材料, 在降低热噪声的同时, 提高注入干涉仪的激光功率, 降低散粒噪声对灵敏度的限制. 硅的透明窗口对应激光的波段为1.5—2.2 μm[13], 因此需要设计全新的激光源来满足大型引力波探测装置的需求.

      当前, 地基引力波探测装置普遍采用等臂迈克尔逊激光干涉仪结构, 同时在干涉仪中设置法布里-珀罗腔(Fabry-Perot cavity, F-P腔)以延长等效臂长[14-16]. 理想情形下, 等臂迈克尔逊干涉仪是一个共模抑制比为无穷大的光学器件, 灵敏度与激光源的参数无关. 然而, 实际的激光干涉仪受诸多因素的限制共模抑制比有限, 包括干涉仪控制系统中人为引入的臂长偏移及臂腔线宽差异、腔镜反射率差异、腔镜质量差异、分束器不平衡等. 基于以上原因, 激光源的噪声不可避免的耦合到干涉仪探测端, 因此引力波探测装置的灵敏度紧密依赖于激光源噪声. 1997年, Sigg等[17]基于LIGO天文台参数分析了激光源振幅和频率噪声与干涉仪噪声的耦合关系. 随后, 2004年, Rana[18]分析了LIGO天文台中激光振幅和频率噪声向装置噪声的传递, 给出在特定的参数要求下差模臂长(differential arm length, DARM)的表达式, 并指出振幅噪声传递函数与干涉仪的臂长差相关, 频率噪声的传递函数与干涉仪对比度缺陷(contrast defect)相关. 2006年, Somiya等[19]进一步完善了激光源噪声传递模型, 提出激光源噪声还可以通过辐射压力差驱动测试质量发生位移耦合到探测端. 2015年, Izumi等[20-22]分析了激光振幅和频率噪声对差模臂长的耦合传递函数, 包括由辐射压力差异和对比度缺陷引起的耦合. 2021年, Cahillane等[23-24]分析了高阶模耦合对干涉仪灵敏度的影响. 以上研究工作从振幅和频率噪声两个指标入手, 分析了LIGO天文台对激光源的指标需求, 为LIGO天文台的建造和升级提供了理论支持. 因此, 噪声分解是“基于量子光源的引力波探测大型地基观测装置”设计和建造过程中首要解决的问题.

      本文从“基于量子光源的引力波探测大型地基观测装置”出发, 根据干涉仪的设计灵敏度和参数指标, 全面分析了激光源与实际干涉仪的噪声耦合及传递, 给出由实际两臂臂腔线宽(约0.3%)、腔内功率(约0.2%)、臂腔反射率(约0.003%)等差异引起的对比度缺陷耦合、辐射压力差耦合、高阶模耦合与DARM的传递函数, 进一步给出上述参数对DARM的总传递函数. 在此基础上, 量化了该装置对激光源指标参数的要求, 为设计和建造符合要求的激光源提供了重要前提.

    2.   山西大学引力波探测装置
    • 引力波探测器光路图如图1所示, 为了获得最高的探测灵敏度, 选择“L型”结构, 主体是双循环法布里-珀罗迈克尔逊干涉仪(dual-recycled Fabry-Perot Michelson interferometer, DRFPMI). 干涉仪公里级长度的臂腔由输入测试质量(input test mass, ITM)和输出测试质量(end test mass, ETM)构成, 功率循环镜(power recycling mirror, PRM)和ITM构成功率循环腔(power recycling cavity, PRC)增大干涉仪中循环的激光功率, 信号循环镜(signal recycling mirror, SRM)和ITM构成信号循环腔(signal recycling cavity, SRC)拓宽探测器带宽[25], 分束器(beam splitter, BS)和ITMs构成迈克尔逊干涉仪(Michelson Interferometer, MICH).

      Diagram of the optical layout of Gravitational wave detection.

      Figure 1.  Diagram of the optical layout of Gravitational wave detection.

      X臂和Y臂包含参数相同的F-P腔, 迈克尔逊干涉仪可以测量共模臂长和差模臂长. 共模臂长(common arm length, CARM)信号可以通过BS反射端探测, 而DARM信号则通过BS透射端探测. 干涉仪臂长通过臂长稳定系统(arm length stabilization, ALS)精确控制, 其包括5个长度自由度, DARM = $ ({L_x} - {L_y})/2 $; CARM = $ ({L_x} + {L_y})/2 $; 功率循环腔腔长PRCL(power recycling cavity length) = $ {l_{{\text{pr}}}} + ({l_x}{\text{ + }}{l_y})/2 $; 信号循环腔腔长SRCL(signal recycling cavity length) = $ {l_{{\text{sr}}}} + ({l_x}{\text{ + }}{l_y})/2 $; 迈克尔逊长度MICH(Michelson interferometer length) = $ ({l_x} - {l_y})/2 $. CARM和DARM是最基本的长度自由度本征模. 当在干涉仪X臂中引入微小偏移量$ \Delta {L_{{\text{DC}}}} $时($ {L_x} = n\lambda /2 + \Delta {L_{{\text{DC}}}} $), Y臂偏移量与X臂偏移量大小相同, 符号相反($ {L_y} = n\lambda /2 - \Delta {L_{{\text{DC}}}} $), 因此DARM易受偏移量的影响, 而CARM几乎不随偏移量而变化. 因此, 引力波信号的探测通过DARM来实现.

    • 理想的等臂迈克尔逊干涉仪可以抑制各种技术噪声, 这一特性被称为“共模抑制(common-mode rejection)”. 采用DRFPMI的配置, 两个臂腔初始设计相同, 但由于制造公差, 测试质量的质量和反射率等参数会存在微小差异, 因此会降低干涉仪的共模抑制特性. 山西大学筹建的基于量子光源的地基引力波探测装置的设计灵敏度为10–24 Hz–1/2, 其他关键参数如表1所示.

      参数符号表示数值
      臂长$ L $10 km
      激光波长$ \lambda $1550 nm
      激光频率$ {\nu _0} $1.94 × 1014 Hz
      ITM, ETM质量$ M $200 kg
      约化质量$ \mu = \dfrac{{{m_{\text{i}}}{m_{\text{e}}}}}{{{m_{\text{i}}} + {m_{\text{e}}}}} = \dfrac{M}{2} $100 kg
      $ \delta \mu = \dfrac{{{\mu _x} - {\mu _y}}}{2} $–0.001 kg
      ITM透射率$ t_{\text{i}}^{\text{2}} $1.4%
      ETM透射率$ t_{\text{e}}^2 $5 × 10–6
      PRM透射率$ t_{\text{p}}^2 $3%
      SRM透射率$ t_{\text{s}}^{\text{2}} $20%
      激光功率$ {P_{{\text{in}}}} $200 W
      功率循环腔增益$ g_{\text{p}}^2 = {\left( {\dfrac{{{t_{\text{p}}}}}{{1 - {r_{\text{p}}}{r_{\text{a}}}}}} \right)^2} $120
      信号循环腔增益$ g_{\text{s}}^{\text{2}} = {\left( {\dfrac{{{t_{\text{s}}}}}{{1 + {r_{\text{s}}}{r_{\text{a}}}}}} \right)^2} $0.06
      臂腔增益$ g_{{\text{arm}}}^{\text{2}} = {\left( {\dfrac{{{t_{\text{i}}}}}{{1 - {r_{\text{e}}}{r_{\text{i}}}}}} \right)^2} $284
      臂腔反射率$ {r_{\text{a}}} = \dfrac{{ - {r_{\text{i}}} + {r_{\text{e}}}}}{{1 - {r_{\text{i}}}{r_{\text{e}}}}} $0.99929
      $ \delta {r_{\text{a}}} = \dfrac{{{r_{{\text{a}}x}} - {r_{{\text{a}}y}}}}{2} $31 × 10–6
      反射率导数$r_{\text{a} }' = \dfrac{ {t_{\text{i} }^{\text{2} }{r_{\text{e} } } }}{ { { {\left( {1 - {r_{\text{i} } }{r_{\text{e} } } } \right)}^2} } }$283.5
      臂腔线宽$ {f_c} = \dfrac{c}{{4{\text{π }}L}}\lg \left( {\dfrac{1}{{r_{\text{i}}^{\text{2}}r_{\text{e}}^{\text{2}}}}} \right) $14.6 Hz
      $ \delta {f_c} = \dfrac{{{f_{cx}} - {f_{cy}}}}{2} $0.05 Hz
      臂腔精细度$ F = \dfrac{{{\text{π }}\sqrt {{r_{\text{i}}}{r_{\text{e}}}} }}{{1 - {r_{\text{i}}}{r_{\text{e}}}}} $445.5
      总损耗$ T $2.24 %
      臂腔功率$ {P_{\text{a}}} = \dfrac{1}{2}{P_{{\text{laser}}}}g_{\text{p}}^{\text{2}}g_{{\text{arm}}}^2 $3.4 MW
      $ \delta {P_{\text{a}}} = \dfrac{{{P_{ax}} - {P_{ay}}}}{2} $–6.5 kW
      CARM腔
      线宽
      ${f_{cc} } = \dfrac{c}{ {4{\text{π } }L} }\lg \left( {\dfrac{ {1 + {r_{\text{i} } }{r_{\text{p} } } }}{ { {r_{\text{i} } }{r_{\text{e} } } + {r_{\text{p} } }{r_{\text{e} } }\left( {t_{\text{i} }^{\text{2} } + r_{\text{i} }^{\text{2} } } \right)} } } \right)$0.06 Hz
      DARM腔
      线宽
      $ {f_{{\text{rse}}}} = \dfrac{c}{{4{\text{π }}L}}\lg \left( {\dfrac{{1 - {r_{\text{i}}}{r_{\text{s}}}}}{{{r_{\text{i}}}{r_{\text{e}}} - {r_{\text{s}}}{r_{\text{e}}}\left( {t_{\text{i}}^{\text{2}} + r_{\text{i}}^{\text{2}}} \right)}}} \right) $131 Hz
      Schnupp
      不对称
      $ {l_{{\text{sch}}}} = {l_x} - {l_y} $0.08 m
      DARM偏移$ \Delta {L_{{\text{DC}}}} $10–13 m
      高阶模耦合$ {q_{{\text{HOM}}}} $1 × 10–7 W/RAN
      $ {k_{{\text{HOM}}}} $8 × 10–17 m/Hz

      Table 1.  Parameter of Shanxi University gravitation waves detection interferometer.

      引力波强度用无量纲振幅$ h = {{\Delta L} \mathord{\left/ {\vphantom {{\Delta L} L}} \right. } L} $表示, 当信噪比为1时, $ h $也可以视作噪声的无量纲振幅. 基于法布里珀罗迈克尔逊干涉仪, 以等效无量纲振幅表示的散粒噪声应变灵敏度为

      其中,τs=2FL/πc为臂腔贮存时间。因此, 为满足灵敏度目标, 且考虑10%的安全裕度(safety margin), 要求1550 nm激光的功率约为105 W. 如此高功率的低噪声单频激光源实现面临挑战.

      本文采用DRFPMI结构, 干涉仪探测灵敏度受散粒噪声的限制,其与激光输入功率的关系如下:

      根据表达式(2), 当输入激光功率大于200 W, 且功率循环腔增益为120, 臂腔增益为284时, 可以达到设计灵敏度10–24 Hz–1/2, 如图2所示. 其中$ \eta = {\eta _{{\text{mm}}}}{\eta _{{\text{FI}}}}{\eta _{{\text{re}}}} $, 其中$ {\eta _{{\text{mm}}}} $ = 95%为激光和功率循环腔的模式匹配效率, $ {\eta _{{\text{FI}}}} $ = 95%为法拉第隔离器效率, $ {\eta _{{\text{re}}}} $ = 95%为剩余光学元件透过率, 实际注入干涉仪的功率小于激光源的输出功率. F-P腔取代简单的迈克耳逊干涉仪后, 高频量子噪声中散粒噪声得到改善, 也降低了对输入功率的要求.

      Detection sensitivity as a function of laser power.

      Figure 2.  Detection sensitivity as a function of laser power.

    3.   激光源需求分析
    • 为了满足第三代地基引力波探测装置对低噪声激光源的要求, 需要综合考虑光源的各项参数指标对装置灵敏度的影响[26-27]. 1)激光源的波长会影响测试质量热噪声, 需要根据测试质量的性质选取. 2)激光源的振幅噪声直接和引力波信号在探测器上耦合; 另一方面, 激光源的振幅噪声引起两个干涉臂上辐射压力的差异(两个臂腔内的循环功率不可能严格相等), 使引力波信号淹没在辐射压力噪声中, 降低探测灵敏度. 因此, 需要抑制激光源的振幅噪声. 3)激光源的频率噪声通过不等臂迈克尔逊干涉仪耦合到干涉信号的强度上, 降低干涉仪的测量灵敏度. 由于实际的迈克尔逊干涉仪不可能完全等臂, 激光源的频率噪声是低噪声激光干涉仪设计必须考虑的问题. 4)激光源的指向波动引起干涉仪功率循环腔耦合效率的变化, 进一步导致腔内循环功率的波动, 从而降低测量灵敏度. 因而, 激光源的指向噪声是必须考虑的关键指标之一. 5)激光源的高阶横模不会耦合到干涉仪中, 而是从功率循环镜反射到控制探测器中, 产生额外噪声, 降低控制环路性能. 因此, 需要尽可能提升激光源的基模光束纯度.

      通过抑制激光源噪声, 使激光源噪声的等效应变噪声在探测频带(10 Hz—10 kHz)内不大于预期设计灵敏度的10%(安全裕度). 由于各个噪声源是相互独立的, 噪声增大10%只会使干涉仪总噪声基底增大0.5%: (12+0.12)1/2 = 1.005(信噪比降低不超过0.5%). 相比于PRCL, SRCL和MICH, DARM对引力波信号更灵敏, 因此在本文的分析计算中, 仅考虑臂腔内辐射压力噪声对测试质量产生的微小位移, 而忽略激光源噪声耦合对其他镜片的影响, 如BS, PRM和SRM.

    • 表2对比了熔融石英和硅基底材料的物理性质. 由涨落-耗散定理(fluctuation-dissipation theorem, FDT)可知, 热噪声的功率谱密度与镜片的杨氏模量成反比, 与镜片的泊松比、机械损耗成正比. 因此选用硅作为镜片基底, 较高密度和较高折射率会减小为实现高反射率所需的镀膜层数和膜层厚度; 在温度123 K附近硅材料有较高热导率、较低热膨胀系数[28-29]、较低吸收、较低机械损耗角(高Q值), 可显著降低镜片的热梯度, 降低高功率激光场引起的热透镜效应和热光噪声[30]; 另外, 硅材料有较大杨氏模量和较小泊松比, 可显著降低高功率引起的镜片热畸变, 防止镜片因热弹性形变使曲率半径发生变化. 因此, 我们选用硅作为干涉仪测试质量的材料. 由于硅的透明窗口位于1.5—2.2 μm激光波段[13], 综合材料成熟度和技术成熟度等因素, 决定采用1.5 μm波段激光源用于引力波探测装置建造.

      比较参数硅(~123 K)熔融石英(~300 K)
      密度/(g·cm3)3.432.21
      折射率(@1.5 μm)~3.48411.445
      热膨胀系数/K–10.001 × 10–65.5 × 10–7
      热导率/
      (W·(m K)–1)
      598.31.38
      吸收系数/cm–11.11×10 @1064 nm4×10–6@1064 nm
      3.2×10–8 @1550 nm2×10–5@1550 nm
      机械损耗角/rad0.00139 × 10–61 × 10–4
      杨氏模量/GPa131.173
      泊松比0.2790.17

      Table 2.  Comparison of physical properties of fused silica and silicon materials.

    • 根据工程实际, 干涉仪中臂腔反射率差异约为0.003%、臂腔功率差异约为0.2%、臂腔线宽差异约为0.3%、测试质量的质量差异约为0.005%. 这些差异均会引起激光源振幅噪声向DARM传递, 降低干涉仪的测量灵敏度. 激光源振幅噪声的耦合传递函数[21]可以表示为

      其中, 所用参数均为表1中所列出的数值, Laplace表示$ {s_c} = {\text{i}}{f \mathord{\left/ {\vphantom {f {{f_c}}}} \right. } {{f_c}}} $, $ {s_{cc}} = {\text{i}}{f \mathord{\left/ {\vphantom {f {{f_{cc}}}}} \right. } {{f_{cc}}}} $, $ {s_{{\text{rse}}}} = {\text{i}}{f \mathord{\left/ {\vphantom {f {{f_{{\text{rse}}}}}}} \right. } {{f_{{\text{rse}}}}}} $, 机械响应$ s_\mu ^2 = - \mu {f^2} $. 第1项表示由DARM偏移$ \Delta {L_{{\text{DC}}}} $引起的直接耦合; 第2项表示由臂腔反射率差异$ \delta {r_{\text{a}}} $、臂腔线宽差异$ \delta {f_c} $和Schnupp不对称$ {l_{{\text{sch}}}} $导致的对比度缺陷引起的耦合; 第3项表示由臂腔功率差异$ \delta {P_{\text{a}}} $、约化质量差异$ \delta \mu $、臂腔反射率差异$ \delta {r_{\text{a}}} $、臂腔线宽差异$ \delta {f_c} $产生的辐射压力差异引起的耦合; 第4项表示由高阶模引起的耦合.

      激光源振幅噪声的耦合传递函数如图3(a)所示. DARM偏移$ \Delta {L_{{\text{DC}}}} $和辐射压力差异引起的振幅噪声耦合呈线性关系. 辐射压力差异引起的振幅噪声耦合(也称光机耦合)在低频(100 Hz以下)显著, 而高阶模引起的耦合在高频(100 Hz以上)占主导地位. 对比度缺陷引起的振幅噪声耦合在高频(100 Hz—10 kHz)影响较大, 臂腔线宽差异$ \delta {f_c} $引起的耦合大于臂腔反射率差异$ \delta {r_{\text{a}}} $引起的耦合, Schnupp不对称$ {l_{{\text{sch}}}} $引起的耦合较小. 图3(b)所示为由臂腔功率差异$ \delta {P_{\text{a}}} $、线宽差异$ \delta {f_c} $、反射率差异$ \delta {r_{\text{a}}} $、测试质量的质量差异$ \delta \mu $产生的辐射压力差异引起的振幅噪声耦合, 其中臂腔功率差异$ \delta {P_{\text{a}}} $引起的耦合占主导地位. 由于工程实际, 干涉仪X臂和Y臂测试质量的质量存在微小差异, 当辐射压力相同时, 测试质量产生的微小位移不同; 同时辐射压力与腔内功率的波动成正比$ {F_{{\text{rp}}}} = {{2\delta {P_{\text{a}}}} \mathord{\left/ {\vphantom {{2\delta {P_{\text{a}}}} c}} \right. } c} $. 综上, 应最大可能地使X臂和Y臂测试质量的质量相同, 臂腔中的循环功率相同.

      Coupling transfer function of laser amplitude noise: (a) Amplitude noise coupling due to DARM offset, radiation pressure difference and contrast defect; (b) amplitude noise coupling due to radiation pressure difference.

      Figure 3.  Coupling transfer function of laser amplitude noise: (a) Amplitude noise coupling due to DARM offset, radiation pressure difference and contrast defect; (b) amplitude noise coupling due to radiation pressure difference.

      对于臂长为10 km的干涉仪, 要达到并突破10–24 Hz–1/2的设计灵敏度, 考虑10%的安全裕度, 在100 Hz时, 振幅噪声总耦合传递函数为10–13 m/RAN, 激光源振幅噪声要优于10–8 Hz–1/2. 在10 Hz时, 振幅噪声总耦合传递函数为10–11 m/RAN, 激光源振幅噪声要优于10–10 Hz–1/2, 与100 Hz相比, 激光源振幅噪声需提高两个数量级, 如图4所示. 山西大学引力波探测装置的灵敏度可以表示为

      Detection sensitivity as a function of laser amplitude noise.

      Figure 4.  Detection sensitivity as a function of laser amplitude noise.

      式中, RAN为激光源振幅噪声, 此灵敏度是干涉仪第一阶段的预期结果.

    • 激光干涉仪具有较好的共模抑制特性, 大多数频率噪声不会到达探测端, 但仍会不可避免的有少部分频率噪声通过迈克尔逊干涉仪的不对称性耦合至DARM, 降低干涉仪的测量灵敏度. 引力波探测装置主要通过前置稳频激光系统(Pre-stabilized laser, PSL)、输入模式清洁腔(input mode cleaner, IMC)和完整干涉仪(in the full interferometer, ITF)抑制激光频率噪声. 本文从第三代引力波探测器的灵敏度出发, 通过计算ITF对频率噪声的传递函数, 获得了激光源频率噪声的需求. 由于激光频率噪声耦合与振幅噪声耦合的影响因素相同, 故采用前述3.2节同样的分析方法进行阐述. 激光频率噪声的耦合传递函数可以表示为[21]

      激光频率噪声耦合的第1项表示由DARM偏移$ \Delta {L_{{\text{DC}}}} $导致辐射压力差异引起的耦合; 第2项表示由臂腔反射率差异$ \delta {r_{\text{a}}} $、臂腔线宽差异$ \delta {f_c} $、Schnupp不对称性$ {l_{{\text{sch}}}} $导致对比度缺陷引起的耦合; 第3项表示由高阶模引起的耦合. 频率噪声耦合传递函数如图5所示. DARM偏移$ \Delta {L_{{\text{DC}}}} $导致辐射压力差异引起的频率噪声耦合在低频(20 Hz以下)显著, 随着频率的增大耦合系数降低, 因此应使DARM偏移较小. 臂腔线宽差异$ \delta {f_c} $引起的耦合在20 Hz以上影响显著, Schnupp不对称性$ {l_{{\text{sch}}}} $引起的耦合影响最小. 高阶模引起的耦合在1 kHz以上占主导地位.

      Coupling transfer function of laser frequency noise.

      Figure 5.  Coupling transfer function of laser frequency noise.

      对于臂长为10 km的干涉仪, 要达到并突破10–24 Hz–1/2的探测灵敏度, 并考虑10%的安全裕度, 可获得激光频率噪声需求:在100 Hz时, 频率噪声总耦合传递函数为10–15 m/Hz, 激光频率噪声要优于10–6 Hz/Hz1/2. 在10 Hz时, 频率噪声总耦合传递函数为10–13 m/Hz, 激光频率噪声要优于10–8 Hz/Hz1/2, 与100 Hz相比, 激光频率噪声需提高两个数量级, 如图6所示. 为了获得该激光频率噪声, 需要高带宽和高增益的控制环路, 将频率噪声抑制 8~10个数量级, 因此, 激光源频率噪声100 Hz/Hz1/2@100 Hz满足引力波探测需求. 山西大学引力波探测装置的灵敏度可以表示为

      Detection sensitivity as a function of laser frequency noise.

      Figure 6.  Detection sensitivity as a function of laser frequency noise.

      式中, δν为激光源频率噪声, κ(f)为频率噪声抑制因子,此灵敏度是干涉仪第一阶段的预期结果.

    • 激光源的指向噪声通过改变基模与臂腔的耦合, 引起干涉仪功率循环腔耦合效率的变化, 导致腔内循环功率的波动; 其次, 由于输入测试质量镀膜不均匀, 功率损耗存在差异, 也会通过破坏干涉臂的对称性将指向噪声耦合到探测端; 最后, 光束指向噪声会降低光束与输出模式清洁器(output mode cleaner, OMC)的耦合效率. 光束在进入干涉仪之前首先经过输入模式清洁器被动过滤指向噪声(部分衰减), 剩余的指向噪声被干涉仪臂腔转化为振幅噪声, 因此指向噪声对测量灵敏度的影响可以统一到振幅噪声的模型中分析.

      高阶模的叠加严重降低引力波探测器输出信号的干涉对比度. 通常情况下干涉仪存在准直误差或模式不匹配, 因此高阶模也会在臂腔中共振, 使测试质量表面的光强分布发生改变, 产生附加的热噪声; 其次高阶模的振幅存在很大的不确定性, 会使测试质量受到的辐射压力发生变化, 产生附加的辐射压力噪声; 最后当臂腔中高阶模的模式与测试质量基底的弹性模(elastic mode)匹配时, 就会发生参量不稳定性[31](parametric Instability, PI), 降低干涉仪的锁定稳定性. 因此需要尽可能提升激光源的基模光束纯度. 综合考虑灵敏度要求和激光源实现面临的技术挑战, 一般要求激光源的基模纯度大于95%.

    4.   总 结
    • 山西大学地基引力波探测装置的设计灵敏度为10–24 Hz–1/2@100 Hz. 全文从该装置对激光源的需求出发, 分析了激光振幅噪声、频率噪声、指向噪声、基模纯度等与DARM的噪声耦合传递函数. 结果表明, 在傅里叶频率100 Hz处, 要达到并突破设计灵敏度, 激光源的输出功率应大于200 W, 振幅噪声要优于10–8 Hz–1/2, 频率噪声要优于100 Hz/Hz1/2. 下一步, 通过优化升级各个分系统的参数, 包括激光器性能、镜片质量、吸收等, 探测灵敏度有望达到10–25 Hz–1/2量级. 本文为设计用于下一代地基引力波探测装置的激光源指明了方向, 将有力推动山西大学地基引力波探测装置的建造进程, 驱动我国在第三代地基引力波探测的研究中实现从追赶到与国际同步发展.

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