Seismic Noise
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2015年度 天体素粒子物理学 特論II （重力波：その導出・発生・検出・応用） 2015年6月23日~7月14日（火曜）3限 三代木伸二 東京大学宇宙線研究所 宇宙物理学研究部門 重力波推進室 Contents 1. General Introduction of Challenge to Direct Detection of Gravitational Waves (GWs). 2. Expected Sources of Gravitational Waves. 3. Small signal measurement (General Knowledge) 4. Interferometric Technology for GW Detection (1). • Power Recycled Fabry-Perot Michelson Interferometer using Resonant Sideband Extraction Technique • practical noise sources and their suppression 5. Interferometric Technology for GW Detection (2). • Interferometer Control as GWDs 6. Data Analyis 7. New Fields Driven by the GW detection technique. 微小信号計測基礎 重力波望遠鏡にとってほとんどすべてが「雑音」 （重力波以外なら何でも受けられる！といわれるほど） 目指す残存雑音要素 Shot Noise 輻射圧雑音 除くのが難しい雑音 地面振動 熱雑音 ? Noise Shot Noise 「雑音」の特性 や種類を知って おくことは重要 測定と雑音 参考資料：三尾典克 先生講義資料 測定器 外力雑音 センサー 𝑥𝑓 Pre-Amplifier 𝑺 位置測定対象 𝑥𝑠 センサー雑音 （原理的な雑音） 𝑮 Amplifier 𝑻 𝑽 データ収集装置 𝑛𝑎 アンプ雑音 （付加的雑音） 𝑉 = 𝑇 𝑆 𝑥0 + 𝑥𝑠 + 𝑥𝑓 + 𝑛𝑎 𝑛𝑎 等価雑音 = + 𝑥𝑠 𝑆 が小さいことが望ましい 信号対雑音比を理解する準備 数学的に扱える雑音 不規則に変動する統計事象 → 確率過程 𝑥 =𝑥 統計的期待値 時間平均 ∞ 𝑝 𝑥 𝑑𝑥 = 1 −∞ （例）ガウス分布（ほぼすべてがこの特性を持つ） 𝑥2 𝑝 𝑥 = exp − 2 2𝜎 2𝜋𝜎 1 しかし、重力波検出器では、これに従わない「非定常雑音」を扱わねばならない。 信号対雑音比を理解する準備 パワースペクトル 有限区間に限る ∞ 𝑋 𝜔 = Fourier変換 𝑥 𝑡 𝑇 2 𝑒 −𝑖𝜔𝑡 𝑑𝑡 −∞ 1 𝑥 𝑡 = 2𝜋 Parsevalの定理 1 2𝜋 ∞ −∞ 𝑥 𝑡 𝑒 −𝑖𝜔𝑡 𝑑𝑡 𝑋𝑇 𝜔 = −𝑇 2 ∞ 𝑋 𝜔 𝑒 −𝑖𝜔𝑡 𝑑𝜔 𝑥 𝑡 = −∞ 1 2 𝑋 𝜔 𝑑𝜔 = 2𝜋 𝑥 𝑡 𝑡 <𝑇 2 0 𝑡 >𝑇 2 ∞ 𝑥 𝑡 2 𝑑𝑡 → ∞ −∞ 𝑥 𝑡 が無限に続くと パワースペクトル密度の定義 𝑆 𝜔 = lim 𝑇→∞ 1 2 𝑥 𝑡 = lim 𝑇→∞ 𝑇 𝑋𝑇 𝜔 2𝜋𝑇 𝑇 2 2 ∞ 𝑥 𝑡 2 𝑑𝑡 = −𝑇 2 𝑆 𝜔 𝑑𝜔 −∞ Parsevalの定理利用 信号対雑音比を理解する準備 自己相関関数（GWDではノイズに埋もれた周期的信号の存在を判定） 定義 𝑅 𝜏 = 𝑥 𝑡 𝑥 𝑡+𝜏 𝑅𝑚𝑎𝑥 = 𝑅 0 = 𝑥 𝑡 𝑇 2 𝑋𝑇 𝜔 = 𝑥 𝑡 𝑒 −𝑖𝜔𝑡 𝑑𝑡 を −𝑇 2 1 1 𝑆 𝜔 = lim 2𝜋 𝑇 → ∞ 𝑇 𝑇 2 𝑆 𝜔 = lim 𝑇→∞ 𝑋𝑇 𝜔 2𝜋𝑇 2 2 に代入 𝑇 2 𝑥 𝑡 𝑥 𝑡′ 𝑒 −𝑖𝜔 𝑡−𝑡′ 𝑑𝑡𝑑𝑡′ −𝑇 2 −𝑇 2 𝑡 − 𝑡′ = 𝜏 相関関数とパワースペクトルはFourier変換の関係にある 信号対雑音比を理解する準備 線形応答とパワースペクトル 𝑌 𝜔 𝐻 𝜔 = 𝑋 𝜔 𝑥 𝑡 ℎ 𝑡 𝑦 𝑡 𝑋 𝜔 𝑆𝑥 𝜔 𝐻 𝜔 𝑌 𝜔 𝑆𝑦 𝜔 1 𝑦 𝑡 = 2𝜋 𝑥 𝑡 =𝛿 𝑡 に対する インパルス応答 パワーススペクトル の関係 ∞ ∞ 𝐻 𝜔 𝑋 𝜔 𝑒 𝑖𝜔𝑡 𝑑𝜔 = −∞ 1 ℎ 𝑡 = 2𝜋 ℎ 𝑡′ ℎ 𝑡 − 𝑡′ 𝑑𝑡′ −∞ ∞ 𝐻 𝜔 𝑒 𝑖𝜔𝑡 𝑑𝜔 −∞ 𝑆𝑦 𝜔 = 𝐻 𝜔 2𝑆 𝑥 𝜔 信号対雑音比を理解する準備 Langevin方程式 確率過程を含む外力による運動を表す方程式がある：確率微分方程式 （アインシュタインのブラウン運動の論文などが初出） 𝑚𝑥 + 𝑚𝛾 𝑥 = 𝑓𝐵 𝑡 𝑓𝐵 𝑡 𝑓𝐵 𝑡′ 例えば熱による揺動力 = 2𝐵𝛿 𝑡 − 𝑡′ という自己相関関数で表せる性質 つまり、時間が違えば、無相関 パワースペクトルは 𝑆 𝜔 = lim 𝑇→∞ 𝑋𝑇 𝜔 2𝜋𝑇 2 から 𝑆𝐵 𝜔 = 𝐵 𝜋 すべての周波数成分を等しく含む雑音 （白色雑音） 信号対雑音比を理解する準備 Langevin方程式 𝑚𝑥 + 𝑚𝛾𝑥 = 𝑓𝐵 𝑡 1 𝑢 𝑡 = 𝑚 の一般的な定常解は 𝑡 𝑓𝐵 𝑡′ 𝑒 −𝛾 𝑡−𝑡′ 𝑑𝑡′ −∞ 自己相関は、 𝑢 𝑡 𝑢 𝑡+𝜏 1 = 2 𝑚 𝑡 𝑡+𝜏 𝑑𝑠 𝑓𝐵 𝑠 𝑓𝐵 𝑠′ 𝑒 −𝛾 𝑑𝑠′ −∞ 熱平衡状態では、 𝑒 −𝛾 𝑡+𝜏−𝑠 −∞ 𝑓𝐵 𝑡 𝑓𝐵 𝑡′ 𝐵𝑒 −𝛾𝜏 = 𝑚2 𝛾 1 𝑚𝑢 𝑡 2 𝑡−𝑠′ 2 = 2𝐵𝛿 𝑡 − 𝑡′ 𝑘𝐵 ∶ 1.38 × 10−23 J/K 1 = 𝑘𝐵 𝑇 → 𝐵 = 𝑚𝛾𝑘𝐵 𝑇 2 信号対雑音比を理解する準備 Langevin方程式 𝑚𝑥 + 𝑚𝛾𝑥 = 𝑓𝐵 𝑡 のフーリエ変換は 𝑈 𝜔 𝑚 𝑖𝜔 + 𝛾 𝑈 𝜔 = 𝐹𝐵 𝜔 → 𝐹𝐵 𝜔 ≡𝐻 𝜔 1 = 𝑚 𝑖𝜔 + 𝛾 応答相関 パワースペクトルは 𝑆𝑢 𝜔 = 𝐻 𝜔 2𝑆 𝐵 𝐵 𝜔 = 𝜋𝑚2 𝜔 2 + 𝛾 2 𝐵 = 𝑚𝛾𝑘𝐵 𝑇 雑音源を用動力として扱い、その系の応答から、出力雑音 のスペクトルを求められる。 信号雑音比 Signal to Noise Ratio (SNR：信号雑音比) and Bandwidth（信号帯域） 信号： 雑音： 𝑥𝑠 𝑡 𝑥𝑛 𝑡 𝑥𝑠 𝑡 𝑆𝑁𝑅 ≡ 𝑥𝑛 𝑡 2 線形インパルス応答:ℎ 𝑡 2 𝑥𝑠 𝑡 𝑋𝑠 𝜔 ℎ 𝑡 𝑦𝑠 𝑡 𝑌𝑠 𝜔 𝑥𝑛 𝑡 𝑋𝑛 𝜔 𝐻 𝜔 𝑦𝑛 𝑡 𝑌𝑛 𝜔 𝑦𝑠 𝑡 𝑆𝑁𝑅 ≡ 𝑦𝑛 𝑡 2 2 = 1 𝑦 𝑡 = 2𝜋 2 ∞ 𝑖𝜔𝑡 𝐻 𝜔 𝑋𝑠 𝜔 𝑒 𝑑𝜔 −∞ ∞ 2 𝑋 𝜔 𝑑𝜔 𝐻 𝜔 𝑛 −∞ ∞ ∞ 𝐻 𝜔 𝑋 𝜔 𝑒 𝑖𝜔𝑡 𝑑𝜔 = −∞ ℎ 𝑡′ ℎ 𝑡 − 𝑡′ 𝑑𝑡′ −∞ 信号雑音比 正弦波の信号（𝜔0 = 2𝜋𝑓0 ）を考え、その周波数付近だけを通過するフィル ターを考える。 𝑦𝑠 𝑡 2 ≈ 𝐻 𝜔0 ∞ 𝑦𝑛 𝑡 2 = 𝐻 𝜔 2 −∞ 2𝑥 𝑠 𝑡 2 1 𝐺𝑛 𝜔 𝑑𝜔 ≈ 𝐻 𝜔0 2𝜋 2 𝐺𝑛 𝜔0 Δ𝜔 𝐺𝑛 𝜔 ∶ 片側スペクトル 1 Δ𝑓 ≡ 2𝜋 𝑥𝑠 𝑡 2 𝑆𝑁𝑅 = 𝐺𝑛 𝑓0 Δ𝑓 ∞ 0 𝐻 𝜔 𝐻 𝜔0 2 𝑑𝜔 2 帯域幅 帯域幅を狭めればいくらでもSNRをあげ られるように見えるが、 信号雑音比と最適フィルター 最適フィルター（ Matched Filtering for known signals） 𝑦𝑠 𝑡 𝑆𝑁𝑅 ≡ 𝑦𝑛 𝑡 2 ∞ 𝑖𝜔𝑡 𝐻 𝜔 𝑋 𝜔 𝑒 𝑑𝜔 𝑠 −∞ ∞ 2 𝑋 𝜔 𝑑𝜔 𝐻 𝜔 𝑛 −∞ 2 2 = 𝐴 𝜔 = 𝐻 𝜔 𝑋𝑛 𝜔 1 2 𝐵 𝜔 =𝐻 𝜔 𝑒 𝑖𝜔𝑡 𝑋𝑛 𝜔 − 1 2 Cauchy-Schwarz の不等式 2 ∞ 𝐴 𝜔 𝐵 𝜔 𝑑𝜔 ∞ ≤ −∞ ∞ 𝐴 𝜔 −∞ ∞ 𝑆𝑁𝑅 ≤ −∞ 𝑋𝑠 𝜔 𝑋𝑛 𝜔 2 𝑑𝜔 𝐵 𝜔 2 𝑑𝜔 −∞ 2 𝑑𝜔 等式成立条件 𝐴 𝜔 = 𝐵∗ 𝜔 𝑋𝑠∗ 𝜔 𝑒 𝑖𝜔𝑡 既知波形にしか使えない。あるパ 最適フィルターは 𝐻 𝜔 = ラメータがわからない場合は、 𝑋𝑛 𝜔 SNR最大化パラメータを選べばよ い。 基本的な雑音 抵抗（電気回路）の熱雑音 散逸（エネルギーのロス） → 𝐵 𝑆𝐼 𝜔 = ∶ 抵抗Rに発生する熱起電力 𝜋 𝐶∶ 等価回路における並列容量 エネルギー等分配則 (平衡状態) 1 1 2 𝐶 𝑣𝑅 𝑡 = 𝑘𝐵 𝑇 2 2 熱的な搖動力の発生 1 𝑆𝑅 𝜔 = 𝑆 𝜔 1 + 𝜔𝐶𝑅 2 𝐼 𝐵 1 = 𝜋 1 + 𝜔𝐶𝑅 2 𝑇 ∶ 温度 ∞ 𝑣𝑅2 𝐵 𝑘𝐵 𝑇𝑅 𝑡 = 𝑆𝑅 𝜔 𝑑𝜔 = → 𝑆𝐼 𝜔 = 𝐶𝑅 𝜋 −∞ 熱雑音を帯域 Δ𝑓で観測した時のスペクトルは 𝑉𝑅 𝑓 = 4𝑘𝐵 𝑇𝑅Δ𝑓 1/ Hz 基本的な雑音 光の散射雑音 ショットキーの定理：ランダムで相関のない電子が単位断面積を通過し、平均 Ｉの電流となった時の電流の揺らぎに関する定理 パワースペクトル 𝐺𝐼 𝜔 = 2𝑒𝐼: 𝑒 ∶ 素電化 光検出器に発生した光電流でも全く同じ。 1/f ゆらぎ 自然界にもよくあらわれる、原因不明のゆらぎ。パワーが周波数にほぼ半比例 する。 𝐺𝑓 𝜔 ∝ 1 , 𝑛 𝑓 (𝑛 ~ 1) 微弱信号検出での高SNR信号検出 ロックインアンプ 対象時系列データのターゲット周波数成分の振幅・位相に関する情報を、既知 のターゲット周波数信号を書けることにより読み取る。 𝑉𝑜𝑢𝑡 𝑡 𝑉𝑖𝑛 𝑡 𝑉𝑖𝑛 cos 𝜔𝑖 𝑡 + 𝜑 𝑅 𝐵𝑃𝐹@𝜔𝑟 𝐶 𝑉𝑟 𝑡 𝑉𝑟 cos 𝜔𝑟 𝑡 𝑉𝑜𝑢𝑡 𝑡 ∝ 𝑉𝑖𝑛 𝑉𝑟 cos 𝜔𝑖 − 𝜔𝑟 𝑡 + 𝜑 + cos 𝜔𝑖 + 𝜔𝑟 𝑡 + 𝜑 𝜔𝑖 + 𝜔𝑟 の成分をLPFで除去し、𝜔𝑖 − 𝜔𝑟 の成分から𝑉𝑖𝑛 の情報を得る。 信号は、cos 𝜑 に比例する。 𝑉𝑜𝑢𝑡 2 𝑡 ∝ 𝑅𝐶 𝑡 𝑡−𝑠 exp − cos 𝜔𝑟 𝑠 𝑉𝑖 𝑠 𝑑𝑠 𝑅𝐶 −∞ 微弱信号検出での高SNR信号検出 ロックインアンプ 等価雑音帯域幅： 2 𝑉𝑜𝑢𝑡 = 𝑛2 ∆𝑓 [V 2 ] 𝑉𝑤ℎ𝑖𝑡𝑒 𝑡 𝑉𝑤ℎ𝑖𝑡𝑒 𝜔 = 𝑛 [V/ Hz] 𝑅 𝐶 𝑉𝑟 𝑡 𝑉𝑟 cos 𝜔𝑟 𝑡 になるような ∆𝑓 で 1 ∆𝑓 = 4𝑅𝐶 位相を見るには、 𝑉𝑟 sin 𝜔𝑟 𝑡 でも掛け算し、二つの出力から振幅と位相を計算。 𝑅 𝑉𝑖𝑛 𝑡 𝑉𝑥 𝑡 𝐶 𝑉𝑟 cos 𝜔𝑟 𝑡 𝑉𝑟 sin 𝜔𝑟 𝑡 𝑅 𝑉𝑦 𝑡 微弱信号検出での高SNR信号検出 変調法（低雑音帯域の選択） 測定対象信号の周波数帯域の中で、雑音成分の少ない、かつ測定に影響しない 帯域の特定の周波数で信号を変調し、同じ周波数で同期検波することで、高 SNRで信号を得る。 例：鏡の透過率（～数十 ppm レベル）特性測定 Laser 強度変調器 LPF 𝑉𝑀 LPF 𝑉𝑛𝑜𝑀 dP/P 鏡 𝑉𝑟 cos 𝜔𝑟 𝑡 Laser 強度変調器 𝑉𝑟 cos 𝜔𝑟 𝑡 ~ DC 𝜔𝑟 𝑉𝑀 Mirror Transmittance = @ 𝜔𝑟 𝑉𝑛𝑜𝑀 f 微弱信号検出での高SNR信号検出 変調法（低雑音帯域の選択） 𝛿𝑃𝜔2 ∝ 𝐺𝑃𝐷 𝑘𝑇 𝜔 + 2 2 dP/P 𝐺𝛿𝑃 𝐷𝐶 ∆𝑓 検出器の雑音の周波数𝝎の成分 𝐺𝑃𝐷 𝜔 光源の雑音のDC付近の成分 𝐺𝛿𝑃 𝐷𝐶 検出器の雑音 < 光源の雑音 の場合、フィードバック制 御で、𝐺𝛿𝑃 𝐷𝐶 を低減する。 𝜔𝑟 ~ DC 制御回路 𝐺𝑃𝐷 𝜔 Laser LPF 強度変調器 𝐺𝛿𝑃 𝐷𝐶 鏡 𝑉𝑟 cos 𝜔𝑟 𝑡 𝑉𝑀 f フィードバック制御 元信号の一部、あるいは安定な信号との残差を取り出し、適当な処理を行った 後、元信号に変えることで、安定化、線形性、応答の改善を図る。 𝑚 𝑋𝑠 𝜔 𝑋𝑟𝑒𝑠 𝜔 𝑚 駆動器 𝐹 𝜔 𝑉 𝑉 𝐻 𝜔 光検出器 𝑉 𝑍𝑠 𝜔 𝑋𝑟𝑒𝑠 = 𝑋𝑠 − 𝑋𝑟𝑒𝑠 𝐻𝐶𝐹 𝑋𝑠 𝑋𝑠 𝜔 𝑋𝑟𝑒𝑠 = = 1 + 𝐻𝐶𝐹 1 + 𝐺 𝜔 𝐺 = 𝐻𝐶𝐹 ∶ Open Loop TF 𝑌𝑠 𝜔 𝑉 𝐶 𝜔 電気回路 𝐺 𝜔 ≫ 1なら、もとのゆらぎ𝑋𝑠 を 1 小さくできる。 𝐺 𝜔 𝑋𝑠 𝜔 を𝑌𝑠 𝜔 や𝑍𝑠 𝜔 などから逆算 できる。 フィードバック制御と雑音 𝑋𝑟𝑒𝑠 𝜔 𝑋𝑠 𝜔 𝐻 𝜔 光検出器 𝑚 駆動器 𝑌𝑠 𝜔 𝐹 𝜔 𝑁𝑃𝐷 𝑍𝑠 𝜔 𝐶 𝜔 電気回路 𝑁𝐹 𝑁𝑐 𝑋𝑟𝑒𝑠 = 𝑋𝑠 − 𝐹𝐶𝐻𝑋𝑟𝑒𝑠 + 𝐹𝐶𝐻𝑁𝑃𝐷 + 𝐹𝐶𝑁𝑐 + 𝐹𝑁𝐹 𝑋𝑟𝑒𝑠 𝑋𝑠 − 𝐺𝑁𝑃𝐷 − 𝐹𝐶𝑁𝑐 − 𝐹𝑁𝐹 = 1+𝐺 𝐺𝑋𝑠 + 𝑁𝑃𝐷 − 𝐹𝐶𝑁𝑐 − 𝐹𝑁𝐹 𝐻 1+𝐺 𝐻𝐺𝑋𝑠 − 𝐻𝐹𝑁𝐹 + 𝑁𝑐 + 𝐻𝑁𝑃𝐷 = 𝐶 1+𝐺 𝑌𝑠 = 𝐻 𝑋𝑟𝑒𝑠 + 𝑁𝑃𝐷 = 𝑍𝑠 = 𝐶 𝑌𝑠 + 𝑁𝑐 𝑆𝑁𝑅 𝑋𝑠 ∶ 𝑁𝑃𝐷 𝑁𝑐 𝑁𝐹 ∶ ∶ 𝐺 𝐻 𝐶𝐻 𝑁𝑃𝐷 𝑁𝑐 𝑁𝐹 𝑋𝑠 ∶ ∶ ∶ 𝐺 𝐻𝐺 𝐶𝐻 安定参照を用いた安定化 𝑋0 − 安定参照 𝐻 𝜔 𝑋𝑟𝑒𝑠 𝜔 𝑋𝑠 𝜔 光検出器 − 駆動器 𝐹 𝜔 𝑍𝑠 𝜔 𝐶 𝜔 電気回路 𝑋𝑟𝑒𝑠 = 𝑋𝑠 − 𝐺𝑋𝑟𝑒𝑠 −𝑋0 𝑋𝑟𝑒𝑠 𝑋𝑠 −𝑋0 = 1+𝐺 𝐺 ≫ 1 → 𝑋𝑠 ~𝑋0 𝑌𝑠 𝜔 信号が混合している制御 𝐹1 𝑋1 𝑋2 − 𝐶1 + 𝑘1𝐵 𝐴 𝑋𝑟𝑒𝑠 𝑘2𝐴 𝐵 𝑋𝑟𝑒𝑠 − + 𝐹2 = 𝑋2 − 𝐴 𝑋𝑟𝑒𝑠 = 𝐵 𝑋𝑟𝑒𝑠 𝐵 𝐺2 𝑋𝑟𝑒𝑠 + 𝑘1𝐵 𝑋1 − 𝐻2 𝐶2 𝐴 𝐴 𝐵 𝑋𝑟𝑒𝑠 = 𝑋1 − 𝐺1 𝑋𝑟𝑒𝑠 + 𝑘2𝐴 𝑋2 − 𝐺2 𝑋𝑟𝑒𝑠 𝐵 𝑋𝑟𝑒𝑠 𝐻1 𝐴 𝐺1 𝑋𝑟𝑒𝑠 𝑘1𝐵 , 𝑘2𝐴 ≪ 1 𝑖𝑠 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑋1 + 1 − 𝑘1𝐵 𝑘2𝐴 𝐺2 𝑋1 − 𝑘2𝐴 𝑋2 𝑋1 + 𝐺2 𝑋1 𝑋1 ≈ ~ 1 + 𝐺1 + 𝐺2 + 1 − 𝑘1𝐵 𝑘2𝐴 𝐺1 𝐺2 1 + 𝐺1 𝐺2 𝐺1 𝑋2 + 1 − 𝑘1𝐵 𝑘2𝐴 𝐺1 𝑋2 − 𝑘1𝐵 𝑋1 𝑋2 + 𝐺1 𝑋2 𝑋2 = ≈ ~ 1 + 𝐺1 + 𝐺2 + 1 − 𝑘1𝐵 𝑘2𝐴 𝐺1 𝐺2 1 + 𝐺1 𝐺2 𝐺2 𝐺1 , 𝐺2 ≫ 1 レーザー干渉計重力波望遠鏡と それに現れる様々な雑音 Michelson Interferometer (MI) GW signal Extraction using Michelson Interferometer Arm : 3000m On the Earth Mirror1 Mirror2 BS Laser (Bright port) Photo-detector (Dark Port) Free falling mirrors and BS are simulated as pendulums on the Earth over the pendulum resonance frequency. ~ 4 km used to be max length on the Earth because of gravity direction difference. (~10km is planed in ET) Antenna Pattern of MI MI type GW detector has a antenna pattern • • • • Assuming MI arms are aligned along X, Y axis. MI has max sensitivity for GWs propagating along Z axis. MI has no sensitivity for GWs propagating along 45, 135, 225, 315 degree directions. Observed signal is ℎ𝑜𝑏𝑠 = 𝐹+ 𝜃, 𝜙, 𝜓 ℎ+ + 𝐹× 𝜃, 𝜙, 𝜓 ℎ× 𝐹+ 𝜃, 𝜙, 0 𝐹× 𝜃, 𝜙, 0 𝐹+ 𝜃, 𝜙, 𝜓 2 + 𝐹× 𝜃, 𝜙, 𝜓 2 Antenna Pattern of MI MI Antenna Pattern ℎ𝑜𝑏𝑠 = 𝐹+ ℎ+ + 𝐹× ℎ× 1 + cos 2 𝜃 𝐹+ ℎ+ = − cos 2𝜙 cos 2𝜓 − cos 𝜃 sin 2𝜙 sin 2𝜓 2 1 + cos 2 𝜃 𝐹× ℎ× = cos 2𝜙 cos 2𝜓 − cos 𝜃 sin 2𝜙 cos 2𝜓 2 All Sky average • SNR > 8 is used for express the GWD’s detectability. However SNR strongly depends on GWD antenna pattern, GW radiation pattern and polarization . • It is convenient to define a spherical GWD’s detectable distance (that is effective observable distance = Reff) 4𝜋 3 4𝜋 𝑅𝑒𝑓𝑓 ≡ 𝐹𝑟𝑚𝑠 𝑅 3 3 Sky average 𝑝𝑜𝑙,𝑛−𝑝𝑜𝑙 𝐹𝑟𝑚𝑠 = 3 𝟏 𝟓 𝐹𝑟𝑚𝑠 = 𝐹+ 2 + 𝐹× 2 ℎ𝑜𝑏𝑠 𝐹= ℎ+ + ℎ× For one polarized GWs and no-polarized GWs Antenna Pattern of MI 𝐹+2 or𝐹×2 𝑑𝜃𝑑𝜙𝑑𝜓 𝑝𝑜𝑙 𝐹𝑟𝑚𝑠 = 4𝜋 𝜋 2𝜋 2 𝑑𝜙 𝐹+,× 𝑑𝜃 0 𝑛−𝑝𝑜𝑙 𝐹𝑟𝑚𝑠 = 0 = 2𝜋 2 2 𝐹 or𝐹 𝑑𝜓 × + 0 4𝜋 = 𝟏 𝟓 1 4 50 2 = 𝜋 102 + cos 2𝜓 ± cos 4 2𝜓 here (+, +)(×, −) 128 5 3 1 2 𝐹+ 𝜃, 𝜙 +𝐹×2 𝜃, 𝜙 𝑑𝜃𝑑𝜙𝑑𝜓 2 4𝜋 𝐹 𝜃, 𝜙 𝐹 𝜃, 𝜙 2 2 ℎ𝑜𝑏𝑠 = ℎ + + ℎ× 2 𝐹+ ℎ+ + 𝐹× ℎ× = ℎ+ + ℎ× 2 2 + 𝐹 2 ℎ2 𝐹+2 ℎ+ 𝐹+2 ℎ2 + 𝐹×2 ℎ2 1 2 × × 2 = = = 𝐹 + 𝐹 + × ℎ+ + ℎ× 2 2ℎ 2 ℎ+ = ℎ× ≡ ℎ ℎ+ ℎ× = 0 Michelson Interferometer (MI) GW signal Extraction using Michelson Interferometer Arm : 3000m On the Earth Mirror1 Interfered fringe at PD was controlled to be dark fringe by adjusting the Mirror1 and/or 2 position according to PD signal. Heterodyne or homodyne technique are used for differential length signal extraction. GW signals are obtained in the differential feedback signal. Feedback Signal Mirror2 BS Laser (Bright port) Photo-detector(PD) (Dark Port) Quantum Noise in MI - Shot noise and Radiation Pressure Noise - GW signal Extraction using Michelson Interferometer Arm : 3000m On the Earth MI sensitivity should be limited only by “Shot Noise (SHN)” and “Radiation Pressure Noise (RPN)” that are originated by the uncertainty principle of photon. Standard Quantum Limit (SQL) is same with the crossing point between SHN and RPN. Theoretically, SQL can be beatable because of the correlation between SHN and RPN. Mirror1 Mirror2 BS Laser (Bright port) Photo-detector (Dark Port) Quantum Noise in MI - Shot Noise and Radiation Pressure Noise ●Heisenberg Uncertainty Principle ● SHN (Sensing Noise) and RPN (Force Noise) Photon number (N) in a FP cavity has fluctuation of (√N) Fig : Pro.Chen’s (Caltech) Slide If enhance the power, SHN decreases, while RPN increase. Quantum Noise in FPMI - Shot Noise and Radiation Pressure Noise Shot Noise ℎ𝑠ℎ𝑜𝑡 = 𝑓𝑐𝑢𝑡 ∶ Cut off Frequency of Interferometer ℎ𝑏𝑎𝑟 𝜆𝑓𝑐𝑢𝑡 1 + 𝑓 𝑓𝑐𝑢𝑡 2𝐿𝑃𝑐𝑎𝑣𝑖𝑡𝑦 = 2.5 × 10−24 3km 𝐿 𝑃𝑐𝑎𝑣𝑖𝑡𝑦 : Laser power inside one FP arm cavity 2 𝐿: FP cavity arm length 𝜆: Laser wave length 0.597MW 𝑃𝑐𝑎𝑣𝑖𝑡𝑦 1 2 𝑓𝑐𝑢𝑡 200Hz 1 2 𝜆 1064nm 1 2 m: Mirror Mass 1 + 𝑓 𝑓𝑐𝑢𝑡 2 [1 Hz] Radiation Pressure Noise ℎ𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 = = 1.6 × 10 2𝜋ℎ𝑏𝑎𝑟 𝑃𝑐𝑎𝑣𝑖𝑡𝑦 4 1 𝐿𝑚𝜔 2 𝜆𝐿𝑓𝑐𝑢𝑡 1 + 𝑓 𝑓𝑐𝑢𝑡 −22 3km 𝐿 30kg 𝑚 𝑃𝑐𝑎𝑣𝑖𝑡𝑦 0.597MW 1 2 200Hz 𝑓𝑐𝑢𝑡 1 2 2 1064nm 𝜆 1 2 10Hz 𝑓 2 1 1 + 𝑓 𝑓𝑐𝑢𝑡 2 Standard Quantum Limit ℎ𝑆𝑄𝐿 = 1 8ℎ𝑏𝑎𝑟 3km = 2.8 × 10−24 𝐿𝜔 𝑚 𝐿 30kg 𝑚 1 2 100Hz 𝑓 [1 Hz] [1 Hz] Michelson Interferometer (MI) GW signals and Dark/Bright Port in the dark fringe locking Mirror1 Mirror2 Bright Port Photon that is not interacted with GWs will come. Discarded if it is not recycled . Photo-detector (Dark Port) BS Laser (Bright port) Dark Port Photon that is interacted with GWs will come. GW signals !! Michelson Interferometer (MI) Michelson Interferometer 10-19 Strain [1/rHz] 10-20 Arm : 3000m On the Earth Mirror1 Photo-detector (Dark Port) Mirror2 BS Laser (Bright port) Power : P [W] 10-21 Optical Noise Shot Noise P 10-22 10-23 10-24 10 100 1k 10k Frequency [Hz] Problem 1 Short light path Less GW Effects Solution Insert Fabry-Perot Cavity for multi path FP Michelson Interferometer - FPMI - Michelson Interferometer Finesse = 200 ~ 1550 Ultra-low loss mirrors are required ! 10-20 Strain [1/rHz] FP cavity (over couple) 10-19 Optical Noise 10-21 Shot Noise 10-22 Radiation Pressure Noise 10-23 Amplitude Reflectance Amplitude (r2) Reflectance (r1) r1 r2 0.999990 is selected (Over Coupled) 10-24 10 100 1k 10k Frequency [Hz] Finesse : Multi Reflection parameter 𝜋 𝑟1 𝑟2 ℱ= 1 − 𝑟1 𝑟2 2ℱ 𝑁= 𝜋 Effective multi reflection number : FP Michelson Interferometer - Fabry-Perot Cavity Effect - Fabry-Perot Cavity Response for GWs GW Effect is proportional to traveling path length below frequency (fc) that corresponds to optical storage time. While, GW effect cancellation will be dominant above fc. 𝑐 𝑓𝑐 = 4𝐿ℱ fc Michelson Interferometer (MI) GW signals and Dark/Bright Port in the dark fringe locking Bright Port Photon that is not interacted with GWs will come. Discarded if it is not recycled . Mirror2 Mirror1 BS Bright fringe can be regarded as reflected laser power from a “Compound” mirror whose actual entity is MI !! Let’s make a FP cavity with a new mirror in front of a laser and “compound” mirror to enhance the effective injected laser power to MI!! Photo-detector (Dark Port) Laser (Bright port) Dark Port Photon that is interacted with GWs will come. GW signals !! FP Michelson Interferometer -Recycled MI- GW signals and Dark/Bright Port in the dark fringe locking Bright Port Photon that is not interacted with GWs will come. Discarded if it is not recycled . Bright fringe can be regarded as reflected laser power from a “Compound” mirror whose actual entity is MI !! Let’s make a FP cavity with a new mirror in front of a laser and “compound” mirror to enhance the effective injected laser power to MI. Enhancement of effective laser power is defined as Recycling Gain. 10 – 50 gain is practical because reflectance of the Compound mirror cannot be so high. Compound Mirror as MI PR Cavity Power Recycling Mirror Power Recycled FPMI - PR-FPMI- Power Recycled Fabry-Perot Michelson Interferometer (PR-FPMI) 10-19 ETMx ITMx ITMy BS 10-20 Strain [1/rHz] ETMy 10-21 Optical Noise Radiation Pressure Noise Shot Noise 10-22 P 10-23 10-24 10 100 1k 10k Frequency [Hz] Power Recycling Mirror Better Sensitivity Quest - What is GW signal proportional to ? - What is GW signal proportional to ?? 10-19 Arm length x Finesse x Input Power x Recycling Gain 10-20 All parameters reach best effort, except Finesse. However, high finesse just enhanced the sensitivity at low frequency that is dominated by other noise such as seismic, gravity gradient and RP noise, because the FP cavity cancelling effect that was referred in the previous page. Is there any method that can individually set the storage time of photons that are not interacted with GWs (go to PRM) and that are interacted (go to dark port)? Answer : Resonant Sideband Extraction (RSE) Mirror !! Strain [1/rHz] !! Photon number stored in a FP cavity !! Optical Noise 10-21 10-22 10-23 10-24 10 100 1k 10k Frequency [Hz] PR-FPMI with RSE technique PR-FPMI With Resonant Sideband Extraction 10-19 Strain [1/rHz] 10-20 Set Finesse ~ 1500 Optical Noise 10-21 10-22 10-23 Power Recycling Mirror Resonant Sideband Extraction Mirror 10-24 Signal Extraction Gain is also defined (practically ~ 10) 10 100 1k 10k Frequency [Hz] This is one of example of sensitivity using RSE technique. RFPMI with RSE technique RFPMI With Resonant Sideband Extraction ETMx ITMx ETMy ITMy PRM RSEM ETM Compound Mirror consisting of RSEM and ITM RSE Cavity GW sidebands (photons that are interacted with GWs) are generated in the FP cavity. For these photons, reflectance of ETM and a compound mirror, which consists of RSEM and ITM, decide the optical storage time ! RSE Cavity optical storage time can be set freely (longer and shorter than arm FP cavity storage time ) by RSE mirror reflectance and position. On the other hand, optical storage time of photons that are NOT interacted with GWs is decided by FP arm cavity itself. Laser - as a ruler for precise length measurement Invention of principle of a laser in 1953 by Towns and Schawlow is epochmaking. Development of laser realized interferometric GWD idea by R.Weiss (MIT), instead of Resonant Type GWD, at that time. Ar+ gas laser(514nm) was used for 80’ GWD development. • Unstable • High energy mirror burn-in due to hydro-carbon in vacuum Nd3+:YAG laser (1064nm) as a solid state laser (NPRO style) by Byer made revolution in 90’ • Narrow line width • Easy handling • Stable and High Efficiency (100 times better) • IR region avoid burn-in • Easy power-up by amplifier After 2010’, a fiber laser will be main !? Laser - requirements Requirement for 1st and 2nd generation GWD (default : 1064nm) • Single mode (TEM00) should be additionally stabilized • Low Frequency noise (dn/n) should be additionally stabilized • Low Amplitude noise (dP/P) should be additionally stabilized • Low transverse mode (M2 ~ 1) • Stable and tunable of the laser frequency (PZT on a crystal, Heater and Outer EOM) • ~200W CW power • Optics for the laser can be high quality (low loss, low absorption, low scattering, high damage threshold ) • Because GWD is a “phase sensor”, a short wavelength laser seems to be preferable. However, optics cannot meet GWD quality . • Two applicants • Solid state seed laser + Bulk Amps (MOPA) • Solid state seed laser + Fiber Amps (MOFA) Laser - High Power laser development Injection Locking Laser Laser oscillation of a slave laser consisting of bow tie FP cavity that includes two YAG crystal inside is synchronized with a master NPRO laser. Frequency noise is dominated by a master laser, while amplitude noise is dominated by the slave laser. In 2000, SONY produced 10W, 1064nm laser Fiber LDs YAG YAG Mirror with PZT Reso EOM NPRO master Laser - MOPA or MOFA Requirement for 1st and 2nd generation GWD • • Solid state seed laser + Bulk Amps (MOPA) Solid state seed laser + Fiber Amps (MOFA) Fiber Laser Idea LZH 200W Laser for Adv-LIGO Mitsubishi 150W FP cavity Mirrors as “free falling mass”, “photon handler”, “thermal noise origin” Size Requirement : Large is better to reduce SQL, thermal noise • ~ 50kg is practical limit. Diameter becomes 35cm ~ 22cm. • practical means substrate production, polishing, coating. Optical Requirement : Low loss (scattering and absorption) and homogeneity • Multi-layered optical coating is inevitable for 1064nm on substrate • Smooth surface (polishing) is required • Loss spoils shot noise, recycling gain, squeezing effect and increases scattered light noise. • Less than ~ 50 ppm loss for 30cm diameter is required. • ~ 2km Radius Of Curvature (ROC) required Advanced-LIGO mirror 1.2nm RMS micro roughness FP cavity Mirrors as “free falling mass”, “photon handler”, “thermal noise origin” Substrate Requirement : Low loss (absorption and mechanical) and homogeneity • Absorption results in heat lensing spoils GWD optical performance. • For substrate (SiO2 or Sapphire, Silicon in 3rd GWD) : f < 10-8 • For optical coating (Ta2O5/SiO2) : f < 10-4 (mainly comes from Ta2O5) • Thermal noise due to optical coating limits the 2nd GWD sensitivity !! Subjects Requirement Subjects Requirement Substrate SiO2 : Adv-LIGO, AdvVIRGO, GEOHF Al2O3 : KAGRA Surface Micro Roughness < 0.1 nm rms Surface Waviness < l /500 Coating Mechanical loss f < 10-4 Substrate loss f < 10-8 ROC ~ 3km ROC error ~ 10m Size D:35cm ~ 22 cm t : ~15cm Substrate Loss < 0.1ppm (SiO2), <20ppm(Al2O3) Coating Loss < 40 ppm (< 0.5ppm for absorption) FP cavity Mirrors - how to make or what company should we order to ? - Mirror manufacturer are strictly limited in the World Polish and Coating LMA Lyon in France CSIRO in Australia REO ATF Polish ZYGO in USA In Japan… Japan Aviation Electronics (not now) Sigma KOKI (small optics ?) Showa Koki Tokai Kogaku Canon (sapphire ??) Inevitable Seismic Noise Introduction Mirrors are suspended like a pendulum on the Earth as free mass X [m/rHz] Single Pendulum f0 [Hz] A mirror is inevitably excited by seismic noise through the pendulum transfer function (TF). TF has isolation effect for seismic noise ! Y [m/rHz] Transfer Function (TF) Y f0 X f 2 Y f0 X f 2N For more isolation… • Multi-stage • Lower pendulum resonance frequency Trade-off : Many resonance mode (excited by Q) damping is necessary. Better Seismic Noise Isolation For more isolation… • Multi-stage • Lower pendulum resonance frequency 10 1 10 0 10 10 10 10 Reduction of peak height –1 –2 –3 2 10 2 10 1 10 1 10 0 10 0 10 10 10 –4 10 10 –1 10 0 10 Frequency 1 Degraded isolation 10 2 10 Better Isolation ratio –1 –2 –3 f -4 f –4 10 Isolation Ratio 2 Isolation Ratio Isolation Ratio 10 Low reso. freq. Low-freq. cut-off → Better isolation Multi stage More steep reduction → Better isolation Damping Reduce Q-value → improve stability –1 10 0 10 Frequency Resonant peaks 10 10 -2 1 10 10 2 10 Better Isolation ratio –1 –2 –3 –4 10 –1 10 0 10 Frequency 1 10 2 Drift by environment Ando’s (U Tokyo) View Graph Seismic Noise Variation on the Earth - 0.1 Hz ~ 100 Hz - Seismic Noise ( f > 0.1Hz) that can be isolated by technology Micro seismic noise peak around 0.2 Hz due to ocean waves Seismic Noise Model Above 1 Hz 2 1 f seis A [m / Hz ] f By Rana (LIGO) A 10 7 city area A 10 9 underground Seismic Noise Variation on the Earth - Low Frequency Range (< 0.1 Hz )- Seismic Noise ( f < 0.1Hz) that cannot be isolated by technology Tidal motion (~ 12 hours) Local trend (~ months) Underground water (~1 month) Snow fall (~ 1 month) Rain, Wind, Air Pressure (~ months) Free Oscillations of the Earth (~ 10 minutes) Permanent step motion due to large earthquake far away (permanent) Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Passive Isolation and damping (Simple) Metal wire pendulum (~ 1 Hz resonance Frequency) for Horizontal Classical blade spring (~ 5 Hz resonance Frequency) for Vertical Eddy Current Damping (no active damping) Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Passive Isolation and damping Example in CLIO (japan) 6 stages seismic noise isolation pendulum in CLIO 4 stages blade spring, 2 stages wire pendulums Orange stages (Copper Plate) motion are damped by NdBFe magnets those are set in themselves and magnet Base. No active damping. Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Passive Isolation and damping Example in CLIO (japan) Motorized mirror alignment stage 3 stage blade springs ~ 2m height magnet base Cryo base Upper Stage Upper masses and magnet base Sapphire Mirror Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Passive Isolation and Active damping in VIRGO Super Attenuator Inverted Pendulum (IP) (~ 30 mHz resonance Frequency) for Horizontal Geometrical Anti Spring (GAS) (~ 200 mHz resonance Frequency) for Vertical Active damping using LVDT sensors and “Digital” feedback to magnet-coil actuators Recoil-mass introduction not to introduce actuating point noise Compressing Force Gravity point is set upper area IP GAS Anti-Spring 30mHz Resonance Frequency Spring (Metal Rod) Gravity (Blade Spring) ~ 200 mHz Resonance Frequency Gravity Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Principle to obtain low resonance frequency : Set the residual spring constant to be small with combination of spring and anti-spring. Problem and Trade-off : 1. Spring constant sensitive to temperature Balanced position change. 2. Internal buckling results in instability. 3. Isolation effect lost at ~100 times resonance frequency because of percussive effect. Compressing Force Gravity point is set upper area IP GAS Anti-Spring 30mHz Resonance Frequency Spring (Metal Rod) Gravity (Blade Spring) ~ 200 mHz Resonance Frequency Gravity Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Recoil-Mass Introduction Coil Magnet If the actuators are fixed on the structure that is not isolated, the seismic noise fluctuates the mirror through this coupling. For this remedy, the actuators are also isolated like a pendulum. (Counter recoils mass arrangement ) 共振ダンプ倒立維持装置 Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Active Isolation introduction and Active damping in GEO and Advanced-LIGO HEPI Low frequency active isolation using accelerometer. Electromagnetic active isolation system. Seismic Noise Isolation -Active and Passive Isolation and Resonance Damping - Active Isolation introduction and Active damping in GEO and Advanced-LIGO 4 stage passive isolation using classical blade springs and wire suspensions. Recoil-mass introduction not to introduce actuating point noise Mirror Actuators - Magnet Coil Actuator with optical shadow sensor- Mirror and Upper Masses Motion Sensors and Actuators OSEM : a set of shadow sensor and magnet-coil actuator in one holder Magnet are glued on the back of mirrors and OSEMs are set in the recoil-mass for mirror motion sensing and actuation Force is linear in ~ mm range. Electrical noise coupling is large. Mirror Actuators - Electro Static Actuator High Voltage Low Noise ! Non linear force, Naive Treatment (distance control is strict : ~ sub mm) Electro Static Mirror Actuator Force Mirror Mass Gravity Gradient Noise (GGN) - gravity force fluctuation from the ground due to seismic noise - What is GGN ? • Gravitational coupling due to seismic density fluctuations • Cannot be shielded • Expected noise model is proportional to f -4 • Limiting noise sources for the 3rd generation GWDs • A few solutions to minimize GGN are • Go underground • Subtraction using some sensors TF f GGN f 4 2 Gravity Force Mirror Seismic Motion Seismic f 2 Gravity Gradient Noise (GGN) - gravity force fluctuation from the ground due to seismic noise - GGN estimation P.R.Saulson et al PRD 1984 Kip Thorne et al PRD 1999 Gravity Gradient Noise (GGN) - gravity force fluctuation from the ground due to seismic noise - GGN estimation in ET project Sound Noise Even the main optics are housed in vacuum tanks, we noticed that sounds deteriorated the sensitivity. Input optics (laser and its injection optics) and output optics (PDs) isolation from sounds were effective. The transfer function from sounds to sensitivity might depend on each GWD environment, so it might be difficult to derive some universal laws. Anyway, sounds shield is important. Residual Gas Noise Residual gas introduces refraction index fluctuation and it results in path length fluctuation as noise in GWD. Requirement : 10-7 [Pa] (for KAGRA) assuming safety factor of 10 for ~ totally 6~8km in length and 1.2~0.8m in diameter volume (~8000 m3). Theoretical Model of “Residual Gas Noise” 2 𝑥𝑣𝑎𝑐 = 3 𝑇0 2 𝐿 8 2 𝑛0 − 1 𝑝 4𝜋 𝜋 𝐴0 𝑉0 𝑢0 𝜆𝐿 𝑝0 𝑇 [m/ Hz] 𝐿 ∶ Cavity length, 𝜆 ∶ Laser wave length, 𝑛0 ∶ Reflactive index, 𝑢0 ∶ Particle mean velosity, 𝑉0 ∶ 1 mol volume,𝑉0 ∶ Avogadro number, 𝑇 ∶ Temperature, 𝑇0 ∶ 273.15 K , 𝑝 ∶ Pressure, 𝑝0 ∶ 1 [atom], Keep -7 10 [Pa] in ~8000 3 m How to obtain 10-7 [Pa] LIGO, VIRGO 1 : Surface of vacuum tube has no special treatment, except normal polishing. 2 : 4km~3km arm tube are connected with welding. 3 : Rough evacuation by TMPs, then evacuation with Baking. 4 : maintain with Cold Trap using liquid Nitrogen and Ion pump. LIGO H1 4km welded tube covered by heater LIGO H1 Mirror Tank in Hanford VIRGO Keep -7 10 [Pa] in ~7000 How to obtain 10-7 [Pa] KAGRA 1. 2. ECB surface for KAGRA Because of tunnel, welding and LN2 drop off Surface passivation by electro-polish followed by baking” • • 3. 3 m Outgassing rate: 10-8 Pa m3 m-2 s-1, or lower surface roughness Rmax 3 mm, Ra 0.5 mm ITMX 3000 m Mirror finish by Electro-Chemical Buffing (tubes in the mid 800-m region)” ETMX • Surface roughness; Rmax 0.2 mm, Ra 0.03 mm 4. 3 m /h >> dry-pump Flange connection with metal O-ring (silver 600 2000 L/s >> TMP 500 L/min >> TMP fore-line pump plated)” • 5. Erosion proof by humidity test 1000 L/s >> IP Rough Evacuation by TMPS and maintain by Ion-pumps TMP IP DRY P FL P Laser Frequency Noise If Interferometer is ideal (perfect balance in each arm length and power) : Laser Frequency noise don’t spoil the GWD sensitivity. Laser Frequency Noise couples with imperfection of the interferometer with a factor of Common Mode Rejection Ratio(CMRR). 𝛿𝜙𝑀𝐼 𝛿𝜈 Δ𝐿 = 2𝜋 → 𝐿 × 𝐶𝑀𝑅𝑅 × 𝑐 𝛿𝜈 [m/ Hz] 𝜈 Δ𝐿 Δℱ 1 𝐶𝑀𝑅𝑅 = + ≈ → 𝛿𝜈 < 10−8 [Hz/ Hz] 𝐿 ℱ 300 MI 𝐿 ∶ Cavity length, Δ𝐿 ∶ Cavity length difference, ℱ ∶ Finesse, Δℱ ∶ Finesse Difference, ν ∶ 𝐿𝑎𝑠𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦, 𝛿𝜈 ∶ Laser Frequency Noise RFPMI-RSE Laser Frequency Noise Stabilization - Concept Original laser frequency noise (~ 1kHz line width) is not enough at all for GWD. 100 Hz/rHz @ 1ooHz original noise should be 10-8 Hz/rHz !! 4~3 km arm FP cavity whose mirrors are well isolated from seismic noise can be a good frequency reference. Laser PBS EOM λ/2 𝛿𝑥 = 𝛿𝐿𝐹𝑃 λ/4 Photo Detector Ref : r1 Tra : t1 Loss : A1 = 0 𝐿𝐹𝑃 r2 + t 2 = 1 Phase Shifter Mixer Oscillator (wm) Feedback to several laser frequency tuning port Demodula ted Signal 𝛿𝐿𝐹𝑃 𝛿𝜈 = 𝐿𝐹𝑃 𝜈 Ref : r2 Tra : t2 Loss : A2 = 0 Laser Frequency Noise Stabilization - Concept Because it is difficult to stabilize the original F-noise to FP revel, two FP cavities are prepared for pre-stabilization. 1. 2. 3. Pre-Mode Cleaner FP (~ 1m) : Rigid triangle or bow tie cavity Mode Cleaner FP (~ 30m) : Moderately isolated triangle cavity FP arm Cavity (~ 3km) : of course, mirrors are isolated by multi-stage pendulums Laser Intensity Noise Laser Intensity Noise also couples with imperfection of the interferometer 𝛿𝑃 𝑥𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑥𝑅𝑀𝑆 [m/ Hz] For MI 𝑃 𝛿𝑃 For FPMI 𝑥𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑘𝐵𝑆 𝑥𝑅𝑀𝑆 [m/ Hz] 𝑃 𝛿𝑃 < 10−8 𝑃 𝑥𝑅𝑀𝑆 𝑥𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 1 𝑘𝐵𝑆 ~ ℱ 𝑃 ∶ laser power, 𝛿𝑃: Laser power fluctuation, 𝑥𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = : Mirror Fluctuation under feedback control, 1+𝐺 ∶ Original mirror fluctuation before control before control, 𝐺 ∶ Loop gain, 𝑘𝐵𝑆 ∶ coupling ratio of 𝑙− signal with 𝐿− signal Laser Intensity Noise Intensity Noise contribution in GW signal monitor port (MI case) GW Signal Monitor Port [V] 𝑥0 𝑥𝑟 𝛿𝑃 1+ 𝑃 𝐻𝑃𝐷 𝐶 𝐻𝑆 [m/V] @ mirror actuator [V/m] @ PD [V/V] Circuit 𝐺 = 𝐻𝑃𝐷 𝐶𝐻𝑠 𝐺 1 𝑥0 𝛿𝑃 𝑥0 + GW Signal Monitor Port ≅ 1 + 𝐺 𝐻𝑆 1+𝐺 𝑃 Laser Amplitude Noise Reduction Stabilization concept is simple • • • Split a beam in two portions and detect them by different PDs. One PD output compared with stabilized DC voltage, and the residual is fedback to laser current control port or AOM on the beam axis. The intensity from the other PD is used for intensity stabilization evaluation. Annoying Results for 10 years • • The intensity noise from the PD that is used for stabilization is of course well stabilized according to control loop gain. While, the intensity noise from the other PD was not stabilized as well as in loop one ??? Laser AOM BS PD2 for out-loop PD1 for In-loop DC source Current Control Spectrum Analyzer Laser Amplitude Noise Reduction A mystery has been solved by a Mode Cleaner. • • F.Seifert found that intensity stabilized level discrepancy between in and out loop by using a beam after a mode cleaner. This implies that beam jitter noise including higher mode mixing affects the discrepancy. F.Seifert et al MPQ(AEI) and Univ. Hanover Beam Jitter Noise (Beam lateral shift originated from higher order transverse mode in a laser and/or seismic noise) and (the fluctuation in yaw/pitch motion of a beam splitter) couple and generate Beam Jitter Noise. M2 𝑥𝑗𝑖𝑡𝑡𝑒𝑟 = 2 2𝛿𝛼 𝛿𝑥 [m/ Hz] δ𝛼:Lateral Shift of Beam 𝛿𝑥:BS Angular Fluctuation L2 Laser Just calculate L2 – L1 BS L1 M1 Reduction for Beam Jitter Noise • Isolated Mode Cleaner to reduce transverse mode and seismic noise reduction. • BS seismic noise isolation. Scattered Light Noise Mirror surface could not be perfect flat and have a few defects in the coating films. Substrate has also imperfection of crystal structure and impurity. These produce scattered light toward the vacuum tube and small fraction of them decouples with the main beam. Scattered light Noise If there is no scattered light from the mirror, we cannot monitor the spots on mirrors as the bottom picture Scattered Light Noise Scattered light noise model Two types • • Non-up-converted (can be quantitatively estimated ) Up-converted (should not appear !! It reveals big motion of components) Decoupling process is so complicated because of many multi path reflection that theoretical prediction is based on simple models. Assuming Non-up-converted Scattered Light… (xseis < l/2) hGW k Pifo fGW Reflector of Scattered Light (SL) fGW : GW signal phase change, xsctr A f xseis [m Pifo : Power in IFO, xseis : SL Reflector Fluctuation , We define κ kL Pifo Hz ] A f : Excitation factor of Reflector, Scattered Light Reflection 2 xseis Phase Change of SL : fsctr 2 Scattered Light Noise : d sctr Lk Precom fsctr l l : wave length, Precom fsctr Pifo L : Arm length, Scattered Light Noise Reduction Scattered Light Noise [m/rHz] For narrow angle SL Effective Baffle Numbers in one arm (KAGRA) Baffle Rings in arm ℎ𝐿 𝑁= 𝑑𝑅 h:Baffle height d:baffle interval L:Arm length R:Duct radius LIGO VIRGO KAGRA R [m] 1.2 1.2 0.8 H [cm] 6 10 5 Angle [d] 55 45 50 ?? Coating Oxide AR DLC number 222 18 250 Scattered Light Noise Reduction For wide angle scattered light For large size mirrors Thermal Noise - Final barrier to reach targeted quantum noise level - Thermal Noise Source in GWD Thermal Noise Fluctuation of a harmonic oscillator (HO) induced by thermal bath. (1) HO Mirror LIGO Φ250 SiO2 Motion Equation of a harmonic oscillator 𝑚𝑥 + 𝛾𝑥 + 𝑘𝑥 = 𝑓𝑡ℎ (1)SiO2 (2)Sapphire(20K) (3)Silicon (cryo) Thermal Force Fluctuation-dissipation theorem 2 𝑓𝑡ℎ = 2𝑘𝐵 𝛾𝑇 White noise 𝛾 = 𝑚𝜔0 𝜙 Thermal Noise Spectrum ∝ 𝜙𝑇 𝜙: Mechanical loss 𝑇 : Temperature In order to reduce thermal noise, reduce 𝑇 and 𝜙 𝟏 𝑸. There are many kinds of origin of𝜙 Possible substrate (2) HO Pendulum (Final Stage) CLIO Φ100 Sapphire Possible material (1)Piano Wire (2)SiO2 (3)Sapphire(20K) (4)Silicon(cryo) Thermal Noise of Mirror - structure damping model - Internal structure damping Model • Internal friction of material is the origin of energy loss • In the case of mirror, Noise spectrum below resonance frequency is expressed… 𝑥𝑚𝑖𝑟 (𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒) 2 4𝑘𝐵 𝑇𝑚 1 − 𝜎 2 = [1 𝐿 𝜋𝑄𝑚𝑖𝑟 𝐸0 𝜔0 𝜔 5.4 × 10−25 3km 𝐿 108 𝑄𝑚𝑖𝑟 1 2 Hz] 𝑇 20K 1 2 3.5cm 𝜔0 1 2 4 1 11 × 10 Pa 2 𝐸0 𝑄𝑚𝑖𝑟 ∶ Q of Mirror, 𝑇 ∶ Temperature, 𝜎:Poisson Ratio, 𝜔0 ∶ Beam spot size, 𝑓 ∶ Frequency, 𝐸0 :Young′s modulus 100Hz 𝑓 1 2 Thermal Noise - structure damping model - To Reduce mirror Thermal Noise … Mechanical Q (1/f) factor (1)SiO2 :~107 (300K) (2)Sapphire :3x106 →2x108 (5K) (3)Silicon :4x107 →4x108 (6K) (4)CaF2 :~1x107 (300K) Generally, cryogenic temperature increase Q, except SiO2. Mechanical Q temperature dependence Thermal Noise of Mirror -Thermo-elastic noise - Thermo-elastic Noise (300K) Loss due originated from temperature gradient relaxation triggered by inhomogeneous elastic compression and expansion. 𝑥𝑚𝑖𝑟 (𝑡ℎ𝑒𝑟𝑚𝑜) = = 8.8 × 10−24 8𝛼𝑇 1 + 𝜎 𝜌𝐶𝐿𝜔 3km 𝐿 𝛼 5 × 10−6 /K 𝑘𝐵 𝜅 [1 𝜋𝜔03 𝑇 300 K Hz] 𝜅 40 W/m/K 1 2 3 4 g/cm 𝜌 3.0cm 𝜔0 3 2 790 J/kg/K 𝐶 1 4 100Hz 𝑓 𝛼 ∶ Thermal expansion ratio, 𝑇 ∶ Temperature, 𝜎:Poisson Ratio, 𝜅 :Thermal conductivity, 𝜔0 ∶ Beam spot size, 𝑓 ∶ Frequency, 𝐸0 :Young′s modulus, 𝐶: Specific heat Thermal Noise of Mirror -Thermo-elastic noise - Thermo-elastic Noise (20K) Loss due originated from temperature gradient relaxation triggered by inhomogeneous elastic compression and expansion. 𝑥𝑚𝑖𝑟 (𝑡ℎ𝑒𝑟𝑚𝑜) = 9.4 × 10−25 3km 𝐿 8𝛼𝑇 1 + 𝜎 = 𝐿 𝛼 5 × 10−6 /K 𝑘𝐵 𝜋𝜅𝜌𝐶𝜔 𝑇 20 K [1 Hz] 1.57 × 104 W/m/K 𝜅 1 4 4 g/cm3 𝜌 1 4 3.0cm 𝜔0 3 2 0.69 J/kg/K 𝐶 1 4 100Hz 𝑓 𝛼 ∶ Thermal expansion ratio, 𝑇 ∶ Temperature, 𝜎:Poisson Ratio, 𝜅 :Thermal conductivity, 𝜔0 ∶ Beam spot size, 𝑓 ∶ Frequency, 𝐸0 :Young′s modulus, 𝐶: Specific heat 1 4 Thermal Noise of Mirror - Brownian noise due to HR optical coating - Brownian noise due to HR (Ta2O5/SiO2 layered) optical coating • This thermal noise limits the present GWD sensitivity. • Ta2O5 is identified to be main mechanical loss source. 𝑥𝑐𝑜𝑎𝑡(𝑠𝑡𝑟𝑢𝑐𝑡) 4 2𝑘𝐵 𝑇 1 + 𝜎 1 − 2𝜎 𝑑𝑐 𝜙𝑐 = [1 𝜔0 𝐿 𝜋𝜔𝐸0 = 1.1 × 10−24 3km 𝐿 𝜙𝑐 4 × 10−4 1 2 𝑑𝑐 5 μm 1 2 𝑇 20 K 1 2 Hz] 3.5cm 𝜔0 11 4 × 10 Pa 𝐸0 1 2 100Hz 𝑓 1 2 𝜙𝑐 : Loss of coating, 𝑑𝑐 : Coating thikness, 𝑇 ∶ Temperature, 𝜎:Poisson Ratio, 𝜅 :Thermal conductivity,𝜔0 ∶ Beam spot size, 𝑓 ∶ Frequency, 𝐸0 :Young′s modulus, Thermal Noise of Mirror - Brownian noise due to HR optical coating - Brownian noise due to HR (Ta2O5/SiO2 layered) optical coating • This thermal noise limits the present GWD sensitivity. • Ta2O5 is identified to be main mechanical loss source. Many ideas to reduce loss of HR coating • • • • Reduce thickness of Ta2O5, keeping same HR reflectivity Use Al2O3 instead of Ta2O5 Dope Ti in Ta2O5 Most possible solution Use not TEM00 mode beam but LG03 mode (doughnut mode) All of them results in 1.5 improvement (so small !!) Cryogenic mirror is a straightforward method ! But… • There are many difficulties of cryogenic techniques. Thermal Noise of Pendulum - structure damping model - Internal structure damping Model • Internal friction of material is the origin of energy loss. • Noise spectrum above resonance frequency. 𝑠𝑡𝑟𝑢𝑐𝑡 𝑥𝑝𝑒𝑛𝑑 4𝜔𝑝 𝜔 = 𝐿 5 −2 = 1.8 × 10−25 𝑘𝐵 𝑇 [1 𝑚𝑄𝑝 3km 𝐿 9.4 Hz] 1 7 2 × 10 𝑄𝑝 𝑇 20 K 1 2 30 kg 𝑚 1 2 𝑓𝑝 0.7 Hz 1 2 100Hz 𝑓 𝑄𝑝 ~ 1 𝜙𝑝 : Quality factor of pendulum, 𝑇 ∶ Temperature, 𝑓𝑝 ∶ Pendulum resonance frequency, 𝑓 ∶ Frequency, 5 2 Thermal Noise of Pendulum - structure damping model - To Reduce pendulum Thermal Noise … Mechanical Q (1/f) factor (1)Metal : ~ 106 (2)SiO2 : ~107-8 (300K) (3)Sapphire : 1x105 →1x108 (20K) (4)Silicon : ?? Generally, cryogenic temperature increase Q, except SiO2. S.Reid et al @ GEO600 Hydroxy-Catalysis Bonding not to increase loss. Actual Thermal Noise Measured in CLIO Suspension Thermal Noise Tangsten Wire Suspension Thermal Noise Sapphire Mirror Thermo Elastic Noise P~100mW Shot Noise Mirror Therm-oelastic Noise Seismic Noise Summary PR-FPMI with RSE is the ultimate style to obtain the targeted strain sensitivity as GWD. Ultimate Sensitivity should be dominated only by “Quantum Noise”. There are many noise sources that should be suppressed and many techniques were investigated. Presentation Files http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-1.pptx http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-2.pptx http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-3.pptx http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-4.pptx
