# LIGO：レーザー干渉法（波長> $10 ^ {-7}$ m）は、腕の長さの変化<$10 ^ {-18}$ mをどのように検出できますか？

LIGO干渉計の感度を理解しようとしています。2つの検出器間のノイズキャンセレーションを管理し、非常に純粋なレーザー信号を達成する方法、干渉計アームの有効長を1000km以上に変更するための多くの反射、およびその他の印象的な技術のトリックについて、多くの議論を続けてきました。この驚くべき測定の偉業を達成します。それでも頭が動かないのは、ノイズがまったくない完璧な世界でさえ、位相シフトが$$10 ^ {-7} - 10 ^ {-11} サイクル程度（実施している感度の主張に応じて）は信号として表示されます。$$

これがどのように機能するかについての計算の詳細がかなり複雑になることを感謝しますが、ここで感度の課題を理解するための最良の方法についていくつかのポインタを本当に感謝します：

• これは、結合されたレーザー信号の小さな振幅変動を測定する場合ですか？
• これらの振幅変動のサイズ（振幅の比率として）は、長さ変動のサイズ（反射のx数およびレーザー波長の比率として）と同じですか？
• 上記のそれぞれに当てはまる場合、私は彼らがレーザー振幅の発振を$$10 ^ {-10}$$倍の標準的な振幅まで検出していると思いますか？
• 最初の2つのポイントが「いいえ」の場合、できる限り真っ直ぐに設定してください。よろしくお願いします。

The LIGO interferometer uses a homodyne detection technique. Basically, the light travelling in each arm of the interferometer is derived from the same laser source and is combined in the output channel and falls onto a photodiode.

The interferometer is operated so that when there is no gravitational wave (GW) passing through the instrument, the beams combine to produce a dark fringe (i.e. they are set to destructively interfere). There is a small offset from this, but basically the phase difference between the combiniing beams is close to $$\pi$$.

The phase difference caused by a GW, due to the changing length of one arm with respect to the other, can be derived as $$\Delta \phi \simeq 2\pi \left(\frac{2hL}{c}\right) \left(\frac{c}{\lambda}\right) = \frac{4\pi}{\lambda} hL \ ,$$ where $$L$$ is the length of the arms, $$\lambda$$ is the laser wavelength and $$h$$ is the strain amplitude of the fravitational wave signal. Actually, it is a little bit more complex than this, since the arms act as Fabry-Perot resonators which means the light effectively travels backwards and forwards many times in the arms (about 300 for LIGO, i.e. $$L$$ is effectively 1200 km).

For a typical dimensionless GW strain of $$h \sim 10^{-21}$$, $$\lambda = 1064$$ nm, then $$\Delta \phi \sim 10^{-8}$$ and is modulated at the frequency of the GW (typically 20-2000 Hz).

The problem then reduces to combining $$E_{\rm tot} = E_0\sin (\omega_l t) + E_0 \sin (\omega_l t + \alpha + \Delta \phi)\ ,$$ where $$E$$ is the electric field in each arm, $$\omega_l$$ is the angular frequency of the laser and $$\alpha$$ is the offset phase between the arms (close to $$\pi$$).

Using the identity $$\sin a + \sin b = 2 \cos[(a-b)/2] \sin[(a+b)/2$$ and squaring the total E-field to get an intensity: $$I = 4E^2 \cos^2[(\alpha + \Delta \phi)/2]\, \sin^2[\omega_l t +(\alpha + \Delta \phi)/2]$$

Since $$\omega_l$$ is much greater than the GW frequency and much higher than can be sampled by any photo-sensitive detector, then the second term in the product above can be replaced by its time-average of $$1/2$$. If we now identify the total power $$P_{\rm in}=E^2$$ as the average input power to each arm of the interferometer and note that $$\cos^2 (a/2) = (\cos(a)+1)/2$$ and $$\Delta \phi \lll 1$$ $$I = P_{\rm in} \left[1 + \cos(\alpha + \Delta \phi ) \right] \simeq P_{\rm in} \left[1 + \cos(\alpha) -\Delta \phi \sin(\alpha)\right] = 2P_{\rm in} \left[ \cos^2 (\alpha/2) - \frac{\Delta \phi}{2}\sin \alpha \right]\ .$$

It is the second term inside the bracket that contains the signal of the GW. That signal is proportional to the power in the interferometer and the phase difference between the arms. Note that although the signal-to-(shot) noise is mathematically maximised when $$\alpha=\pi$$, this would mean the SNR was 0/0 ! In practice there is always some other noise present so $$\alpha$$ is shifted a little bit away from $$\pi$$ - Fricke et al. (2012) suggests that $$\alpha \sim \pi+ 6\times 10^{-5}$$ is used.

The power input into each arm is about 600 W (the 100s of kW Steve Linton mentions in a comment is after accounting for the Fabry-Perot resonator, which I did above by talking about an "effective $$L$$"). In the absence of other forms of noise then photon counting (shot noise) becomes the limiting factor and is proportional to the square root of the power.

The output signal is the modulated GW signal discussed above which is recorded by detecting photons with the photodiodes. The response function that translates the photodiode signal into a strain is determined by acting on the test masses/mirrors with precisely calibrated lasers modulated at GW frequencies that can produce monochromatic phase shifts in the arm lengths.