Resonant Sideband Extraction
2.2 Practical Limitations
2.2.1 Power Recycling
Power recycling is the method used to make the most efficient use of the laser light as possible.[39] When the Fabry-Perot Michelson is thought of as a compound mirror, the concept of power recycling is fairly easy to understand. The carrier is optimally coupled into the interferometer when the transmittance of the power recycling mirror is equal to the losses of the Fabry-Perot Michelson. This also corresponds to the maximum power both stored in the power recycling cavity, and incident on the arms.
Hence, the amount of power in the arms is maximized.
One of the freedoms afforded by adding the signal mirror is the ability to store more light in the arm cavities. This tends to increase the energy stored in the in- terferometer. Without the signal mirror, the only parameter with which to tune the frequency response of the interferometer is the ITM transmittance. This sets the total arm losses, which in turn sets the power recycling gain to one fixed value. However, with the signal mirror, the ability to make the signal bandwidth much broader than the arm cavity bandwidth alone frees up the choices for the arm cavity ITM. Since the largest sources of loss in the interferometer are the anti-reflection (AR) coatings of the ITM substrates and the beamsplitter,5 all of which are in the power cavity, the
51n the case of sapphire ITMs, the loss associated with substrate absorption is also very high.
total losses can be decreased by reducing the power recycling gain. More light is in the arms as a result, which can be seen in Figure 2.9. This curve is generated using
1.8
~ 1.6 ....... .
i
;;; 1.4 .. ···'···'···'··'·<·< .-·.·.-··· .. •... ; ....•. :~
8.
0.8
0.6
. : .. ; . ~ .:. ; . ... ; .... , .. , .. , ...... , . · :• ..
. .. . . . ... . :, .... ·. ··' . .-.. · .. ,· . .
..
0 • • • • • •
T1TM Transmittance
. ... .
..
. : . . : .. ~ . ~ . ~ . :
Figure 2.9: Arm cavity power as a function of transmittance. Assumed are average losses of 25 ppm, AR coating losses of 300 ppm, sapphire substrates with 480 ppm loss, and a power mirror matched to 1% carrier reflectance.
Eq. (2.15). The power is the square of this, doubled for the total in both arms.
(2.27)
where Pine is the incident carrier power. Each arm individually has half this amount.
The amplitude gains for the power recycling cavity as well as the arm cavity are given by
iprm g~ = ---~~----
1 - r ~mr Michelson titm
9arm
= ---
1-ritmretm
(2.28) (2.29)
The reflectivity parameters used are found in the following way. The power cavity
mirror is calculated based on the model of the power recycling cavity as a cavity whose end mirror is the Fabry-Perot Michelson. For the carrier and ideal optics, this is simply the reflectivity of the arm cavities. This reflectivity is then modified by the losses of the substrates and AR coatings in the much the same way as in Eq. (2.2).
(2.30)
Given the desired 1% power reflectivity, 6 the power mirror reflectivity can be calcu- lated as
JD.OI +
TMichelsonAprm r prm = - /{)"i\11
+
Y 0.01rMichelson(2.31)
Figure 2.9 shows the effect of decreasing the transmittance of the ITM, and the effect it has on the power available for gravitational wave signal extraction. Clearly, decreasing the transmittance of the ITM increases the amount of power stored in the arm cavities.
There is one significant effect which has been ignored to this point. Losses have been assumed to be a constant, independent of incident power. In fact, it's well known that those losses heat the mirrors, causing deformations and lensing, which further increases the losses. Two reasons justify ignoring lensing at this point.
First, this non-linear effect is somewhat mitigated by the arm/power coupled cavity. [29] The insensitivity of the carrier to thermal lensing in the power recycling cavity is due to the fact that the resonance of the carrier in the power cavity depends on the resonance in the arms. The arm cavities are over-coupled, which generates a sign flip upon reflection. The power cavity alone has to be an anti-resonant cavity for the carrier, which attains resonance only when the carrier resonates in the arms.
When the optics are lensed in the power cavity, they change the spatial mode structure of the cavity. Ordinarily, changing the mode structure of the power cavity would cause more carrier light to be scattered into higher order modes, which effectively are
6This number has two purposes. One, some carrier light is needed in reflection to give adequate control signals in the reflection port. Second, the optimal coupling point is not robust to changing losses in the interferometer. Choosing a point slightly on the over-coupled side of optimal coupling balances the need for high power with significant decrease in sensitivity to parameter drift.
losses. The arms, however, are very stable cavities with very little lensing, and they effectively enforce their mode structure on the power cavity, with little regard for the thermal lensing. There is a decrease in power, though, due to the initial coupling of the input light into the interferometer, but the scatter which would occur in the
power cavity can't happen because the spatial higher order modes aren't resonant.
Second, LIGO II plans to use some type of thermal compensation on the beam- splitter and ITMs. [12] These take the form of either passive heating rings, which would act to reduce the thermal gradients in the masses, or active heating, in which a C02 laser beam is scanned through the volume of the mass, heating it. A Schack- Hartmann wavefront sensing scheme is used to measure the thermal gradients, and feedback is applied to the scan of the beam to smooth them out. While it's not currently known which method would be used, there is enough confidence in at least the passive technique that a factor of 10 reduction in lensing could be achieved.