Implications

Resonant Sideband Extraction

2.2 Practical Limitations

2.2.3 Implications

The two preceding sections suggest that Tsem should be high, and 7itm should be low. Of course, a proper numerical optimization should be done for whatever type of signal the interferometer is designed to detect, and the effects of loss will naturally be built into the optimization routine. However, it's worthwhile to try and draw some conclusions about the result of these effects and how they might impact the range of optics that might reasonably be used, given the intuitive understanding of this interferometer.

In the case that the desired detector bandwidth is wide, for both detuned and broadband operation, the reflectivity of the signal cavity must be relatively low.

Assume a broadband bandwidth of 300Hz. The free spectral range is about 37.5 kHz for 4 km arms, so the finesse of the arm cavity for signals would need to be about 65.

This implies

65 ^{;:::j _ _} rr_

1-Tsec Tsec ;:::j 95%

(2.34) (2.35)

Figure 2.11 shows the signal cavity reflectivity (on resonance) for an ITM transmit- tance of 7itm

=

0.01%, 7itm

=

0.1%, and Titm

=

1.0%, as a function of the signal mirror transmittance. For a reflectivity of 95% and Titm = 0.1 %, the signal mirror can have a transmittance around 4%, which is an acceptable number for the losses expected.

An interesting feature of Figure 2.11 is that the functional shape of the signal cavity reflectivity remains the same for different ITM transmittances- only the posi- tion moves back and forth. It can be concluded, then, that in order to maintain the same signal cavity reflectivity, and hence the same signal bandwidth, a decrease in the ITM transmittance must be accompanied by a similar fractional decrease in the signal mirror transmittance. Using the example in the previous paragraph, if Titm is reduced by an order of magnitude to 0.01 %, the signal mirror transmittance which

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Figure 2.11: Signal cavity reflectivity as a function of signal mirror transmittance.

Three ITMs are modeled: Titm

=

0.01 %, T;,tm

=

0.1 %, and Titm

=

1.0%.

gives 95% reflectivity likewise drops an order of magnitude to about 0.4%. The note- worthy result of this is that Figure 2.10 indicates a decrease in the signal of about 30% due to signal cavity losses, but Figure 2.9 doesn't indicate a similar increase in the power stored in the arms. Overall, the sensitivity will decrease with increasing arm cavity finesse.

The broadband RSE interferometer is considered as an example of the signal and arm cavity loss tradeoff. Non-power recycled RSE will be compared to the case where the power recycled RSE interferometer is optimized for greatest sensitivity.

This comparison addresses an early promise of RSE, which suggested that it may be possible to do away with the power recycling mirror altogether. In both cases, the optical parameters are chosen to allow 1% of the carrier light to return to the laser, and the signal mirror is chosen such that the internal thermal noise is equal to the shot noise at the 3 dB point of the shot noise sensitivity. The ITM substrates are assumed to be sapphire, with 500 ppm loss, and the AR coating loss of the ITMs and beamsplitter is assumed to be 300 ppm each. There is some uncertainty regarding

the coating losses of the arm cavity optics - numbers between 10 and 25 ppm have been quoted.[40] The 10 ppm case is summarized in Table 2.1. Clearly, the sensitivity

II

^{With PR}

I

Without PR

I

T;tm 1.0% 55 ppm

Tsem 42% 0.20%

Tprm 1.7% None

Coating losses 10 ppm 10 ppm

DC sensitivity (h/

JHz)

o.s9 x 10-²⁴ 1.40 x 10-²⁴

Table 2.1: Comparison between broadband RSE with and without power recycling.

10 ppm coating losses are assumed.

without power recycling suffers considerably due to signal cavity losses, being about a factor of 1.6 times less than power recycled RSE. Cubing this factor indicates that the volume of space that the interferometer can probe is nearly a factor of four greater with power recycling.

This factor, however, is very sensitive to the actual losses of the arm cavities, as demonstrated by the case when the coating losses are 25 ppm. These results are summarized in Table 2.2. The difference between RSE with and without power

II

^{With PR}

I

Without PR

I

Titm 1.0% 75 ppm

Tsem 50% 0.37%

Tprm 3.1% None

Coating losses 25 ppm 25 ppm DC sensitivity (h/VHz) 1.os x 10-²⁴ 1.22 x 10-²⁴

Table 2.2: Comparison between broadband RSE with and without power recycling.

25 ppm coating losses are assumed.

recycling isn't nearly so pronounced, due to the increase in arm cavity losses. The volume of space probed is 1.4 times greater with power recycling, which isn't to be sneezed at, but it certainly isn't as compelling as the factor of four difference with the 10 ppm coating losses. Clearly, better knowledge of the coating losses is needed. If it's expected that the optics will be very good, power recycling becomes a necessity.

For high frequency response, both the under-coupled and over-coupled signal cav- ities are considered. It's first noted that the tendencies might favor an under-coupled signal cavity, since the effects outlined suggest such an arrangement of mirrors (Tsem high, Titm low). However, in the under-coupled case, Eq. (2.13) shows that all detuned frequencies come below the broadband RSE bandwidth frequency. High detuning fre- quencies imply a higher RSE bandwidth at resonance. For sensitivity at 1 kHz, a 5kHz RSE bandwidth will probably be necessary, implying an effective finesse of the arm/signal cavity of about 4. The signal cavity reflectivity which permits this wide a bandwidth is about 20%. To keep the signal mirror above 1%, the ITM transmittance must be at least 0.7%. At this point, the stored power isn't degraded much. However, modeling such configurations has shown that the high frequency bandwidths in these cases still tend to be fairly large.

The other option is the over-coupled signal cavity, in which case Tsem is low and Titm is high. As the transmittance of the ITM increases, the losses of the carrier increase, as in Figure 2.9, due to the re-tuning of the power mirror. However, the effect of the losses in the signal cavity decreases, because its finesse is decreasing.

This particular case, as noted earlier, is more amenable to high frequency response, due to the fact that the peak frequency at RSE is already at half the arm free spectral range. Compared to the under-coupled case, the over-coupled signal cavity probably will generate better high frequency signal response.