RF Sidebands

Laser Noise Couplings in RSE

5.5 Transmission of Light

5.5.3 RF Sidebands

of interest, then, the noise sidebands are about a factor of 100 smaller than the DC term.

The RSE response is given m Eq. (5.34), and has two zeros at 100 Hz, the arm/power coupled cavity pole at 1 Hz, and the arm/signal cavity pole, which is characteristic of the frequency response of the arm/signal coupled cavity system.

This is typically much larger than the arm cavity pole frequency in broadband. Even for a detuned interferometer, this will typically be larger than the arm cavity pole frequency, since the signals of interest are in the several hundred Hz range. The response is then expected to roll off at 1 Hz, as before, but at the arm cavity pole frequency, the two zeros cause the response to increase. The details of the signal cavity determine the overall coupling in the bandwidth of interest. In the broadband RSE case, this will be a simple pole at some fairly high frequency. For a detuned interferometer, the pole will be complex, and a somewhat more complicated response is expected.

An example using the same parameters as in the previous section is displayed in Figure 5.8, for a broadband RSE interferometer, and one detuned by </Jdt

=

0.2 radians. In each case, the gain reduction of roughly 0.16 is evident, as well as the presence of the extra zero in the RSE response. In the broadband case, the noise sidebands are actually enhanced over the LIGO I case above 600 Hz. It should be pointed out that, in all likelihood, this example interferometer is not practical, because the bandwidth is much too high to be of any use. However, the detuned case is perhaps more disturbing. The peak frequency here is around 900 Hz, which is also exactly where the carrier noise sidebands are being most effectively transmitted. It is expected that the carrier noise terms might be more of an issue in RSE than in LIGO I, where they were insignificant.

10¹ ~~~~~~~~~~~~~~~TT~~~~~~~~~ _:·::· :: :·::: :::::· ....

.. . ,., ... ... , ..

·. : . . ....

-~ ·-:--·.-:·- ·-· -·

... ; .

: :: : ~ : : :: :::

10' ~~~~~~~>-n~~~~~~r-0~0 ^.. '-, ~~~~~~~rn

... , ........ , ..... , ... ,., ..

Frequency (Hz)

Figure 5.8: Carrier noise sideband transmission gain for RSE ( c/Jdt

=

0, top) and a detuned RSE (¢dt

=

0.2, bottom) and LIGO I. The same optics are used, Titm

=

3%,

Tprm

=

1%, and ^Tsem

=

10%.

transmission to the dark port. Hence, first order terms due to imperfections are very small compared to the zeroth order term. Also, the calculation of the measured noise will use products of carrier and RF sideband terms. Since only first order terms exist in the carrier equations, any first order terms in the RF sideband equations would generate second order predictions, which are not being considered here. As a result, the interferometer imperfections and phase offsets can be ignored.

Second, since the RF sidebands are non-resonant in the arm cavities, all references to arm cavity phases and mismatches are neglected, and the arm cavity reflectivity is set to unity.⁵ However, the physical Michelson asymmetry included in the Michelson degree of freedom will keep the sines and cosines in the Michelson reflectivity and

5This approximation breaks down at frequencies in the many kHz, as the frequencies get closer to a resonant condition (18.75 kHz for 4 km arms).

transmissivity terms. Eq. (5.15) simplifies in the following way.

Defining the phases for the RF sidebands requires more care. At issue is the method of optimizing the transmission of one RF sideband to the dark port when the interfer- ometer is detuned. As the detuning changes, there are two methods of re-optimizing the RF sideband transmission - either the length of the signal cavity can be changed to null the effect of the detuning, or else the frequency of the modulation can be changed, to re-optimize the RF sideband transmission.

It will be assumed that a configuration is chosen such that the RF sidebands resonate within the power and signal cavities, individually, at the broadband RSE tuning. The RF sideband frequency for the broadband interferometer is Obb, so that

(5.41)

will be the nominal broadband phase for the three degrees of freedom that the RF sidebands are sensitive to: ¢+,

cp_,

^{and ¢}8 • The length l then refers to the degree of freedom length, and w₀ is the carrier frequency. Detunings will be accommodated by introducing a phase rJ, such that

'rJ

=

2l dO

+

20bb dl

c c (5.42)

where a length change is indicated by dl and a frequency shift in the RF modulation frequency is indicated by dO. This allows for either of the methods for optimization of RF sideband power at the dark port to be used in the equations. The general RF sideband phase will then be written as

± 2l ±

c/J = - ( Wo ± Obb)

+

'rJ

c (5.43)

The broadband RF sideband phases are then (modulo 27r)

4>! =

±7r

¢-;- =

±27r

4>~ =

^±a

=

± nbbo

c where 8 is the macroscopic asymmetry

it -

l_{2 .}

(5.44) (5.45) (5.46)

The power and signal cavities are both relatively short (of order 10 m), which corresponds to a free spectral range of roughly 10 MHz. Even with an unlikely finesse of 1000, the bandwidth of these cavities wouldn't be much less than 10kHz (for typical numbers, the bandwidth of the power/signal coupled cavity tends to be roughly 100 kHz). This allows the assumption that the transmission for the DC RF sidebands and their noise sidebands is constant over the bandwidth of interest, so the dependence on the frequency of the noise sideband

±w

can be ignored.

The transmissivity for the RF sidebands is derived in Appendix B.3.

t±

=

±tprmAbstsem sin(a')gi}o (5.47)

± e-i(tl>dt+'11'+'1~)/2

9ijo

=

t

D±

(5.48)

(5.49)

where the primed a is defined to include the TJ phase as

(5.50)

The approach used in this thesis is to modify the length of the signal cavity to

match the detuning phase for the lower RF sideband, that is

_ _ A.. _ -0.20lsec

'fls - -'f'dt - C

+ A.. +f220lsec

"' _s = ^{+ 'f'dt} = - - - -c

All other rt's are zero.