RSE Transfer Functions

Resonant Sideband Extraction

2.1 Frequency Response

2.1.2 RSE Transfer Functions

-~ 6

- 4>d,=O - - 4>d,=nt4 0.9 - 4> dt=n/2 - 4>d,=3rrl4 0.8 - 4>d,=n

0.7

II

)I:',. '·' ... •···· ..

1·1 II II I I

li II I I •J I

·n·'·l' , .. ;, .. , ... .. .

1

':\' ' I I :/ I

: I l 1 I

····+

_/:tt

..

¹_I^·!_I ^{· ·'} ₁ ^... ₁^f _: ^{.. l ..}

'E: 0.6

I

^{I \ )}

^{~ i·}

^{I·· ... ····· .. . .}

"' ^c:

~ ^{·\ I, I I : t}

( . .11; . .

1 , . .

^t

I., I ^'I

. . , .... · I

ⁱ

I ' ' , ^~\

.. ... / :' : II/

\i

. ^{I I} ,.-: .. . , ^..

Ok---~---L---~---L---~---L----~~--~

-2 -1.5 -1 -0.5 0 0.5 1.5 2

Frequency (Hz) X 1 o•

Figure 2. 7: Transmissivity of a three mirror coupled cavity as a function of frequency and detuning. The signal cavity is over-coupled. The arm length is 4000 m, the signal cavity is 6 m. The optics are the same as in Figure 2.6.

the mirror is

~ rEiei²k1

(1

+

ikol cos(wt)) (2.14)

~ rEiei²k1(1

+

ikolj2eiwt

+

ikolj2e-iwt)

The approximation is a Taylor series assuming a very small modulation of the mirror position (ol

«

k). In this limit, the phase modulation of the light adds small signal sidebands on either side of the incident frequency, with amplitudes irkol/2 relative to the input field. These occasionally are referred to as "audio sidebands," since the frequencies of interest tend to be in the 10 to 1000 Hz bandwidth. In terms of strain, the amplitude ol/2

=

h larm·

Eq. (2.14) shows that the size of the signal sidebands is directly proportional to the amplitude of the light incident on the arm cavity end mirror. This is a product of the light incident on the arm cavity times the amplitude gain of the arm. The light incident on the arm is itself a product of the amplitude gain of the power recycling cavity times the light incident from the laser.

(2.15)

where E1 is the incident amplitude of the laser light on the interferometer. The quantity Earm is the amplitude in a single arm, and the factor of 1/

v'2

^{comes from}

the beamsplitter.

Thus, signal sidebands are generated which then propagate through this coupled cavity system via Eq. (2.1) at plus and minus the frequency of the input signal.

There are several readout methods that can be used. One method uses a homodyne readout, in which some carrier light is also present along with the audio sidebands at the output photodiode. The power measured by the square-law photodiode detector is proportional to

(2.16)

where it's been assumed that the power in the signal sidebands is negligible. The signal of interest is the second part, and the signal sideband fields can be expressed in terms of Eq. (2.1) and Eq. (2.14), where it has been assumed that the mirror reflectivity r etm ~ 1.

E+f

=

iklarmh [Earm[tcc(

+f)

E_f

=

iklarmh [Earm[tcc(-f)

(2.17) (2.18)

Substitution into the second term of Eq. (2.16) and some algebra gives a measured signal voltage as

(2.19) where ^Zimp is the transimpedance of the photodiode in amps to volts, TJ is the quantum efficiency of the photodiode in number of electrons to number of photons, e is the electric charge, hv0 is the energy of a carrier photon. The transfer function

ii

(f) is defined as

(2.20) where ¢o is the phase of the carrier field E_{0 .}

If the phase of the carrier can be chosen arbitrarily, it can be seen that the output at a particular frequency can be maximized by choosing the carrier phase such that both terms have the same overall phase. This can be found to be 2¢₀

=

arg( tee(+ f))+

arg(tee(-f))+ 1r. This carrier phase defines a maximum transfer function

(2.21)

If tee(+!)*

=

-tee(- f), then this holds true for all frequencies. This condition is met at resonance (RSE or dual-recycling), but fails for all detunings. Hence, all real transfer functions will be less than or equal to Eq. (2.21). The transfer function Hmax(f) defines a theoretically ideal transfer function which, in general, is not be reached at all frequencies.

The shot noise current spectral density is given by the standard formula

(2.22)

where Si (f) is the one-sided current noise power spectral density and

l

is the average current from the photodiode. The shot noise voltage spectral density includes a factor of the transimpedance. The average current is due to the carrier local oscillator, assuming no other light is present at the dark port.

(2.23)

A little bit of algebra gives the strain spectral density equivalent to shot noise as

h(f)

=

_ll

arm

fie).. 1

27rrJParms IH'(f)l ^(2.24)

where the normalized transfer function H'(f) has been defined as

(2.25)

and ^Parms

=

2IEarml² is the total power in the arm cavities.

For practical readout schemes, there are two main classes, homodyne and het- erodyne detection. If a homodyne scheme such as offset locking is implemented, the carrier "local oscillator" is generated by offsetting the arms from resonance by some very small amount, allowing a small bit of light to leak out the dark port. In this case, the phase of the Eo field isn't controllable.³ The frequency

f

at which the trans- fer function Eq. (2.25) is maximized is dependent on the phases of the transmission functions.

3The field generated in this fashion is typically orthogonal to the leakage field due to arm mis- match, which is a result to be shown in Chapter 5. In this way, one could conceivably control the phase of the local oscillator by changing the amplitude of the offset field, since the local oscillator is then a combination of the (unintentional) mismatch field and the orthogonal offset field. Another possibility is outlined in [35], where output optics are added to shift the phase of the signal in such a way as to cancel the phase of the transmission.

Heterodyne detection, in which RF sidebands present at the dark port are used as local oscillators for the signal sidebands, is discussed in Chapter 3. The measured photovoltage is electronically demodulated at a phase (3, and subsequently low-pass filtered. The output is given in Eq. (3.19), where the carrier field ^Eo is set to zero for

the dark port case (the notation is modified for this context).

(2.26)

This is similar to Eq. (2.20) with the substitution Eo -+ (E_e-if3

+

^E+eif3).4 In this case, the demodulation phase of this local oscillator is controllable, and the signal can be maximized at a particular frequency to match Eq. (2.21). One downside of this technique is that the signal to noise in a heterodyne readout is typically larger than in the homodyne case, such that the achievable signal to noise isn't as good.[36, 37, 38]

Ultimately, the details of a particular readout scheme need to be analyzed properly, which would include the efficiency of the readout and the noise characteristics of the readout as well. However, it's simpler and sufficient in this context of comparison, as well as a first pass at optimization, to utilize an ideal transfer function of Eq. (2.21). Likewise, the ideal case of Eq. (2.24) will be used when referring to a "shot noise limited sensitivity."