Laser Noise Couplings in RSE
5.1 Conceptual Motivation
The use of phasor diagrams, which was outlined in Section 3.1.3, is very helpful to conceptually understand the mechanisms for the generation of the measured noise.
5.1.1 Balanced Sidebands
It's easiest to first consider the case when the RF sidebands are equal. The summed resultant RF sideband phasor is a swinging length phasor along the horizontal axis.
The dark port is a special case where the carrier phasor is ideally null. The only light present is due to the two counter-rotating RF sideband phasors. A small differen- tial phase modulation of the arms generates carrier light at the dark port which is orthogonal to the carrier axis, as shown in Figure 5.1. If the signal phasor is along the horizontal axis, the demodulation phase which generates the largest measured signal is the one at which the two RF sideband phasors are parallel, hence their sum is maximum. This is the inphase demodulation.
Carrier signal
:_)
f---' ~
)
: Counter-rotating : sidebands
L sultant sideband
J
Figure 5.1: Phasor representation of gravitational wave signal generation.
When considering noise couplings, first imagine that the arm cavities are mis- matched in reflectivity, due to differing losses in the cavities. This results in a con- trast defect of carrier light at the dark port, since the two beams summed at the beamsplitter cannot completely cancel if their magnitudes are different. The relative phase of this "mismatch carrier defect" phasor is parallel to the carrier axis. In the static case, this still presents no signal, since the defect and resultant RF sideband vectors are orthogonal. Phase noise on the carrier and RF sidebands is decomposed into the nominal fixed phasor plus an orthogonal swinging length phasor, generating a measured signal as shown in the right-hand side of Figure 5.2. It can be easily shown
: Carrier defect :
...----
____ _t
!~-_____,.
-s·
Resultant sideband
rNoR:ors
___ ] ,t ,L _
Signal= l- - x
- I+ [8]
Figure 5.2: Phasor representation of frequency noise generation. The top left shows the static case, the top right shows the noise sidebands generated by frequency noise.
The bottom diagram indicates the vector products which generate the measured noise.
that if the phase fluctuation of the carrier defect and resultant RF sideband were equal, the measured noise terms indicated in the bottom of Figure 5.2 would cancel.
However, this typically is not the case. The carrier and RF sidebands propagate very differently through the interferometer. Hence, the relative amplitude of the phase fluctuations in the carrier and RF sidebands which reach the dark port tends to be
different, and these terms do not cancel. For LIGO I, this is the dominant form of coupling for frequency noise. [59]
Amplitude noise wiggles the length of each of these phasors. The noise phasors generated are parallel to the source phasors. For the mismatch carrier defect and the resultant RF sidebands, these remain orthogonal, and there is no measured signal due to amplitude noise. A different coupling mechanism is required. If either the Michelson ( rp_) or the arm cavity differential mode ( ci> _) are offset, there will be a small bit of "offset carrier defect" present which is parallel to the RF sidebands (similar to the measurement of the gravitational wave signal, but at DC). This bit of offset carrier defect provides a local oscillator for amplitude noise on the RF sideband, as shown in Figure 5.3. The amplitude noise of the carrier generates measured noise as well through the modulation of the offset carrier defect, although this is typically a much smaller term.
Mismatch carrier defect
"'-. , Differential
_ :_t--.-~
offset~~~~~t
!Resultant sideba)
(i~ors
_____ b - . -
-=: _. .
Signal = L l ==========:_x_-__J +
1 -
x -Figure 5.3: Phasor representation of amplitude noise generation.
The amplitude modulation of mismatch carrier defect doesn't generate a signal.
However, if any of the cavities are off resonant, then the resultant sideband vector can be rotated relative to the carrier. In this case, there will be a component of the RF resultant parallel to the mismatch carrier defect, which can now couple amplitude noise from the arm mismatch carrier defect as in Figure 5.4. This second order effect would tend to be the largest to be considered, due to the fact that the mismatch carrier defect tends to be much larger than any other coupling mechanism. Earlier work considered this coupling, which was typically found to contribute to the total
\s~atch carrier defect
---~---
!Resultant sideba:d
c-No~ :
---~ ----
Signal =
~ ---
x t +t
X - -Figure 5.4: Phasor representation of second order amplitude noise generation.
noise by no more than 10%. [60] Since the point of this analysis is to set a specification, rather than determine an accurate prediction of measured noise, this mechanism will not be considered here.
Clearly, one could imagine all sorts of couplings involving various rotations, offsets, etc., but most will be second order and will be ignored.
As the demodulation phase is varied from in phase to quadrature, the length of the resultant RF sideband goes from maximum to zero. At the inphase demodulation, these couplings are maximum, while at the quadrature phase, other mechanisms would come into effect, for example differential amplitudes of the RF sidebands. In general, though, the size of the these signals will be much smaller (second order) than in the inphase demodulation. For this reason, quadrature demodulation will not be considered in the balanced sideband case.
5.1.2 Unbalanced Sidebands
In the detuned RSE configuration, the RF sidebands become unequal when the inter- ferometer is adjusted to maximize the transmission of one of the RF sidebands to the dark port. Since propagation of the upper and lower RF sidebands is unequal, the resultant RF sideband vector is no longer a swinging length vector along a fixed axis, but is rather a rotating vector which traces out an ellipse at the frequency of the RF modulation. In the reflected and pickoff ports, this forces the choice of demodulation phase such that there is no DC offset of an optical origin. However, at the dark port,
where there (ideally) is no carrier, the choice of demodulation phase is arbitrary.
The detuning of the signal cavity causes the upper and lower noise sidebands to propagate unequally as well. The resultant noise phasor also traces out an ellipse, but at the noise sideband frequency. Since demodulation can be thought of as the scalar product of the resultant RF sideband vector with the resultant noise sideband vector, the signal is seen to be the projection of the noise ellipse onto the axis defined by the demodulation phase. In general, since there is no axis where either the RF or the noise resultant phasor is null, the measured noise doesn't vanish at any demodulation phase. The analysis will have to include both quadratures.
Mismatch carrier defect typically has the largest magnitude of all the noise cou- pling mechanisms. Amplitude noise is free from this coupling mechanism in the balanced sideband case due to the orthogonality of the resultant phasors. This is no longer the case for unbalanced sidebands. An example of this is shown in Figure 5.5.
In this case, the noise ellipse of the RF sidebands is projected onto the carrier mis-
Noise sidebands
':.t:.a::~ J Signal~ I I
X~ I
RF sidebands
Figure 5.5: Amplitude noise coupling via mismatch carrier defect with unbalanced RF sidebands.
match contrast defect. The details of the amplitudes of all the phasors, their relative phases, as well as the appropriate demodulation phase, will need to be derived, of course. It should be quite apparent, however, that the description of coupling when the RF sidebands are unbalanced will be fairly complicated.