BALANCED HOMODYNE DETECTION
5.3 Noise requirements on fields
This chapter makes references to the "two-photon formalism", developed in [62,63].
Since an electric field can be represented as a complex-valued quantity, it can be expressed as the sum of two modulations, one∝cos(ωt)and one∝ sin(ωt)- in the literature, these are assigned the names "amplitude quadrature" and "phase quadra- ture", because an amplitude (phase) modulation applied to a hypothetical ideal, noiseless input field will create modulation sidebands exclusively in the amplitude (phase) quadrature (the detailed derivation may be found in Sec. 2.2.2 of [73] for example). This distinction is significant because it represents two different physical mechanisms of creating modulation sidebands - furthermore, an interferometer is capable ofconvertingamplitude modulations to phase modulation, and vice-versa, due to radiation pressure effects and complex-valued reflectivities of optical cav- ities. While exact calculations are required to derive accurate noise models and couplings, there is a very useful diagrammatic way of representing the quadrature
= DARM signal
= Contrast defect
= LO field
= Homodyne phase
Figure 5.1: Quadrature fields for a DRFPMI operated in ESR mode. a1 is usually referred to as the "amplitude quadrature", and a2 the "phase quadrature". The lengths of the arrows denote the coherent amplitude∝ √
PiwherePiin theith field, not drawn to scale here since typicallyPLO Psig. The shaded circles at the ends of each arrow denote uncorrelated amplitude (axis along the arrow) and phase noise (axis perpendicular to the arrow) on each field - again, the sizes of the circles are for illustration purposes only. The angleζ, drawn here following Eq. 2.26 of [54]
determines the projections of various signal and noise components to the readout.
fields as phasors, which provide some intuition and serves as the starting point for a detailed calculation. An example of such a diagram for the case of a DRFPMI operated in the Signal-Recycled (φSRC =0◦) mode [54], with some important fields, is shown in Figure 5.1. This configuration is sometimes also referred to in the literature using the acronym ESR, which stands for Extreme Signal Recycling.
In a practical BHD implementation, the photocurrent readout scheme is different from what is stated in Eq. (5.1) - the signal and LO fields are optically mixed on a 50:50 beamsplitter, and the photocurrent at each output port, i1,2 are read out. Following the derivation in Appendix A of [74], the difference photocurrent, i− ≡i1−i2, has the form
i− ∝2ELOEsigcosζ+2ELOδEζ
sig+2EsigδEζ
LO, (5.2)
where the LO and signal fields have been decomposed into a DC component,ELO,sig, and a time-dependent component, δEζ
LO,sig, with the angle ζ defined as before to be the relative phase between the two fields (so ζ "picks out" the quadrature noise components ofδELO,sigthat get amplified by the coherent amplitudes ELO,sig). In a perfectly balanced interferometer,Esig = 0, and so only the second term in Eq. (5.2) survives, with the signal field getting amplified by the coherent amplitude of the LO field.
86
10
1010
910
8RI N [1 / Hz ]
PCD= 0.1mW PCD= 1.0mW PCD= 5.0mW PCD= 10.0mW Shot noise RIN of 100mW
10
110
210
310
4Frequency [Hz]
10
810
7LO Ph as e N ois e [ ra d / Hz ]
Wanser phase noise for 20m fiber
LO requirements for the 40m Ponderomotive Squeezing BHD readout
Figure 5.2: Requirements on the LO field noises for measuring optomechanical squeezing at the 40m. An arbitrarily chosen safety factor of 10 is assumed in these plots (i.e. if these noise levels are achieved, the LO noise contribution to the readout will be< 10% of the next highest noise contribution).
In practice, small asymmetries can lead to a non-zero power in the signal beam.
For concreteness, let us consider the example of the "contrast defect" field shown in Figure 5.1 - this can arise due to reflectivity imbalances between the two arm cavities of the DRFPMI. If the coherent amplitude of this field is non-zero, then its projection onto the "LO field" phasor can amplify amplitude and phase fluctuations in the LO field, and the third term in Eq. (5.2) can become non-negligible. If we were only interested in the "DARM signal", which for the ESR configuration shows up entirely in thea2quadrature, we would chooseζsuch that the red and blue arrows are collinear. In this case, the contrast defect field amplifiesphase fluctuations in the LO field and adds a noise term to the readout. If, on the other hand, we want to make a measurement very close to thea1 quadrature and choose ζ accordingly, then the contrast defect would amplify amplitude fluctuations in the LO field and contribute noise to the readout.
The requirements for the case of phase quadrature readout ζ = 0◦ is analyzed in detail in [69], so I will focus here on the near-amplitude-quadrature readout case, with ζ ≈ 88◦, which is what is proposed for the optomechanical squeezing experiment at the 40m. Results from numerical modeling are shown in Fig.5.2. The requirement was that the contribution of the LO noises to the readout was a factor of 10 below the next-dominant noise source at each frequency, which was assumed to be coil-driver noise at mid frequencies and quantum noise at high frequencies - this is why the graph has a discontinuous profile around≈ 200 Hz, as the envelope of the next-dominant noise source follows this profile. Assuming we have 1 mW of contrast defect light, and focusing on a frequency of≈ 200 Hz (which is where we expect to be able to be sufficiently sensitive to have any chance of measuring a squeezed output field from the interferometer), the requirements on the LO field may be summarized as RIN< 2×10−9/√
Hz andδφ . 3×10−8rad/√ Hz.
These, and in particular the requirement on the RIN, are extremely challenging. Any intensity stabilization servo (ISS) will be limited to stabilizing the intensity to the level of the sensing photodiodes, and to achieve the required stabilization, we need to sense≈ 100 mW of light (dashed grey line in upper plot of Fig. 5.2). Handling such high power levels with low-noise sensing photodiodes will require considerable engineering. One possible way to relax the requirements on an external ISS is to pick off the LO field from the PRC - once the interferometer is locked, the laser frequency is stabilized to the CARM DoF. Thepassivefiltering offered by the CARM cavity, whose linewidth is≈20 nm at C1, has a 1/f corner frequency of≈ 100 Hz.
As a result, the stabilization of intensity fluctuations due to the passive filtering action at C1 is insufficient to meet the intensity noise requirements on the LO field - an active stabilization servo will be needed to supplement the passive filtering.
For comparison, the aLIGO interferometers with 4 km long arm cavities move this corner frequency down by a factor of 100, and the CARM linewidth is≈0.7 Hz - so there is significantpassivefiltering even at the low end of the detector’s bandwidth of≈ 20 Hz. Nevertheless, even at H1 and L1, an active intensity stabilization servo will be needed to meet the requirements on the main interferometer beam’s intensity noise.
It is expected that the phase noise requirement will be easier to satisfy (at least, at 200 Hz), provided all the optics are suspended. 30 nrad/
√
Hz corresponds to≈ 3× 10−15m/√
Hz of displacement noise, and the isolation provided by the suspensions and passive seismic isolation stacks available at the 40m are expected to provide
> 200 dB of isolation from ambient ground motion at 200 Hz. A dashed grey line labelled "Wanser phase noise for 20m fiber" is included in the lower plot in Fig.5.2 - an in-air length of optical fiber was being considered as an option to deliver the LO field to the photodetection chain, but as this modeling shows, it is not a feasible option for any reasonable amount of contrast defect light expected at the 40m.
Furthermore, it is extremely unlikely that the fundamental thermo-optic noise of the optical fiber quantified by the Wanser model [21, 22] will in fact be the dominant noise source if such a delivery mechanism were pursued - acoustic vibrations due to imperfect shielding is likely to manifest at a much higher level. This reveals an important advantage of other length sensing techniques, such as heterodyne PDH locking or DC readout [34], which inherently have the LO field and signal field co-propagate along the same optical path to a sensing photodiode - no significant effort has to be made to stabilize therelativephase between the LO and signal fields.
One of the questions being studied in detail at the time of writing is whether all the extra engineering complexities brought about by changing the readout scheme of the aLIGO interferometers from DC readout to BHD is justified1, and offers a measurableperformance improvement.