Balanced homodyne readout for quantum limited gravitational wave detectors

Peter Fritschel; Matthew Evans; Valery Frolov

doi:10.1364/OE.22.004224

1. Introduction

The Laser Interferometer Gravitational-Wave Observatory (LIGO) is part of a global effort to directly detect gravitational waves [1–4]; this emerging field has the potential to revolutionize our understanding of both astrophysics and fundamental physics. Advanced LIGO is the second generation of gravitational wave detectors, currently being installed and beginning operation at the LIGO observatories [5]. To realize the full potential of gravitational wave observation, significant improvements in sensitivity will be needed beyond Advanced LIGO and other advanced detectors currently under construction [6].

Interferometric gravitational wave detectors are designed to convert a gravitational wave induced phase shift into a measurable signal at the detector output. Most commonly this is done using a Michelson interferometer operating with a minimum of intensity at the anti-symmetric port (AS port; see Fig. 1), and a maximum of intensity returning to the laser source; the AS port is said to be at a “dark fringe”. A small differential phase shift between the arms of the Michelson then produces a small electric field at the AS port, linear in the phase shift. The readout scheme is the technique by which this small electric field, or signal field, is translated into a measurable signal. The design of the readout scheme is critical to determining the performance of a detector and ultimately its astrophysical reach and scientific output.

Fig. 1 A simplified interferometer with DC readout.

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In general, all readout schemes involve interfering the small signal field with a “local oscillator” field, to convert the signal field into a linear amplitude modulation of the AS port light. The first generation of interferometric detectors used a heterodyne readout scheme, where the local oscillator fields were radio frequency (RF) sidebands around the main light carrier frequency [7]. This had the advantage of mixing the signal up to RF frequencies at which the intensity fluctuations of the laser are typically low. The detectors then transitioned to using a homodyne readout scheme, where the local oscillator field is directly a sample of the main carrier field [8]. While such detection is then susceptible to baseband fluctuations of the laser intensity, the technique is made practical by two factors: a multi-kilometer long interferometer forms a very low frequency (1 Hz or below) low pass filter for the laser light, so intensity fluctuations are greatly filtered; and along with the homodyne readout, an output optical filter cavity is employed to spatially filter out all but the fundamental interferometer mode, greatly reducing the laser power that must be detected. Compared to the previously used RF heterodyne scheme, homodyne readout is less susceptible to a number of technical noise couplings, but its primary benefit is in lower quantum noise [9]. Another significant benefit of homodyne readout is that it provides a clearer path to further reductions of quantum noise through the use of squeezed light [10–12].

The method used to generate the local oscillator field for homodyne readout in Advanced LIGO is quite simple: the Michelson is held slightly off the dark fringe, by offsetting the arm cavities slightly from their resonance conditions, to produce a static carrier field at the AS port. This technique has several advantages: it requires no additional interferometer components; the local oscillator is automatically mode-matched and co-aligned with the signal field; and the amplitude of the local oscillator can be adjusted by changing the dark fringe offset. This particular method of implementing homodyne readout we refer to as “DC readout”.

While DC readout offers significant advantages over RF heterodyne readout, it has its own set of disadvantages. In particular, the presence of the static carrier field at the AS port creates several technical problems. For example, any amount of static carrier field that is sent back into the interferometer–either unintentionally through backscattering, or intentionally if signal recycling is employed–can create noise at the main output. To control these noise paths, stringent requirements on vibration isolation and light backscattering are required. Additionally, the AS port static carrier field impacts the interferometer alignment control system. In this system, segmented photodiodes detect a sample of the AS port light to measure misalignments of the interferometer arm optics. With DC readout, the static carrier produces a static signal in each of these segments that must be carefully subtracted out to uncover the alignment signal. A final type of disadvantage arises from the asymmetry of operating the arm cavities slightly off of their resonance; through radiation pressure, this increases the coupling of laser intensity fluctuations to the output [13], potentially increasing the requirements on laser noise.

This article examines an alternative method of generating the local oscillator in order to implement a balanced homodyne readout (BHR). In BHR, the LO is a sample of the field stored in the interferometer, and is interfered with the AS port in a balanced detection scheme. This technique is similar to the BHR schemes used to measure squeezed states of light [14,15]. By removing the need for a carrier field at the AS port and eliminating the dark fringe offset, BHR can avoid many of the disadvantages of DC readout. Balanced homodyne readout can also be considered as a form of external modulation, where the modulation frequency is zero [16].

For any homodyne readout technique, an important parameter is the readout angle, or homodyne phase: this is the relative phase between the LO field and the AS port signal field. While DC readout potentially offers some limited ability to control the homodyne phase, BHR provides complete freedom in choosing the readout angle. For interferometer designs like Advanced LIGO, the ability to optimize the homodyne phase can produce minor quantum noise improvements. For future interferometer designs, however, in which more exotic forms of quantum non-demolition (QND) readout schemes may be used, the ability to choose the homodyne readout angle will be critical. In most analyses of QND readouts the homodyne angle is assumed to be a free parameter, though no provision is made for producing the needed field [17–21]. Among potential candidates for a QND interferometer include variational readout [17] and speed meters [19], both of which are incompatible with DC readout either because the field produced by DC readout is in the wrong quadrature, or because no DC field can be produced. The proposed balanced homodyne readout approach is thus a fundamental piece of any QND readout scheme which may be employed in future gravitational wave detectors.

2. DC readout

In this section we use a simplified detector, in which the optical configuration has been reduced to a standard Michelson interferometer as shown in Fig. 1, to develop a description of the fields used in DC readout. The same description will be used in the following section to describe balanced homodyne readout. In both cases, this is a simplified analysis that aims at illuminating the essential points of the two types of readout. In the last section, we mention how this analysis should be extended to a more complete analysis that includes imperfections found in real interferometers.

A passing gravitational wave produces a differential change in the light travel time from the beam splitter (BS) to the end mirrors (EX and EY) and back, thereby changing the interference condition on the BS and causing some of the light in the interferometer to exit at the AS port. This can be seen by writing the field at the AS port as

A_{A S} = r_{B S} e^{i ϕ_{X}} A_{E X} - t_{B S} e^{i ϕ_{Y}} A_{E Y} = \frac{A_{I N}}{2} (e^{i ϕ_{X}} - e^{i ϕ_{Y}})

where in the second equality it is assumed that

r_{B S} = t_{B S} = 1 / \sqrt{2}

such that the fields arriving at both ends are equal

A_{E X} = A_{E Y} = A_{I N} / \sqrt{2}

. A small differential phase shift ϕ_GW caused by a passing gravitational wave thus produces an AS port field component,

A_{G W} = \frac{A_{I N}}{2} (e^{i ϕ_{G W}} - e^{- i ϕ_{G W}}) ≃ i ϕ_{G W} A_{I N}

If this is the only field at the AS port, however, the power measured there will be only second order in A_GW (i.e.,

P_{A S} \propto A_{G W}^{2}

).

Introducing a small differential offset in the interferometer arm lengths, as in ϕ_X = −ϕ_Y = ϕ_DC + ϕ_GW, sends a static field A_DC to the AS port, where

A_{A S} ≃ i (ϕ_{D C} + ϕ_{G W}) A_{I N} = A_{D C} + A_{G W} .

The presence of A_DC causes the power at the AS port to show a linear response to A_GW proportional to the leaked field amplitude

P_{A S} = {| A_{D C} |}^{2} + A_{D C}^{*} A_{G W} + A_{D C} A_{G W}^{*} = {| A_{D C} |}^{2} + 2 ℛ e (A_{D C} A_{G W}^{*})

where P_AS is the power measured at the AS port, and we neglect the term second order in A_GW.

The challenges associated with the current DC readout scheme are essentially all associated with the fact that the gravitational wave signal is produced with the help of a static carrier field at the AS port. Intensity noise on the LO field can be included by adding a modulation term to the LO field as A_DC = (1 + ε)Ā_DC, such that

P_{A S} = {\bar{P}}_{A S} + 2 ℛ e ({\bar{A}}_{D C} {(A_{G W} + ε {\bar{A}}_{D C})}^{*})

where terms second order in ε have been dropped, and the average power at the AS port is

{\bar{P}}_{A S} = {\bar{A}}_{D C}^{2}

.

From Eq. (5) we can see that amplitude modulation due to technical noise on the LO is indistinguishable from the gravitational wave signal in DC readout. Among potential sources of amplitude modulation are alignment fluctuations in the interferometer, beam jitter as it enters the interferometer (coupled via some optical defect or misalignment), and simple intensity noise of the laser source.

The preceding equations show that the phase of the LO field (A_DC), relative to the field produced by the gravitational wave signal (A_GW), is fixed. This relative phase is zero in our simple interferometer example, but will in general not be zero for more realistic interferometers. In principle, some control over the LO phase is possible if the field amplitudes from the two arms are not identical; i.e. if A_EX ≠ A_EY. Such an imbalance will produce a static AS port field in quadrature to the A_DC given in Eq. (3), and the LO will be the sum of these two static fields. By varying the amplitude of the differential offset, the phase as well as the amplitude of the LO field can therefore be adjusted. In practice, however, the static AS field due to amplitude imbalance is typically small compared to the desired LO amplitude. The LO will thus be dominated by the differential offset field, and the achievable phase variation of the LO will be small.

3. Balanced homodyne readout for Gravitational Wave detectors

In balanced homodyne readout, the carrier field used to produce a power modulation from the gravitational wave signal at the AS port does not co-propagate with the signal. Instead, the LO field is interfered with the AS port field on another 50/50 beam splitter placed in the AS port path, and both outputs of this readout beam splitter are detected. Figure 2 indicates one way the LO field could be generated in a Michelson interferometer, namely by using the light reflected off the back surface of the main beam splitter (the beam labelled PO, for pick-off, in Fig. 2). In this type of balanced detection, all of the noises which enter with the LO field appear in the linear combination of photodiode signals P_A + P_B, while the gravitational wave signal appears in P_A − P_B. That is,

A_{P O} = (1 + ε) {\bar{A}}_{P O} and {\bar{P}}_{P O} = {\bar{A}}_{P O}^{2}

P_{A} = {\bar{P}}_{P O} / 2 + ℛ e (e^{i ϕ} {\bar{A}}_{P O} {(A_{G W} + ε e^{i ϕ} {\bar{A}}_{P O})}^{*})

P_{B} = {\bar{P}}_{P O} / 2 + ℛ e (e^{i ϕ} {\bar{A}}_{P O} {(- A_{G W} + ε e^{i ϕ} {\bar{A}}_{P O})}^{*})

P_{A} + P_{B} = {\bar{P}}_{P O} (1 + 2 ε)

P_{A} - P_{B} = 2 ℛ e (e^{i ϕ} {\bar{A}}_{P O} A_{G W}^{*})

where the complex phase e^iϕ is included explicitly to highlight the fact that the LO phase is no longer tied to the gravitational wave signal phase as it is in DC readout.

Fig. 2 A simplified interferometer with balanced homodyne readout.

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Equation (9) highlights two main advantages of BHR. First, the ability to change the phase of the LO relative to the gravitational wave signal can be used to change the detector’s homodyne readout phase, and thus to optimize the sensitivity to a given source [20]. Second, the gravitational wave signal readout is in principle insensitive to noise on the LO beam (with some limitation in practice).

Balanced homodyne readout has its own set of technical challenges. One of these is that since the LO field no longer follows the same path to the readout as the signal field, the LO phase must be measured and controlled. An identical problem is present in experiments which measure squeezed states of light, and the solution used in these experiments is applicable here; small radio-frequency (RF) sidebands on the input light, and which co-propagate with the gravitational wave signal field, can be used to measure the phase difference between the LO and the signal [22]. The beat signal between the LO and RF sideband fields can be detected by the homodyne detectors (PD_A and PD_B in Fig. 2) [23]. The relative phase can then be controlled by a servo loop which actuates on an optic which is present only in the LO propagation path (e.g., the optic labeled ϕ in Fig. 2).

In quantum optics, various technical noise aspects of BHR have been addressed in the context of measuring squeezed states of light [24], [25]. Those challenges, and the techniques developed to deal with them to measure highly squeezed states of light, are also applicable in this proposed application of BHR. For example, with careful balancing, common mode rejection of LO noise typically reaches 80 dB. Also, compared to DC readout, BHR has the disadvantage that the LO field is not automatically co-aligned and mode-matched to the signal field. This can be addressed using a mode-cleaning cavity in the LO path, as mentioned in [25] and discussed in the next section.

4. Mode cleaning cavities

DC readout, as implemented in Enhanced LIGO and planed for advanced detectors, is somewhat more complicated than simply measuring the total power at the AS port. Any real interferometer has a non-zero contrast defect: the AS port contains some power in higher-order optical modes, even at the dark fringe, due to imperfect optics. To prevent the contrast defect from increasing shot noise on the gravitational wave readout detectors, an “output mode cleaner” cavity (OMC) is placed between the AS port and the readout detectors [8]. The OMC cavity is typically constructed in a monolithic fashion with the readout detectors, to prevent relative motion of the OMC output beam and the photodiodes, which otherwise is a potential source of beam jitter noise. Especially for high power operation of the interferometer, an AS port OMC is a necessary component of any homodyne readout scheme.

In DC readout, length and alignment control of the OMC are achieved using the LO field contained in the AS port; essentially, the OMC is controlled by maximizing the LO amplitude in transmission of the OMC. The presence of higher-order modes at the AS port can complicate this optimization. Control schemes that use an audio frequency signal field from the interferometer, rather than the static LO field, have been designed to deal with this, and achieving improved performance [26].

For BHR, there is of course no LO field in the AS port to use for the OMC control. However, the improved control methods just mentioned could be applied to OMC control along with BHR. Furthermore, the absence of the static LO field at the AS port may be beneficial in removing unwanted modulation products.

With BHR, an additional mode cleaner in the PO path (LMC in Fig. 3), while not in principal necessary, would provide significant practical benefits. The LMC would ensure near-perfect spatial overlap between the gravitational wave signal transmitted through the OMC and the PO (i.e., LO) field. Mode overlap is critical for the use of squeezed vacuum, since mode mismatch is equivalent to a reduction in quantum efficiency and thus presents a limit to the noise reduction which can be achieved with squeezing. An LMC could ensure > 99% mode overlap to be achieved independent of the LO beam geometry, which could vary with interferometer alignment and thermal state. If constructed in a monolithic fashion along with the AS port OMC, the LMC would also largely suppress relative pointing fluctuations (jitter) between the LO and signal fields. BHR is otherwise vulnerable to beam jitter through imperfections in the readout optics and photodiodes [25].

Fig. 3 An interferometer with signal mirror (SRM) for recycling or signal extraction, combined with balanced homodyne readout. The BHR scheme includes an output mode cleaner (OMC) and a similar LO mode cleaner (LMC) which pass the gravitational wave signal and LO fields respectively, while rejecting unwanted light in other spatial modes. The OMC and LMC are shown associated with the readout beam splitter (BS2) and photodiodes (PD A and B, in the blue shaded area) to indicated that these parts may be rigidly mounted together to ensure optimal overlap between the LO and signal field. The SRM in this figure can be replaced with a variety of QND readout schemes, such as variational readout [17] and speed meters [19], which depend on being able to choose the homodyne readout angle independent of the DC response of the interferometer.

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5. BHR in Advanced LIGO

In Advanced LIGO, balanced homodyne readout could be implemented by using the beam reflected from the back side of the beam splitter as the LO (i.e., the beam labeled PO in Fig. 2). This surface is anti-reflection coated with a typical reflectivity of 50–100 ppm. Over the anticipated interferometer input power range of 25 – 125 W, the LO power would be in the range 25 – 250 mW. In Advanced LIGO, this PO beam already propagates through some of the same telescoping optics as the AS beam, before it is captured by an in-vacuum beam dump. To use the beam as an LO for BHR, the beam dump would be replaced by a few beam directing optics to bring the beam into the output vacuum chamber, where the new LMC would be located along with the existing OMC.

In this arrangement the LO is derived from a source that is essentially free of high-order spatial modes, unlike the AS port field used in standard DC readout [8]. Furthermore, compared to an LO derived from the interferometer input beam, the PO field has the advantage of being highly stable in frequency, intensity, and spatial mode content. The required changes are relatively minor, and plausible in the context of a near-term upgrade to Advanced LIGO.

Scattered light must be carefully controlled in any of these readout schemes. In particular, backscattered light in the output paths can produce parasitic interference that easily exceeds shot-noise of the detected light. In the Advanced LIGO DC readout scheme, the dominant source of backscattered light is expected to be from the LO field in the OMC. To control this light, a Faraday isolator is included in the AS port, between the OMC and the signal recycling mirror. For a BHR scheme, backscattered light in the LO path would be attenuated by the same anti-reflective surface of the BS from which the LO is derived. This attenuation (≤ 10⁻⁴) is at least as large as that provided by a Faraday isolator; thus no additional isolator should be required in the LO path. Furthermore, since the AS port does not contain a large LO field, a Faraday isolator may not be required in this path either.

Looking to the future when squeezed light will be implemented on the interferometer, the measurement of quantum noise in a squeezed vacuum field presents a significant technical challenge with DC readout–a challenge that BHR can avoid. Assume that the photo-current in DC readout passes through a resistor R to produce a measurable voltage, and that the thermal voltage noise of that resistor is the dominant source of readout electronics noise. The resistor voltage noise must be smaller, by a safety factor α, than the quantum noise to be measured:

ν_{n} = 2 \sqrt{k_{B} T R} \leq ε R \sqrt{2 h ν P_{A S}} / (α F_{S Q Z})

where k_BT = 4.1 × 10⁻²¹ J at room temperature, hν = 1.8 × 10⁻¹⁹ J for 1064 nm photons, and a high quality photo-diode converts these photons to electrons with ε ≃ 0.85A/W. The introduction of squeezing would reduce the shot-noise component by a factor in the range 2 < F_SQZ < 3. The constant voltage at the output of a trans-impedance amplifier, designed with α = 10 such that thermal noise is well below quantum noise, turns out to be independent of the detected power:

V_{D C} = ε R P_{A S} = \frac{2 k_{B} T}{ε h ν} F_{S Q Z}^{2} ≃ 5 V F_{S Q Z}^{2} .

This is a reasonable value in the absence of squeezing, but becomes technically challenging with

F_{S Q Z}^{2} ~ 10

(i.e., 10 dB of squeezing). The balanced homodyne readout, however, can employ a current subtracting amplifier, in which the two photo-currents are directly subtracted before passing through a trans-impedance resistor [27]. The need to support a high constant voltage is thus avoided.

6. Conclusion

The improvements afforded by DC readout in gravitational wave detectors–reduced quantum noise and a clear path to further improvement through squeezed states–also introduce several technical noises and interferometer control problems. These problems arise from the presence of an LO field at the AS port, and from the asymmetry introduced to produce this field. As our simplified analysis shows, balanced homodyne readout offers a means of eliminating these problems while maintaining the advantages of DC readout. Future work will include a more detailed analysis, examining noise propagation in the presence of the various imperfections found in real interferometers.

In addition to improving readout noise in current gravitational wave interferometers, the proposed BHR scheme represents a necessary piece of future QND interferometers. By separating the homodyne readout angle from the DC detector response, QND readout schemes are given the freedom to choose the optimal readout quadrature. These features make balanced homodyne readout essentially imperative in future gravitational wave detector design.

References and links

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Balanced homodyne readout for quantum limited gravitational wave detectors

Abstract

1. Introduction

2. DC readout

3. Balanced homodyne readout for Gravitational Wave detectors

4. Mode cleaning cavities

5. BHR in Advanced LIGO

6. Conclusion

References and links

Cited By

Figures (3)

Equations (12)

Optics Express