Impact of upconverted scattered light on advanced interferometric gravitational wave detectors

David J Ottaway; Peter Fritschel; Samuel J. Waldman

doi:10.1364/OE.20.008329

1. Introduction

The first generation, long baseline gravitational wave detectors have reached design sensitivity. The LIGO, Virgo and GEO detectors have completed their first observing campaigns, collecting more than a year of coincident data and observing with high duty factor [1, 2]. Following the success of the first generation detectors, the installation of the advanced second generation detectors began in 2010 [3, 4]. The new detectors increase the sensitivity by a factor of 10 and extend the low frequency bandwidth down to 10 Hz, increasing the predicted gravitational wave detection rate by a factor of 1000. The push towards increased sensitivity and lower frequencies introduces a variety of new, low frequency noise sources. Further, advanced second generation detectors such as Japan’s Large Cryogenic Gravitational wave Telescope(KAGRA) [5] and third generation detectors such as Europe’s Einstein Telescope (ET) [6] are being built deep underground to increase the sensitivity at even lower frequencies, requiring even higher tolerances on these low frequency noise sources.

In this paper, we present an analysis of the impact of non-linear, up-converted scattered light on the performance of gravitational wave detectors. We discuss the scattered light contributions to detector noise using actual seismic spectra recorded for the Advanced LIGO detector (aLIGO) located in Livingston Parish, Louisiana, USA as an example.

Figure 1 schematically illustrates the optical configuration of a second generation gravitational wave interferometer. The interferometer uses a series of nested cavities to increase the shot noise limited sensitivity to the displacement of its test masses and hence a gravitational wave [7]. Up to 750 kW of circulating power is stored in the 4 km long Fabry-Perot arm cavities, supplied by a narrow-linewidth, 180 W laser [3]. To achieve this level of build-up, each optic must scatter less than 35 ppm per reflection, requiring state of the art super-polished substrates with an RMS surface figure of less than 0.3 nm. Even at this level, the scattered light can have a significant impact on the noise performance of the interferometer if it is allowed to re-scatter back into the cavities after reflecting from a moving surface.

Fig. 1 Schematic of an advanced gravitational wave detector.

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The effect of scattered light can be considered in two regimes differentiated by the total RMS motion of the surface. If the total relative motion between the surface and the arm cavity optics is significantly less than the laser wavelength, the surface imparts a linear signature on the light at the same frequency as that of the motion [8–10]. In this situation it is only necessary to consider vibrations that have frequencies in the sensitive band of the instrument, which for aLIGO is between 10 Hz and 10 kHz. If, however, the relative motion is larger than the wavelength of interferometer light, so called “fringe-wrapping” upconverts low frequency motion to higher frequencies [9]. In this paper we consider this second effect for realistic vibrational motions and determine the level of optical attenuation required to ensure the performance of second-generation gravitational wave detectors.

The primary source of low frequency, large amplitude ground motion is the so called “oceanic microseism” which is caused by the motion of ocean waves, varies seasonally, and peaks between 0.1 and 0.4 Hz at the LIGO sites. The amplitude of the microseism varies dramatically dependent on the local metrological conditions and often has a peak to peak motion many times the wavelength of the light. The North Atlantic generated microseism is occasionally large enough to prevent interferometer operation. The microseism is coherent over length scales comparable to the LIGO interferometers [11].

For scattered light to be an issue, three scattering events must occur. First the light must be scattered out of the arm cavity mode off the high quality test mass surface; then, it needs to hit an external surface and be scattered back in the direction of the test mass; and finally, it must be scattered off of the test mass surface back into the arm cavity mode. Although these three scattering events provide a significant amount of attenuation, in this paper we will show that it is necessary to have in excess of 24 orders of magnitude of optical attenuation to prevent upconversion being a dominant noise source.

In the following sections, we present an analytic mathematical framework used to evaluate the impact of scattered light. Next, we estimate the scattering coefficient for aLIGO ion-beam milled optics and stray light control baffles. Finally we use the measured ground motion and the estimated scattering to determine the likely impact on the noise of the aLIGO detector at the LIGO Livingston Observatory.

2. Theory

The light circulating in the arms of the interferometer can be represented as a complex electric field of the form:

E (x, t) = E_{0} exp [i (ω t - k z + ϕ (t))]

where

P_{0} = E_{0}^{2}

is the stored arm cavity power and field, k = 2π/λ is the laser wave number, ω is the angular frequency, ϕ(t) is a time varying phase resulting from an external stimulus, and z is the distance from an arbitrary reference plane. In the following, we assume z = 0. Assuming a sinusoidal phase modulation of amplitude ϕ_m and frequency ω_m, neglecting terms of constant phase, and employing the Bessel function expansion identity, the electric field at a reference plane within the cavity can be written as:

E (t) = E_{0} exp [i ω t] \sum_{n = - \infty}^{\infty} i^{n} J_{n} (ϕ_{m}) exp [i n ω_{m}]

In the long wavelength approximation appropriate for low frequencies, a gravitational wave strain, h, appears in an interferometer with arm cavities of length L as a differential test mass displacement x_G = hL [12]. The displacements create a phase shift, ϕ_G = 2kx_G which for plausible gravitational waves, obeys ϕ_G ≪ 1. In this limit, only the n = 0,±1 terms contribute and the electric field generated by a sinusoidal gravitational wave at frequency ω_G can be approximated as a phase modulation:

E_{G} (x, t) ≃ E_{0} exp [i ω t] (1 + i k x_{G} exp [\pm i ω_{G} t]) .

The phase modulation sidebands resonate in the coupled arm and signal-recycling cavities with an amplitude gain of Γ = 8.3 [13]. The aLIGO detectors use a DC readout scheme –a form of homodyne detection – in which the modulated field is detected by interference with a static carrier field on an audio frequency photodetector [14]. By appropriately tuning the carrier phase, the readout is only sensitive to phase modulation sidebands such as those in Eq. (3).

For simple sinusoidal motion, a scattered field has the form of Eq. (2) with an additional amplitude transfer coefficient, A,

E_{s} (t) = A E_{0} exp [i ω t] \sum_{n = - \infty}^{\infty} i^{n} J_{n} (2 k x_{s}) exp [i n ω_{s}] .

Here, x_s is the amplitude of the scatterer’s motion at frequency ω_s. When 2kx_s ≪ 1, the scattered field reduces to the phase modulation of Eq. (3). For larger amplitudes, Eq. (4) includes both phase and amplitude components at frequencies nω_s. For 2kx_s ≫ 1, the Bessel functions, J_n, are of order unity up to n ≈ 2kx_s, setting an upper limit on the highest frequency sideband of max(ω) ≈ 2kx_sω_s. Thus, the sources of scattered light with the highest velocity dominate the noise spectra.

The frequency dependent transfer function from field amplitude to the interferometer readout can be expressed in terms of the effective GW displacement, x_eff, required to generate the same signal. Only the fraction of the scattered field in the differential degree of freedom, one half the total field, will resonate in the interferometer. Fields in the common mode degree of freedom will be suppressed by a high bandwidth servo. The phase modulation fields (|n| = 1,3,5,... in Eq. (4) are directly detected by the interferometer’s DC readout. The phase transfer function can be written as Eq. (5), the additional factor of 1/2 is due to the split between common and differential arm motion.

\frac{x_{eff}}{Φ} = \frac{1}{2} \frac{λ}{4 π} [\frac{m}{rad}] .

The amplitude quadrature of the scattered field is weakly detected by the DC readout. However, the differential component of the field resonantly builds up by the factor Γ and interferes with the static arm cavity field. The resulting power fluctuations at each arm cavity mirror exert a radiation pressure force. The force is filtered by the mechanical transfer functions of the mirrors’ suspension [15]. Altogether, the transfer function for amplitude noise expressed as Relative Intensity Noise, RIN, to the effective displacement is

\frac{x_{eff}}{RIN} = 2 \frac{Γ}{M} \frac{1}{Ω^{2} - ω^{2}} \frac{2}{c} P_{0} [m] .

Here Γ is the coupled cavity gain, M = 40 kg the mirror mass,

\frac{Ω}{2 π} = 0.45

Hz the suspension eigenfrequency, and c the speed of light. The leading factor of 2 includes the contributions from 4 mirrors and the assumption that 1/2 the field is in the differential quadrature. Because the radiation pressure arises from an interference with the arm cavity field, the transfer function is proportional to the arm cavity power P₀ and can be reduced by decreasing the laser power.

3. Scattered light control for aLIGO

For large diameter mirrors figured with ion-beam milling, the scattering into 2π solid angle, including point scatterers, is predicted to be ≃ 35 ppm. In aLIGO, approximately 2 ppm of this light falls onto a baffle located at the far end of the 4 km arm cavity. The baffle has a toroidal geometry with inner and outer radii of 17 cm and 40 cm, respectively. Of the light incident on the baffle, 1.4 ppm travels directly from the optic, while 0.6 ppm scatters off the optic and specularly reflects off the other mirror. Ray tracing shows that this light reflects multiple times from the reflecting arm cavity mirrors before hitting the baffle. We therefore consider the worst case scenario in which we simply add these contributions. To reduce back reflections, the baffle is constructed from oxidized stainless steel with a Bi-directional Reflectivity Distribution Function (BRDF_b) of 0.02 sr⁻¹ for large angles. To create noise in the interferometer, the light from the scatterer must reflect back into the solid angle of the arm cavity mode.

The total cross section for the three scattering steps is [10]

\frac{δ I}{I} = {(\frac{λ}{R})}^{2} {B R D F}_{m}^{2} {B R D F}_{b} δ Ω_{b} \equiv A^{2} .

The power scattering coefficient, δI/I, is a function of the wavelength (λ), the distance between the mirror and the scatterer (R), the BRDF of the mirror and the baffle (BRDF_m and BRDF_b), and the solid angle subtended by the baffle, (δΩ_b). The BRDF_m of the aLIGO test masses is estimated from the measured surface figure errors before coating. The power spectrum of the errors for an un-coated, ion-milled aLIGO optic is approximately

P S D (k_{s}) = 10^{- 16} k_{s}^{- 1} {cm}^{3}

for spatial frequencies k_s(cm⁻¹). The BRDF is estimated from the power spectrum using:

B R D F (k) = {(4 π / λ^{2})}^{2} D \times P S D (k_{s}) / k_{s} .

Here D ≃ 1 is a geometric parameter that weakly depends on the spectral shape of surface PSD. For the 1/k power spectrum, D = 1/(2π) [16]. Assuming the ion-beam surface figure, the four aLIGO test masses have a total power scattering coefficient of δI/I = 0.9 × 10⁻²⁴, with a corresponding amplitude transfer coefficient of ≈ 10⁻¹². This exceptionally low surface figure may be degraded by the dielectric coatings that must be added to the substrate to form a mirror. Therefore for comparison we also consider the surface figure of the coated Initial LIGO optics [17], which has a total power transfer coefficient ten times larger.

4. In-situ, high motion scattering

The two LIGO Observatories are located in Livingston Parish, Louisiana and Hanford, Washington. Of these two locations, the Livingston Observatory suffers from microseismic motion with significantly higher amplitudes and frequencies. To quantify the impact of seismic motion on the sensitivity of the LIGO interferometers we measured the ground displacement, x_s(t), using Guralp CMG-40T three-axis seismometers. We sampled the ground motion between Nov. 19, 2005 and Oct. 15, 2007 and between Feb. 7 2009 and Oct. 19, 2010, corresponding to the LIGO S5 and S6 science runs and including total time of 956 days. The effective displacement noise spectrum, S_s(ω), is estimated using the Fourier transform of the ground displacement and the transfer functions of Eq. (5) and Eq. (6):

\begin{array}{l} S_{s} (ω) = A [\frac{x_{eff}}{Φ}] [\int_{- \infty}^{\infty} sin (\frac{4 π x_{s} (t)}{λ}) exp (i ω t) d t] + \\ A [\frac{x_{eff}}{RIN}] [\int_{- \infty}^{\infty} cos (\frac{4 π x_{s} (t)}{λ}) exp (i ω t) d t] \end{array}

In practice, the Fourier spectra in Eq. (9) are estimated from discrete time series, each 2048 seconds long, using Welch’s method with Hanning windowing and 1/32 Hz bandwidth. The amplitude spectrum of the original seismic driving signal is estimated in the same fashion. The N = 40,349 displacement and seismic noise spectra are stored in two-dimensional histograms with 200 logarithmically spaced bins for each frequency. The solid contours in Fig. 2 are the Amplitude Spectral Densities (ASD) of the seismic motion, ≃ ∫ x_s(t) exp(iωt)dt. The dashed contours are the ASD of the upconverted scattered field, ≃ ∫ cos(4πx_s(t)/λ)exp(iωt)dt. The colors represent the RMS amplitude for which 50%, 90% and 95% of the 2048-second segments have an equal or lesser amplitude.

Fig. 2 956 days of the seismic spectra at the LIGO Livingston Observatory. Each contour represents the fraction of time for which the ground motion was smaller than the contour.

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As Fig. 2 shows, the RMS displacement is dominated by the microseism between 0.1 and 0.4 Hz, and varies by a factor of 4 from the 2 μm_rms of the 50% contour to 8 μm_rms at 95%. For these large excitations, the Bessel function approximation of Eq. (4) implies upconversion sidebands up to 15 Hz, which can be seen in the “scattering shoulder” of the dashed curves; the amplitude remains constant while the frequency increases. Implicit in our calculation is the assumption that the scatterer is moving with the full amplitude of the microseism. In practice, seismic isolation systems may decrease the relative motion between cavity field and scatterer, with a corresponding nonlinear effect on the upconverted noise.

To calculate the amplitude of the scattered field incident on the test mass, we use the amplitude scattering coefficient of A = 1 × 10⁻¹² and A = 8 × 10⁻¹², corresponding to ion-milled optics and Initial LIGO optics, respectively. These coefficients include the contributions of four optics in quadrature. The scattering spectra for the 50%, 90% and 95% amplitude percentiles are shown in Fig. 3, together with the aLIGO requirements.

Fig. 3 The predicted magnitude of upconverted scattered light noise for the aLIGO Livingston interferometer. “aLIGO” is the design sensitivity for a zero-detuned, 120 W interferometer and “Aux. Noise” is the auxiliary noise limit described in the text. The noise depends linearly on the total amplitude reflectivity, here taken as A = 1×10⁻¹² appropriate for ion-milled optics (solid lines), and A = 8 × 10⁻¹² for Initial LIGO (dashed lines).

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Many sources contribute to the noise floor of the aLIGO interferometers. Two mirror displacement noise terms set the ultimate design limits on the aLIGO interferometers: the thermal noise of the mirror suspensions and of the mirrors dielectric coatings [3]. To guarantee that the other individual noise sources do not significantly contribute to the final sensitivity, each term must contribute less than 1/10th the sum of the suspension and coating noises. This target noise limit is shown in Fig. 3 as the design requirements for auxiliary noise.

The interferometer is configured with a zero-detuned signal recycling cavity and 120 W of input power, as described in Ref [18]. With the ion milled optics and baffles, scattered light will not contribute to the auxiliary noise budget except during the times with the highest microseism amplitude. The noise exceeds the requirements at the very lowest frequencies between 10 and 15 Hz. Because of the quadratic dependence on the mirrors BRDF in Eq. (8) and the likely degradation of the surface figure by the coating process, the upconverted noise shown in Fig. 3 is likely an underestimate of the actual noise. The Initial LIGO spectra shown use the measured surface maps of Initial LIGO optics to establish a worst-case upper limit on the scattering noise contribution of a coated optic. Note that in all cases, the induced radiation pressure noise from the amplitude quadrature dominates the scattered light noise.

5. Conclusion

We have evaluated the impact of scattered light on the performance of a second generation gravitational wave detector. We have shown that if the surface figure of ion-milled surfaces is not realized in Advanced LIGO and scattered light attenuation is not improved, the performance of aLIGO will be compromised at low frequencies during the highest microseism periods. Once significant observation with aLIGO is completed, it is anticipated that technical upgrades such as non classical light sources and optics with reduced coating thermal noise will be implemented. For these upgrades to improve the noise performance below 10 Hz, it will be necessary to improve the performance of the scattered light control. Alternatively, the baffles position should be controlled to follow the motion of the interferometer, significantly reducing their relative motion and eliminating the upconversion of scattered light.

Third generation detectors aim to push to lower frequencies than Advanced LIGO by locating underground where the seismic noise around 1 Hz is significantly reduced compared with the surface. However, recent measurements have shown that the magnitude of the motion at the microseism frequencies is virtually unaffected by moving deep underground [19]. Therefore, sensitivity at lower frequency is to be achieved in third generation detectors, the scattered light control will need to be significantly improved.

Acknowledgments

The authors are grateful for illuminating discussions with R. Adhikari, M. Landry, R. Schofield, D. Shoemaker and M. Evans. These results were validated with the Optickle simulation tool developed by M. Evans. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0107417.

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Impact of upconverted scattered light on advanced interferometric gravitational wave detectors

Abstract

1. Introduction

2. Theory

3. Scattered light control for aLIGO

4. In-situ, high motion scattering

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (9)

Optics Express