Abstract
Most second-generation gravitational-wave detectors employ an optical resonator called an output mode cleaner (OMC), which filters out junk light from the signal and the reference light, before it reaches the detection photodiode located at the asymmetric port of the large-scale interferometer. The optical parameters of the OMC should be carefully chosen to satisfy the requirements to filter out unwanted light whilst transmitting the gravitational-wave signal. We use the simulation program FINESSE and realistic mirror phase maps that have the same surface quality as the KAGRA test masses to find out a proper design of the KAGRA OMC.
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1 Introduction
Gravitational waves are ripples of space–time predicted by Einstein [1]. The historical first detection of a gravitational wave is expected to be realized with the second-generation detectors that will begin taking science data over the next few years. KAGRA is the Japanese second-generation detector being constructed underground in the Kamioka mine [2]. The detector is based on a Michelson interferometer locked on the dark fringe, with Fabry–Perot optical resonators in each arm and additional resonators in the symmetric and anti-symmetric ports of the interferometer (see Fig. 1). The mirrors in the arm cavities are made of sapphire and cooled down to 20 K to reduce thermal noise. The underground facility and cryogenic operation are the two main characteristics of KAGRA. With the subsequent reductions in seismic noise and thermal noise, quantum noise limits the sensitivity in the observation band. The primary target of KAGRA is gravitational waves from binary neutron stars [3]. For such signals, the gravitational-wave frequency will gradually increase until \(\sim\)1.5 kHz when the two stars collide. Since the noise level is high at low frequencies and the signal is weak at high frequencies, the detector is most sensitive to the inspiral signal at around 100 Hz and it is the most efficient to improve the sensitivity at around the frequency. The observation range will be around 120–150 Mpc depending on the operation scheme of the interferometer.
While the optical resonator at the symmetric port of the interferometer, called the power-recycling cavity, resonates the carrier light to increase the input power to the interferometer, the role of the optical resonator at the anti-symmetric port, called the signal-recycling cavity, is to resonate the gravitational-wave signal sidebands exiting the Michelson interferometer to optimize the sensitivity spectrum [4]. The signal-recycling cavity length can be detuned from the resonance or anti-resonance of the carrier light to maximize gravitational-wave signals from binary neutron stars expected at specific frequencies, and so maximize the observation range for such events.
A gravitational wave will modulate the phase of the carrier light in the arm cavities to create the upper and lower sidebands at the gravitational-wave frequency. Without the detuning of the signal-recycling cavity, the signal sidebands are balanced. While the signal readout scheme used in first-generation detectors was a radio frequency heterodyne readout scheme [5], second-generation detectors aim to use a homodyne readout scheme using carrier light leaked into the anti-symmetric port as a reference light to beat with gravitational-wave signals, which is called DC readout [6]. There are two ways in which some of the carrier light in the arm cavities can leak into the anti-symmetric port. One is via a reflectivity imbalance between the two arm cavities. This component appears in the amplitude quadrature. The other is via differential offsets to the operating point of the arm cavities. This component appears in the phase quadrature as a constant signal at DC. The combination of the two components determines the readout phase of the reference field. In the tuned configuration, the gravitational-wave signal is extracted in the most efficient way with a DC readout phase of 90\(^\circ\). However, quantum radiation pressure noise is high in the phase quadrature and so the readout phase should be optimized for the target source even for zero detuning of the signal-recycling cavity. This becomes more complicated in a detuned configuration. The upper and lower signal sidebands are no longer balanced. To select the readout phase, one should compare the spectra and observation ranges for different combinations of detuning and readout phase. For KAGRA, a detune phase of 3.5\(^\circ\) has been selected, with a corresponding DC readout phase of 132\(^\circ\) [2]. The observation range for zero detuning and with a readout phase of 90\(^\circ\) is 128\(^\circ\) while that with 3.5\(^\circ\) detuning and the optimal readout phase is 148 Mpc. The optimization of the readout phase is important in KAGRA where the sensitivity is limited by quantum noise.
The laser beam propagating in the interferometer can be described as a sum of the Hermite–Gauss beams. In this study, the fundamental mode shall be set as the nearest fundamental mode of the signal field and higher order Gauss beams are regarded as junk light. The OMC plays an important role in filtering out the higher order spatial modes that are generated in the arm cavities and the radio frequency (RF) sidebands used for the interferometer control, before the output light reaches the detection photodiode. In this study, 515 W is assumed for the laser power in the power-recycling cavity [2]. The radius of curvature of the test masses is 1.9 km and we assume ±1 % error between the two arm cavities. The arm cavity finesse is 1,550 and the round-trip optical loss is 100 ppm. Here we also assume a ±10 % loss imbalance between the two arm cavities. The round-trip Gouy phase shifts in the power and signal-recycling cavities are 33\(^\circ\) and 35\(^\circ\) respectively [7], and the incident angles at the folding mirrors in the recycling cavities are about 0.6\(^\circ\). The power reflectivity of the signal-recycling mirror is 85 %.
As a result of a simulation without a mirror phase map included, the power of the DC light that is to be used as reference light is about 1 mW before the OMC, achieved by applying a differential offset to the arms of the main interferometer. The KAGRA OMC must not degrade the shot noise level by more than 5 % compared with the case of an ideal OMC that filters out all the unnecessary light. If the DC light power is 1 mW, a possible breakdown of an acceptable degradation after the OMC is as follows; the signal loss is to be 2 % or less, the residual spatial higher order modes are to be 10 \(\upmu\)W or less, and the residual RF sidebands are to be 20 \(\upmu\)W or less. This requirement can be satisfied by selecting a proper parameter set; namely, the finesse, round-trip Gouy phase, and length of the OMC.
The finesse of an optical cavity is a number which represents how many times the light circulates in the cavity and is determined by the reflectivity of the cavity mirrors. In the case of a 4-mirror ring cavity with mirror reflectivities \(r_1\), \(r_2\), \(r_3\), and \(r_4\), the finesse is given by
To maximize transmission of the signal through the OMC we use equally partially transmissive mirrors OMC1 and OMC3 (i.e., \(r_1=r_3<1\)) and nearly perfectly reflective mirrors for OMC2 and OMC4 (\(r_2=r_4\simeq 1\)). The optical loss \(\mathcal{L}\) for each OMC mirror is expected to be about 40 ppm per bounce [8]. If the finesse is too high, the light circulates too many times and the total optical loss of the signal would not satisfy the requirement. To keep the signal loss at \(<\)2 %, the finesse should not be higher than 800.
The OMC filters out higher order spatial modes, with the filtering efficiency of each mode determined by the round-trip Gouy phase of the OMC cavity or by the radius of curvature of the OMC mirrors:
Here \(g\) is the g-factor of the cavity, \(L\) is half the round-trip length of the OMC cavity and \(R\) is the radius of curvature of the curved mirror (note that only OMC1 is curved whilst the other mirrors are flat). The OMC power transmittance, of the mth spatial mode is given by
If the Gouy phase shift accumulated by a higher order mode is close to an integer multiple of 2\(\pi\), the mth mode nearly resonates in the OMC and cannot be filtered out. The ratio of the distance to the closest resonance to the cavity bandwidth determines the filtering efficiency for each higher order mode. It is important to select a proper combination of the Gouy phase and the cavity length, paying attention to the expected power in each higher order mode before the OMC, which should be numerically calculated using a simulation tool.
The power of an RF sideband transmitted by the OMC is given by
Here \(\omega _{\rm {RF}}\) is the radio frequency. In the case of KAGRA, \(\pm 16.875\) MHz sidebands exit the anti-symmetric port of the interferometer. To filter out the RF sideband to the requirement of \(<\)20 \(\upmu\)W after the OMC, \(L\) should be longer than 75 cm.
FINESSE is a simulation tool for modeling interferometers based on the modal model [9, 10]. FINESSE can analyze waves propagating in a user-defined interferometer not only for a plane wave approximation but also using a Hermite–Gauss modal expansion to describe the shape and spatial properties of light fields in a given setup. In this paper, we use FINESSE to simulate the expected beam distortions due to the nanometer-scale mirror surface errors of the test masses. Figure 2 shows one of the mirror maps used in the simulations documented here.
A mirror phase map is an array of numbers representing the height of a mirror over its surface. The phase maps were generated so that their PSDs (power spectral density) are consistent with the KAGRA mirror PSD specification, determined based on the loss requirement [11]. We create four independent mirror maps with the same PSD and apply them to the four test masses of the interferometer. There are 24 combinations and the results provide a statistical distribution of the higher order modes exiting the interferometer. In this study, we use mirror maps only for test mass surfaces. Inhomogeneities of the light absorption and the refractive index in the sapphire substrate have been reported [12] but the effect is not included in the current simulation. The aberrations of the beam splitter and the signal-recycling mirrors could also contribute to the generation of higher order modes but we leave them as a future work.
In a past study, Waldman calculated the distance between each higher order mode and the nearest OMC resonance for the first eight modes to select the most appropriate optical design of the OMC [13]. While Waldman treated all the higher order modes equally, Arai implemented a weighting factor on each mode according to the beam distortions observed in a similar interferometer [14]. For the KAGRA OMC, we can predict proper weighting factors according to simulation results with realistic mirror maps. The left panel of Fig. 3 shows the total power of the mth mode at the output of the interferometer before the OMC, normalized by the power in the fundamental mode for the 24 different mirror map combinations. We take the average of these values and use these as the weighting factors. We then multiply the weighting factors by the OMC transmittance \(T_{\text{omc}}\) for each mode. The results are compared with different Gouy phases and cavity lengths and we find that \(\eta =45^\circ\) and \(L=80\) cm is the most appropriate parameter set for the KAGRA OMC (Fig. 3, the right panel). As is shown in Table 1, the higher order modes and the RF sidebands are well suppressed with this KAGRA OMC design. Here, we use the 5th worst combination of the mirror maps since the fundamental mode beam power is less than 0.2 mW in the cases with even worse combinations (in such unlucky cases, we will have to increase the DC offset, descoping the readout strategy).
In the end, we calculate the shot noise limited sensitivity of KAGRA and compare the results for a lossless high-finesse OMC and with the realistic KAGRA OMC. The difference of the shot noise limited sensitivities is 3.2 % at 100 Hz, which satisfies the requirement of the KAGRA OMC.
Note that the higher order mode power distribution shown in the left panel of Fig. 3 is different from the previous work by Arai carried out with a measurement result of a first-generation detector. For a second-generation detector, a better mirror surface quality is expected so that the power distribution is rather flat. In our simulation, the 10th mode comes out to the anti-symmetric port the most since its distance to the neighboring resonance is as small as 6\(^\circ\), which is the closest among the first 20 modes in KAGRA.
References
A. Einstein, Ann. Phys. (Berlin), 49 (1916) [in German]
K. Somiya and the KAGRA collaboration, Classical Quantum Gravity, 29, 124007 (2012)
A. Maselli, V. Cardoso, V. Ferrari, L. Gualtieri, P. Pani, Phys. Rev. D 88, 023007 (2013)
J. Mizuno, K.A. Strain, P.G. Nelson, J.M. Chen, R. Schilling, A. Ruediger, W. Winkler, K. Danzmann, Phys. Lett. A 175, 273 (1993)
K. Somiya, Y. Chen, S. Kawamura, N. Mio, Phys. Rev. D 73, 122005 (2006)
T. Fricke, N. Smith-Lefebvre, R. Abbott, R. Adhikari, K. Dooley, M. Evans, P. Fritschel, V. Frolov, K. Kawabe, J. Kissel, B. Slagmolen, S. Waldman, Classical Quantum Gravity 29, 065005 (2012)
Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, H. Yamamoto, Phys. Rev. D 88, 043007 (2013)
K. Arai, Presented at Sept. LVC Meeting 2013, LIGO-G1301001-1 (2013). https://dcc.ligo.org
A. Freise, D. Brown, C. Bond. arXiv:1306.2973
A. Freise, G. Heinzel, H. Lueck, R. Schilling, B. Willke, K. Danzmann, Classical Quantum Gravity, 21, S1067 (2004). http://www.gwoptics.org/finesse
H. Yamamoto, private communications
E. Hirose, D. Bajuk, G. Billingsley, T. Kajita, B. Kestner, N. Mio, M. Ohashi, B. Reichman, H. Yamamoto, L. Zhang, Phys. Rev. D 89, 062003 (2014)
S. Waldman, Tech. Rep., LIGO-T1000276-4 (2011). https://dcc.ligo.org
K. Arai, S. Barnum, P. Fritschel, J. Lewis, S. Waldman, Tech. Rep., LIGO-T1000276-5 (2013). https://dcc.ligo.org
Acknowledgments
The authors would like to thank Daniel Brown and Andreas Freise for their kind help on the use of FINESSE, and Hiro Yamamoto for his kind support in creating realistic mirror maps. Special thanks also go to Jumpei Kato, Kazushiro Yano, Sho Atsuta, Yuu Kataoka for their kind support. This work is partially supported by the Specially Promoted Research and the Core-to-Core Program, A. Advanced Research Networks of the Japan Society for the Promotion of Science (JSPS).
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Kumeta, A., Bond, C. & Somiya, K. Design study of the KAGRA output mode cleaner. Opt Rev 22, 149–152 (2015). https://doi.org/10.1007/s10043-015-0028-2
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DOI: https://doi.org/10.1007/s10043-015-0028-2