Aaron Spector Deutsches Elektronen-Synchrotron (DESY)

Laser Interferometers (for GW detection)

Aaron Spector Deutsches Elektronen-Synchrotron (DESY) Hamburg, Germany Transmission Field Cavity Power This conﬁguration additionally avoids co-resonances of End Mirror (via Transmission Port) [North] higher-order cavity modes (HOM) with the fundamental mode. The Gouy phase shift for one-way propagation down the inter- - DARM ferometer arm is Motion Ly~39m L Gouy arctan = 0.806 rad (46.2) (6) ⌘ ZR Lx~39m ! Power Recycling End Mirror Any Hermite-Gauss mode, H , attains a round-trip phase ex- =1064nm Mirror [East] mn Input Field cess of

2 kW Laser - mn = 2(1 + m + n) Gouy (7) 1 Watt compared to the propagation of unfocused spherical wave- REFL PD Field - Lp= 20cm fronts. When 2(m+n)Gouy is an integer multiple of 2⇡, the Hmn Reflection Beam DARM Splitter Motion mode will be co-resonant with the fundamental H mode. With Port 00 1.61 rad (92 ) of phase separation between each mode order, Anti-Symmetric 200 mW the lower-order HOM resonances are well-separated from the Photodiode (dark) Port fundamental. The fourth-order mode wraps back near the fundamental, but the 0.17 rad (10) phase separation is >100 times Figure 1: Signal flow diagram of the interferometer static fields, represent- the cavity linewidth. ing the linear system of equations for the interferometer response. The fields are sourced by Elaser. Each reflection and transmission coecient is given by The power recycling cavity provides not only the resonant r or t, subscripted by its respective optic. F is the optical carrier frequency power enhancement, but also filters noise sidebands from the and the arm lengths are Lx and Ly. The round nodes represent internal states laser. Because of the higher-order-mode separation, di↵erent of the physical system whereas the square nodes are output fields observed transverse modes have di↵erent filtered noise spectra, which in photodetectors, such as the anti-symmetric port field EAS. To fully model the contrast-defect, additional copies of this diagram must be added to rep- beat with residual light at the Michelson anti-symmetric port resent additional transverse modes, with small transfer coecients arising by readout. The modes then show as noise peaks at the HOM res- defects in the end mirror and beamsplitter shapes. The signal sidebands from onance frequencies in the readout spectra, confirming estimates the end-mirrors are signified with red and blue arrows, which source (unshown) copies of the graph at the modulation frequencies. At each port the optical field of the arm length and Gouy separation. These noise measure- at carrier and sideband frequencies beat together to model the full frequency- ments are described in 6.4.1. FHdependent workshop response. on § 2.2. Instrument Response gravitational waves and With power recycling, the response of the interferometer to 2.1. Power-Recycling Cavity time-varying path length displacements is complicated by the particleThe Michelson physics interferometer, formed by the beamsplitter storage time of the recycling cavity, which imposes a band- and two end mirrors, forms an e↵ective mirror where the losses width limit of approximately 350 Hz on the cavity response. are determined by the fraction of light escaping to the AS port At frequencies 350 Hz, the response at the anti-symmetric ⌧ and the remainder is reflected. The addition of a power recy- port to arm length displacements reflects both a change in the cling mirror (PRM) forms a cavity with this e↵ective mirror. Michelson fringe o↵set and in the cavity storage power. How- The power-recycled interferometers are designed to be nearly ever, at frequencies 350 Hz, arm length displacements occur confocal resonators, folded by the 45 incidence beamsplitter on a shorter time scale than the cavity can respond. In this so that each arm forms a flat-curved cavity. They have an arm limit, the power-recycled interferometer responds equivalently length of L = 39.2 m and an end mirror radius of curvature of to a single-pass Michelson interferometer of the same optical R = 75.1 m, which is matched between the two arms to within power. 10 cm. The resulting waist, with radius Fig. 2 shows the numerically-calculated transfer function of the power-recycled interferometers at several fringe o↵sets. The Holometer operates at an o↵set of approximately 1 nm, and w0 = 2L(2R 2L) 3.57 mm (4) r2⇡ ⇡ all science and calibration signals are measured at frequencies p 1 kHz. At this o↵set, the deviation from a single-pass Michel- lays at the position of the flat PRM. The end mirrors are each son response is 2% above 1 kHz. The third plot, indicating located nearly one Rayleigh range away,  the optical sensitivity, shows this di↵erence at the calibration 2 line frequency from the asymptotic response, which is small ⇡w0 ZR = 37.6 m (5) for the 1nm o↵set. Thus, for calibration purposes (see 7.2) ⌘ § the instrument can be modeled as an equivalent high-power, where the beam half-width has grown to w 5 mm. The devi- single-pass interferometer. Neglecting the cavity correction un- 1 ⇡ ation from a pure confocal configuration satisfies the resonator derestimates the instrumental sensitivity. The degradation in stability criterion, R < 2L (for a review of laser resonators, see sensitivity would be relevant for smaller interferometers with [17]). higher bandwidth recycling. For instance in tabletop versions 4 3 Page 6 of 217 LivingRevRelativ(2016)19:3

Fig. 1 Gravitational waves are transverse quadrupole waves. If a wave passes through the ring of test particles that is oriented perpendicular to the direction of wave propagation, the distances between the particles would change periodically as shown in this sketch

Fig. 2 Simpliﬁed layout of a Michelson interferometer. The laser provides the input light, which is split into two beams by the central beamsplitters. The beams reﬂect off the end mirrors and recombine at the beamsplitter. The light power on the main photo detector (PD) changes when the difference between the arm length ∆L L L changes = X − Y

The measurable length change induced by a gravitational-wave depends on the total length being measured. For gravitational waves with wavelength much larger than the Principles of GW detection detector size we get:

Strain • GWs stretch and squeeze space-time ∆L hL, (1.1) = with L the length of the detetor and h the strain amplitude of the gravitational wave. This scaling of the change with the base length led to the construction of interferometers with arm length of several kilometres. Gravitational-wave detectors strive to pick out signals carried by passing gravitational waves from a background of self-generated noise. This is challenging because of the extremely small effects produces by the gravitational waves. For example, the

ﬁrst gravitational wave detected in September 2015 by the LIGO2 detectors (Abbott et al. 2016b), which is considered to be a strong event, reached a strain amplitude of 21 10− .ThissignalcouldnothavebeenmeasuredwithasimpleMichelsoninterferom-

123 3 Page 6 of 217 LivingRevRelativ(2016)19:3

The measurable length change induced by a gravitational-wave depends on the total length being measured. For gravitational waves with wavelength much larger than the detectorPrinciples size of we GW get: detection

Interferometers ∆L hL, (1.1) • Phase changes induced by length changes in arms = • Sensitivity determined by response, displacement noise, sensing noise with L the length of the detetor and h the strain amplitude of the gravitational wave. This scaling of the change with the base length led to the construction of interferometers with arm length of several kilometres. Gravitational-wave detectors strive to pick out signals carried by passing gravitational waves from a background of self-generated noise. This is challenging because of the extremely small effects produces by the gravitational waves. For example, the ﬁrst gravitational wave detected in September 2015 by the LIGO detectors (Abbott 3 et al. 2016b), which is considered to be a strong event, reached a strain amplitude of 21 10− .ThissignalcouldnothavebeenmeasuredwithasimpleMichelsoninterferom-

123 Principles of GW detection

[Ringwald, Tamarit, Welling]

GWs in SMASH

GWs from quantum uctuations during ination

GWs from inaton fragmentation during preheating

GWs from thermal uctuations after preheating

4 Principles of GW detection Space Based (LISA) Ground Based (aLIGO, VIRGO, ET) HF Ground Based • 0.1 mHz - 0.1Hz • 10 Hz - 1 kHz • 1 MHz - 100 MHz km Earth 2.5 million

19 – 23° 60° 1 AU (150 million km) Sun 1 AU Sun 5 ± ± V !"! ∞ =

µ ! ! z H pm √ ) ! ( ! ! ! ○ ) ± ( Space based GW detection km Earth 2.5 million Lower frequencies ➜ Longer arms 19 – 23° 60° • Increased response • 2.5 million km arms 1 AU (150 million km) Sun • Telescope transfer of laser signals • Small power sample at far spacecraft • Michelson formed by measuring phases 1 AU Sun 6 ± ± V !"! ∞ =

pm Hz! ! ! ! z ! ! y !θ nm s! !! nrad s! ! ! x !! fm s! Hz! ! !!! µ ! ! ! x Space based GW detection [Danzmann et al., 2017] Upper GRS Caging No seismic noise Vacuum Housing • Test masses in drag-free orbits

• Must suppress spurious forces on TM Gravitational Balance Masses • Charges, residual gas/eddy currents, magnetic fields, SC coupling, IFO, UV Illumination TM and coupling, etc. electrode Lower Caging housing and (not visible) Vent Duct µ

JPL / NASA 7

µ ! ! !! µ z H pm pm Hz √ √ ) ! ( ( ! ) ! ! ! !! ! ○ ○ ) ± ( ( ± ) !! µ pm Hz √ ( ! ) Space based GW detection Sensing Noise • Low received!! !power ➜ higher shot noise ○ ( ± ) • Low frequencies ➜ temperature effects • Path length noise • Pointing effects • Measurement noise

! LISA Pathfinder Launched December 2015 • Single test mass in a drag free orbit • 2nd test mass as witness sensor !! nrad Hz • Successful demonstration of √ acceleration noise ! α x! x! ! nm Hz √ ! 9

Ground Based GW detectors

More complex optical system for high response • Arm Cavities • Power and signal recycling

10 Ground Based GW detectors

Test mass motion • Seismic noise • Radiation pressure noise • Coating/suspension thermal noise

Sensing noise • Photon counting statistics

11 aLIGO Noise Spectrum

12 2 Ackley, Adya, Agrawal et al.

1 INTRODUCTION 22 10 NEMO Total Newtonian Gravity Gravitational-wave astronomy is reshaping our under- Coating Brownian Quantum Vacuum Coating Thermo-Optic Seismic standing of the Universe. Recent breakthroughs include Excess Gas Substrate Brownian ITM Carrier Density Substrate Thermo-Elastic the detection of many gravitational-wave signals from 23 Hz] 10 ITM Thermo-Refractive Suspension Thermal

binary black hole collisions (Abbott et al., 2019a) leading to an enhanced understanding of their population / properties (Abbott et al., 2019d), measurement of the 24 Hubble parameterExtending (Abbott et al. the, 2017d Second; Hotokezaka Generation10 Band et al., 2019), unprecedented tests of Einstein’s theory of Strain [1 General Relativity, including constraints on the speed NEMO [Ackley et al., 2020] of gravity (Abbott et al., 2017e) and hence the mass 25 10 NEMO Total CE of the graviton (Abbott et al., 2017b, 2019b), to name 20 • 4 km L shaped Michelson 10 A+ ET a few. Plans for building the next generation of obser- binary neutron star signal • Relaxed requirements on seismic isolation vatories are afoot. The United States National Science 10 21 Foundation, Australian• Higher Research power in Council, arm cavities and (4.5 British MW) government have ﬁnanced an upgrade to Advanced LIGO • Cooled test masses (silicon) 22 (aLIGO) known as A+, which will increase the sensitivity 10 of the current detectors by a factor of 2-3 dependent on 23 the speciﬁc frequency of interest (Miller et al., 2015). Re- 10

search and development is ongoing for third-generation Characteristic Strain 24 observatories, the Einstein Telescope (Punturo et al., 10 2010a) and Cosmic Explorer (Abbott et al., 2017a): 101 102 103 104 broadband instruments with capabilities of hearing black Frequency [Hz] hole mergers out to the dawn of the Universe.

Third-generation observatories require substantial, Figure 1. Noise budget and indicative gravitational-wave signal global financial investments and significant technolog- from a binary neutron star collision. Top panel: we show the13 ampli- ical development over many years. To bridge the gap tude spectral density of the various noise components that make between A+ and full-scale, third-generation instruments, up the total noise budget shown as the black curve. Bottom panel: The black curve is the same total noise budget as the top panel, it is necessary to explore smaller-scale facilities that will now shown as the noise amplitude hn = fSn(f),whereSn(f) not only produce significant astrophysical and funda- is the power-spectral density. This curve is shown in comparison mental physics outcomes, but will simultaneously drive to design sensitivity of A+ (blue), the Einstein Telescope (ET; technology development. In this spirit, we introduce a green), and Cosmic Explorer (CE; pink). Also shown in red is the Neutron star Extreme Matter Observatory (NEMO): a predicted characteristic gravitational-wave strain hc for a typical binary neutron star inspiral, merger, and post-merger at 40 Mpc, dedicated high-frequency gravitational-wave interferom- where the latter are derived from numerical-relativity simulations. eter designed to measure the fundamental properties of nuclear matter at extreme densities with gravitational waves. We envision NEMO as a specialized, detector print on the gravitational waveform, which becomes in- with optimum sensitivity in the kHz band operating creasingly important at higher frequencies 0.5 4 kHz. ≥ ≠ as part of a heterogeneous network with two or more Mergers produce remnants, some of which collapse to A+ sensitivity observatories. The A+ observatories pro- black holes, and some of which survive as long-lived, mas- vide source localization while a NEMO measures the sive neutron stars. Up to 79% of all binary neutron star ¥ imprint of extreme matter in gravitational-wave signals mergers may produce massive neutron star remnants from binary neutron star mergers. To maximise scien- that emit strong gravitational-wave signatures (Mar- tific impact, a NEMO must exist simultaneously with galit & Metzger, 2019). The precise nature of the rem- 2.5-generation observatories, but before full-scale third- nant is strongly dependent on the details of nuclear generation instruments are realised. physics, which is encoded in the neutron star equation Neutron stars are an end state of stellar evolution. of state (e.g., see Bernuzzi, 2020, and references therein). They consist of the densest observable matter in the Measuring gravitational waves at these high frequen- Universe, and are believed to consist of a superfluid, su- cies therefore oers a window into the composition of perconducting core of matter at supranuclear densities. neutron stars, not accessible with other astronomical Such conditions are impossible to produce in the labora- observations or terrestrial experiments. We show that tory, and theoretical modelling of the matter requires detection rates of gravitational waves from post-merger extrapolation by many orders of magnitude beyond the remnants with a network of only two A+ observatories point where nuclear physics is well understood. As two is between one per decade and one per century, while neutron stars coalesce, their composition leaves an im- adding a NEMO to the network increases this to more Third Generation Ground Based Observatories

Einstein Telescope and Cosmic Explorer • ET: Triangle configuration with 10 km arms • CE: 40 km L shaped Michelson • More advanced seismic isolation • Cooled test masses

14 High Frequency Interferometric Detectors

Correlation detectors • Identical interferometers in close proximity and same orientation • At high frequencies limited by shot noise (must suppress common noise sources) • Can be resonant [Akutsu et al., 2008]

Challenges going to Higher Frequencies • Arm length vs. frequency response • Overlap function

FIG. 3: (color online) Overlap reduction function in the caseoffourdetectorconﬁgurations. Each setup is (a) ideal, ∆X! =0,β =0,(b)T-shaped,∆X! =0,β = π/2, (c) crossed, ∆X! = (L/2,L/2, 0), β = π,(d)stacked,∆X! =(0, 0,L/2), β =0.The”exact”meansthecalculation with arm response function and the ”long wavelength limit” =1.Thelatterisnotvalid T T around 100 MHz, but merely plotted for comparison. Note that the sign of γ(f)in(b)isinversed for convenience of comparison.

spectral density of noise [32] around 100 MHz is

4 2 − 1.60 10 1W − P (f) 4.65 10 42 × Hz 1, i=1, 2. (25) i ≈ × α"(f) I ! " ! 0 " The factor α" is called the optical amplification factor in a cavity and gives α" 1.6 104 with ≈ × 2 the reflectivity of the recycling mirror, RF =0.99996, and the reflectivity of the other three mirrors, R2 =(0.99998)3. The bandwidth is 2 kHz with these reflectivities. Substituting E ∼ Pi(f)andγopt into Eq. (17), and assuming that observation time is T =1yrandthat

Ωgw(f) has a ﬂat spectrum around 100 MHz (which is suﬃcient for practical purposes [29]), one can calculate the sensitivity of two SRIs to GWB and obtain h2 Ω 1.4 1014. 100 gw ≈ × 13 High Frequency Interferometric Detectors

[Chou et al., 2017]

Transmission Field Cavity Power This configuration additionally avoids co-resonances of Fermilab Holometer End Mirror (via Transmission Port) [North] higher-order cavity modes (HOM) with the fundamental mode. The Gouy phase shift for one-way propagation down the inter- - DARM ferometer arm is • 39 m Dual Michelson interferometers Motion Ly~39m L Gouy arctan = 0.806 rad (46.2) (6) ⌘ ZR (0.635 m separation) Lx~39m ! Power Recycling End Mirror Any Hermite-Gauss mode, H , attains a round-trip phase ex- =1064nm Mirror [East] mn • 1 MHz - 13 MHz Input Field cess of 2 kW Laser - mn = 2(1 + m + n) Gouy (7) 1 Watt ➜ compared to the propagation of unfocused spherical wave- • Power recycling: 1 W injected REFL PD Field - Lp= 20cm fronts. When 2(m+n)Gouy is an integer multiple of 2⇡, the Hmn Reflection Beam DARM Splitter Motion mode will be co-resonant with the fundamental H mode. With 2 kW circulating Port 00 1.61 rad (92 ) of phase separation between each mode order, Anti-Symmetric 200 mW the lower-order HOM resonances are well-separated from the Photodiode (dark) Port fundamental. The fourth-order mode wraps back near the fundamental, but the 0.17 rad (10) phase separation is >100 times Figure 1: Signal flow diagram of the interferometer static fields, represent- the cavity linewidth. ing the linear system of equations for the interferometer response. The fields The power recycling cavity provides[Akutsu not only the et resonant al., 2008] Akutsu Interferometer are sourced by Elaser. Each reflection and transmission coecient is given by r or t, subscripted by its respective optic. F is the optical carrier frequency power enhancement, but also filters noise sidebands from the and the arm lengths are Lx and Ly. The round nodes represent internal states laser. Because of the higher-order-mode separation, di↵erent of the physical system whereas the square nodes are output fields observed transverse modes have di↵erent filtered noise spectra, which • 75 cm Dual synchronous recycling in photodetectors, such as the anti-symmetric port field EAS. To fully model the contrast-defect, additional copies of this diagram must be added to rep- beat with residual light at the Michelson anti-symmetric port resent additional transverse modes, with small transfer coecients arising by readout. The modes then show as noise peaks at the HOM res- interferometers (10 cm separation) defects in the end mirror and beamsplitter shapes. The signal sidebands from onance frequencies in the readout spectra, confirming estimates the end-mirrors are signified with red and blue arrows, which source (unshown) copies of the graph at the modulation frequencies. At each port the optical field of the arm length and Gouy separation. These noise measure- at carrier and sideband frequencies beat together to model the full frequency- ments are described in 6.4.1. • 100 MHz dependent response. § 2.2. Instrument Response With power recycling, the response of the interferometer to 2.1. Power-Recycling Cavity • Finesse 100 time-varying path length displacements is complicated by the The Michelson interferometer, formed by the beamsplitter storage time of the recycling cavity, which imposes a band- and two end mirrors, forms an e↵ective mirror where the losses width limit of approximately 350 Hz on the cavity response. are determined by the fraction of light escaping to the AS port At frequencies 350 Hz, the response at the anti-symmetric ⌧ and the remainder is reflected. The addition of a power recy- port to arm length displacements reflects both a change in the 16 cling mirror (PRM) forms a cavity with this e↵ective mirror. Michelson fringe o↵set and in the cavity storage power. How- The power-recycled interferometers are designed to be nearly ever, at frequencies 350 Hz, arm length displacements occur confocal resonators, folded by the 45 incidence beamsplitter on a shorter time scale than the cavity can respond. In this so that each arm forms a flat-curved cavity. They have an arm limit, the power-recycled interferometer responds equivalently length of L = 39.2 m and an end mirror radius of curvature of to a single-pass Michelson interferometer of the same optical R = 75.1 m, which is matched between the two arms to within power. 10 cm. The resulting waist, with radius Fig. 2 shows the numerically-calculated transfer function of the power-recycled interferometers at several fringe o↵sets. The Holometer operates at an o↵set of approximately 1 nm, and w0 = 2L(2R 2L) 3.57 mm (4) r2⇡ ⇡ all science and calibration signals are measured at frequencies p 1 kHz. At this o↵set, the deviation from a single-pass Michel- lays at the position of the flat PRM. The end mirrors are each son response is 2% above 1 kHz. The third plot, indicating located nearly one Rayleigh range away,  the optical sensitivity, shows this di↵erence at the calibration 2 line frequency from the asymptotic response, which is small ⇡w0 ZR = 37.6 m (5) for the 1nm o↵set. Thus, for calibration purposes (see 7.2) ⌘ § the instrument can be modeled as an equivalent high-power, where the beam half-width has grown to w 5 mm. The devi- single-pass interferometer. Neglecting the cavity correction un- 1 ⇡ ation from a pure confocal configuration satisfies the resonator derestimates the instrumental sensitivity. The degradation in stability criterion, R < 2L (for a review of laser resonators, see sensitivity would be relevant for smaller interferometers with [17]). higher bandwidth recycling. For instance in tabletop versions 4 Interferometric GW Detector Spectrum

[Ringwald, Tamarit, Welling]

GWs in SMASH

GWs from quantum uctuations during ination

GWs from inaton fragmentation during preheating

GWs from thermal uctuations after preheating

17 101 ALPSII Optical System Measurment Exponential Fit

Cavities 100 • Both together increase regenerated photon rate by factor of ~3×108

• Optical gain of ~16,000 in each cavity 10-1 Transmitted Power [a.u.]

• τd ~ 5.4 ms (2x world record) • Can we go even further with upgrades? 10-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Time [ms]

Production Cavity (PC) Regeneration Cavity (RC)

Laser

Detector Wall Magnet String 18 101 ALPSII Optical System Measurment Exponential Fit

Cavities Fig.100 3. Decay of the light transmitted from the cavity after switching off the laser frequency t/τ locking system. The decay is fitted with the exponential function a+ be− d ,andgivesfor the decay time τ = 2.70 0.02 ms. • Both together increase regenerated d ± 8 photon rate by factor of ~3×10 using full power we observe amplitude instabilities in the cavity output, and also a smaller ratio Pt = 0.21. By reducing the input power the output becomes stable and we obtain a higher Pin • Optical gain of ~16,000 in each cavity coupling.10-1 This behavior can probably be explained with thermal lensing effects on the mirrors [16, 17]: when using 1.2 W as input, the power circulating in the cavity is Pc 100 kW, and Transmitted Power [a.u.] 2 2 " the average intensity on each of the mirrors is Pc/πwm = 2.7MW/cm .Thisvalueisbelowthe • τd ~ 5.4 ms (2x world record) damage threshold for the mirrors, but it can cause lensing deformation of the reflecting surface. With a lower input, the power on each mirror surface is reduced to 1.85 MW/cm2,andthisis • Can we go even further with upgrades? sufficient to avoid instabilities and obtain a better geometrical coupling. 10-2 Table-1.5 1. Summary-1 of a few-0.5 Fabry Perot0 cavities with0.5 longest1 decay time1.5 ever realized,2 2.5 3 together with the highest finesse for λ = 1064 nm andTime the highest[ms] finesse in absolute. The coherence length is defined as !c = cτd.

Production Cavity (PC) Regeneration Cavity (RC) 3 Extremely long decay time optical cavity Cavity Length (m) τd (ms) Finesse δνc (Hz) λ (nm) !c/10 m VIRGO [18] 3000 0.16 50 1000 1064 48 PVLAS [19] 6.4 0.905 144 000 176 1064 272 1 1 2 2 2 F. Della Valle, E. Milotti, LaserA. Ejlli, U. Gastaldi, G. Messineo, L. LIGO [20] 4000 0.975 220 163 1064 293 2 2 3 3, Piemontese, G. Zavattini, R. Pengo, and G. Ruoso ∗ BMV [10] 2.27 1.28 530 000 125 1064 384 1Dip. di Fisica and INFN - Sez. di Trieste, Via A. Valerio, 2 - I - 34127 Trieste, Italy This work 3.303 2.7 770 000 59 1064 810 2 Dip. di Fisica e Scienze della Terra and INFN - Sez. di Ferrara, Via G. Saragat, 1 - I - 44122 This work 0.017 0.0143 789 000 11 100 1064 5.1 Ferrara, Italy 3INFN - Laboratori Nazionali di Legnaro, Viale dell’Universit`a, 2 - I - 35020 Legnaro, Italy J. Millo et al.[21] 0.1 800 000Detector 1064 G. Rempe et al.[12] 0.004 0.008 ≈1900000 20 000 850 2.4 ∗[email protected] Wall Magnet String 19 Abstract: We report on the resonant Fabry Perot cavity of the PVLAS (Po- Table 1 lists some of the most performing cavities ever realized. Our work represents an larization of the Vacuum with LASer) experiment operating at λ = 1064 nm improvement of a factor larger than two with respect to previous results, and it is the biggest with a record decay time of 2.7 ms, a factor more than two larger than any previously reported optical resonator. This corresponds to a coherence length of 8.1 105 m. The cavity length is 3.303 m, and the resulting ﬁnesse · #206039 - $15.00 USD Received 11 Feb 2014; revised 27 Mar 2014; accepted 27 Mar 2014; published 6 May 2014 is 770 000. (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011570 | OPTICS EXPRESS 11575 ©2014OpticalSocietyofAmerica OCIS codes: (120.2230) Fabry-Perot; (120.3180) Interferometry.

References and links 1. F.DellaValle,U.Gastaldi,G.Messineo,E.Milotti,R.Pengo,L.Piemontese,G.Ruoso,andG.Zavattini, “Measurements of vacuum magnetic birefringence using permanent dipole magnets: the PVLAS experiment,” New J. Phys. 15,053026(2013). 2. R. Battesti and C. Rizzo, “Magnetic and electric properties of a quantum vacuum,” Rep. Prog. Phys. 76,016401 (2013). 3. R. Bähre, B. Döbrich, J. Dreyling-Eschweiler, S. Ghazaryan, R. Hodajerdi, D. Horns, F. Januschek, E.A. Knabbe, A. Lindner, D. Notz, A. Ringwald, J. E. von Seggern, R. Stromhagen, D. Trines, and B. Willke, “Any light particle search II - Technical Design Report,” Journal of Instrum. 8,T09001(2013). 4. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye “A sub-40- mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6,687–692(2012). 5. H. S. Margolis, “Optical frequency standards and clocks,” Contemp. Phys. 51,37–58(2010). 6. T. Isogai, J. Miller, P. Kwee, L. Barsotti, and M. Evans, “Loss in long-storage-time optical cavities,” Opt. Express 21,30114–30125(2013). 7. M. Evans, L. Barsotti, P. Kwee, J. Harms, and H. Miao, “Realistic filter cavities for advanced gravitational wave detectors,” Phys. Rev. D 88,022002(2013). 8. G. M. Harry, “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quant. Grav. 27, 084006 (2010). 9. The Virgo Collaboration, “Advanced Virgo Baseline Design,” Virgo Technical Report VIR-0027A-09 (2009). 10. R. Battesti, P. Berceau, M. Fouché, G. L. Rikken, and C. Rizzo, “Quantum vacuum magneto optics,” Comptes Rendus Physique 14,27–38(2013). 11. H. Kogelnik and T. Li, “Laser beam and resonators,” Appl. Opt. 5,1550–1567(1966). 12. G. Rempe, R. J. Thompson, H.J. Kimble, and R. Lalezari, “Measurement of ultralow losses in optical interferometer,” Opt. Lett. 17,363–365(1992). 13. M. Bregant, G. Cantatore, S. Carusotto, R. Cimino, F. Della Valle, G. Di Domenico, U. Gastaldi, M. Karuza, V. Lozza, E. Milotti, E. Polacco, G. Raiteri, G. Ruoso, E. Zavattini, and G. Zavattini, “Limits on low energy photon- photon scattering from an experiment on magnetic vacuum birefringence.” Phys. Rev. D 78,032006(2008). 14. A. C. Nilsson, E. K. Gustafson, and R. L. Byer, “Eigenpolarization theory of monolithic nonplanar ring oscilla- tors,” IEEE J. Quantum Electron. 25,767–790(1989) 15. G. Cantatore, F. Della Valle, E. Milotti, P. Pace, E. Zavattini, E. Polacco, F. Perrone, C. Rizzo, G. Zavattini, and G. Ruoso, “Frequency locking of a Nd:YAG laser using the laser itself as the optical phase modulator,” Rev. Sci. Instrum. 66,2785–2787(1995) 16. J. Degallaix, C. Zhao, L. Ju, D. Blair, “Simulation of bulk absorption thermal lensing in transmissive optics of gravitational waves detectors,” Appl. Phys. B77,409-414(2003)

#206039 - $15.00 USD Received 11 Feb 2014; revised 27 Mar 2014; accepted 27 Mar 2014; published 6 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011570 | OPTICS EXPRESS 11570 101 ALPSII Optical System Measurment Exponential Fit

Cavities 100 • Both together increase regenerated photon rate by factor of ~3×108

• Optical gain of ~16,000 in each cavity 10-1 Transmitted Power [a.u.]

• τd ~ 5.4 ms (2x world record) • Can we go even further with upgrades? 10-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 • τd > 15 ms??? Time [ms]

Production Cavity (PC) Regeneration Cavity (RC)

Laser

Detector Wall Magnet String 20 Conclusions

• New detectors coming online • Technologies exist to probe a GWs in SMASH range of frequencies

GWs from quantum uctuations during ination

GWs from inaton fragmentation during preheating

GWs from thermal uctuations after preheating

21 Limits Due to Shot Noise

J. Mizuno (1995)

22 ALPS II Optical Cavities

Individual cavities Technology Demonstrations • Length: 122 m - 245 m • Circulating power ~ 50 kW • Input power: 70 W amplified NPRO (1064 nm) • Power build up factor ~ 33,000 • Circulating power: 150 kW - 2 MW • Laser frequency and cavity length stabilization • Finesse 40,000 - 120,000 • Picometer relative length control • FSR: 600 kHz - 1.2 MHz • Optical path length stability of COB • Linewidth: 30 Hz down to 5 Hz

Production Cavity (PC) Regeneration Cavity (RC)

Laser

Detector Wall Magnet String

23 Principles of GW detection

Living Rev Relativ (2016) 19:3 Page 47 of 217 3 Detector Response • Geometry, arm length • Arm cavities • Power/signal recycling Spurious Test Mass Motion • Coupling to environment • Noise intrinsic to test mass • Noise induced by interferometer Sensing Noise • Photon counting statistics • Pathlength noise • Measurement noise Fig. 32 A Michelson interferometer shown with three types of light field: the ‘carrier’, representing24 the undistorted laser input field, ‘laser phase noise sidebands’, which enter the interferometer with the carrier, and ‘signal sidebands’, which are phase modulation sidebands caused by differential arm length motion. All three fields leave the interferometer through both output ports (here only the detector in the South port is shown). The graph shows the amplitude of the three light fields in the South port as a function of the Michelson tuning (differential arm length change). At 0◦ the Michelson is on a bright fringe and at 90◦ on a dark fringe

and maintain an operating point). For an interferometer in a steady state it is possible to describe and analyse the behaviour using a steady state model,describingthelight field coupling in the frequency domain and making use of the previously introduced concept of sidebands, see Sect. 3.1. Consider a Michelson interferometer which is to be used to measure a differential arm length change. As an example for a signal to noise comparison we consider the phase noise of the injected laser light. For this example the noise can be represented by a sinusoidal modulation with a small amplitude at a single frequency, say 100 Hz. Therefore we can describe the phase noise of the laser by a pair of sidebands superim- posed on the main carrier light field entering the Michelson interferometer. Equally the change of an interferometer arm represents a phase modulation of the light reflected back from the end mirrors and the generated optical signal can be represented by a pair of phase modulation sidebands, see Sect. 5.5. In order to get an estimation of the signal to noise ratio we can trace the individual sidebands through the interferometer and compute their amplitude in the output port. Figure 32 shows the setup of a basic Michelson interferometer, indicating the insertion of the noise and signal sidebands. It also provides a plot of the sideband amplitude in the South output port as a function of the differential arm length of the Michelson

123 Principles of GW detection

Power Recycling 3 Page 80 of 217 LivingRevRelativ(2016)19:3 • Amplifies carrier in Michelson

Living Rev Relativ (2016) 19:3 Page 81 of 217 3

Fig. 49 Optical layout of a Michelson interferometer with arm power recycling

of the shot noise and quantum noise discussed in Sect. 6.Insteadwewillcomputeonly the transfer functions of the signal to the photo detector using the sideband picture. We will ignore radiation pressure noise and shot-noise contributions from any light field Fig. 50 This graph shows the signal sideband amplitude for a differential arm lengthbut change, the local as detected oscillator. Thus the amplitude of the signal sidebands in the detection in the anti-symmetric output port, as a function of the frequency of the signal. TheLivingportsolid Rev give Relativ red trace a (2016) goodat 19:3 an figure of merit for the shot-noise limited Page 47 sensitivity of 217 3 of the detector. amplitude of 1 refers to the case without power recycling. The other two traces show the increased amplitude 25 for different reflectivity’s of the power-recoiling mirror. Compare this plot also with Figs. 52 and 54 7.1 Michelson interferometers with power recycling recycling mirror equal to the round trip losses of the power-recyclingThe cavity Michelson and the interferometer, when held on the dark fringe and ignoring internal gain becomes losses, reflects all the incoming light back into the laser port; seen from the laser it 1 GPR F acts like a highly(7.2) reflective mirror. It was soon realised we can utilise this fact to = TPRM ≈ π increase the light power inside the interferometer: an additional mirror inter the input The power in the signal sidebands is proportional to the carrierport, power the and so-called thus power-recycling mirror (PRM), will generate an optical cavity scales with the power-recycling gain as well. The amplitudes plottedwith in Fig.the Michelson50 thus interferometer acting as a second ‘mirror’. This scheme which show values of √4/0.1 6.32 and √4/0.01 20. is now called power recycling was first proposed in 1983 independently by Billing Power-recycling has≈ further advantages: the= cavity effect can beet used al. (1983 to reduce)andDrever et al. (1983). The newly formed cavity is often called power- beam jitter and to filter laser frequency noise. The disadvantage is thatrecycling another cavity mirror.Theopticallayoutofapower-recycledMichelsoninterferometeris position needs to be carefully maintained by a feedback control system.shown In in addition, Fig. 49.Figure50 shows the amplitude of signal sidebands for different levels of power recycling, as a function of the frequency of the signal. We will compare this the increase in circulating power also increases the laser power within the substrate of to similar plots for other techniques described below. the beam splitter which can cause thermal distortions leading to higher-optical losses. As we have discussed in Sect. 5.1,thepowercirculatinginsideacavitycanbemuch In practise this often limits the achievable power-recycling gain. Fig.higher 32 A Michelson than the interferometer injected shown light with power. three types The of light power field: the enhancement ‘carrier’, representing is given the by the finesse of undistorted laser input field, ‘laser phase noise sidebands’, which enter the interferometer with the carrier, andthe ‘signal cavity sidebands’, which which is are given phase modulation by the sidebandsoptical caused losses by differentialin the interferometer arm length motion. and the reflectivity All threeof the fields power-recycling leave the interferometer through mirror. both When output ports the (here losses only the inside detector in the the South Michelson port interferometer 7.2 Michelson interferometers with arm cavities is shown). The graph shows the amplitude of the three light fields in the South port as a function of the Michelsonare negligible tuning (differential the arm cavity length change). formed At 0◦ bythe Michelson the Michelson is on a bright fringe andand the at 90 power-recycling◦ on mirror a dark fringe Another way to employ cavities to enhance the light power circulatingis over-coupled in the interfer- and the power enhancement in the interferometer arms, also called computes as ometer arms is to place optical cavities into these arms, as so-calledpower-recyclingarm cavities gain, as shown in Fig. 51.Thisopticalconfigurationsometimesreferredtoasand maintain anFabry– operating point). For an interferometer in a steady state it is possible to describe and analyse the behaviour using a steady state4 model,describingthelight2 Perot–Michelson interferometer.Similartopower-recyclingthefinesseofthecavity GPR F (7.1) field coupling in the frequency domain and making= useTPRM of the≈ previouslyπ introduced determines the enhancement of the light power. concept of sidebands, see Sect. 3.1. The arm cavities have another effect on the detector sensitivity: theyConsiderwith affect athe Michelson not finesse only interferometer of the power-recycling which is to be used cavity. to measure a differential the power of the circulating carrier field, but also that sidebands generatedarm lengthWhenF by change. a the length As optical an example losses for can a signal not to be noise ignored comparison the maximum we consider the power-recycling gain phase noise of the injected laser light. For this example the noise can be represented change. This results in a further increase of the sensitivity for signalsby acan with sinusoidal be a reachedfrequency modulation by impedance with a small amplitude matching, at a i.e., single setting frequency, the say transmission 100 Hz. of the power- within the linewidth of the arm cavities but to a decrease in sensitivityTherefore regarding we can signals describe the phase noise of the laser by a pair of sidebands superim- with frequencies that fall outside the linewidth of the cavities.posed This on can the main be shown carrier light field entering the Michelson interferometer. Equally the again very clearly with the sideband amplitudes detected at the interferometerchange123 of an interferometer output arm represents a phase modulation of the light reflected back from the end mirrors and the generated optical signal can be represented by a as shown in Fig. 52.Wecancomparethisresultstothepower-recyclingcase(Fig.pair of phase modulation50): sidebands, see Sect. 5.5. when the reflectivity of the PRM and ITMs is set to R 0.99, theIn expected order to get gain an estimation for of the signal to noise ratio we can trace the individual = sidebands through the interferometer and compute their amplitude in the output port. Figure 32 shows the setup of a basic Michelson interferometer, indicating the insertion of the noise and signal sidebands. It also provides a plot of the sideband amplitude in the South123 output port as a function of the differential arm length of the Michelson

123 Living Rev Relativ (2016) 19:3 Page 83 of 217 3

Fig. 52 This graph shows the signal sideband amplitude for a Fabry–Perot–Michelson interferometer. The signal is a differential arm length change detected in the anti-symmetric output port, as a function of the frequency of the signal. The solid red trace at an amplitude of 1 refers to the case without arm cavities. PrinciplesThe other two tracesof GW show thedetection increased amplitude for different reﬂectivities of the cavities’ input mirrors. Compare this also with Figs. 50 and 54

Signal Recycling Fig. 53 Optical layout of a •MichelsonAmplifies interferometer side-bands in with signalMichelson recycling

3 Page 84 of 217 LivingRevRelativ(2016)19:3

Living Rev Relativ (2016) 19:3 Page 47 of 217 3 Fig. 54 This graph shows the signal sideband amplitude for a Michelson interferometer with different the signal sidebands. The optical layout of a signal-recycled Michelson interferometer26 signal-recycling configurations. For all 4 traces the reflectivity of the signal-recycling mirror was set to R 0.9, the interferometeris shown arm in length Fig. is 4 km.53 The. red trace shows the tuned case in which the signal- recycling= cavity is resonant for the carrier light and thus maximises signals around DC. The other red traces show different detunings,It is microscopic somewhat offsets to counterintuitive the longitudinal positions of that the signal-recalling placing mirror. a highly-reflective mirror in front of The maximum amplitude and bandwidth of the trace is the same in all four cases, just the frequency of the peak sensitivitythe is shifted photo by the detector detuning. Compare would this the increase plots for arm the cavities power in Fig. 52 detectedand power on the same photo detector. This recycling, Fig. 50is because the signal sidebands are created within the interferometer, and thus within cavities havethe a very signal high finesse recycling and the signal-recycling cavity, by a mirror parametric is tuned to oreffect, near the in which light is transferred from a anti-resonant operating point, thus effectively increasing the bandwidth of the detector for the signalmuch sidebands. larger An analysis reservoir, of the different the carrier techniques field. can Gerhard be found in the Heinzel provides, in Appendix D of thesis of Mizunohis thesis(1995). ( ItHeinzel is interesting 1999 to note), a that clear for and all variants compact of the mathematical signal overview of a two-mirror recycling thecavity total integrated including gain remains this effect. constant. For example, the areas under curves for the different detunings shown in Fig. 54 are constant.10 This means that signal-recyclingWhen is used to bothshape the recycling response function techniques of the detector are used with respect together, to power recycling for enhancing the signal-to-shot-noisethe carrier ratio. power and signal recycling for increasing the signal interaction time, the The main interferometer of an Advanced LIGO detector is based on a Michelson interferometercombination with arm cavities of plus the power two and methods signal recycling. isFig. called 32 ThisA Michelson configurationdual interferometer recycling shown with three.Itwasactuallytheconcept types of light field: the ‘carrier’, representing the undistorted laser input field, ‘laser phase noise sidebands’, which enter the interferometer with the carrier, is most commonlyof dual called recyclingdual-recycled which Fabry–Perot–MichelsonMeers (1988and ‘signal interferometer)proposed,andthiswasdemonstratedfirstasa sidebands’, which areeven phase modulation sidebands caused by differential arm length motion. though the signal recycling mirror is here used in the resonantAll three sideband fields leave the extraction interferometer through both output ports (here only the detector in the South port table-top experiment by the Glasgowis shown). group The graph shows in the1991 amplitude ( ofStrain the three light and fields in Meers the South port as 1991 a function). of the mode, see Fig. 130 for a schematic of this layout. Michelson tuning (differential arm length change). At 0◦ the Michelson is on a bright fringe and at 90◦ on The combination of arm cavitiesa dark and fringe a signal-recycling mirror is sometimes also 7.4 Sagnac interferometercalled resonant sideband extraction (Mizuno et al. 1993). The difference between and maintain an operating point). For an interferometer in a steady state it is possible signal-recycling and resonant sidebandto describe and extraction analyse the behaviour is that using a steady in the state model latter,describingthelight case the arm Another interferometer type which has a similar-looking opticalfield layout coupling to in the the Michel- frequency domain and making use of the previously introduced son interferometer is the Sagnac interferometer, see Fig. 55.concept Originally of sidebands, proposed see Sect. by 3.1. Sagnac (1913a, b)itbecameofinterestedtothegravitational-wavecommunityasaConsider a Michelson interferometer which is to be used to measure a differential arm length change. As an example for a signal to noise comparison we consider the possible alternative to the Michelson interferometer: in 1995 successfulphase noise of experimental the injected laser light. For this example the noise can be represented123 tests of a zero-area Sagnac demonstrated a different mode ofby operation, a sinusoidal modulation in which with it a small amplitude at a single frequency, say 100 Hz. becomes insensitive to rotation but sensitive to mirror motion (SunTherefore et al. we 1996 can describe). Further the phase noise of the laser by a pair of sidebands superim- posed on the main carrier light field entering the Michelson interferometer. Equally the investigations into the performance and technical limitations ofchange a Sagnac of an interferometerinterferomet- arm represents a phase modulation of the light reflected ric gravitational-wave detector have been undertaken (Mizunoback et al. from 1997 the; endPetrovichev mirrors and the generated optical signal can be represented by a pair of phase modulation sidebands, see Sect. 5.5. In order to get an estimation of the signal to noise ratio we can trace the individual 10 The tuned case is slightly special, because the integrated area is half comparedsidebands to through the others, the interferometer because and compute their amplitude in the output port. the plot shows only the positive half of the total linewidth seen by signal sidebands.Figure 32 shows the setup of a basic Michelson interferometer, indicating the insertion of the noise and signal sidebands. It also provides a plot of the sideband amplitude in the South output port as a function of the differential arm length of the Michelson 123 123 3 Page 82 of 217 LivingRevRelativ(2016)19:3 Principles of GW detection

Living Rev Relativ (2016) 19:3 Page 47 of 217 3 Michelson with Arm Cavities • Sideband amplitude amplified by cavity transfer function • Optimal response when signal is within cavity bandwidth

• Can take advantage of higher Living Rev Relativ (2016) 19:3 Page 83 of 217 3

frequency resonances as well Fig. 51 Optical layout of a Michelson interferometer with arm cavities (not shown in plot) the carrier field inside the cavities must be the same and equal to 400, assuming an • Must consider SR and PR over-coupled case. At low frequencies the signal sidebands will experience the same enhancement, namely by a factor of 400 in power. Thus the total enhancement for the signal sidebands in the Michelson with arm cavities is 16 000, which gives the amplitude of 400 shown for sideband amplitude in Fig. 52.Thereforethearmcavities also change the detector response function in a way that limits the possible sensitivity increase. The limited bandwidth of the arm cavities is a disadvantage when compared to the power-recycling technique; however, the arm cavitiesFig. 32 haveA Michelson the significant interferometer advantage shown with three types of light field: the ‘carrier’, representing the of not increasing the light power in the beam splitterundistorted substrate. laser input In field, practise ‘laser the phase two noise sidebands’, which enter the interferometer with the carrier, and ‘signal sidebands’, which are phase modulation sidebands caused by differential arm length motion. techniques are commonly used together, with theAll finesse three fields of the leave arm the cavities interferometer and thethrough both output ports (here only the detector in the South port Fig. 52 reflectivityThis graph shows of the the power-recycling signal sideband amplitude mirror the for result a Fabry–Perot–Michelsonis shown). of a trade-off The graph analysisshows interferometer. the amplitudebetween The of the three27 light fields in the South port as a function of the signal isthe a differential bandwidth arm reduction length change of the detected arm cavities in the and anti-symmetric theMichelson light power tuning output (differential increase port, as in a arm function the length beam of change). the At 0◦ the Michelson is on a bright fringe and at 90◦ on frequencysplitter of the substrate. signal. The Suchsolid an red optical trace at layout an amplitude is also calledofa dark 1 refers fringepower-recycled to the case without Fabry–Perot– arm cavities. The otherMichelson two traces interferometer show the increased. amplitude for different reflectivities of the cavities’ input mirrors. Compare this also with Figs. 50 and 54 and maintain an operating point). For an interferometer in a steady state it is possible Fig. 53 7.3Optical Signal layout recycling, of a dual recycling and resonantto describe sideband and extraction analyse the behaviour using a steady state model,describingthelight Michelson interferometer with field coupling in the frequency domain and making use of the previously introduced signal recyclingSoon after the development of power recycling in whichconcept an of additional sidebands, mirror see is Sect. used3.1 to. ‘recycle’ the laser light leaving the Michelson interferometerConsider athrough Michelson the symmetric interferometer which is to be used to measure a differential port, Brian Meers recognised that it would be of interestarm length to employ change. a similar As an technique example for a signal to noise comparison we consider the in the anti-symmetric port. In the ideal Michelson interferometerphase noise of on the the injected dark fringe, laser the light. For this example the noise can be represented carrier light and the signal sidebands become separatedby a sinusoidal at the central modulation beam splitter with and a small amplitude at a single frequency, say 100 Hz. leave the interferometer though different ports. MeersTherefore(1988)suggestedtheadditionof we can describe the phase noise of the laser by a pair of sidebands superim- a signal-recycling mirror at the anti-symmetric port,posed to form on the a signal-recyclingmain carrier light cavity field entering the Michelson interferometer. Equally the with the Michelson interferometer. In a similar mannerchange to of the an power-recycling interferometer arm cavity represents a phase modulation of the light reflected the signal-recycling cavity could resonantly enhanceback the from light the circulating end mirrors within, and i.e., the generated optical signal can be represented by a pair of phase modulation sidebands, see Sect. 5.5. 123 In order to get an estimation of the signal to noise ratio we can trace the individual sidebands through the interferometer and compute their amplitude in the output port. the signal sidebands. The optical layout of a signal-recycled Michelson interferometer Figure 32 shows the setup of a basic Michelson interferometer, indicating the insertion is shown in Fig. 53. of the noise and signal sidebands. It also provides a plot of the sideband amplitude It is somewhat counterintuitive that placing a highly-reflectivein the South output mirror port as in a function front of of the differential arm length of the Michelson the photo detector would increase the power detected on the same photo detector. This is because the signal sidebands are created within the interferometer, and thus within the signal recycling cavity, by a parametric effect, in which light is transferred from a 123 much larger reservoir, the carrier field. Gerhard Heinzel provides, in Appendix D of his thesis (Heinzel 1999), a clear and compact mathematical overview of a two-mirror cavity including this effect. When both recycling techniques are used together, power recycling for enhancing the carrier power and signal recycling for increasing the signal interaction time, the combination of the two methods is called dual recycling.Itwasactuallytheconcept of dual recycling which Meers (1988)proposed,andthiswasdemonstratedfirstasa table-top experiment by the Glasgow group in 1991 (Strain and Meers 1991). The combination of arm cavities and a signal-recycling mirror is sometimes also called resonant sideband extraction (Mizuno et al. 1993). The difference between signal-recycling and resonant sideband extraction is that in the latter case the arm

123 Akutsu Proposal

Synchronous recycling IFO • Determined to be optimal configuration

28 Akutsu Interferometer

Initial Results • 75 cm arm length optimized sens at 100 MHz • Finesse 100, 0.5 W injected power

29 Fermilab Holometer

39 m Michelson • Dual interferometers targeting 1 MHz - 13 MHz • Seperated by 0.635 m • Power recycling: 1 W injected -> 2 kW circulating

Transmission Field Cavity Power This conﬁguration additionally avoids co-resonances of End Mirror (via Transmission Port) [North] higher-order cavity modes (HOM) with the fundamental mode. The Gouy phase shift for one-way propagation down the inter- - DARM ferometer arm is Motion Ly~39m L Gouy arctan = 0.806 rad (46.2) (6) ⌘ ZR Lx~39m ! Power Recycling End Mirror Any Hermite-Gauss mode, H , attains a round-trip phase ex- =1064nm Mirror [East] mn Input Field cess of

2 kW Laser - mn = 2(1 + m + n) Gouy (7) 1 Watt compared to the propagation of unfocused spherical wave- REFL PD Field - Lp= 20cm fronts. When 2(m+n)Gouy is an integer multiple of 2⇡, the Hmn Reflection Beam DARM Splitter Motion mode will be co-resonant with the fundamental H mode. With Port 00 1.61 rad (92 ) of phase separation between each mode order, Anti-Symmetric 200 mW the lower-order HOM resonances are well-separated from the Photodiode (dark) Port fundamental. The fourth-order mode wraps back near the fundamental, but the 0.17 rad (10) phase separation is >100 times Figure 1: Signal flow diagram of the interferometer static fields, represent- the cavity linewidth. ing the linear system of equations for the interferometer response. The fields 30 are sourced by Elaser. Each reflection and transmission coecient is given by The power recycling cavity provides not only the resonant r or t, subscripted by its respective optic. F is the optical carrier frequency power enhancement, but also filters noise sidebands from the and the arm lengths are Lx and Ly. The round nodes represent internal states laser. Because of the higher-order-mode separation, di↵erent of the physical system whereas the square nodes are output fields observed transverse modes have di↵erent filtered noise spectra, which in photodetectors, such as the anti-symmetric port field EAS. To fully model the contrast-defect, additional copies of this diagram must be added to rep- beat with residual light at the Michelson anti-symmetric port resent additional transverse modes, with small transfer coecients arising by readout. The modes then show as noise peaks at the HOM res- defects in the end mirror and beamsplitter shapes. The signal sidebands from onance frequencies in the readout spectra, confirming estimates the end-mirrors are signified with red and blue arrows, which source (unshown) copies of the graph at the modulation frequencies. At each port the optical field of the arm length and Gouy separation. These noise measure- at carrier and sideband frequencies beat together to model the full frequency- ments are described in 6.4.1. dependent response. § 2.2. Instrument Response With power recycling, the response of the interferometer to 2.1. Power-Recycling Cavity time-varying path length displacements is complicated by the The Michelson interferometer, formed by the beamsplitter storage time of the recycling cavity, which imposes a band- and two end mirrors, forms an e↵ective mirror where the losses width limit of approximately 350 Hz on the cavity response. are determined by the fraction of light escaping to the AS port At frequencies 350 Hz, the response at the anti-symmetric ⌧ and the remainder is reflected. The addition of a power recy- port to arm length displacements reflects both a change in the cling mirror (PRM) forms a cavity with this e↵ective mirror. Michelson fringe o↵set and in the cavity storage power. How- The power-recycled interferometers are designed to be nearly ever, at frequencies 350 Hz, arm length displacements occur confocal resonators, folded by the 45 incidence beamsplitter on a shorter time scale than the cavity can respond. In this so that each arm forms a flat-curved cavity. They have an arm limit, the power-recycled interferometer responds equivalently length of L = 39.2 m and an end mirror radius of curvature of to a single-pass Michelson interferometer of the same optical R = 75.1 m, which is matched between the two arms to within power. 10 cm. The resulting waist, with radius Fig. 2 shows the numerically-calculated transfer function of the power-recycled interferometers at several fringe o↵sets. The Holometer operates at an o↵set of approximately 1 nm, and w0 = 2L(2R 2L) 3.57 mm (4) r2⇡ ⇡ all science and calibration signals are measured at frequencies p 1 kHz. At this o↵set, the deviation from a single-pass Michel- lays at the position of the flat PRM. The end mirrors are each son response is 2% above 1 kHz. The third plot, indicating located nearly one Rayleigh range away,  the optical sensitivity, shows this di↵erence at the calibration 2 line frequency from the asymptotic response, which is small ⇡w0 ZR = 37.6 m (5) for the 1nm o↵set. Thus, for calibration purposes (see 7.2) ⌘ § the instrument can be modeled as an equivalent high-power, where the beam half-width has grown to w 5 mm. The devi- single-pass interferometer. Neglecting the cavity correction un- 1 ⇡ ation from a pure confocal configuration satisfies the resonator derestimates the instrumental sensitivity. The degradation in stability criterion, R < 2L (for a review of laser resonators, see sensitivity would be relevant for smaller interferometers with [17]). higher bandwidth recycling. For instance in tabletop versions 4