Feasibility Study of Beam-Expanding Telescopes in the Interferometer Arms for the Einstein Telescope

Feasibility study of beam-expanding telescopes in the interferometer arms for the Einstein Telescope

Samuel Rowlinson1, Artemiy Dmitriev1, Aaron W. Jones2, Teng Zhang1, and Andreas Freise1,3,4 1School of Physics and Astronomy, and Institute of Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom 2OzGrav, University of Western Australia, Crawley, Western Australia, Australia 3Department of Physics and Astronomy, VU Amsterdam, De Boelelaan 1081, 1081, HV, Amsterdam, The Netherlands and 4Nikhef, Science Park 105, 1098, XG Amsterdam, The Netherlands (Dated: November 6, 2020) The optical design of the Einstein Telescope (ET) is based on a dual-recycled Michelson interferometer with Fabry-Perot cavities in the arms. ET will be constructed in a new infrastructure, allowing us to consider diﬀerent technical implementations beyond the constraints of the current facilities. In this paper we investigate the feasibility of using beam-expander telescopes in the interferometer arms. We provide an example implementation that matches the optical layout as presented in the ET design update 2020. We further show that the beam-expander telescopes can be tuned to compensate for mode mismatches between the arm cavities and the rest of the interferometer.

I. INTRODUCTION between the beamsplitter and the recycling mirrors [6], and in Advanced Virgo similar telescopes are part of the The Einstein Telescope (ET) is a proposed third- input-output optics outside the main interferometer [7]. generation gravitational wave detector [1]. Once con- However, in the ET the beam sizes on the main mirrors structed, ET will provide an unprecedented level of sensi- are signiﬁcantly larger, requiring a very large substrate tivity enabling: precise tests of General Relativity, stud- for the central beamsplitter, larger than the main op- ies of compact binary coalesces involving both interme- tics, due to the angle of incidence of 60 deg. In this pa- diate black holes and neutron stars, and will be sensi- per we investigate the feasibility of an alternative layout tive enough to test several dark matter candidates [2]. with beam-expander telescopes located between the main ET combines a unique layout and design combining well- beamsplitter (BS) and the arm cavities. Such telescopes proven concepts from current gravitational-wave detectors with new technology. Figure 1 shows a sketch with the basic features of the layout: the ET observatory is composed of three detectors that together form an equi- lateral triangle. Each detector consists of two interferometers, one low-frequency detector (ET-LF) with its sensitivity optimised for low frequencies from 3 Hz to 30 Hz and another high-frequency detector (ET-HF) with its sensitivity optimised for high frequencies from 30 Hz to 10 kHz. Similarly to current generation gravitational wave detectors, i.e. Advanced LIGO [3] and Advanced Virgo [4], each interferometer in ET is a Fabry-Perot Michelson interferometer with power and signal recycling cavities (PRC & SRC) for arm cavity power enhancement and shaping the signal response [5], respectively. This interferometer conﬁguration presented in [1] represents the

arXiv:2011.02983v1 [astro-ph.IM] 5 Nov 2020 initial detector anticipated to be installed, with upgrades and refinements to be implemented over several decades. The details of the design of the initial detector will be prepared, in sync with research and development of the required technology, over the next years. The Fabry-Perot cavities in the arms of the detec- FIG. 1. The diagram on the top left shows a general overview tors are designed to have large beam sizes on the test of the ET observatory layout, with 3 detectors forming a equi- mass mirrors, i.e. the input test mass (ITM) and end lateral triangle of 10 km length. Bottom left is a sketch that test mass (ETM), for reducing the impact of thermal shows that each detector consists of two interferometers, one noise of the optics on the detector sensitivity. Beam- optimised for high frequencies (HF) and one for low frequen- expander telescopes are used to match input light with cies (LF). The core interferometer layout is based on a Michel- smaller beam diameters to the larger beams in the arm son interferometer with Fabry-Perot cavities in the arms and cavities. In Advanced LIGO such telescopes are located recycling. 2 would provide smaller beam sizes in the central interferometer formed by the beamsplitter, power recycling TABLE I. A summary of the key parameters of the ET-LF and ET-HF arm cavities. Note that we have assumed sym- mirror (PRM) and signal recycling mirror (SRM), allow- metric arm cavities for simplicity here. ing for using much smaller optical components. This not only reduces the cost and complexity of these optics and ET-HF ET-LF their suspension systems, but also simplifies the mitiga- Wavelength (λ) 1064 nm 1550 nm tion of secondary reflections and scattered light, and reduces the effect of beam jitter [8]. Figure 2 shows a sketch Cavity length (L) 10 km 10 km of the optical layout in the lower left corner of the ET FSR (∆ν) 15 kHz 15 kHz triangle, including possible locations for the beam expan- ITM/ETM diameter (Md) 62 cm 45 cm sion telescopes. An additional advantage of positioning ITM/ETM curvature (RC ) 5070 m 5580 m the telescopes between the beamsplitter and arm cavity Beam radius on ITM/ETM (w) 12.0 cm 9.0 cm is that Z-shaped telescopes provide flexibility in beam Beam radius at cavity waist (w0) 1.42 cm 2.90 cm steering, for example they provide the possibility to steer Rayleigh range (zR) 591 m 1702 m the ET-HF beam around the suspension system of the Distance to waist from ITM (z ) 5 km 5 km ET-LF ITM, and they decouple the angle of incidence 0 on the main beamsplitter from the angle between the Cavity stability factor (g) 0.95 0.63 ◦ ◦ long interferometer arms [9, 10]. Round-trip gouy phase (ψRT) 333 285 In this paper we analyse different arm telescope designs Mode separation frequency (δf) 1.1 kHz 3.1 kHz with the target of achieving a 6 mm large beam waist at the main beamsplitter. The constraint on the size of this waist (and thus the spot size on the beamsplitter) is rationalised in §II, where other constraints stemming with R as the mirror radius [14]. The resulting beam sizes from the design of the central interferometer and the arm and corresponding mirror radii of curvature for ET-HF cavities [1] are discussed as well. Regions of interest in and ET-LF that satisfy the requirement of clipping loss of our parameter searches, covered in §III for ET-LF and 1 ppm are listed in Table I. Note that the diameters of the in §IV for ET-HF, are defined as regimes where the tele- mirrors shown in this table were taken directly from [1, scope configuration gives this 6 mm waist positioned at 15] but do not necessarily represent final design values. the beamsplitter, whilst the SRC is stable. In §V, we The arm cavity parameters shown in Table I are used to present an analysis on the sensitivity of the ET-LF tele- define the arm cavity model serving as the starting point scope solution to each free parameter. Following on from of the beam propagation analyses in §III. this, a preliminary study quantifying the necessary ac- tive changes (for mode matching) to the telescope mirror curvatures in the presence of thermal lensing in the A. Telescope parameter constraints ITM, is given. The results in this paper were obtained using beam parameter propagation [11] via the symbolic A simplified schematic of one beam-expander telescope ABCD matrix capabilities of our open-source simulation is shown in Fig. 3. In order to reduce the impact of software FINESSE 3 [12]. The methodology involved grid- thermal noise on the sensitivity, ET-LF will make use of based searches of the parameter space to find regions sat- cryogenics to cool down the test masses to ∼ 10 to 20 K. isfying the aforementioned requirements. In this context, Cryoshields of around 40 m length will be used along the grid-based searches are defined as analyses of high dimen- beam before and after the cryogenic mirrors [15], placing sional scans over any given free parameters. An indepen- a lower limit on the distance between the telescope mir- dent verification of the results was performed using a new rors and the ITM. Thus, commensurately setting a lower analytical framework for optimisation of beam expansion limit for the SRC length for ET-LF. In §III B and §IIIC telescopes in coupled cavities [13]. this distance is kept fixed at the current design value of 52.5 m. II. ARM CAVITY EIGENMODE AND The picture is different for ET-HF, where a signal recy- TELESCOPE CONSTRAINTS cling cavity length of 100 m or less is preferred in order to improve the quantum-noise limited higher-frequency sensitivity [16]. ET-HF does not use cryogenics, so that The maximal beam sizes on test masses is set via the the lower limit on the SRC length is given only by the tolerance on power lost through clipping at the mirror minimum distance allowed between the vacuum tanks. edge. The relation between the maximum beam size As stated in §I, we targeted a waist size of 6 mm at wmax to achieve a minimum clipping loss lclip is given by: the beamsplitter. The final design for the size of the beam on the central beamsplitter will be based on a v u 2 trade-off study including the following considerations. In u wmax = t R, (1) Advanced LIGO the main beamsplitter sits between the log 1 lclip telescopes and the arm cavities [6] — resulting in a spot 3

Top view of left lower corner of one tunnel HF vacuum system in the triangle, showing ITMs of one detector, LF vacuum system and ETMs of another cryo shields HF PRM ﬁlter cavity vacuum HF ﬁlter cavity fused silica optics HF BS HF SRM silicon optics LF BS 70 m LF PRM

LF telescope HF telescope LF ETM

HF ETM LF ITM HF ITM LF SRM 40 m 40 m 10 km

LF ﬁlter cavities

Side views of vacuum tubes in tunnels at different points, as seen from this central cavern

FIG. 2. This is a sketch of the lower left corner of the triangle, showing an example implementation of the optical layout, the vacuum system for the main optics and the corresponding cavern layout. In particular this shows the possible location of Z-shaped telescope systems for ET-LF and ET-HF detectors. In this example the telescopes have been placed to achieve an angle of incidence of 45 degrees on the ET-HF beamsplitter. size on the central beamsplitter comparable to the beam to other arm size on the ITMs; and thus requiring a large beamsplitter to PRM ZM1 to avoid clipping losses. In contrast, in this work, we pro- pose placing the telescopes between the beamsplitter and BS ITM ETM the arms, to allow use of a smaller beamsplitter, an idea ZM2 ITM briefly discussed in [17]. Assuming a beamsplitter radius SRM L = 10 km of 15 cm and 60 degree intersection angle of two incident lens beams, sub part per million clipping losses are achieved FIG. 3. Schematic of an ET arm telescope with a lens at the with beam sizes smaller than ∼10 mm, setting an upper input test mass (ITM). This type of configuration is used for bound on the acceptable beam size at the beamsplitter. the analyses in this section, where the Z mirrors are flat in §III A whilst the lens has an infinite focal length in §III B. Similarly to GEO600 [18], ET-HF will operate with ∼ kW levels of power on the central beamsplitter [15]. As such, this circulating power will induce thermal lensing in the beamsplitter substrate, as is the case for III. ET-LF ARM TELESCOPE DESIGN GEO600 [19]. This thermal lensing causes an undesir- able excitation of higher-order modes and reduces the interferometric visibility. The strength of this lensing In this section we analyse potential arm telescope con- is related to the beam intensity and thus reduced with figurations focusing on achieving a stable SRC whilst larger beams at the central beamsplitter. For consistency keeping a waist size of w0 ∼ 6 mm near to the beam- between both solutions, a waist size of 6 mm at the cen- splitter. The analyses in this section are performed for tral beamsplitter was targeted for both ET-LF and ET- ET-LF. In this and the following sections we will describe HF. It is important to note, however, that, in the case beam-expander telescopes with curved mirrors with a of ET-HF, this may contribute undesirably to scattering non-normal angle of incidence. With the commonly used into higher-order modes. The strength of this effect will spherical mirrors, such telescopes would suffer from astig- require further studies. matism which would reduce the mode matching in the interferometer. Note that the following computations as- Note that although the targeted beam size on the sume spherical mirrors with small angles of incidence and beamsplitter is 15 to 20 times smaller than that on the negligible astigmatism. The schematic layout shown in test masses of ET-LF and ET-HF, respectively, the ther- Figure 2 however implies relatively large angles of inci- mal noise contribution from the main beamsplitter is still dence; requiring non-spherical mirrors. If the final opti- much smaller than that from arm cavity mirrors, taking cal design will include significant angles of incidence, our into account the arm cavity finesse, at around 900, and results provide a good starting point for designing the re- the fact that the beamsplitter will have fewer coating quired non-spherical surfaces, based on the desired beam layers and a smaller substrate volume. parameters along the optical path. 4

A. Flat Z mirrors the Gouy phase is calculated as, z The simplest beam-expander configuration possible is ψ = arctan . (2) one based on a lens at the ITM whilst keeping the Z mir- zR rors flat (i.e. they are just steering mirrors). Using our Secondly, the main Gouy phase contribution comes from target waist size, we can deduce an ITM lens focal length the distance from SRM to the telescope, because the value for this setup, which is shown in Figure 4. Also dis- Gouy phase changes faster near to the beam waist po- played in that figure are the distances to the beam waist sition, see Eq. 2. For our analyses, a very short distance from the ITM. This figure demonstrates that a distance from the waist to the SRM of 10 m is assumed. Finally, of ∼ 1 km is necessary for maintaining a waist, of the a minimal distance ∼ 100 m between the beam-expander appropriate size, at (or near to) the beamsplitter. Given mirrors helps to reduce the overall SRC length to ∼ 250m the impracticality of this distance (implying a compara- whilst retaining a 6 mm beam waist. ble SRC length), we can reject this type of configuration. The results shown in Figure 5 were obtained after a wide parameter space search. This figure shows that a stable SRC is possible, however, a relatively long SRC B. Curved telescope mirrors, and no lens at ITM is required – on the order of 250 m on average - whilst the RoC of ZM1 is small relative to the ZM2 RoC. We Allowing the mirrors in the Z-configuration to have can trade off the achievable SRC length and the required some curvature gives another type of beam-expander con- curvature of ZM1, i.e. a shorter SRC can be achieved figuration which can be explored for feasibility. Based on by reducing the radius of curvature of this mirror – this the results in the single lens case we expect ZM2 to have ultimately comes down to a design choice, based on other a positive RoC for converging the beam rapidly and ZM1 design parameters outside the scope of this work. to have a negative RoC in order to achieve a small beam size on the beamsplitter over a short distance. Another criterion comes from the stability of the SRC. C. Curved telescope mirrors with a lens at the The setup has significant number of free degrees of free- ITM dom, i.e. RoCs of ZM1 and ZM2 , the telescope distance (the distance between ZM1 and ZM2) and other The previous solutions, shown in Figure 5, lead to a free spaces in the SRC, which determine the round trip relatively long SRC, > 100 m. In this section we will Gouy phase. But the basic behaviour of this system can investigate how adding a lens at the ITM can be used to be understood intuitively when combining a basic under- reduce the length of the SRC and relax the requirements standing of beam propagation and our simulation results. on the radius of curvature of ZM1. Firstly, the accumulated Gouy phase contribution from We produced a set of animated plots of the round- ITM to ZM2 can be ignored, since this distance is much trip Gouy phase to gain an intuition of how the solution smaller than the Rayleigh range of the beam from the arm cavities which is ∼ 1.7 km as shown in Table. I and ZM2 RoC [m] 220 217 213 209 205 202 160 317 2520 f = 656 m 140 292 w0 = 6 mm 2160 → z = 1090 m 120 268 10 → 1800 1440 100 243 1080 80

in SRC [deg] 219 5

720 SRC ψRT Length of SRC [m] Waist size [mm] 60 195 Round-trip Gouy phase 360 SRC length 40 170 0 0 40 35 30 25 20 15 Distance to waist from ITM [m] − − − − − − 200 400 600 800 1000 ZM1 RoC [m] ITM lens focal length [m] FIG. 5. ZM1 and ZM2 RoC combinations yielding a 6 mm FIG. 4. The waist-size (blue) and distance to waist from ITM waist size where the position of this waist is less than 150 (red) of a beam matched to the arm cavity for different ITM m from ZM1. The blue trace shows the round-trip Gouy lens focal lengths. Highlighted on this plot is the focal length phase, ψRT, in the SRC. All the values in this trace satisfy ◦ ◦ value which corresponds to our targeted waist size. Note that the condition 20 ψRT 160 so that every solution shown even with some flexibility on the waist size of a few millimeter, in this figure represents≤ ≤ a stable SRC. The red trace gives the required distance from the ITM to the waist is still on the the corresponding length of the SRC for each of the RoC order of at least 0.5 km. combinations. 5 regions evolve. Similarly to §III B, the content shown ZM2 RoC [m] here is based on these wide parameter searches and thus -73 -76 -80 -83 -87 only shows the conclusions. We find short focal lengths 160 282 (f / 100 m) that result in a real waist beyond the ZM1 140 260 optic give solutions satisfying our requirements. A focal 120 238 length of f = 75 m was chosen for a more in-depth analysis. Note that, any focal length comparable to this value 100 216 will result in similar telescope behaviour but with slightly 80 194 different solutions for the ZM1 and ZM2 curvatures and in SRC [deg] 60 172 distances to the waist. SRC ψRT Length of SRC [m]

Round-trip Gouy phase 40 150 Using this focal length, and a distance between the Z SRC length mirrors of 50 m, Figure 6 was produced – giving the SRC 20 127 120 100 80 60 40 round-trip Gouy phase over the ZM RoCs, with contours − − − − − for the 6 mm waist size and distances to the waist (from ZM1 RoC [m] ZM1) of 50 and 150 m overlaid on the plot. These contours, along with the color-map, frame the region which FIG. 7. The key take-away from this plot is that by introduc- provides potential configurations for achieving a stable ing a lens, shorter SRC lengths can be obtained along with SRC of a suitable length. It is immediately apparent the option of less demanding (i.e. larger) ZM1 RoCs. Taking ZM1 RoC = 40 m as an example data point, we see from from this figure that the range of possible ZM1 curvatures this plot that− this corresponds to an SRC length of approx- which give solutions is much larger than for the configura- imately 170 m and a round-trip Gouy phase of approx. 110 tion with no lens at the ITM. Here the lens takes the role degrees. Compare this to the same ZM1 RoC value on Fig- of focusing the beam such that both ZM1 and ZM2 have ure 5 which gives around 310 m and 155 degrees for these negative RoCs and act together to collimate the beam quantities, respectively. from the arm cavity. By inspecting Figure 6 we find that this solution region approximately corresponds to ZM1 RoC ∈ [−130 m, −30 m], ZM2 RoC ∈ [−70 m, −90 m]. This region is shown in Figure 7. We can use Figure 7 to pick a reference solution for the telescope parameters. One such solution set is given in 50 Table II, where the curvature combinations were chosen such that a relatively short SRC is obtained (important 0 for ET-HF, see §II A) whilst the edges of the solution 50 ITM range are avoided (i.e. avoiding a near-unstable SRC). Beam size [mm] − The resulting g-factor of the signal recycling cavity for lens ZM2 ZM1 BS SRM this solution set is g ∼ 0.37. In addition, this solution

40 65 360 − 20 70 z − = 150.0 m 270 Gouy phase

accumulation [deg] 0 75 w − 0 = 6.0 mm 180 0 50 100 150 z 80 = 50.0 m Distance [m] − in SRC [deg] ZM2 RoC [m] 90 85 − FIG. 8. The ET-LF telescope design from the arm cav- Round-trip Gouy phase ity to the SRM is shown; where the beam size and accumu- 90 0 − 150 125 100 75 50 25 lated Gouy phase are plotted over the distance propagated − − − − − − through the SRC. Note that, whilst the beam sizes and tele- ZM1 RoC [m] scope length diﬀer slightly for ET-HF, the beam remains collimated in the ZM1 to SRM path; this is an important con- FIG. 6. SRC round-trip Gouy phase for an ITM lens of focal sideration for Figure 9. length f = 75 m and a distance between the Z mirrors of 50 m. Note that the solutions now require the curvatures of ZM1 and ZM2 to both be negative - this is because the ITM lens, of the focal length used here, is responsible for focusing the beam to a waist from the arm cavity. ZM1 collimates the beam going towards the beamsplitter, whilst ZM2 acts as a yields > 99.9% mode matching between the SRC and arm “beam expander” to prevent the beam (as propagated from cavity. A discussion of the ET-HF results, also stated in the arm cavity) from focusing down to a waist too quickly. Table II, is given in §IV. 6

TABLE II. Parameters of the telescopes chosen using Figure 7, with values for the beam size and accumulated Gouy phase associated with these given at each optic in the conﬁguration. Note that the computed values have been given to 2 s.f. to avoid unnecessary precision at this stage. Where appropriate, values for LF and HF have been given separately. See Figure 8 for a visual representation of these data for ET-LF. The focal length of the ITM lens, in both cases, is 75 m.

Optic SRM BS ZM1 ZM2 LF -9410 -82.5 RoC [m] inf -50 HF -630 -63.2 LF 6.1 6.2 8.9 30 Beam radius [mm] HF 6.3 6.4 8.3 38 Space SRM-BS BS-ZM1 ZM1-ZM2 ZM2-ITM LF 50 Length [m] 10 70 52.5 HF 80 LF 7.5 39 5.3 0.6 Total accumulated 52 Gouy phase [deg] HF 4.8 26 4.9 0.2 Gouy phase [deg] 36

IV. ET-HF ARM TELESCOPE DESIGN V. PARAMETER SENSITIVITY AND MODE MATCHING

Taking the results found in the end of §III C as our baseline configuration, we can determine the critical pa- The ET-LF solution given in Table II was used as a rameters of, for example, our ET-HF telescope design. starting point for a similar analysis on the ET-HF arm These can be determined by deviating the key free pa- telescope. The results are then given in the same ta- rameters of the system to observe the effect on the SRC ble, denoted with HF to distinguish the values from LF waist size, round-trip Gouy phase and mode matching where appropriate. The larger beam size (impinging on to the arm cavity. Figure 10 displays the results of such the arm cavity mirrors, see Table I), and shorter wave- an analysis, where the left plots give the aforementioned length of ET-HF, result in the requirement for a longer target parameters as a function of the distances between telescope length when considering the same waist size the optics whilst the right plots are based on deviations target of 6 mm. This increased length requirement can in the radii of curvature of the Z mirrors and the focal potentially be relaxed by decreasing this waist size tar- length of the ITM lens. get, however the required trade-off study is beyond the The mode-matching quantity shown in Figure 10 is scope of this paper. The solution given for ET-HF in defined by the “overlap” (O) figure-of-merit [20], Table II is optimised given this waist (and stable SRC) requirement. Note, also, that the ET-HF solution uses the same ITM lens focal length (f = 75 m) as the ET- 4|={q1} ={q2}| O = ∗ 2 , (3) LF solution found in §III C. The resulting g-factor of the |q1 − q2| signal recycling cavity for this solution set is g ∼ 0.65. The solution stated also yields > 99.9% mode matching between the SRC and arm cavity.

100 ψRT (LF) wBS (LF) 6.4 ψ w RT (HF) BS (HF) 6.3 Given that both the solutions for ET-LF and ET-HF 80 6.2 result in a beam that is roughly collimated between ZM1 and the SRM (see Figure 8), we can alter the distance be- 6.1 60 6.0

tween BS and ZM1 without aﬀecting the beam size on the in SRC [deg] beamsplitter by much more than a few hundred microns. 5.9 Beam size on BS [mm] This is demonstrated in Figure 9. Even by reducing this Round-trip Gouy phase 40 5.7 distance signiﬁcantly, e.g. to 10 m, a stable SRC can still be obtained for both ET-LF and ET-HF (with ψRT ≈ 60 10 20 30 40 50 60 70 and ψRT ≈ 40 degrees, respectively); where the beam size Distance from ZM1 to BS [m] on the beamsplitter would then be around 5.7 mm and 5.9 mm for ET-LF and ET-HF, respectively. Of particu- FIG. 9. The accumulated round-trip Gouy phase in the lar importance to ET-HF, this could allow for a nominal SRC and radius of the beam impinging on BS for both ET- reduction in the SRC length from 210 m to around 150 m LF and ET-HF where the distance between the ZM1 and BS whilst keeping the other telescope parameters, shown in is decreased from its nominal value given in Table II. All other Table II, constant. telescope parameters are kept constant. 7

BS-ZM1 7.5 7.5 ZM1 RoC ZM1-ZM2 ZM2 RoC [mm] ZM2-ITM [mm] 5.0

Waist size 5.0 ITM lens f Waist size 2.5

RT 100 RT 200 ψ ψ [deg] [deg] 100 SRC 50 SRC 1.0 1.00

0.75 Mode Mode overlap

0.5 overlap

100 50 0 50 100 40 20 0 20 40 − − − − Deviation in distance [mm] Deviation in ZM RoCs or ITM lens focal length [µD] (a) (b)

FIG. 10. SRC waist-size, round-trip Gouy phase and mode overlap with the arm cavity as functions of the key distances between optics, (a), and the curvatures of the telescope optics, (b). Each deviation is given in relative terms where a value of zero corresponds to the (ET-HF) baseline value given in Table II. From (a) we can see that the distance between the ITM and ZM2 is the most critical length. Whilst, in (b), we ﬁnd that the focal length of the ITM lens is the critical parameter in terms of the optic geometries. Note that the mode matching values in the lower sub-plot were computed via Equation (3).

where q1 and q2 are the beam parameters being com- are required [22, 23]. This is one of the tasks which could pared. In this case q1 represents the eigenmode of the be performed by the arm telescopes of the ET detectors, SRC propagated to the arm cavity and q2 is the arm cav- thus in this section we will quantify the required devi- ity mode itself. This quantity returns values O ∈ [0, 1], ations to the telescope mirror curvatures, for recovering where unity indicates a full mode match between the mode matching to the arm cavities, in the presence of two beam parameters and zero gives complete mode mis- varied thermal lens focal lengths. In this case the ET- match. HF telescope (plus arm cavity) configuration is used, as From Figure 10 we can deduce, in terms of the optic this detector is designed to operate at high power [1, 15] geometries, that this telescope configuration is most sen- where thermal lensing will be more prevalent. sitive to the focal length of the lens at the ITM. Thus the Figure 11 quantifies the necessary modifications to the next step of this analysis will focus solely on the ITM lens radii of curvature of ZM1 and ZM2 in order to recover focal length changes a result of thermal aberrations in the > 99.9% mode matching of the SRC to the arm cavity; beam. Thermal lensing is investigated, in particular, due for an assumed range of thermal lens focal lengths of to it, potentially, being responsible for the largest effec- fth ∈ [100 km, 15 km]. Note that for a strong focal length tive changes to the ITM lens focal length — as shown at of 15 km from the thermal lens, the effective focal length the end of the next section. of the ITM reduces to f ≈ 74.63 m; i.e. a deviation of about 0.5% from the target value of 75 m noted in §V. This focal length distortion results in a mode mismatch, A. Mode matching in the presence of thermal between the arm cavity and signal recycling cavity, of lensing approximately 20%. At this extreme thermal lens, the modifications to the Z mirror radii of curvature indicated Surface deformation and refractive index differentials, by Figure 11 reduce this mode mismatch to effectively caused by temperature distributions in the mirror sub- 0%. strates, lead to thermal aberrations (lensing) in the beam in ground-based GW detectors [21]. This thermal lensing results in mode mismatches between the arm and VI. CONCLUSION recycling cavities. In terms of the optics present in our configuration, the thermal lensing acts to modify the ef- In this paper we investigated arm telescope configu- fective focal length of the ITM lens, thereby altering the rations, for both ET-LF and ET-HF, which are suitable geometry of the beam in the signal recycling cavity (see for the optical layout in the ET 2020 design update [1]. the solid traces in Figure 10 for this effect in a broad These telescope configurations are motivated by smaller sense). To minimise these distortions, adaptive optics beams on the optics in the central part of the interferome- 8

We demonstrated that it is possible to achieve a stable ZM1 RoC change SRC, of a sensible length, with telescopes in the arms ZM2 RoC change 1.5 of both the ET-LF and ET-HF interferometers. Reduc- ing the length of SRC, in accordance with [16], can be attained via the introduction of a lens at the ITM for 1.0 pre-focusing the beam from the arm cavity. Our baseline solutions for such a configuration are given in Ta- ble II. Further reductions to the length of the SRC, whilst 0.5 changing the spot size on the beamsplitter by only a few hundred microns, are possible via decreasing the distance from ZM1 to BS - see Figure 9 for details. Required change in0 ZM. RoC0 [%] 20 40 60 80 100 Our baseline configuration for ET-HF was analysed in V where we found that the focal length of the ITM Thermal lens focal length [km] § lens is the critical parameter in terms of the sensitivity for mode matching and SRC stability. However, in §V A, FIG. 11. Simultaneous changes in Z mirror RoCs required we found that effective changes in this focal length due for recovering “complete” mode matching from the SRC to to thermal lensing can be compensated with actuation the arm cavity. At the extreme of fth = 15 km on this plot, the required deviation in the RoC of ZM1 is approximately on the curvatures of the telescope mirrors. In particular, 1.4% whilst for ZM2 it is 1.9%. we saw that the mode mismatch (of approximately 20%) due to a strong thermal lens, with fth ∼ 15 km, can be fully corrected with changes of approximately 1.4% and ter. The beam expanders also provide the ability to steer 1.9% in the RoCs of ZM1 and ZM2, respectively. the ET-HF beam around the ET-LF ITM suspension sys- The results presented here provide evidence for the fea- tems; with the added benefits of decoupling the angle of sibility of beam-expander telescopes in the interferometer incidence on the beamsplitter from the beam axes in the arms of ET and provide essential input for trade-off stud- interferometer arms. Our requirements for this telescope ies of the optical layout. Further studies are required to can be summarised as targeting a 6 mm waist size po- study other aspects of this setup, in particular the effects sitioned at the main beamsplitter whilst maintaining a of astigmatism in specific telescope implementations, and stable SRC (quantified approximately as having a length the possible negative impact on the contrast defect due of the same order as the Rayleigh range of the beam). to having the telescopes in a configuration that allows Further details on these requirements were given in §II A. differential beam tuning.

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